tag:blogger.com,1999:blog-23509970940830297152024-03-14T06:54:29.739-07:00Bekaar BokbokNever leave for tomorrow what can be done the day after!Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.comBlogger31125tag:blogger.com,1999:blog-2350997094083029715.post-87465816624782364162017-04-09T16:57:00.001-07:002017-04-09T16:57:50.665-07:00Of Ants and Universes<div align="center" class="MsoNormal" style="text-align: center;">
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<span style="font-family: "georgia" , "serif";">An ant is sitting at one end of an
elastic string, which is one metre long.<br />
It starts walking towards the other end at a speed of one centimetre per
second.<br />
At the end of every second, the string is instantaneously and uniformly stretched
by an additional one metre.<o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";">Question: Does the ant ever reach the
other end of the string?</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKizF7WJ7peXILuDx4iCwAcquvKu1S0wk_Q8bDZD1d0UcKaMbrJvF-P2QQW5h0ADHqH9o9OBx4WoMRucgJIsVjufL0Nog-EmtVGD4Wg34Qomykzmz-PstEQgQ565LGjx9TM_9rGywTshiq/s1600/ant_string.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="152" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKizF7WJ7peXILuDx4iCwAcquvKu1S0wk_Q8bDZD1d0UcKaMbrJvF-P2QQW5h0ADHqH9o9OBx4WoMRucgJIsVjufL0Nog-EmtVGD4Wg34Qomykzmz-PstEQgQ565LGjx9TM_9rGywTshiq/s640/ant_string.jpg" width="640" /></a></div>
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<span style="font-family: "georgia" , "serif";"><br />
<br />
At first sight, the answer seems an obvious “No”. After all, the ant moves only
a centimetre every second and then its destination recedes by a metre, so clearly
not possible.<br />
But here’s the rub – notice the phrase “uniformly stretched”. This changes
everything. <o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "georgia" , "serif";">For instance, after one second the ant is
1 cm away from the starting point. But when the string is stretched, its length
doubles. <o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";">Since the stretching is uniform, it means
the ant gets pulled an <i>additional</i> 1
cm, which means it is now <i>two</i>
centimetres from the starting point.<o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";">So, while the destination keeps moving
away, the uniform stretch also pulls the ant somewhat further along each time. </span></div>
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<span style="font-family: "georgia" , "serif";">So, which effect wins?<br />
<br />
The quickest way to solve this is to work out the <i>fraction</i> of the string’s length traversed by the ant at the end of
each second.<br />
Why ? <i>Because uniformly stretching the
string leaves this fraction unchanged !</i> <i><br />
</i>If the fraction ever becomes 1, the ant has reached the other end.<br />
<br />Let’s work it out.<br />
At the end of 1 second, the ant has covered 1/100 of the string.<br />
At the end of 2 seconds, it has covered another 1 cm, but the string is now 2
metres long – so the additional fraction is only 1/200. Similarly, at the end
of 3 seconds, it covers another 1/300.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Thus, the cumulative fraction of covered
by the ant looks like 1/100 + 1/200 + 1/300 +…<o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";">Does this ever reach 1 ?</span></div>
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<span style="font-family: "georgia" , "serif";">
For this we need a little help from the mathematician, who informs us that the
sum of the series (1 + 1/2 + 1/3 +…. ) <i>diverges</i>,
which means that if you add up the series far enough, it will exceed any number
you can think of – a hundred, a million, a googolplex, whatever.<br />
<!--[if !supportLineBreakNewLine]--><br />
<!--[endif]--><o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";">How does this help ? Well, notice that the fraction 1/100 + 1/200
+ 1/300 +… is simply<o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";">1/100*(1 + 1/2 + 1/3 +….) and we need
this to reach 1.<br />
But this just means that the term in brackets should eventually reach 100, and
our resident mathematician has assured us that it will.<br />
So we are done – the ant <i>does</i> reach
the other end.<br />
<br />
How long does it take ?<o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";">This can also be worked out with some
effort and the answer is – about 2.7*10<sup>43</sup> seconds, which is about
9*10<sup>35</sup> years or in other words 90 million billion billion times the
lifetime of the known universe.<br />
Hey, I never said this would be quick.<br />
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<!--[endif]--><o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";">What if the ant were moving at one <i>millimetre</i> per second ? It still gets
there, except that it now takes about 9*10<sup>350</sup> years, which I won’t
even bother to convert to millions and billions.<br />
What if the string stretches by a kilometre every second ? <o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";">Still the same, just longer. (A <i>lot </i>longer)<br />
What if the string was a trillion kilometres long at the start ? Same
conclusion.<br />
<br />
Now let’s go back to the original problem. Since we know the ant does reach the
other end, the new question is: What is the total distance the ant has covered
?<br />
<br />
A no-brainer, you say.<br />
The ant takes 2.7*10<sup>43</sup> seconds to reach, it moves 1 cm per second,
so a quick multiplication gives 2.7*10<sup>43</sup> centimetres, right ? Wrong
!!<o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";"><br />
As mentioned before, the ant gets an extra shove forward every time the string
stretches. <br />
So, the correct approach would be to calculate the <i>length of the string</i> when the ant reaches because we know the ant
has traversed this entire distance.<br />
Well, the string stretches 1 metre every second so the correct answer is 2.7*10<sup>43</sup>
<i>metres</i> – a full one hundred times
bigger than the answer we previously got.<br />
In other words, the <i>vast majority</i> of
the total distance covered comes not from the ant’s own motion, but the
stretching of the string !<br />
<br />
Game for more ? Why not ?<br />
Let’s imagine a second ant starts walking exactly one second after the first
ant at exactly the same speed – 1 cm per second. In other words, when it starts
out, the second ant is 1 cm behind the first.<br />
Question: How far apart are they by the time the first ant reaches the other
end ?<br />
You may want to think of this for a bit before seeing the answer…<br />
<br />
Solved ? No ? Okay then.<br />
The trick here is to realize that since the ants walk at the same speed, the
distance between them <i>only</i> changes
when the string is stretched.<o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";">By how much ? Easy problem.<o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";">Since the stretching is uniform:<o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";"><br /></span><span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><i>Distance between Ants after Stretch <b>/ </b>Distance between Ants before Stretch<br />= Length of String after Stretch </i>/<i> Length of String before Stretch</i></span></div>
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<br /></div>
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<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Applying the same reasoning, if we define
“the end” to be “when the first ant reaches its destination”, we get:<o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br /></span><span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><i>Distance between Ants at the End <b>/</b> Distance between Ants at the Start</i></span></div>
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<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">= <i>Length of String at the End <b>/</b> Length of String at the Start</i></span><span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">= </span><span style="font-family: "georgia" , "serif";">2.7*10<sup>43<br />
<br />
</sup>In other words, the ants which started out just a centimetre apart will
be 2.7*10<sup>43</sup> centimetres, or about <i>3 million trillion light years apart</i> at the end !!</span></div>
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<span style="font-family: "georgia" , "serif";">
Just one more and we are done.<o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";">Consider a string which stretches in a
different manner.<br />
Rather than extending by a fixed amount every second, it <i>doubles its length</i>.<o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif";">The ant still moves at the same speed.
Now what ?<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Revisiting our approach of calculating
the fraction of the string covered by the ant after each second, we now get the
series – 1/100 * (1 + 1/2 + 1/4 + 1/8 + ….)<br />
But now our mathematician friend has new information.<br />
This time the series in brackets <i>converges
to 2</i> – in other words, as you add more and more terms you get ever closer
to 2, but never reach or exceed it.<br />
Which means that 1/100 * (1 + 1/2 + 1/4 + 1/8 + ….) is always less than 1/50 –
however long it walks, our hapless ant never even covers 2% of the string.<br />
<br />
What if the string were longer or shorter ? <o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Unlike what happened before, now it <b><i>does</i></b>
make a difference.<br />
A quick calculation shows that if the string was 2 cm or longer, the ant never
reaches.<br />
If it was any shorter, even by a zillionth of a millimetre, it always does.<br />
<br />
Enough about the ants, when do we get to universes ?<br />
Just one more point to clear up first.<o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br />
When thinking about this problem, we typically visualize the starting end of
the string to be stationary while the other end recedes every time.<br />
Now consider the perspective of an observer at the destination point. From her
viewpoint, the ant is always walking towards her, but every time the string
stretches, <i>the starting point recedes
further away and the ant gets pulled back</i>.<br />
<br />
Neither perspective is intrinsically “more correct” than the other, and more
importantly, all the conclusions we derived so far remain valid. Now we are
ready.</span></div>
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<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br />
<br />
<b>Cosmic Expansion<o:p></o:p></b></span></div>
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<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Everybody knows the universe is
expanding. (If you didn’t till now, well, I just told you!)<br />
This is typically presented as a somewhat mysterious phenomenon, frequently
with misleading analogies.<br />
But the problem about the ants above offers the best way I know of
understanding how it works.<br />
<br />
Let’s take the flipped perspective I mentioned at the end.<br />
The observer at the destination end of the string is an astronomer on earth.<br />
The starting end is a distant galaxy. <o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">The ant is light from the galaxy speeding
towards the earth.<br />
The stretching string? That is the space between the earth and galaxy which is
expanding.<br />
<br />
<i>This</i> is the correct way to understand
cosmic expansion. <br />
Many popular articles often have statements like “The galaxies are receding
from us because they were thrown apart by the explosion of the Big Bang”.<br />
Nothing of the sort ! The Big Bang wasn’t at all like the explosion of a bomb
throwing shrapnel everywhere.<br />
Cosmic expansion happens because <i>space
itself</i> is stretching, just like our elastic string.<o:p></o:p></span></div>
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<br /></div>
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<br /></div>
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<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">What if you want to look at a different
galaxy?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Just consider a different stretching
string with a different starting length.<br />
Does the starting length matter ? Now we need to get into details….<br />
<br />
We started with a string stretching uniformly by a metre every second.<br />
This corresponds to the case where space expands <i>linearly with time</i>.<br />
<br />
(Okay, I am cheating a bit. The expansion of the string was “discretized” as an
instantaneous stretch at the end of every second, while cosmic expansion is
continuous in time. <o:p></o:p></span></div>
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<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">However, all the conclusions go through -
we just use calculus rather than summing series.)<br />
<br />
Our first result was that the ant always reaches its destination, irrespective
of the starting length of the string.<br />
Translated into cosmology this means:<br />
<i>If space expands linearly with time (or
slower), then the light from <b>any</b>
galaxy, however distant, will eventually reach the Earth.</i></span></div>
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<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><i><br /></i></span></div>
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<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">In other words, with linear expansion of space, our cosmic horizon grows ever
larger as increasingly distant galaxies come into view.<br />
An interesting corollary is that if the universe as a whole is finite, then we
would eventually see all of it – and possibly observe multiple images of the
same galaxy.<br />
<br />
Now here’s my get-rich scheme: When you read popular articles on astronomy in
future, please send me a dollar every time you read something like - “Since the
universe is about 13.8 billion years old, the most distant objects we can see
are 13.8 billion light years away.”<br />
<br />
See what happened there?<br />
This is <i>precisely</i> the error we made
when we tried to calculate the length of the string at the end by multiplying
the ant’s speed by the time taken – in reality, the string would generally be a
<i>lot</i> longer.<br />
<br />
This is, in fact, what happens.<br />
Although the Big Bang happened about 13.8 billion years old, the most distant
objects we can see in any direction are currently about <i>46.5 billion light years</i> away.<br />
In other words, the diameter of the observable universe is currently about 93
billion light years and growing.<br />
<br />
What about the third example where two ants get spread far apart by the
stretching string ?<br />
This is the analogue of <i>cosmological
redshift</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Light from a distant galaxy is an
electromagnetic wave and the ants in the analogy represent two successive
crests of the wave – the distance between them is the wavelength of light.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">The expansion of the universe stretches
out light waves, increasing their wavelength.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Longer wavelength means redder light,
hence the term “redshift”.<br />
<!--[if !supportLineBreakNewLine]--><br />
<!--[endif]--><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">This is no mere academic exercise.<br />
Light from the most distant galaxies seen by the Hubble space telescope was
emitted about 12 billion years ago when the universe was much smaller and as a
result, they appear deep red.<br />
In fact, the most ancient galaxies are invisible to Hubble because their light
has been stretched all the way to the infrared spectrum !<br />
The future James Webb telescope, designed with the goal of observing the
earliest stars and galaxies is, therefore, equipped to detect infrared
radiation.</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjpm2XQJFPxty3wlq3-jSWamwZo-ocSNDdR7nTsar7qgmPFcMQ1gyBvJNYN0U8NzWuE6GrqlnkEvakxj2yvkN5dDX5xq3YFmZUIBUnxZ1Lc6m-6SwDiT25Y7nOcYSHuYq2-TYdKab-HWoi/s1600/Dist_galax_redshft.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="448" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjpm2XQJFPxty3wlq3-jSWamwZo-ocSNDdR7nTsar7qgmPFcMQ1gyBvJNYN0U8NzWuE6GrqlnkEvakxj2yvkN5dDX5xq3YFmZUIBUnxZ1Lc6m-6SwDiT25Y7nOcYSHuYq2-TYdKab-HWoi/s640/Dist_galax_redshft.jpeg" width="640" /></a></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br />
<br />
The most ancient light we can detect – the Cosmic Background Radiation – was
emitted as visible light when the universe was about 380,000 years old.<br />
This light has been stretched even beyond the infrared spectrum, we see it as
microwave radiation – a consequence of the fact that the universe has expanded
by a factor of about 1,000 between then and now.<br />
<br />
Finally, what about the string which doubles every second ?<br />
<!--[if !supportLineBreakNewLine]--><br />
<!--[endif]--><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">This represents a universe where space is
expanding <i>exponentially</i>.<br />
In this universe, every observer has a <i>fixed</i>
cosmic horizon beyond which everything is forever invisible. (This is analogous
to the fact that ants 2 cm or more from the end of the string never reach it)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Even worse, since the universe keeps
expanding, points which start off within the horizon get pulled beyond it and
disappear.<br />
<!--[if !supportLineBreakNewLine]--><br />
<!--[endif]--><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">In stark contrast to a linearly expanding
universe where ever more distant galaxies keep coming into our view, the skies
in an exponentially universe empty out with time and eventually, every galaxy
or galaxy cluster not tightly bound together by gravity ends up completely
isolated.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">So, what kind of universe do we live in ?<br />
Well, right until the end of the 20<sup>th</sup> century, all evidence pointed
to a universe expanding at a rate even slower than linear. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Then in 1998, the world of cosmology was
overturned when a team led by Saul Perlmutter established that cosmic expansion
was, in fact, “accelerating” – ie, the expansion rate was <i>faster</i> than linear.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Currently, the best theoretical
explanation for this is the presence of a “cosmological constant” which is,
crudely put, a kind of “energy of empty space”. (Nowadays, it’s fashionable to
use the term “dark energy” instead, but given that we know nothing about what it
really is, we might as well call it “The Force”.)<br />
<br />
However, the upshot of having a cosmological constant is that space expands
exponentially, so we appear to be stuck with a cosmic horizon and emptying
skies. <br />
In fact, here is something even more striking.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Let’s revisit the problem of the doubling
string with an ant starting 1 cm away from the endpoint. <br />
Now this ant reaches its destination in 1 second, but within that time, it
starting point is now 2 cm away – which is the horizon.</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXwB3-4T-jYmOvKmkBFewtG8D9STgDSSpOjcgzq6kw_GBXhBXplVTQckYfuHYuiPmJWh0wuf74YsEMSg6N8usAlVGRBP8NHWF0IY7I9_mZBHetVm7xwN2MoTaAyhWxPmxSoC_UN2008zll/s1600/expansion-of-the-universe-accel.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="459" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXwB3-4T-jYmOvKmkBFewtG8D9STgDSSpOjcgzq6kw_GBXhBXplVTQckYfuHYuiPmJWh0wuf74YsEMSg6N8usAlVGRBP8NHWF0IY7I9_mZBHetVm7xwN2MoTaAyhWxPmxSoC_UN2008zll/s640/expansion-of-the-universe-accel.gif" width="640" /></a></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br />
<!--[if !supportLineBreakNewLine]--><br />
<!--[endif]--><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Translated into cosmology, this means that
by the time the light from a galaxy reaches the earth, the galaxy itself may
have already crossed the cosmic horizon.<br />
So, while we shall see it shining in our night sky, it is already “gone for
good” – beyond the reach of any spaceship or signal we send, even one
travelling at the speed of light.<br />
<!--[if !supportLineBreakNewLine]--><br />
<!--[endif]--><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Does this really happen? <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Not only does this happen, but it is
estimated that <i>97% of the galaxies we see</i>
are already beyond the horizon.<br />
<!--[if !supportLineBreakNewLine]--><br />
<!--[endif]--><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">So is this what lies ahead for us? <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">An inexorable thinning out of the
universe, as the galaxies vanish over the cosmic horizon and the skies grow
black and empty?<br />
Alas, that is indeed our best understanding of the future of the universe, but
don’t get too depressed because this is very likely not “the end of everything”.<br />
All evidence so far, not clearly understood yet but fairly consistent, strongly
suggests that what I am calling <i>the</i>
universe is merely <i>our</i> universe – <a href="https://www.forbes.com/sites/startswithabang/2016/01/30/ask-ethan-the-multiverse-and-the-road-not-traveled/#14c1595b6f63" target="_blank"><span style="color: black;">an infinitesimally small fragment</span></a> of “All that is, or ever was, or ever will be”…<br />
<br />
<br />
<!--[if !supportLineBreakNewLine]--><br />
<!--[endif]--></span><span style="font-family: "georgia" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<br /></div>
<br />
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";"><br />
<br />
<br />
<br />
<br />
<br />
<!--[if !supportLineBreakNewLine]--><br />
<!--[endif]--><o:p></o:p></span></div>
Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com6tag:blogger.com,1999:blog-2350997094083029715.post-6845270368488837422016-07-11T16:08:00.000-07:002016-07-11T16:20:56.826-07:00The Game of Hats<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: "georgia" , serif;">Lord Shamsay Snow, Warden of the North in
the grim land of Pesteros delights in playing games with his prisoners.
Especially games which end with said prisoners being skinned alive. Nothing
enhances delight like holding out some hope of escape, then brutally
extinguishing it.</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Having rounded up a hundred of the
miserable wretches, Shamsay announces:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">“You will all play the Game of Hats –
where you either go free or you die.”<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">The game will proceed as follows:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">All prisoners will be lined up facing the
dungeon wall.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Prisoner 100 stands right against the
wall, prisoner 99 stands behind him and so on. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Shamsay’s trusty servant, Meek, will
place a hat – which may be coloured Black or White – on their heads. Obviously
they won’t know the colour of the hat. All they will see is the hats of
everyone in front of them.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Now, starting from Prisoner 1, right at
the back, Shamsay will ask each one of them the colour of the hat on their
heads, one by one.<br />
The only words they can say are “Black” or “White” – <i>which they will shout out for everyone to hear.</i><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Any other word or phrase leads to <i>everyone</i> being fed to the hounds. (This
is what Shamsay is secretly hoping for….)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Now, if the word they shout out
corresponds to the colour of their own hat, they go free. Otherwise, they are
skinned alive. Such is the Game of Hats.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUQwHCIFYVqNqL7-xnLZa_ctdcD3xPCiAbw5CMzn0g2VOew4txCY4UhzT71L01tdswZmEzUFQcI7-06VSpsZWtWXgTf_M78qCfxihRp7EtV25vaNtksopeqoTqzILqno12MLxz20egwZEF/s1600/Prisoner_hats.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="288" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUQwHCIFYVqNqL7-xnLZa_ctdcD3xPCiAbw5CMzn0g2VOew4txCY4UhzT71L01tdswZmEzUFQcI7-06VSpsZWtWXgTf_M78qCfxihRp7EtV25vaNtksopeqoTqzILqno12MLxz20egwZEF/s640/Prisoner_hats.gif" width="640" /></a></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">As the prisoners stare at him, hope and
apprehension mingling in their eyes, Shamsay offers them a sporting chance –
they have till next episode to figure out any way they can to save their sorry
selves.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">As the camera pans onto Shamsay’s leer
and the credits start to roll, can you anticipate what follows? Will the next
episode show a procession of prisoners hobbling off to freedom, or will carnage
abound? Now may be a good time to go and think….<br />
<br />
SPOILER ALERT !!!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="color: #c00000; font-family: "georgia" , "serif";">At most one prisoner dies, all the rest go free.<o:p></o:p></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">How in Pesteros did that happen? <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Well, due to the ingenuity of an
extremely short prisoner of very high intelligence, who then insisted on
standing right at the front of the line.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">The solution involves “Arithmetic mod 2”
which is much simpler than it sounds.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">The basic idea is that every number is
replaced by its remainder when divided by 2 - so every even number becomes 0,
every odd number is 1.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">This applies to negative numbers as well - <b>-</b>2,
<b>-</b>12, <b>-</b> 1000 are all 0, while -1, -25, -117 are all 1.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Now the cool thing is, you can add,
subtract and multiply “mod 2” as well, hence “Arithmetic mod 2”.<br />
All you need to do is add, subtract or multiply as you normally do, then divide
by two and take the remainder.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">To illustrate: <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">13 + 16 mod 2 = 29 mod 2 = 1</span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEcifPGM5BgKekMwYadsgddYIJMpFcQEmLJCDmGjP5eYu5JAts1imhxTx8jqFMwg6kM-HZt1LTvsJgtOJos4eUmltvXLJEALqK6VODA9My78qckKL7K2X0-VimRpG28UpMtIvXyR5Gg1w8/s1600/bin_arithm.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEcifPGM5BgKekMwYadsgddYIJMpFcQEmLJCDmGjP5eYu5JAts1imhxTx8jqFMwg6kM-HZt1LTvsJgtOJos4eUmltvXLJEALqK6VODA9My78qckKL7K2X0-VimRpG28UpMtIvXyR5Gg1w8/s200/bin_arithm.jpg" width="173" /></a></div>
<o:p></o:p><br />
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">4 x 3 mod 2 = 12 mod 2 = 0<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">3 – 7 mod 2 = <b>-</b>4 mod 2 = 0<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">All clear? Good. Now on to the hats.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">The strategy starts with assigning a number
to each colour: Black = 0, White = 1.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Prisoner 1 adds up the numbers
corresponding to the 99 hats he sees in front of him mod 2.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">If he gets a 0, he shouts “Black”, if 1,
then “White”.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Prisoner 2 does the same for the 98 hats
in front of her, then subtracts what she gets from Prisoner 1’s answer. This
gives her the colour of her hat and she shouts it out.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">What about the rest, say Prisoner 17 ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Well, prisoner 17 starts off by getting
the total of the 83 hats in front of him. Call it X.<br />
Then he hears Prisoner 1’s shout and gets the corresponding number. Call it Y.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Now as Prisoners 2 to 16 call out their colours,
he subtracts 1 from Y every time he hears “White” and o when he hears “Black”.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">By the time Shamsay gets to him, all the
subtraction leaves him with some number, Z.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">So “What is the colour of your hat ?”.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Calculate (X – Z ) mod 2. If 0, say “Black”,
if 1, say “White”, and he is free !<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Was that confusing? Okay, let me give an
example with ten prisoners rather than a hundred.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Suppose, this is the sequence of hats on
their heads, with prisoner 1 on extreme left and Prisoner 10 on extreme right
(B is black, W is white).<o:p></o:p></span></div>
<div align="center" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; text-align: center;">
<span style="font-family: "georgia" , "serif";"><br />
B W W B B B B W W W<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Now then, Prisoner 1 sees 4 black hats
and 5 white hats.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">So, following the procedure, he gets (4 x
0 + 5 x 1) mod 2 = 5 mod 2 = 1<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Hence, he shouts out “White”. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Alas, his own hat is black, so he dies,
but this is the only casualty.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Prisoner 2 sees 4 black hats and 4 white
hats. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">So she gets (4 x 0 + 4 x 1) mod 2 = 4 mod
2 = 0<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Subtracting from Prisoner 1’s answer
gives 1 – 0 = 1, so she says “White”.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">And notice that this <i>is</i> the colour of her hat!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Now let’s try it for Prisoner 5.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">He sees, 2 black and 3 white hats, so X =
(2 x 0 + 3 x 1) mod 2 = 1<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Now he hears Prisoner 1, shout out
“White”, so Y = 1.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">As Prisoners 2 to 4 shout out their
answers, he keeps subtracting accordingly.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Easy to verify that: Z = 1 – 1 – 1 – 0 = -1<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Now it’s his turn to answer.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Well, X – Z mod 2 = 1 – (-1) mod 2 = 2
mod 2 = 0. And “Black” it is! </span><span style="font-family: "wingdings"; mso-ascii-font-family: Georgia; mso-char-type: symbol; mso-hansi-font-family: Georgia; mso-symbol-font-family: Wingdings;">J</span><span style="font-family: "georgia" , "serif";"><br />
<!--[if !supportLineBreakNewLine]--><br />
<!--[endif]--><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Note that the number of prisoners makes
absolutely no difference. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Be it ten or a hundred or millions,
everyone goes free except maybe Prisoner 1. And there’s a 50-50 chance that
even he or she gets it right.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">So what happens if Shamsay ups the ante
by increasing the number of hat colours, to Black, White, Red and Green ? <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Makes absolutely no difference. At worst Prisoner 1 dies or they all go free.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">If you followed the argument above, you
will quickly see that the solution here is
to use “Arithmetic Modulo 4” where you take remainders by 4 rather than 2.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">So, 13 + 16 = 29 mod 4 = 1, 8 -4 = 4 mod 4 = 0, 11 x 2 = 22 mod 4 =
2 etc.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<br />
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">To spell it out further, assign a number
from 0 to 3 to each colour, say Black = 0, White = 1, Red = 2 and Green = 3.
The strategy is exactly analogous, except you need to do the calculations using
the four numbers corresponding to the four different colours and do all the
arithmetic mod 4.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "georgia" , "serif";">Exercise:</span></i><span style="font-family: "georgia" , "serif";"> With 10 prisoners, work out the
procedure in detail for the hat sequence below:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">B G R R W W B G R G<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="color: #c00000; font-family: "georgia" , "serif";">No matter how many prisoners or hat colours, the worst possible
outcome is one death.</span></i><span style="font-family: "georgia" , "serif";">
Thanks to the power of modular arithmetic.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8iChBdXfmJKqUb8UgUKtiU_KNy3NiOp-3tZD0XBfEWijDKUJPhGyOfGGkM6umWyKMd9BjTSMVo5na8f3-7ntRIfOVFb8gGr-xIhw-ruMNS53uWbH3BWvgE8hA9jErzTTzk7sHVtf713Uf/s1600/Mod_arithm.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8iChBdXfmJKqUb8UgUKtiU_KNy3NiOp-3tZD0XBfEWijDKUJPhGyOfGGkM6umWyKMd9BjTSMVo5na8f3-7ntRIfOVFb8gGr-xIhw-ruMNS53uWbH3BWvgE8hA9jErzTTzk7sHVtf713Uf/s320/Mod_arithm.png" width="310" /></a></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">With his approval ratings plummeting
wildly - “At most one death per episode ? Yawn.” - Lord Shamsay is in despair
until he gets a note from a mysteriously named GRRM.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "georgia" , "serif";">“Don’t
let them shout out the answer, dammit ! Have them whisper it in your ear !”<o:p></o:p></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Carnage ensues. With no shouted answer to
guide them, each prisoner is reduced to making a guess. With two hat colours
about half of them get it wrong, more colours make it worse.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">There is much screaming and bloodshed,
the fans are ecstatic and the show is destined to continue for many more
seasons.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span style="font-family: "georgia" , "serif";">Analytical
Interlude<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Time to tune out the morbidity and engage
in some dispassionate analysis.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">If you think about the “shout version” of
the game, you realize that in principle, the outcome is not too surprising.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Every prisoner knows the colour of the
hats of the people in front, and shouting allows them to communicate
information. Yes, it takes a lot of ingenuity to convey all that information in
a single word, but ultimately it all boils down to each person getting to know
the colour of their hat from the prisoner behind them in a suitably coded form.
<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">It is, therefore, also reasonable that Prisoner 1 is at risk – after all, there is no
one behind him to tell him his colour.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">The “whisper version” of the game destroys
the information channel and each person must now figure out their hat colour
based <i>only </i>on the hats in front of
them. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Is this always impossible? Not
necessarily.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">For instance, if Shamsay put black hats
on everyone, or an alternating sequence of black and white hats, then it’s
quite possible that prisoners may figure out the pattern. But all hope is lost
if he assigns the colours <i>randomly</i>,
say by tossing a coin each time.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Surely, if the hat colours are chosen
randomly, then it is impossible to figure out the colour of your own hat from
the ones in front ? Surely, in such a case, half the prisoners must guess their
colour wrong on average ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">In a finite world, yes.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">But what if we throw infinity into the
mix?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span style="font-family: "georgia" , "serif";">The
Infinite Game<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">In a strange dimension unimaginably
distant from Pesteros, yet all too near, R’Holler, the Lord of Might has an
infinity of demons at his command.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">His command is to play the Game of Hats.
(Why ? Do not question the will of R’Holler, puny mortal !)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">The rules are the same. Hats are either
black or white.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Demon 1 stands behind Demon 2, who stands
behind Demon 3 and so on forever.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">(Math Alert: This implies that the
infinity of demons is a<span style="color: blue;"> <span style="color: #0070c0;"><a href="http://bekaarbokbok.blogspot.co.nz/2015/10/the-ladder-of-infinity.html" target="_blank">countable infinity</a></span></span>.)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">All too aware of Shamsay’s initial
discomfiture in the mortal realm, R’Holler goes straight for the “whisper
version” of the game.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Oh yes, punishments. The penalty for
guessing the wrong hat colour is the worst fate imaginable – being born as a
commoner in Pesteros.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Based on our analysis above, since this
is the whisper version, each demon has a
50% chance of getting their colour wrong. So, is Pesteros headed for a
population explosion of literally infinite proportions?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">In fact, even if the demons work out some
intricate statistical strategy so that the chance of a wrong answer is 1% or
0.0001%, any fraction of infinity is infinity and Pesteros is doomed.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">So, what if I tell you that:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="color: #c00000; font-family: "georgia" , "serif";">There exists a strategy to ensure that </span></i><span style="color: #c00000; font-family: "georgia" , "serif";">only a finite number of
demons<i> get their hat colour wrong – ie, the
proportion of demons getting it wrong can be made precisely zero.</i><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="color: #c00000; font-family: "georgia" , "serif";">All without any communication whatsoever between the demons
during the game.<o:p></o:p></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">How in the seven heavens and hells ???<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">This will take a bit of explanation.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span style="font-family: "georgia" , "serif";">Strings
and Grouping<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Consider any arrangement of black and
white hats chosen by R’Holler.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">This would correspond to an infinite
sequence of the letters B and W.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">We will call such a sequence a “<i>string”</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigP5PjkCZ4pwxTav-mZqwnc8-6agRkWddP_Nwxsg8GKEjwVihfg643oa_hFXeesPqhQLWUrSnAa__Ui4jXZhgotGzLw_dJrGYngWslEj18E7iZWqC4t9bWI7EwbsuAa4bcdcc6wrYIv_XX/s1600/binary_seq.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="298" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigP5PjkCZ4pwxTav-mZqwnc8-6agRkWddP_Nwxsg8GKEjwVihfg643oa_hFXeesPqhQLWUrSnAa__Ui4jXZhgotGzLw_dJrGYngWslEj18E7iZWqC4t9bWI7EwbsuAa4bcdcc6wrYIv_XX/s400/binary_seq.jpg" width="400" /></a></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">We will call two strings “<i>finitely different</i>” if their values
differ in finitely many locations and “<i>infinitely
different</i>” otherwise.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">To illustrate, consider the four strings
below.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">S1: B B B B B B B B ..… all black<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">S2: BWWWBWWB…all black henceforth<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">S3: BWBWBWBW…alternating B and W<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">S4: BBBWWWBW…alternating B and W
henceforth<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">You can verify that S1 and S2 are
finitely different, as are S3 and S4. However, S1 is infinitely different from
both S3 and S4.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">To start things off, here’s an <i>Easy Result:</i><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">If S1 is finitely different from S2 and
S2 is finitely different from S3, then S1 is finitely different from S3.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "georgia" , "serif";">Quick
Proof:</span></i><span style="font-family: "georgia" , "serif";"> Suppose S1
differs from S2 at <i>n</i> different
locations and S3 differs from S2 at <i>k</i>
different locations. </span><br />
<span style="font-family: "georgia" , "serif";">Then S1 and S2 differ at at most <i>k + n</i> different locations.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Now, I am going to break up the set of
all possible strings into <i>groups</i> as
follows.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Pick any string (the string which is all
B for instance), call it S1. Make a group consisting of <i>all</i> strings which are finitely different from S1.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Anything string left over? If so, call it
S2. Make another group consisting of <i>all</i>
strings which are finitely different from S2.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Anything string left over? If so, call it
S3. Make a group…..you get the idea.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "georgia" , "serif";">We
keep repeating this again and again until all possible strings belong to some
group or another.</span></i><span style="font-family: "georgia" , "serif";"> <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Observe two facts about this grouping:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "georgia" , "serif";">Fact
1:</span></i><span style="font-family: "georgia" , "serif";"> Every possible string
belongs to <i>some</i> group.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "georgia" , "serif";">Fact
2:</span></i><span style="font-family: "georgia" , "serif";"> No string belongs to
two different groups.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">How come Fact 2? <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Okay, suppose a string S belongs to two <i>different</i> groups, say group of S1 and
group of S2. This means S is finitely different from S1 and S2 is finitely
different from S.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">But then the Easy result we just proved
implies that S2 is finitely different from S1.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">But then S2 and S1 should be in the <i>same</i> group, which isn’t true. QED.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">(My mathematician friends will realize
that I am partitioning the set of strings into equivalence classes, but enough
of that.)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Fact 1 and Fact 2 taken together can be
combined to give us:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "georgia" , "serif";">Big Fact</span></i></b><i><span style="font-family: "georgia" , "serif";">:
<span style="color: #c00000;">If the set of all possible strings is partitioned
into groups whose members are all finitely different from each other, then
every string will belong to </span></span></i><span style="color: #c00000; font-family: "georgia" , "serif";">exactly one of these <i>groups.<o:p></o:p></i></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span style="font-family: "georgia" , "serif";">Winning
the Infinite Game<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">For what follows, we shall assume that
all demons have infinitely good vision, infinite memory and infinite computing
power.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Any objections ? None ? Very well then…</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , serif;"><br /></span><span style="font-family: "georgia" , "serif";"></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigdzgVskMm8nyFHSB6mPQQCaEnr_vrrNggsLpNNyk6lMfLkJBWtjlI1tiUWau175zPUySNAJpEod61qUxkiEPxy2NL648Fg859KfX9dw1SPtI8e87yfKO7bPJk-CZvOlrwlgD0EWxcIqVK/s1600/axiom+of+choice.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigdzgVskMm8nyFHSB6mPQQCaEnr_vrrNggsLpNNyk6lMfLkJBWtjlI1tiUWau175zPUySNAJpEod61qUxkiEPxy2NL648Fg859KfX9dw1SPtI8e87yfKO7bPJk-CZvOlrwlgD0EWxcIqVK/s1600/axiom+of+choice.png" /></a></div>
<br />
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">As a first step, the demons combine their
infinite computing power to split the set of all possible strings into groups
of finitely different strings as above.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">The next step is to form a <i>Representative Set, R,</i> consisting of
exactly one string from each group. (Each string in the set ‘represents’ its
group, hence the name.)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">At the third step, the infinite memory
comes in handy – the demons simply memorize all the strings in the
Representative Set. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">(Note: Infinitely many different
Representative Sets can be formed. The demons come to a consensus about which
one to use before memorizing it.)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Now the demons line up as commanded and
the Lord of Might places the hats. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">With their infinitely good vision, each
demon can see all the hats in front of them.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "georgia" , "serif";">Theorem:<o:p></o:p></span></i></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="color: #c00000; font-family: "georgia" , "serif";">There will be a unique string, S*, in the Representative Set, R,
which is finitely different from the string of hats on the demons’ heads.<o:p></o:p></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="color: #c00000; font-family: "georgia" , "serif";">Furthermore, each demon can identify S* simply by looking at the
hats in front of it.<o:p></o:p></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "georgia" , "serif";">Proof: </span></i></b><span style="font-family: "georgia" , "serif";">Let S represent the string of hats. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">By the Big Fact stated above, S belongs
to exactly one group of finitely different strings.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">R consists of exactly one string from
each such group.<b><i><o:p></o:p></i></b></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">So, S* will be the string in R which
belongs to the group where S lies.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Given how the groups were formed, S and
S* will be finitely different.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Consider Demon 17, say. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">It can see all but the first 17 entries
of S. (The 17 missing entries being the hat on its own head and the ones
behind.)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Demon 17 simply goes through each string
in R, picks out all but the first 17 letters and compares the result with what
it sees. (Infinite computation power, remember?)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Now, suppose a string in R differs from S
at infinitely many locations.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Removing 17 letters will reduce the
number of different locations by at most 17 – ie, the truncated strings will
still be infinitely different.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Conversely, if a string in R is only
finitely different from S then, after truncation, the number of differences
will either stay the same or decrease – ie, the truncated strings will be
finitely different.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">From what we stated above, <i>only</i> S* is finitely different from S and
everything else in R is infinitely different. Hence, Demon 17 can identify S*
because <i>only</i> the truncated version of
S* will be finitely different from the string of hats it can see.<i><o:p></o:p></i></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Clearly, the argument above applies to
Demon 1, Demon 113274 or any demon in the line.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "georgia" , "serif";">QED<o:p></o:p></span></i></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">So, finally we have it.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">The hats are placed, the demons see
what’s in front of them, and <i>all of them
identify S*</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">When the questioning begins, Demon 1
whispers the first colour in the string S*, Demon 2 whispers the second colour
and so on.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Since S* differs from the actual string S
in finitely many places, all but finitely many of them get their hat colour
correct.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">To really see how surprising this is,
imagine R’Holler walking down the line hearing the answers. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Initially, there are some mistakes, but
at some point he passes the last location where S and S* differ, <i>and from then on every single demon gets its
hat colour correct</i> !!!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">In fact, if you think about it, it
doesn’t matter if the Lord of Might walks down the line from back to front. He
can question them in <i>any order</i> and
the result is the same !!!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">If you think a bit more, you will see
that the same strategy works for any number of hat colours with exactly the
same result !!!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">In fact, even with <i>infinitely many</i> different hat colours, you get the same result !!!!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">If <i>that</i>
didn’t get an exclamation from you, I don’t know what will !!!!!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span style="font-family: "georgia" , "serif";">In
Conclusion<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Did the infinite game bother you ? It
ought to.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">In the finite version of the game played
by Shamsay, blocking the information flow between prisoners by changing from
“shout” to “whisper” led to a devastating change of outcome.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">In the infinite version, the lack of
communication matters not at all – in terms of proportions, precisely 0% of
players guess wrong.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Maybe we can figure out the secret by
trying to apply the winning strategy to a finite game with, say, 100 prisoners.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Very well then, we start off with strings
of B and W of length 100.<br />
Now we lump them into groups of finitely different strings.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Result: Since the strings are all
finitely long, <i>any two strings</i> are
finitely different, so the entire set forms a single group !<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">But let’s go on.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Since there’s only one group, the
representative set, R, consists of just one string.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Basically, all the prisoners agree on one
string and memorize it.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Now they are standing in line. What next
?<br />
Well, there’s no problem of “recognizing S*”, since there <i>is</i> only one S* in the set R<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">And the strategy ? When Shamsay comes up,
simply recite the letter of S* corresponding to your location.<br />
<br />
Does the strategy work ? Yes it does !!<br />
You see, if they are very lucky, the string of hats on their heads is exactly
S*, whereby everyone is right. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Or it could be that only one person gets
it wrong. Or all hundred. Or any number in between.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "georgia" , "serif";">In
every single case above, only finitely many people get their hat colour wrong,
which is precisely what the strategy promised to achieve.<o:p></o:p></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "georgia" , "serif";"><br /></span></i></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilIJIKMUs26srhIcvkgtOnT6Zab8VGGKRca3meYjkK9_JWPjSHF9oyus5oqWIRNv9q7nScF7CdZaBIvtKWw1d6N0JvK9ORo-JpBJ9idwPl5cq-YndnRWEWnICG7BThBlMCosOr7YXp137o/s1600/Inf+Spiral.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilIJIKMUs26srhIcvkgtOnT6Zab8VGGKRca3meYjkK9_JWPjSHF9oyus5oqWIRNv9q7nScF7CdZaBIvtKWw1d6N0JvK9ORo-JpBJ9idwPl5cq-YndnRWEWnICG7BThBlMCosOr7YXp137o/s640/Inf+Spiral.jpg" width="640" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , serif;">So there you have it. A strategy which
worked with almost magical power for infinitely many players is totally
ineffective, even ridiculous, in the finite world.</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Which brings me to a point I have often
made about infinity.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">Most people imagine infinity by starting
with something very big, then making it bigger and bigger and bigger and then it
becomes, you know, infinite.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "georgia" , "serif";">But when you think carefully, the world
of the infinite reveals completely peculiar features with absolutely no finite
analogue.<o:p></o:p></span></div>
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Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com1tag:blogger.com,1999:blog-2350997094083029715.post-13594840991165522722015-10-14T01:27:00.000-07:002015-10-14T01:36:23.267-07:00The Ladder of Infinity<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: Georgia, serif; text-align: left;">Infinity comes in many different sizes.</span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif";">Yes, it’s true. Get over it.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">Or at least, let’s think about <i>why</i> it so difficult to get over it.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">After all, nobody is particularly
bothered when I say “Numbers come in many sizes”. <o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif";">I suspect the reason is that when people
hear “Infinity” they think of the<o:p></o:p></span></div>
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<b><span style="font-family: "Georgia","serif";">Poetic
Definition:</span></b><span style="font-family: "Georgia","serif";"> Infinity is
the Ultimate Maximum<o:p></o:p></span><br />
<span style="font-family: "Georgia","serif";"><br /></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGUzgf32zPjADxgqPY-WSx5irvmhcJu6lXUQX5s5g052Bto8vkUNiZ6O3YoIJO348_fYfmtSeM9TAFa3Y0J_03fsEeJN9tgLfhQB7m8rceGayhUFm0dsw2wOb_t_V7nAl23jZNVCRbabu8/s1600/Infinity+image.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="425" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGUzgf32zPjADxgqPY-WSx5irvmhcJu6lXUQX5s5g052Bto8vkUNiZ6O3YoIJO348_fYfmtSeM9TAFa3Y0J_03fsEeJN9tgLfhQB7m8rceGayhUFm0dsw2wOb_t_V7nAl23jZNVCRbabu8/s640/Infinity+image.jpg" width="640" /></a></div>
<span style="font-family: "Georgia","serif";"><br /></span>
<span style="font-family: "Georgia","serif";">This notion of Infinity as the Absolute,
the Whole, “Something greater than which nothing can be conceived” has a long
history in human culture.<br />
Notions of the Infinite are frequently associated with a Supreme Being.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif";">Viewed in this light, the idea of
different sizes of infinity is indeed very confusing.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">How can there be a larger or smaller
Ultimate Maximum?<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">The scientific study of infinity – a
branch of mathematics known as Set Theory – begins by descending from the lofty
heights of the Absolute and starts with a<o:p></o:p></span></div>
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<b><span style="font-family: "Georgia","serif";">Prosaic
Definition:</span></b><span style="font-family: "Georgia","serif";"> An infinite
set is a collection of objects that does not end<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif";">Viewed in this light, different sizes of
infinity suddenly look more plausible.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">For instance, consider the collections
{0, 1, 2, 3, 4, 5 ...} and {2, 3, 4, 5, …}<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">Both never end, but the latter has two
fewer members than the former.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif";">Now if you look at {3, 5, 7, 9,….} (the
odd numbers greater than 1), this has infinitely fewer members than <i>both</i> the above.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">Surely it is conceivable that, while all
three sets above are infinite, they have different sizes - <o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">just like the numbers 5, 17 and 5000 are
all bigger than 2, but different from each other?<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">But how can we be sure ? <o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">After all, as everyone knows, </span><a href="http://www.thestorypedia.com/top-8-rajnikanth-jokes-will-blow-mind/rajinikanth-has-counted-to-infinity-twice/"><span style="font-family: "Georgia","serif";"><span style="color: blue;">only Rajinikanth can count to infinity</span></span></a><span style="font-family: "Georgia","serif";">. </span><span style="font-family: Wingdings; mso-ascii-font-family: Georgia; mso-char-type: symbol; mso-hansi-font-family: Georgia; mso-symbol-font-family: Wingdings;">J</span><span style="font-family: "Georgia","serif";"><o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">Let’s start at the very beginning – with
the idea of a Set.<o:p></o:p></span></div>
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<b><span style="font-family: "Georgia","serif"; font-size: 12.0pt; line-height: 115%;">The Joy of Sets<o:p></o:p></span></b></div>
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<span style="font-family: "Georgia","serif";">A set is a collection of objects. (Some
technical conditions apply but they can wait.)<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">We take the objects belonging to the
collection and put curly brackets around them.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">{1, 2, 3} is a set. So is {a, b, c} or {Jack,
Jill}.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">The objects belonging to a set are called
its <i>members</i>. <o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">1 is a member of {1, 2, 3} and b is a
member of {a, b, c}, but 1 is not a member of {a, b, c} and vice versa.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">Of particular interest to us will be <b>N</b>, the set of positive integers (called
“Natural Numbers” by mathematicians).<o:p></o:p></span></div>
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<b><span style="font-family: "Georgia","serif";">N</span></b><span style="font-family: "Georgia","serif";"> = {0, 1, 2, 3, 4,….} is an example of an <i>infinite </i>set, as opposed to the other sets we saw above which are <i>finite</i>.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif";">Once you have a few sets in hand, you can
more sets out of them, using a couple of operations.<o:p></o:p></span></div>
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<b><i><span style="font-family: "Georgia","serif";">Union</span></i></b><b><span style="font-family: "Georgia","serif";">:</span></b><span style="font-family: "Georgia","serif";"> Given two sets A and B, their union </span><span style="font-family: Georgia, serif;">A ∪ </span><span style="font-family: Georgia, serif;">B, consists of all the
members that belong to A or to B.</span><br />
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">So, if A = {1, 2, 3} and B = {2, 4, 6},
then </span><span style="font-family: "Georgia","serif";">A ∪</span><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> B = {1, 2, 3, 4, 6}<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Using this idea repeatedly, you can
define the union of any number of sets or even an infinite number of sets.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">So, for instance, if A<sub>1</sub> = {0},
A<sub>2</sub> = {0, 1}, A<sub>3</sub> = {0, 1, 2} and so on, we can take the
union of all the sets to get A<sub>1</sub> ∪</span><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> A<sub>2</sub> ∪</span><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> A<sub>3 </sub></span><span style="font-family: "Calibri","sans-serif"; font-size: 11.0pt; line-height: 115%; mso-ansi-language: EN-NZ; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-text-raise: -4.5pt; position: relative; top: 4.5pt;">∪ ...</span><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> = {0,1, 2, 3, 4, ….} =
Our old friend <b>N<o:p></o:p></b></span><br />
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhL4mE9iIQuTantVA1yTf_LV5bdgosQQ5XGtzUSR-SuzquijE86CJtwcLiQm8pvEog6JlED26oxh_xb1R5YkSKkuaI4csDKtXOAMOSp13lGa9OoXfK4r5H0PDhdZvvmUQtneXEl2iRseNsH/s1600/venn+diagram.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="148" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhL4mE9iIQuTantVA1yTf_LV5bdgosQQ5XGtzUSR-SuzquijE86CJtwcLiQm8pvEog6JlED26oxh_xb1R5YkSKkuaI4csDKtXOAMOSp13lGa9OoXfK4r5H0PDhdZvvmUQtneXEl2iRseNsH/s400/venn+diagram.png" width="400" /></a></div>
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<b><i><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Intersection:</span></i></b><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> </span><span style="font-family: "Georgia","serif";">Given
two sets A and B, their intersection A ∩</span><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> B, consists of all the
members which belong to <i>both</i> A and B<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">So, if A = {1, 2, 3} and B = {2, 4, 6},
then </span><span style="font-family: "Georgia","serif";">A ∩</span><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> B = {2}<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">What if A and B have <i>nothing</i> in common? Let’s say if A = {1, 2, 3} and B = {4, 5, 6}?<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Well, in that case, </span><span style="font-family: "Georgia","serif";">A ∩</span><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> B is defined to be the <b><i>Empty
Set</i></b>, which contains nothing at all.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">The Empty Set is denoted by { }, (there’s
nothing inside the brackets) or the letter ∅</span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Just like unions, you can also talk about
the intersection of infinitely many sets, although we won’t make much use of
them in this post.<o:p></o:p></span></div>
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<b><i><span style="font-family: "Georgia","serif";">Subsets:</span></i></b><span style="font-family: "Georgia","serif";"> A set A is said to be <i>a subset</i> of another set B, if <i>every member</i> of the set A also belongs
to B.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">So for instance {2} and {2, 6} are both
subsets of B ={2, 4, 6}. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">But {2, 5} is not.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">Two important things to note: <o:p></o:p></span></div>
<div class="MsoListParagraphCxSpFirst" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "Georgia","serif"; mso-bidi-font-family: Georgia; mso-fareast-font-family: Georgia;">-<span style="font-family: 'Times New Roman'; font-size: 7pt; font-stretch: normal;">
</span></span><!--[endif]--><span style="font-family: "Georgia","serif";">Any
set is a subset of itself. So {2, 4, 6} is a subset of {2, 4, 6}. (Check that
the definition is satisfied!)<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpLast" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "Georgia","serif"; mso-bidi-font-family: Georgia; mso-fareast-font-family: Georgia;">-<span style="font-family: 'Times New Roman'; font-size: 7pt; font-stretch: normal;">
</span></span><!--[endif]--><span style="font-family: "Georgia","serif";">The
Empty Set, { }, is always a subset of every set.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">These last two facts often confuse
newbies to set theory, but here’s a way to think about it:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">To generate a subset of a given set – go
through every member of the set, one by one, and make a decision about whether
to include that member or not.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "Georgia","serif";">Every
possible set of decisions corresponds to a valid subset.<o:p></o:p></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">So, if A = {3, 4, 5}, you could decide to
include only the first member and exclude the rest.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">This gives you the subset {3}.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">If you decide to include the first two
and exclude the third, you get {3, 4}<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">If you decide to include <i>all</i> the members, that gives you the set
A.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">If you decide to include <i>none</i>, you get the empty set.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "Georgia","serif";">Exercise 1:</span></i></b><span style="font-family: "Georgia","serif";"> If a set, A, has n members, show that
there are 2<sup>n</sup> possible subsets of A<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">With subsets under our belt, we come to a
crucial concept for studying infinity.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Firstly, note that <i><u>a set can have other sets as its members</u></i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">So, for instance: A = { {1, 2} , {3, 4,
5}, {a, b} } is a set.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Note that {1, 2} and {a,b} are <i>members</i> of A. But { {1,2}, {a,b}} is a <i>subset</i> of A.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">However, {1, 2, 5} is <i><u>not a member </u></i>of A. (Neither is it
a subset)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Now, are ready to define:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "Georgia","serif";">Power Set:</span></i></b><span style="font-family: "Georgia","serif";"> The power set of a set A is the set P(A)
consisting of all subsets of A.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Best to illustrate this by example:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">If A = {1, 2, 3}, the power set of A is
given by:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">P(A) = { { }, {1}, {2}, {3}, {1, 2}, {1,
3}, {2, 3}, {1, 2, 3} } <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">(Note carefully the placement of curly
brackets and commas)<o:p></o:p></span><br />
<span style="font-family: "Georgia","serif";"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhip4WITGDRSKem21E5XcFxZcTZHhykQNfnxOdNnE9WICkFPmT0Fu3kNvOuImhX8HTs0r6JMrttJhyMJUgU3BiH5fYVDVYmZIq8uV4j9zt9pjiD21onJ6rY5ZzPnjQsz_mwHfz9eDQvQdUg/s1600/Power-Set.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhip4WITGDRSKem21E5XcFxZcTZHhykQNfnxOdNnE9WICkFPmT0Fu3kNvOuImhX8HTs0r6JMrttJhyMJUgU3BiH5fYVDVYmZIq8uV4j9zt9pjiD21onJ6rY5ZzPnjQsz_mwHfz9eDQvQdUg/s400/Power-Set.jpg" width="400" /></a></div>
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "Georgia","serif";">Exercise 2: <o:p></o:p></span></i></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">(easy) If A has n members, show that P(A)
has 2<sup>n</sup> members<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">(harder) If A = {1, 2}, write down P(
P(A) ) (“power set of power set of A”)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Note that everything we are doing can be
defined for an infinite set like <b>N</b>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">The power set of <b>N</b> will consist of finite subsets like {3, 5, 7} and well as
infinite subsets like<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">{All the odd numbers}<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">So, what’s all this got to do with
different sizes of infinity?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Well, the language of sets allows us to
define <i>sizes of sets</i> – finite or
infinite – in a very precise way. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">In fancy math-speak, the word used for
“size” is “<i>cardinality</i>”, but we’ll
stick with “size”<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span style="font-family: "Georgia","serif"; font-size: 12.0pt; line-height: 115%;">Sizing Things Up<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Imagine you have forgotten how to count.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Is there any way for you to distinguish
the size of two different sets ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">For instance, is there any way for you to
recognize and convey the fact that {1, 2, 3} has just as many members as {a, b,
c}, but fewer members than {a, b, c, d, e} ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Turns out that there is.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">In fact, this is how pretty much
everybody learns to count but then forgets !<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">It’s the very basic idea of <i>pairing things up</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">The idea is, given two sets A and B, I
will try to pair up every member of A with some member of B - with the
stipulation that <i>different members of A
must be paired with different members of B</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">(In fancy math-speak, a pairing as above
is called a </span><a href="https://en.wikipedia.org/wiki/Injective_function"><i><span style="font-family: "Georgia","serif";"><span style="color: blue;">one-to-one
function</span></span></i></a><span style="font-family: "Georgia","serif";">)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">In what follows, I will use the notation
“x -> y” as shorthand for “x is paired with y”<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">So, if A = {1, 2, 3} and B = {a, b, c},
then: 1 –> a, 2 –> b, 3 –> c is
a valid pairing. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">So is: 1 –> b, 2 –> c, 3 –> a. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">But: 1 –> a, 2 –> a, 3 –> b is
invalid because 1 and 2 are both paired with a.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">But if we took, B ={a, b, c, d, e}
instead, then 1 –> a, 2 –> b, 3 –> c or 1 –> b, 2 –> d, 3 –>
e are both valid ways of pairing all members of A with some members of B (there
are many other ways).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">The crucial difference, is that, <i>every</i> member of B = {a, b, c} ended up
getting paired with <i>some</i> member of
{1, 2, 3} but when we took B ={a, b, c, d, e}
instead, then however you try to pair things up, you <i>always end up missing some members of</i> B.
<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Hence, even without knowing how to count,
one can compare sizes of sets, in a way which matches perfectly with our
intuition of one set having “more things in it” than another one.<br />
Let’s make this official.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhR78a-m42AjLjooUTHOwnuB0TQ1MmoU14yt6q8TivtE3xqWUXxhpmjGmSG3iZvQAsUpa3aabx2czTS-mgiNUlXQfzplrj_BkM56h-yOOTbIqJnsfH30z1i2wzxiItEJcnZkyWxPvmkNniN/s1600/one-to-one-function.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="133" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhR78a-m42AjLjooUTHOwnuB0TQ1MmoU14yt6q8TivtE3xqWUXxhpmjGmSG3iZvQAsUpa3aabx2czTS-mgiNUlXQfzplrj_BkM56h-yOOTbIqJnsfH30z1i2wzxiItEJcnZkyWxPvmkNniN/s400/one-to-one-function.png" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<b><i><span style="font-family: "Georgia","serif";">Definition 1:</span></i></b><span style="font-family: "Georgia","serif";"> Given two sets A and B, we say that A ≤
B (Size of A is less than or equal to size of B) if there is <i>some way</i> of pairing up <i>all members</i> of A with <i>some members</i> of B (but not necessarily
all of them)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "Georgia","serif";">Definition 2:</span></i></b><span style="font-family: "Georgia","serif";"> Given two sets A and B, we say A = B (A
and B have the same size), if there is <i>some</i>
way of pairing up <i>all members</i> all members
of A with <i>all members</i> of B<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "Georgia","serif";">Definition 3:</span></i></b><span style="font-family: "Georgia","serif";"> Given two sets A and B, we say A < B
(A is smaller than B or B is greater than A) if:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">-There is <i>some way</i> of pairing up <i>all members</i>
of A with <i>some members</i> of B (but not
all)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">- There is <i><u>no way</u></i> of pairing up all members of A with <i>all members</i> of B<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "Georgia","serif";">Exercise 3:</span></i></b><span style="font-family: "Georgia","serif";"> If A is a subset of B, then show that A
≤ B<o:p></o:p></span><br />
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Notice that proving A = B is relatively
easy because you just have to figure out <i>one
possible way </i>to pair up all the members of the two sets.<br />
Showing that A < B is much trickier. It’s not enough to show <i>some</i> attempt at pairing up the two sets
doesn’t work – <i>you need to show that
every possible attempt at pairing will fail. <o:p></o:p></i></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">This requires a good dose of ingenuity as
we shall see later.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">So, with these definitions in hand, we
can go ahead and say that {1, 2, 3} ≤ {a, b, c} and <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">{1, 2, 3} ≤ {a, b, c, d, e}.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Furthermore, {1, 2, 3} = {a, b, c}, but
{1, 2, 3} < {a, b, c, d, e}<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">“Now this is all very interesting”, you
say, “but the fact is I <i>do know</i> how
to count and I could have told you this much earlier. Why did we take such a
long winded route to this easy conclusion?”<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "Georgia","serif";">Because
– and this was the great insight of George Cantor, father of Set Theory – this
method of comparing sizes of sets by pairing works perfectly well for infinite
sets, whereas counting fails miserably !!<o:p></o:p></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">In another words, we have just managed to
extend the concept of counting from finite to infinite collections.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">So, without further ado, allow me to
introduce…<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span style="font-family: "Georgia","serif"; font-size: 12.0pt; line-height: 115%;">The Smallest Infinity<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Let <b>N</b>
= {0, 1, 2, 3, …..}, be the natural numbers. This is the smallest possible
infinite set.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">What do I mean by that exactly? Well,
this.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "Georgia","serif";">Theorem:</span></i></b><span style="font-family: "Georgia","serif";"> For infinite set A, <b>N</b> ≤ A<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "Georgia","serif";">Proof: </span></i></b><span style="font-family: "Georgia","serif";"> Pick any member from A, pair it up with 1.
Now pick up another member, pair it up with 2. Keep going. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Now, if we run out of members to pick up
at the k-th step for some number k, then A had only k members. But that means A
was finite!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">So, the process must continue until every
member of N gets paired up with some members of A, but not necessarily all.
Hence, by definition, <b>N</b> ≤ A. □<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">But are we really sure of this? After all
remember the sets {2, 3, 4, 5,…} and the odd numbers {1, 3, 5, 7,…} ? We had
suspected that these were <i>smaller</i>
than <b>N</b>, but how is that possible if
N is the smallest infinity?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "Georgia","serif";">The
answer is that both the sets above have the same size as <b>N</b></span></i><span style="font-family: "Georgia","serif";">.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Look at the pairing 0 -> 2, 1 -> 3,
…., k -> k + 2,….<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Convince yourself that using this formula
every single member of {2, 3, 4, 5..} gets paired up with N. Hence they have
the same size.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Similarly, 0 -> 1, 1 ->3, 2 ->
5, …, k -> 2k + 1, …. pairs up N and all the odd numbers.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">So, here we have a property of infinite
sets that has absolutely no analogue in the world of the merely finite:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "Georgia","serif";">It
is possible to remove members from an infinite set – even infinitely many of
them – and still end with a set of the same size !!<o:p></o:p></span></i><br />
<i><span style="font-family: "Georgia","serif";"><br /></span></i>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJqU5yvzzShSgwoe0qQhF3QhbCFq4jD4t_BUjXXh-V_sLOj-_yWKjFychqoNw85XR64Ii0u_3iwaomyeCvluKGOz-vph0y3sgRwDkYLYk09169y3rQ5apxOC8Za7mt8rBQ5hMsV-6PPYBH/s1600/Hilbert+hotel.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJqU5yvzzShSgwoe0qQhF3QhbCFq4jD4t_BUjXXh-V_sLOj-_yWKjFychqoNw85XR64Ii0u_3iwaomyeCvluKGOz-vph0y3sgRwDkYLYk09169y3rQ5apxOC8Za7mt8rBQ5hMsV-6PPYBH/s640/Hilbert+hotel.jpg" width="640" /></a></div>
<i><span style="font-family: "Georgia","serif";"><br /></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">What about the other direction? What if
we <i>add</i> more things to <b>N</b> ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">How about S = {0, 0.5, 1, 1.5, 2, 3, 4,
5, … } where we threw in two more numbers ? Maybe that’s a bigger infinity?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">No such luck.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">0 -> 0, 1 -> 1.5, 2 -> 1, 3
-> 1.5, 4 -> 2, 5 -> 3, k -> k – 2 for all subsequent numbers k,
and we have paired up <b>N</b> and S with
nothing left over.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">In fact, you could have thrown in a
million, a trillion or indeed, <i>any finite
number</i> of extra members to <b>N</b>
without changing its size at all!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Ok, then, let’s get serious.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Let S = {0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5,
…. } (every whole number and whole number + 0.5)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Now we have thrown in infinitely more
things. Could this give something bigger ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">0 -> 0, 1 -> 0.5, 2 -> 1,…., k
-> 0.5k for all numbers k and once again, our hopes are dashed as
everything gets paired up.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">It keeps getting worse. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">You can add a million sets to <b>N</b>, or a gazillion – each having the
same size as <b>N</b> – and still the size
remains stubbornly unchanged!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">In desperation, we go all the way.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Consider an <i>infinite</i> collection of sets, A<sub>0</sub>, A<sub>1</sub>, A<sub>2</sub>
and so on, one for each whole number.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "Georgia","serif";">Let
each of these sets be individually as large as <b>N</b></span></i><span style="font-family: "Georgia","serif";">.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhN_EgLQ1mOZ9syu0xnDJy-VBTtKwPnQP4IybQK1T1hn3lNsf6xKOLXio2qrd_AbOfufQO9ifd9jBaF9VOTAQdM-rkfXoEKM1oq60LFQeIKVcH7jnQ139Uak0CTeRY4RXQWKabY3dcBw4xE/s1600/countable.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="208" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhN_EgLQ1mOZ9syu0xnDJy-VBTtKwPnQP4IybQK1T1hn3lNsf6xKOLXio2qrd_AbOfufQO9ifd9jBaF9VOTAQdM-rkfXoEKM1oq60LFQeIKVcH7jnQ139Uak0CTeRY4RXQWKabY3dcBw4xE/s320/countable.jpg" width="320" /></a></div>
<span style="font-family: "Georgia","serif";">How large is this entire infinite collection
of sets taken altogether?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">The completely stunning – but unequivocal
– answer is:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "Georgia","serif";">Just
as large as <b>N</b>, not one bit more</span></i><span style="font-family: "Georgia","serif";">!!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">At this point, you may legitimately
wonder why we called <b>N</b> the <i>smallest</i> infinity.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Every infinite set which looked like it
might be smaller than <b>N</b> ended up
being the same size. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">On the other hand, every attempt to get
something <i>bigger</i> than <b>N </b>failed as well.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Maybe it’s time to call it quits and
admit that there is only <i>one</i> infinity
– the size of the set <b>N</b>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Not quite.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span style="font-family: "Georgia","serif"; font-size: 12.0pt; line-height: 115%;">A Bigger Infinity<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Ladies and gentlemen, allow me to
present:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "Georgia","serif";">Cantor’s Theorem:</span></i></b><span style="font-family: "Georgia","serif";"> Let A be any set, finite or infinite.
Then A < Power set of A<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><i><span style="font-family: "Georgia","serif";">Proof:</span></i></b><span style="font-family: "Georgia","serif";"> Let’s explicitly write down what we need
to prove, first.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Let P(A) denote the power set of A. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">By definition of A < P(A) we need to
show two things<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">1) There is <i>some way</i> of pairing up <i>all members</i>
of A with <i>some members</i> of P(A) (but
not all)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">2) There is <i><u>no way</u></i> of pairing up all members of A with <i>all members</i> of P(A)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">To illustrate the ideas, I will use the
set <b>N</b>, but the proof works for
absolutely any set.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Step 1) is easy. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Note that P(A) consists of all possible
subsets of A. In particular, for any member x belonging to A, the set {x}
belongs to P(A). So, simply pair x -> {x} for each x and we are done.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">To illustrate using <b>N</b>, we just pair 0 -> {0}, 1 -> {1}, 2 -> {2} and so on….<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Step 2) was Cantor’s stroke of genius.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Remember, we need to show that <i>no conceivable pairing</i> will between members
of A and P(A) will work.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">So, how do we check through all possible
pairings, especially if there are infinitely many of them ? <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">We don’t<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Instead, we show that <i>no matter what pairing recipe was given to
you</i> to link all members of P(A) and A, it couldn’t possibly have worked.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">So, let’s say the World’s Biggest Math
Genius has given you a possible way of pairing them all up. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">As per his formula:<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpFirst" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "Georgia","serif"; mso-bidi-font-family: Georgia; mso-fareast-font-family: Georgia;">-<span style="font-family: 'Times New Roman'; font-size: 7pt; font-stretch: normal;">
</span></span><!--[endif]--><span style="font-family: "Georgia","serif";">Every
member x gets paired with some subset of A, which we will call “Pair of x”<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpLast" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "Georgia","serif"; mso-bidi-font-family: Georgia; mso-fareast-font-family: Georgia;">-<span style="font-family: 'Times New Roman'; font-size: 7pt; font-stretch: normal;">
</span></span><!--[endif]--><span style="font-family: "Georgia","serif";">More
importantly, for <i>any</i> subset of A,
there is some member of A which is paired with this subset.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgn8S6D9a6NdGem6HoA6YrQGdZjLeVuQW-7EszXkCdKseP8iKu-hnKno8eokwYpibaXHoXZph2wqlJsYspFqSE0OnvEuijes78qSdXk4HFKbzR7ZewBSoZxmH5FJtuvsT8869h0IEy6NSnf/s1600/Diagonal_argument_powerset.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="345" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgn8S6D9a6NdGem6HoA6YrQGdZjLeVuQW-7EszXkCdKseP8iKu-hnKno8eokwYpibaXHoXZph2wqlJsYspFqSE0OnvEuijes78qSdXk4HFKbzR7ZewBSoZxmH5FJtuvsT8869h0IEy6NSnf/s400/Diagonal_argument_powerset.png" width="400" /></a></div>
<span style="font-family: "Georgia","serif";">We will use Mr. Genius’ pairing formula
against him, by constructing a Rogue Subset as follows:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">For any x, look at the subset “Pair of x” as per his formula.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Now ask the question, Does “Pair of x”
contain x?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">If the answer is “No”, then include x in
Rogue Subset.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">If the answer is “Yes”, then don’t
include x in Rogue Subset.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">To illustrate, suppose as per Mr.
Genius’s formula:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">0 -> {1, 2, 3}, 1 -> {1, 3, 5, 7,…,
all odd numbers} and 4 -> {1, 6, 11, 13}.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Then, Rogue Subset would include 0 and 4,
but not include 1.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Now take a moment to realize that:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span style="font-family: "Georgia","serif";">The
Rogue Subset cannot have been paired with any member of A.<o:p></o:p></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">How so? <i>Because you intentionally made it that way.</i><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">If the subset paired to a member
contained that member then Rogue Subset wouldn’t contain it and vice versa. So,
it is <i>guaranteed</i> that you won’t find
Rogue Subset anywhere on the list of pairs.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Note that this doesn’t depend on any
particular recipe for pairing.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">If Mr. Genius realized his mistake and
gave you another pairing recipe, then you would just follow the same procedure
for <i>that </i>recipe and end up with <i>another</i> Rogue Subset.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Every possible attempt is doomed to
failure and this completes Step 2)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">This we have shown that A < P(A)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif";">Done !! □<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif";">Let’s just take a deep breath and
appreciate the magnitude of what we just proved.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">For finite sets, this result is not a big
deal because n < 2<sup>n</sup> for any number n (Exercise 1).<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif";">But the true magic of Cantor’s Theorem
shines through when A is an infinite set.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">For at last, we can break through the
tyranny of <b>N</b> and soar upwards to a
provably bigger infinity, the power set of <b>N</b>.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">With that small (or is it infinitely large?)
step, our journey has begun.<o:p></o:p></span></div>
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<br /></div>
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<b><span style="font-family: "Georgia","serif"; font-size: 12.0pt; line-height: 115%;">The Ladder of Infinity<o:p></o:p></span></b></div>
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<span style="font-family: "Georgia","serif";">Start by renaming <b>N</b>. Call it I<sub>0</sub> – the “zero-th infinity”.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">We now know that <b>N</b> < P(<b>N</b>), the power
set of <b>N</b>.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">Call that I<sub>1</sub><o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif";">But what worked once, works again. So, P(<b>N</b>) < P(P(<b>N</b>)). Call that I<sub>2</sub>.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">Repeat. Repeat. Repeat again.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif";">What you get is a chain of infinities,
each bigger than the last – I<sub>0</sub> < I<sub>1</sub> < I<sub>2 </sub><
.... < I<sub>1000</sub> < …<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">In other words, not only are there
different sizes of infinity, you get <i>infinitely
many different sizes</i> - <i>one for every
whole number</i>.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif";">(Quick Question: How do we know there are
no infinities between I<sub>0</sub> and I<sub>1</sub> ?<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";"> Quick Answer: This is the </span><a href="https://en.wikipedia.org/wiki/Continuum_hypothesis"><span style="font-family: "Georgia","serif";"><span style="color: blue;">Continuum Hypothesis</span></span></a><span style="font-family: "Georgia","serif";"> and way beyond the scope of this post.)<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif";">Leaving already? But we are just getting
started.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif";">Remember how we are allowed to take unions
of sets?<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">Well then, once we get our infinity of
infinite sets, each bigger than the last, let’s take the union them all !<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif";">Look at the Jumbo Set, J = I<sub>0</sub> ∪</span><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> </span><span style="font-family: "Georgia","serif";">I<sub>1</sub> ∪</span><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> </span><span style="font-family: "Georgia","serif";">I<sub>2</sub> ∪</span><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> ….∪</span><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> ….<o:p></o:p></span></div>
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<i><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">J
is a strictly bigger infinite set than all the I’s !!<o:p></o:p></span></i><br />
<i><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br /></span></i></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Why ? Okay, suppose, instead, that J = I<sub>k</sub>
for some integer k. <o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">But that can’t be, because J contains all
the members of I<sub>k+1</sub> and I<sub>k</sub> < I<sub>k+1</sub><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">So, I<sub>k</sub> < J for all I<sub>k</sub>
– in other words, J is bigger than all the I’s.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">So, we can think of J as the “infinity-th
infinity”, larger than anything that came before.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Let’s call it I<sub>ω</sub> <o:p></o:p></span></div>
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<br /></div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwSmUoyU0z6REoonLfW8vzURpRFzBty5aLGkk0vxPZn-ZH0nwOfs2NMsPuaGcg5HWJWjT2xzeXTgUFG7HxCX5QOqVtJZhsz0xdktyacSeHWTcRxS7OF7BagkWpEy6TgPC9MII6TeyWWr_x/s1600/Ladder.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="478" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwSmUoyU0z6REoonLfW8vzURpRFzBty5aLGkk0vxPZn-ZH0nwOfs2NMsPuaGcg5HWJWjT2xzeXTgUFG7HxCX5QOqVtJZhsz0xdktyacSeHWTcRxS7OF7BagkWpEy6TgPC9MII6TeyWWr_x/s640/Ladder.jpg" width="640" /></a></div>
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">But now the fun starts all over again –
for nothing stops us from looking at P(J), P(P(J)) and so on to get I<sub>ω</sub> < I<sub>ω+1</sub> < I<sub>ω+2</sub>
< ….<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">And yes, when this chain gets exhausted,
we take the union once again to get a super jumbo sized infinite set I<sub>2ω</sub><o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Power set, power set, power set, union,
repeat, repeat… – I<sub>3ω</sub> , I<sub>4ω
, </sub>I<sub>5ω</sub> , ….<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Union them all and out pops I <sub> ω<sup>2 </sup> </sub></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Plodding ever onwards, we get </span>I <sub>ω<sup>3 </sup> </sub><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> </span><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> </span>I <sub>ω<sup>4 </sup> </sub><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> </span>I <sub>ω<sup>5 </sup> </sub><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> ….. Then eventually </span>I<sub>ω<sup>ω</sup> </sub></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">And beyond that as well…<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">At this point, you may turn around and
ask:<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">“How many different sizes of infinity are
there? How many rungs on the ladder of infinity?”<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Infinitely many, obviously, but now we
have learned to calibrate more finely.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">I<sub>0</sub> different sizes of infinity
? I<sub>1</sub> maybe ? Surely not I<sub>ω</sub> ?<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">It can be proved that, </span><a href="https://en.wikipedia.org/wiki/Ordinal_number"><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><span style="color: blue;">in a very well defined sense</span></span></a><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">, there is an I<sub>1</sub>-th infinity, an I<sub>2</sub>-th
infinity, an I<sub>ω</sub>-th infinity, even an </span>I<sub>ω<sup>ω</sup> </sub> th infinity.</div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">In fact, we can find an infinity - call
it I<sub>Crazy</sub> – such that I<sub>Crazy</sub> is the I<sub>Crazy</sub> th
infinity.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">In other words, I<sub>Crazy</sub> = I <sub>I<sub>Crazy </sub> </sub></span></div>
</div>
<br />
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">That’s crazy ! Yes, I know.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">As for your question, the answer is that
there are more rungs on the ladder of infinity than can be counted by any of
the infinities lying on that ladder !<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Let’s say that once again - not only are
there different sizes of infinity, there are far, far, <i>far </i>more of them than there are numbers. <o:p></o:p></span></div>
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<i><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">In
fact, there are more types of infinity than can be counted by any of the
infinities in our ever increasing ladder.</span></i><br />
<i><br /><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"></span></i>
<br />
<div class="separator" style="clear: both; text-align: center;">
<i><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKbjNW-UCYXH8qpYuH49CJW3r6m2pCsgago2cI01LiFta6m7BpXvH_5RTPYJo0BtmbBusdlJudQXO4wWVyqa4pE53hJrmO_nYNlD93oHhvGhhp0jJ3324wSppU33kQg57902yTol3jIB0B/s1600/Paradox.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="175" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKbjNW-UCYXH8qpYuH49CJW3r6m2pCsgago2cI01LiFta6m7BpXvH_5RTPYJo0BtmbBusdlJudQXO4wWVyqa4pE53hJrmO_nYNlD93oHhvGhhp0jJ3324wSppU33kQg57902yTol3jIB0B/s200/Paradox.png" width="200" /></a></span></i></div>
<i><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><o:p></o:p></span></i></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">But what if we joined them all together
and looked at the “Set of all sets” ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Right away, we have a problem:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Let U be the set of all sets.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">From Cantor’s Theorem, we know that U
< P(U)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">But by definition, U contains <u>all</u>
sets – which means P(U) is a subset of U.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">But then P(U) ≤ U.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Contradiction !<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Something has gone seriously wrong. Maybe
it’s time to slow down and take stock of things.<o:p></o:p></span></div>
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<br /></div>
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<b><span style="font-family: "Georgia","serif"; font-size: 12.0pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">The Universe of Sets<o:p></o:p></span></b></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Infinity has traditionally connoted
something mighty and mysterious, possibly beyond human understanding.<b><o:p></o:p></b></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">But now that we are dealing with
infinities in the plural, it becomes apparent that they behave rather like the
familiar finite (whole) numbers.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Some are bigger than others.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Taking a power set always yields a bigger
infinity, just like adding one always gives a bigger number.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">The enormous profusion of infinities,
with entities like </span>I<sub>ω<sup>ω</sup> </sub><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> or I<sub>Crazy</sub> may
be startling, but perhaps no more so than the existence of </span><a href="http://bekaarbokbok.blogspot.co.nz/2009/09/large-numbers_10.html"><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><span style="color: blue;">really large numbers</span></span></a><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">.</span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">It turns out we can </span><a href="https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic"><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><span style="color: blue;">even do arithmetic with these infinities</span></span></a><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> – add them up, multiply them, take one infinity
to the power of another and so on.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">The fact that there are more different
sizes of infinity than can be counted by any given infinity is precisely
analogous to there being more finite numbers than can be counted by any one
number. <o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Even the problem we saw with our “Set of
all sets” is very similar to what happens when we talk about the “Biggest
possible number” - adding one should give a bigger number which leads to a
contradiction.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">But while it is fairly clear why you
can’t talk about a biggest number, it isn’t quite apparent why we can’t mention
the “set of all sets”.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Isn’t a set just a collection of things ?
So why not the “set of all sets” ?<o:p></o:p></span></div>
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<br /></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Remember how I had qualified my
definition of a set with “technical collections apply” ? <o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">It’s time to apply them now.<br />
<br />
A set is defined as a collection of things which satisfies a set of rules
called the “<a href="https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory"><span style="color: blue;">Zermelo
Fraenkel Axioms</span></a>” – usually abbreviated to ZFC.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">For instance, the ZFC rules say that the
Empty Set is a valid set and taking the power set of a set gives a valid set
(among other things).<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">However, it so happens that the
“collection of all sets” <i>does not</i>
satisfy ZFC and is therefore <i>not</i> a
set.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">So taking its power set and looking at
its size are out of bounds. <o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Contradiction avoided.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">But let’s stick with the analogy just a
little longer.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Once we realize that the finite numbers
never end, it gradually dawns on us that the collection of <i>all</i> finite numbers is not described by some “biggest number”, but
corresponds to something entirely new and incomparably larger. <o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">(Or as your high school teacher would
say, “Infinity is not a number”)<o:p></o:p></span></div>
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<br /></div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisMgwAtVjl54a0GAwUfV9wmFm0tuw7O-eiZprs0dMX1DtCIBzseApc5PWEo4RtUagUM44slEahgk-P4gBgIunEg1zH0Nl_oCShMTGFjClqKttaKhVY23-f5eCZlCrL7hicI7Z3pwO0adKC/s1600/Absolute.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisMgwAtVjl54a0GAwUfV9wmFm0tuw7O-eiZprs0dMX1DtCIBzseApc5PWEo4RtUagUM44slEahgk-P4gBgIunEg1zH0Nl_oCShMTGFjClqKttaKhVY23-f5eCZlCrL7hicI7Z3pwO0adKC/s640/Absolute.jpg" width="640" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">By analogy, although there is never a
largest infinity, can’t we still imagine collecting all sets together to get –
not a “largest possible set”, but “Something Beyond” ?<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Yes, indeed we can.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">What we get is the Universe of Sets.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Associated with the Universe is an
“Infinity beyond all infinities” which Cantor referred to as the Absolute
Infinite.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">We are back to the long neglected “poetic
definition” of Infinity – an ultimate maximum which admits nothing larger. <o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">A deeply religious man, Cantor associated
the Absolute Infinite with God – in contrast to the garden variety infinities
we have been discussing, which he called the “<i>transfinite cardinals</i>”<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">or “transfinite numbers”.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">So, at the end of our excursion into the
set theoretic realm, we are left with:<o:p></o:p></span></div>
<div class="MsoListParagraphCxSpFirst" style="margin-bottom: .0001pt; margin-bottom: 0cm; margin-left: 18.0pt; margin-right: 0cm; margin-top: 0cm; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "Georgia","serif"; mso-bidi-font-family: Georgia; mso-fareast-font-family: Georgia;">-<span style="font-family: 'Times New Roman'; font-size: 7pt; font-stretch: normal;">
</span></span><!--[endif]--><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">The
finite numbers which we all know and love<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "Georgia","serif"; mso-bidi-font-family: Georgia; mso-fareast-font-family: Georgia;">-<span style="font-family: 'Times New Roman'; font-size: 7pt; font-stretch: normal;">
</span></span><!--[endif]--><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">The
transfinite numbers forming our “ladder of infinity”<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "Georgia","serif"; mso-bidi-font-family: Georgia; mso-fareast-font-family: Georgia;">-<span style="font-family: 'Times New Roman'; font-size: 7pt; font-stretch: normal;">
</span></span><!--[endif]--><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">The
Absolute Infinite encompassing and transcending them all<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Or maybe we are thinking too small…<o:p></o:p></span></div>
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<br /></div>
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<b><span style="font-family: "Georgia","serif"; font-size: 12.0pt; line-height: 115%; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Stretching the Universe<o:p></o:p></span></b></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Recall that all sets are required to be
in compliance with the ZFC Axioms.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">It turns out that all finite sets and the
infinite sets – aka transfinite cardinals – we have seen till now can be
explicitly constructed once we have the ZFC Axioms in place.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">For what follows, I’ll call this
collection the “ZFC universe”.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">From here onwards, one possible approach
is the<o:p></o:p></span></div>
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<b><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Conservative
View:</span></b><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> The universe consists
of all sets which may be constructed from the ZFC Axioms.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">In other words, the ZFC universe is everything
you will ever get and you can stop reading now.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Life gets more interesting, however, if
we adopt the<b><o:p></o:p></b></span></div>
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<b><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Liberal
View:</span></b><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"> The universe consists
of all sets which do not contradict the ZFC Axioms.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">To see why, we define a set <b>K</b> with the following three properties.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">a) I<sub>0</sub> < <b>K</b> (where I<sub>o</sub> is <b>N</b>, our smallest infinity)<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">b) If a set A < <b>K</b>, then P(A) < <b>K</b><o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">c) (Union of fewer than <b>K</b> sets which are each smaller than <b>K</b>) < <b>K</b><o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">So, what’s the big deal ? <o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">For starters, since I<sub>0</sub> < <b>K</b>, Property b) implies I<sub>1</sub>
< <b>K</b> (since I<sub>0</sub> = <b>N</b> and I<sub>1</sub> = P(<b>N</b>) ).<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">But repeating the same logic, I<sub>2</sub>
< <b>K</b>, I<sub>3</sub> < <b>K</b> , in fact I<sub>n</sub> < <b>K</b> for every number n.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Now Property c) implies I<sub>ω</sub>
< <b>K</b> since it is just the unions
of all the I<sub>n</sub> <o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Then we use Property b) again….<o:p></o:p></span></div>
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<br /></div>
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<i><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">It
turns out that <b>K</b> is at least as
large as the entire ZFC universe !!!</span></i><br />
<i><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br /></span></i>
<i><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"></span></i>
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimHt_K8vSFxT1tTnAXbUdL2evGHNDfZNg2f5hNgm-2YesYpbEVybdxB90pWnpAVXveTHQthWGl6CWDLX2IgVngKXh1d5LnKoK_kevPNcbohiSeR_nnmVZu8a5t8vlLF5sPx9ObLSyB5Rzs/s1600/Von_Neumann_Hierarchy.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimHt_K8vSFxT1tTnAXbUdL2evGHNDfZNg2f5hNgm-2YesYpbEVybdxB90pWnpAVXveTHQthWGl6CWDLX2IgVngKXh1d5LnKoK_kevPNcbohiSeR_nnmVZu8a5t8vlLF5sPx9ObLSyB5Rzs/s320/Von_Neumann_Hierarchy.png" width="309" /></a></div>
</div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">But hang on, didn’t I just say that the
universe of all sets is too large a collection to be a set ?<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">So am I talking about a set as big as the
universe ? <o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Can I please make up my mind ?<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Well, it all depends on <i>how</i> my mind is made up.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">If I take the Conservative stance, then a
monstrous set like <b>K</b> does not exist,
<i>cannot</i> exist.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">But if I take the Liberal view, then the
ZFC universe is not “The Universe of <i>All</i>
Sets”, it is <i>merely</i> “the collection
of sets which can be constructed from ZFC” and there is no problem at all with
a set as large as <i>this</i> collection, or
even larger.<o:p></o:p></span></div>
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<br /></div>
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<i><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">With
the introduction of <b>K</b>, we have just stretched
the Universe of sets</span></i><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">And what a stretch it is! <o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">To begin with, the once-mighty ZFC
universe, which we were about to associate with Absolute Infinity, deflates to
become the <i>Least Inaccessible Cardinal</i>
- <b>K<sub>0</sub></b><sub><o:p></o:p></sub></span></div>
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<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">And now of course, we can take the power
set of <b>K<sub>0</sub></b>, the power set
of the power set, union of all the power sets, all the good old tricks to get a
new ladder of infinities built atop <b>K<sub>0<o:p></o:p></sub></b></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">But now we have a new trick up our sleeve
!<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Because standing just beyond this new
ladder, we find the <i>next inaccessible cardinal</i>,
<b>K<sub>1</sub></b><sub><o:p></o:p></sub></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">And similarly, we get <b>K<sub>2</sub> K<sub>3</sub></b> and – yes,
you got it –an infinity of inaccessible cardinals.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">And once you got those, triple your fun
by taking the union of them all, then keep going.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Forget the ladder - we are now on the
space elevator of infinity.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">As we casually zoom past the inaccessible
cardinals, it is all too easy to forget that the gap between one and the next
is equivalent to the gulf between the smallest infinity I<sub>0</sub> and the
entire ZFC universe.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">More power sets, more unions, more inaccessibles
– are we stretching the universe of sets to breaking point ?<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Not by a very long shot.</span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Inaccessible cardinals form the bottom
level of the so called <i>Large Cardinal
Hierarchy</i> – the frontier of research in modern set theory.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Inaccessible, Mahlo, weakly compact,
ineffable, measurable, extendible, huge, superhuge… </span><a href="http://cantorsattic.info/Upper_attic"><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><span style="color: blue;">the
chain of Large Cardinals goes on and on</span>.</span></a><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Each step up the hierarchy is a
gargantuan, mindboggling leap, dwarfing the jump from the smallest infinity
(almost invisible from these heights) to the least inaccessible cardinal and
enlarging the set theoretic universe almost beyond comprehension.<o:p></o:p></span><br />
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br /></span>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhibp4jnQtXkaVh6lZWFT_qr6UgnShR7bSe8QeqD-Rt9_KT4JWAo6XfwYkAXPJTcEJFBlANv7Hjp6xOmn7GrFEZR5U4IqxCW-MKk-OKZe89VJ-qs3OwSGSnv-IIYEvtC0_W4N0ZH0KBY4xn/s1600/Galaxy+mountain.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhibp4jnQtXkaVh6lZWFT_qr6UgnShR7bSe8QeqD-Rt9_KT4JWAo6XfwYkAXPJTcEJFBlANv7Hjp6xOmn7GrFEZR5U4IqxCW-MKk-OKZe89VJ-qs3OwSGSnv-IIYEvtC0_W4N0ZH0KBY4xn/s640/Galaxy+mountain.jpg" width="640" /></a></div>
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><br /></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Like ever-frustrated mountaineers, we triumphantly
reach a summit, ready to touch the sky - when the fog rolls back to reveal an
immeasurably higher peak towering overhead and the starry firmament of the
Absolute recedes further than ever.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Does it ever end ? <i>Could</i> it ever end ?<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">These are, after all, mountains of the
mind, unconstrained by any connection to reality.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">What prevents us from spinning out ever
larger cardinals for centuries to come, defining them into existence <i>ad infinitum</i> ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Answer: The ZFC Axioms.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">The only constraint on our “Liberal
worldview” was that our sets should not contradict ZFC. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">It turns out that as we climb the large
cardinal hierarchy, we put ever more pressure on the girdle of ZFC – increasing
the possibility of contradiction – until it finally snaps.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">Bloated with mathematical hubris, we try
to define a Reinhardt cardinal – an ultimate mountain of infinity that “touches
the Absolute” – and </span><a href="http://cantorsattic.info/Kunen_inconsistency"><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><span style="color: blue;">it collapses into inconsistency</span></span></a><span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">.<o:p></o:p></span></div>
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<span style="font-family: "Georgia","serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;">But that is a story for another time.</span></div>
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<br /></div>
Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com1tag:blogger.com,1999:blog-2350997094083029715.post-3893507243916518652015-05-18T18:11:00.001-07:002015-10-14T01:30:36.995-07:00Warp Speed Ahead ? <div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgpFC9XUdyTlFN_rlJiiHf03casRvy56HcCHZGEeV3-jx53MKPs9wqSYi_ghc8xzoDJPL71_aK7chT7d17YDjjkJZL9YjYMO_UGdCCXO_MbtbzZsTKCxP5AljTKppJbJ1VAJJvmMpBV8p1/s1600/Enterprise+at+warp+speed+1440x900.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgpFC9XUdyTlFN_rlJiiHf03casRvy56HcCHZGEeV3-jx53MKPs9wqSYi_ghc8xzoDJPL71_aK7chT7d17YDjjkJZL9YjYMO_UGdCCXO_MbtbzZsTKCxP5AljTKppJbJ1VAJJvmMpBV8p1/s640/Enterprise+at+warp+speed+1440x900.jpg" width="640" /></a></div>
<span lang="EN-NZ" style="font-family: "Georgia","serif";"><br /></span>
<span lang="EN-NZ" style="font-family: "Georgia","serif";">“NASA May Have Accidentally
Created a Warp Field ” went the <a href="http://mysteriousuniverse.org/2015/04/nasa-may-have-accidentally-developed-a-warp-drive/" target="_blank"><span style="color: blue;">breathless proclamation</span></a>.<o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-NZ" style="font-family: "Georgia","serif";">This was on April 24, 2015 on
a site called “Mysterious Universe”. Within a few days the news echoed across
the Internet with dozens of websites and newspapers chiming in and the space
travel fanatics going gaga.</span><br />
<span lang="EN-NZ" style="font-family: "Georgia","serif";"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">About a week or two later,
scientists tried to push back with a splash of <a href="http://www.wired.com/2015/05/nasa-warp-drive-yeah-still-poppycock/" target="_blank"><span style="color: blue;">cold reality</span></a></span><span lang="EN-NZ" style="font-family: "Georgia","serif";">, but by then the genie had
escaped the bottle.</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Google the words “NASA warp
drive 2015” and you might get the impression that we are off to the stars
tomorrow or in a few years at most.</span><br />
<span lang="EN-NZ" style="font-family: "Georgia","serif";"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">But let’s take a step back. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">What is warp drive anyway?
What do we need it for ? And is it really in our grasp ?<o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-NZ" style="font-family: "Georgia","serif";">If science fiction is not your
thing (and you are still somehow reading this post), then for your benefit:
Warp Drive is a staple of the science fiction series Star Trek, which allows
spaceships to travel much faster than the speed of light - putting stars and
galaxies within easy reach and allowing folks to “boldly go where no one has
gone before”.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Now this is science-fiction.<o:p></o:p></span></div>
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<span lang="EN-NZ" style="font-family: "Georgia","serif";">Anyone with some knowledge of
popular science vaguely knows that the speed of light presents some kind of barrier for space travel and its somehow
connected to Einstein.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">But what is the obstacle
exactly? And why do we think we may be able to get around it at all?<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">To begin with, one must
realize that the “speed of light barrier” comes in two forms – local and
global.<o:p></o:p></span></div>
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<br /></div>
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<b><span lang="EN-NZ" style="font-family: "Georgia","serif"; font-size: 12.0pt; line-height: 115%;">Local Speed and the Energy
Barrier<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Since we are in sci-fi land,
imagine a racetrack 3 million kilometres long – which means light takes about
10 second to get from one end to the other.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">You sit at the starting line
in your hyper-tech antimatter propelled rocket car all geared to make a
historic attempt to “break the light barrier”.<o:p></o:p></span></div>
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<br /></div>
</div>
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<span lang="EN-NZ" style="font-family: "Georgia","serif";">A series of speed cameras
lines the track from start to finish, spaced 1 meter apart (so there’s 3
billion of them).<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">At any point on the track, the
speed of the car will be measured by <i><u>the
nearest speed camera</u></i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">This will be called the <b>local speed</b> of the car.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">History will be made if, at
any point, the local speed measured exceeds the speed of light (approximately 300,000
km/second).<o:p></o:p></span><br />
<span lang="EN-NZ" style="font-family: "Georgia","serif";"><br /></span>
<br />
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</div>
</div>
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<span lang="EN-NZ" style="font-family: "Georgia","serif";">Well, I’m sorry to disappoint
you, but as per Einstein’s Special Theory of Relativity (STR) –which has been
verified by thousands of experiments – the effort is bound to fail. <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">The theory is unequivocal: <o:p></o:p></span></div>
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<i><span lang="EN-NZ" style="color: #c00000; font-family: "Georgia","serif";">The local speed of any object is either <u>always less</u> than
light speed, <u>always equal</u> to light speed or <u>always greater</u> than
light speed.<o:p></o:p></span></i></div>
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<br /></div>
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<span lang="EN-NZ" style="font-family: "Georgia","serif";">To put this in racetrack
terms, if <i><u>any</u></i> of the cameras
clocked your car moving slower than light, then <i><u>every other camera must do so</u></i>. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">And since your car was
presumably standing still at the starting line (as measured by the camera there),
its local speed is therefore doomed to be always less than light speed.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">But what exactly enforces this
speed limit? <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">What if your car roared past the
midpoint mark, moving at 99.9999% of light speed and then you exploded a 100
megaton hydrogen bomb right behind it.<o:p></o:p></span></div>
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<span lang="EN-NZ" style="font-family: "Georgia","serif";">Wouldn’t <i>that</i> push the car past the light barrier? <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">If not, where does the
tremendous energy of the H-bomb go?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">(Yes, yes, this is a thought
experiment, so assume the car is made of indestructible material like
Superman’s costume)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSE9g7SSSuan4BDR3QbiVkibzZpdUmh0BzgDHmvR2Ibrc-vxIC5KxgCFkXbQD4nAsqwgPmionrjfo2mxBMoToMw79_zjvHMqYvC-i8xLzxcEV7piiqnrpKIax4juGyiexrj2KKnW1d_jqe/s1600/nae_f104b.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="265" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSE9g7SSSuan4BDR3QbiVkibzZpdUmh0BzgDHmvR2Ibrc-vxIC5KxgCFkXbQD4nAsqwgPmionrjfo2mxBMoToMw79_zjvHMqYvC-i8xLzxcEV7piiqnrpKIax4juGyiexrj2KKnW1d_jqe/s400/nae_f104b.jpg" width="400" /></a></div>
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">The answer is one of the key
subtleties of Relativity.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Let’s assume you exploded your
first H-bomb behind the car at the starting line.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">What would happen, of course,
is a tremendous increase in the velocity of the car.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><u><span lang="EN-NZ" style="font-family: "Georgia","serif";">But
at the same time, there would be a slight increase in the car’s mass.<o:p></o:p></span></u></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Now, when the car is moving at
very close to light speed, say 99.9999%, the effect of the bomb is precisely
the reverse.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">This time, the <i>mass</i> increases tremendously, while the <i>velocity</i> increase is tiny.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">(Incidentally, this is the reasoning
which behind E = mc<sup>2</sup>)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">The closer you get to light
speed, the more the effect of pushing the car harder – by hydrogen bombs or
otherwise – is to primarily increase the mass rather than the velocity.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">In fact, when you work out the
math, the energy required to accelerate the car all the way to light speed
turns out to be <i>infinite</i> – which is
just a polite way of saying that it can’t be done !<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">So to summarize: <i><span style="color: #c00000;">It is impossible
to accelerate any object to reach a local speed equal to, or greater than, the
speed of light because that will require an infinite amount of energy.</span></i><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">However, there is no problem
if the local speed is <i>always</i> the same
as light speed. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">(That’s what happens in the
case of light itself!)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Similarly, it is theoretically
possible to have objects which always move faster than light.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Such hypothetical particles
are called Tachyons. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Remember that name as they
shall return later in the post to cause much weirdness!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span lang="EN-NZ" style="font-family: "Georgia","serif"; font-size: 12.0pt; line-height: 115%;">Global Speed and Magic Rubber Tracks<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">The media hype about your
speed record attempt has spread across the Solar System.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">So much so, that Alison the
astronomer, inhabitant of Mars, has trained her telescope on the race track to
see what happens.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Now Alison doesn’t have the
benefit of the speed cameras. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">However, she can simply
measure the time at which your car passed various points on the track and
calculate your speed based on her observations.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">These calculations give her
the <b>global speed</b> of your car.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">So, for example, if she sees
your car reach the finish line two seconds after starting, then she can
conclude that your global speed was 1,500,000 km/second or 5 times the speed of
light.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">But can she? <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Surely, if the car managed to
cover 3 million kilometres in two seconds, then at least one of the speed cams
would have registered a speed faster than light (which is impossible) ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Well….not if you have a track
made of Magic Rubber.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Imagine your car starting off
at a leisurely 1000 km/second.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Now, one second later, at the
1000 km mark, the track instantly str-r-r-r-e-e-e-e-tches behind the car while
contracting violently in front of it.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Result: The car is instantly
1000 km from the finish line.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Another second later, the
track reverts back to normal.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">But the car is now at the
finish line! <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Thanks to the Magic Rubber
track, the car has crossed the track in a mere two seconds and that’s what our
Martian astronomer sees.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">But what about local speed ?
What about the speed cameras ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Well, notice that the <i>speed cameras get dragged along with the
track</i> as it stretches and squeezes. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Since the velocity of the car <i>with respect to the track</i> <i>and hence, the cameras</i> is always 1000 km/second,
that’s the speed that the cameras record – there is no local violation of the
light speed barrier and relativity theory doesn’t protest.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">So, thanks to the Magic Rubber
track, the local speed, as per speed cam, is a mere 1,000 km/sec, while the
global speed, as per astronomer, is 5 times the speed of light!!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Now if only we knew what this
Magic Rubber really is. Turns out we do.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_byv9U3K2IJD5Uqu4iRG4koMvr-NhvI9THSzENuW3Z9J6v6MWVnclLUHufOgFgCjnHKpcywYGNxLvt88nBl0EASTV15Vi2ao4VMK7qzwlzxvSuruNlbAHlJgPTSwQS8todCCTABCuKw5d/s1600/warpdrive.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="354" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_byv9U3K2IJD5Uqu4iRG4koMvr-NhvI9THSzENuW3Z9J6v6MWVnclLUHufOgFgCjnHKpcywYGNxLvt88nBl0EASTV15Vi2ao4VMK7qzwlzxvSuruNlbAHlJgPTSwQS8todCCTABCuKw5d/s640/warpdrive.jpg" width="640" /></a></div>
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span lang="EN-NZ" style="font-family: "Georgia","serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></b>
<b><span lang="EN-NZ" style="font-family: "Georgia","serif"; font-size: 12.0pt; line-height: 115%;">Warp Drive<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Magic Rubber is Space itself.
Come again?? Okay then…<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">As per Einstein’s General Relativity,
Space is not “mere emptiness”.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">It can bend and warp (as in
black holes), expand and contract (as with the universe itself) and even
oscillate (gravitational waves).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">And, yes, under suitable
conditions, it can even behave precisely like the Magic Rubber track.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">This fact was discovered by
Miguel Alcubierre in 1994 and was the basis of a hypothetical propulsion devise
called the “Alcubierre Warp Drive” – an attempt to ground the warp drive of
Star Trek in genuine science.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">But how does one make space
stretch and squeeze to order?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Aaah, we have a slight problem
there.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Shortly after Alcubierre
proposed the warp drive, other physicists quickly determined it would require
something extremely weird called <i>exotic
matter.</i><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Exotic matter is a completely hypothetical
substance which is very strange indeed.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">It is not to be confused with
matter or even run of the mill antimatter (which is just matter which charges
reversed).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">It is even different from the
dark matter or dark energy you may have heard of lately.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Exotic matter, if it exists at
all, would have negative energy and mass.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">To see how peculiar this is –
suppose you had a ball of regular matter and placed a ball of exotic matter
next to it. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Then the regular ball would
shoot away from the exotic ball due to gravity. But, the exotic ball <i>would follow it</i> (again due to gravity).
And thus you’d have a perpetual chase across the skies!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">In fact, attach the balls to
adjacent spokes of a wheel, and voila! Perpetual motion. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">The attraction-repulsion
effect described above makes the wheel spin forever.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Due to this and other
peculiarities, many scientists doubt that exotic matter is even possible.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">The best candidate for the
title is “Quantum vacuum fluctuations in curved space-time”, according to
relativity expert Caltech professor Dr. Kip Thorne – and even he isn’t quite sure<a href="file:///C:/Users/anindya/Downloads/Warp%20Speed%20Ahead.docx#_ftn1" name="_ftnref1" title=""><span class="MsoFootnoteReference"><!--[if !supportFootnotes]--><span class="MsoFootnoteReference"><span lang="EN-NZ" style="font-size: 11pt; line-height: 115%;">[1]</span></span><!--[endif]--></span></a>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">This is the grey zone where
our understanding of fundamental physics begins to come apart at the seams and
further light can’t be shed until we have a working theory of Quantum Gravity.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">But let’s suppose that exotic
matter does turn out to exist and we got our hands on a big supply.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Would warp drive then become
possible and the Universe open up to us?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Errrm… not so fast (literally).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span lang="EN-NZ" style="font-family: "Georgia","serif"; font-size: 12.0pt; line-height: 115%;">Tachyons and Time Paradoxes<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Remember those faster-than-light
tachyons we encountered earlier? <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">They are back to cause some
havoc.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">So, assume Alice and Bob both
have tachyon transmitters which can send messages at, say, 1000 times the speed
of light.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Also, let’s assume that Alice
is speeding away from Bob at 25% lightspeed.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Now Bob uses his transmitter
to send Alice a lewd message.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">In return Alice transmits back
a virus which causes Bob’s transmitter to self-destruct.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Thus, Bob sends his message,
and shortly afterwards, his transmitter explodes. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">So far, so unremarkable<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">But let’s see the story from
Alice’s viewpoint.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Alice sees Bob speeding away
from <i>her</i> at 25% lightspeed.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">She receives his message,
sends the virus, and shortly afterwards Bob’s transmitter explodes.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Which all seems fine – until
she does some more calculations and finds that Bob sent the message <b><i>after</i></b>
his transmitter blows up !!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Yes, this happens to be an
inevitable consequence of the Lorentz transformations connecting the viewpoints
of observers moving with respect to each other.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">But wait, if Bob’s transmitter
has already exploded, then how did he send the message ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">And if he didn’t send a
message, then what did Alice respond to ? <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Confused ? You may well be, as
what we have on hand is a Time Paradox.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSjPyYsbvRGI-TMw-x4M0eztznHyUKGKcEG3MV4wh1LaWu8pApYJ1xXI4lxiJhFY0cBhmjtqb5kW_o93CHCl6Z7NxR89kN1gRd9GB3-zpEznk2ySdAJMKJRmYrPlTQtaey7IYkYwNhnM-G/s1600/tachyon_paradox.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="292" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSjPyYsbvRGI-TMw-x4M0eztznHyUKGKcEG3MV4wh1LaWu8pApYJ1xXI4lxiJhFY0cBhmjtqb5kW_o93CHCl6Z7NxR89kN1gRd9GB3-zpEznk2ySdAJMKJRmYrPlTQtaey7IYkYwNhnM-G/s400/tachyon_paradox.gif" width="400" /></a></div>
<br />
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">It doesn’t matter whether the
tachyons are moving thousands of times faster than light or just a teensy bit
quicker, the time paradoxes just keep cropping up.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Most scientists take time
paradoxes as <i><u>proof that</u></i><u> <i>faster-than-light tachyons can’t exist</i></u>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Other solutions are possible,
each of them fairly drastic.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">For instance, it is possible
that tachyons exists but they don’t interact in any way with ordinary (slower
than light) matter. So, it wouldn’t be possible to make tachyonic transmitters
or, in fact, construct any means of detecting tachyons either – which means for
all practical purposes they don’t exist.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Alternatively, you may have
tachyonic transmitters, but some hidden law of nature intervenes whenever you
try to set up a time paradox. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">For example, the moment Alice
tries to transmit her virus, she always ends up pressing the wrong button –
although it’s difficult to see how that would work.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Now, where does warp drive
come into all this ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Well, in effect, <i>warp drive works exactly like tachyons</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">For example, instead of
tachyonic trasmitters, Alice and Bob could use warp drive enabled courier
drones to transmit their messages and set up the same contradictory situation
as before. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">So, all the objections to
tachyons would also apply to warp drive.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span lang="EN-NZ" style="font-family: "Georgia","serif";">In
Conclusion<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Where does all this leave us
then ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">For starters, it is impossible
to break the local light speed barrier. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Any method of interstellar
travel based on locally violating the light speed limit is bound to fail, so
don’t even try.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">If you try to break the limit
globally with warp drive, then you will need supplies of exotic matter, which
in all likelihood may not exist.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Even if you could somehow get
hold of this strange stuff, some unknown process might throw a wrench into the
works to prevent time paradoxes.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Interestingly enough, all the
problems above also show up when you try to make Wormholes – an entirely
different method of globally travelling faster than light, made famous by the
movie Interstellar.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";"><br />
All this leads me – Anindya the Unqualified - to propose the following conjectures:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span lang="EN-NZ" style="font-family: "Georgia","serif";">1)
Any conceivable method to travel faster than light – warp drive, wormholes,
hyperspace jumps, whatever – will require the use of exotic matter/energy
(which may not exist)<o:p></o:p></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span lang="EN-NZ" style="font-family: "Georgia","serif";">2)
Any method of travelling faster than light will also permit time travel. A civilization
which develops faster-than-light intergalactic travel will also have the key to
changing the past !<o:p></o:p></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">This suggests that warp drive,
wormholes or any conceivable method of faster than light travel is way, <i>way</i> beyond our current technological
ability, if not outright forbidden by the laws of nature.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">To paraphrase Kip Thorne, the
worldwide authority on wormholes, such technology is “probably much further
beyond the human race’s present technological capabilities than space travel
was for cavemen”.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Regarding warp drive,
cosmologist Sean Carroll estimates “The probability for humanity <i>ever </i>developing warp drive is less than
one percent. The probability of developing warp drive within the current
century is less than one hundredth of a percent”.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">To which I will add that the
probability of having developed warp drive <i>by
accident</i>, as the media articles claim, is less than one millionth of a
percent.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-NZ" style="font-family: "Georgia","serif";">Meanwhile, we can console
ourselves with the fact that while travel to distant galaxies may forever be
impossible, our ever-improving telescopes continue to bring the Universe closer
to us in all its beauty and grandeur. </span><b><span lang="EN-NZ" style="font-family: "Georgia","serif"; font-size: 14.0pt; line-height: 115%;"><o:p></o:p></span></b><br />
<span lang="EN-NZ" style="font-family: "Georgia","serif";"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0JhOrPiFU87Z5iggR4eWWU-lKrcL4evufFPR74npwjUbUcIh2Sh2K1jj-tUxvB7Gt07aCGPjH-lT02a5rK1mv0LQRiGUNVXpSWpLpmZbUDw84ZlNU3fe3Ash5xaVXVYQMdpc0IwKcTF3C/s1600/maxresdefault.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0JhOrPiFU87Z5iggR4eWWU-lKrcL4evufFPR74npwjUbUcIh2Sh2K1jj-tUxvB7Gt07aCGPjH-lT02a5rK1mv0LQRiGUNVXpSWpLpmZbUDw84ZlNU3fe3Ash5xaVXVYQMdpc0IwKcTF3C/s640/maxresdefault.jpg" width="640" /></a></div>
<br />
<div>
<!--[if !supportFootnotes]--><br clear="all" />
<hr align="left" size="1" width="33%" />
<!--[endif]-->
<br />
<div id="ftn1">
<div class="MsoFootnoteText">
<a href="file:///C:/Users/anindya/Downloads/Warp%20Speed%20Ahead.docx#_ftnref1" name="_ftn1" title=""><span class="MsoFootnoteReference"><span lang="EN-NZ"><!--[if !supportFootnotes]--><span class="MsoFootnoteReference"><span lang="EN-NZ" style="font-family: "Calibri","sans-serif"; font-size: 10.0pt; line-height: 115%; mso-ansi-language: EN-NZ; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;">[1]</span></span><!--[endif]--></span></span></a><span lang="EN-NZ"> All Kip Thorne references are from his book “Black Holes and Time
Warps”, Chapter 14<o:p></o:p></span></div>
</div>
</div>
Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com3tag:blogger.com,1999:blog-2350997094083029715.post-72292697272573697522013-10-17T08:29:00.000-07:002013-10-17T10:51:39.133-07:00What Scale Do You Sing In?<div align="center" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm; text-align: center;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";"><span style="font-size: x-large;">I</span>f you sing <i>Rabindrasangeet</i> like, oh, say 85% of all
Bengalis, then this is the first question your musical accompanist will ask
you. If you ask him/her to elaborate, you’ll hear something like, “Yes, what’s
your scale? Is it A or B-flat or C-sharp or.. ?”<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">“Setting the right scale” is
of paramount importance in the “chorus” songs where men and women sing
together, usually accompanied by much grumbling from the women that </span><span style="font-family: Georgia, serif;">“The men are singing way too
high”.</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">At this point, anybody trained
in Western classical music – where the concept of a scale comes from – will be
scratching their heads in perplexity.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">To begin with, things like
B-flat and C-sharp are not scales – they are <i>musical notes</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">As for “men singing too
high”….. well, just keep reading….<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">What follows is “original
research” - the pieces are all there,
but based on my conversations, what I am about to say is certainly not common
knowledge among Indian singers. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Western singers and musicians
use the nomenclature of scales and so on, but the words are used quite
differently in the Indian music context, resulting in much unnecessary confusion
when talking “across cultures”. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">So, let’s first get the
language straight. </span><br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<span lang="EN-US" style="font-family: "Georgia","serif";">Every note in music corresponds to a certain <i>pitch</i> or <i>frequency</i>, measured in Hertz (Hz). A higher frequency gives a
higher note.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Musicians, however don’t refer
to notes by frequency (too unromantic, I guess), but give them names as we
shall see below.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span lang="EN-US" style="font-family: "Georgia","serif";">Absolute
Notes on a Piano<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span id="goog_2049766234"></span><span id="goog_2049766235"></span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEji0ObG6obl-iyOK4f2Kz0QFvvggUVOvsDlGW_6L0usvVZ2eitMMNb2ZI_XbpRn8adWZEOUNjcjJ3ol36onCLzTa0eMsLbQHZLV_3LMeWmrmmygbF8kcivaqnZ6OWJVI1m6pSuSIKM8Oi5B/s1600/1000px-Piano_Frequencies.svg.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="122" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEji0ObG6obl-iyOK4f2Kz0QFvvggUVOvsDlGW_6L0usvVZ2eitMMNb2ZI_XbpRn8adWZEOUNjcjJ3ol36onCLzTa0eMsLbQHZLV_3LMeWmrmmygbF8kcivaqnZ6OWJVI1m6pSuSIKM8Oi5B/s640/1000px-Piano_Frequencies.svg.png" width="640" /></a></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Take a look at the image of a
piano keyboard. Each key sounds a note when pressed.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">The white key on the extreme
left is A0 (pronounced “A zero”).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">It corresponds to a frequency
of 27.5 Hz, which is very close to the lower limit of human hearing (around 20
Hz). In fact, if someone plays an A0, you pretty much <i>feel it as a vibration</i> rather than hearing a sound !<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">The second white key from the
left is B0, with a frequency of 31 Hz (still really low).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">But we start our analysis with
the <i>third</i> white key from the left.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">This is C1 (“C one”) with a
frequency of about 32.75 Hz.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">The 7 successive white keys
starting with C1 are: C1, D1, E1, F1, G1, A1 and B1.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">The next white key is C2 (“C two”)
and the cycle starts all over again with C2, D2 and so on.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">The eight notes going from C1
to C2, form an <i>octave</i>. Every note in
the octave has a fixed relative frequency to the other notes. For example, G1
is about 1.5 times the frequency of C1.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Most importantly, <i>C2 has exactly twice the frequency of C1</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Similarly, D2 is twice the
frequency of D1 and so on. (Also, A1 is twice the frequency of A0).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Now the cycle of keys just
keeps repeating as you move from left to right – with C3, D3, etc, then C4, D4,
etc – until you reach the extreme right of the keyboard.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">That final white key is C8.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">With a frequency of
approximately 4192 Hz, it is well beyond the reach of any human voice. (But nowhere
near the limit of human hearing, which goes up to 20,000 Hz)<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Georgia","serif";"><br /></span>
<img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEji0ObG6obl-iyOK4f2Kz0QFvvggUVOvsDlGW_6L0usvVZ2eitMMNb2ZI_XbpRn8adWZEOUNjcjJ3ol36onCLzTa0eMsLbQHZLV_3LMeWmrmmygbF8kcivaqnZ6OWJVI1m6pSuSIKM8Oi5B/s640/1000px-Piano_Frequencies.svg.png" /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">So much for the white keys.
What about the black ones ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Firstly, note that every black
key lies between two white keys. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">The nomenclature formula is
simple – If X is the white key immediately to the left, then your black key
becomes “X sharp” (written “X#”), if Y
is the white key immediately to the <i>right</i>,
then your black key becomes “Y flat” (written “Yь”).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">The first black key from the
left lies between A0 and B0.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Using our formula, it would
therefore become A#0 (“A sharp zero”) or Bь0 (“B flat zero”).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Similarly, we have notes like
C#1, Eь3 etc.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Two points to note here:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">- The ordering of the keys
starts with C. Hence, A1 and B1 are <i>higher
notes</i> than C1 or G1, but lower than C2 or D2. (This is what happens when
you leave nomenclature to musicians rather than scientists </span><span lang="EN-US" style="font-family: Wingdings; mso-ascii-font-family: Georgia; mso-char-type: symbol; mso-hansi-font-family: Georgia; mso-symbol-font-family: Wingdings;">J</span><span lang="EN-US" style="font-family: "Georgia","serif";">)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">- The sharp and flat notation
has considerable overlap. C sharp is the
same as D flat, B flat is also A sharp and so on. Which name is used depends
purely on convention. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Exercise for the reader:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Rank the following in
ascending order of pitch: C4, B3, G#3, A4, Eь4, D5<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">I hope this clarifies why I
said that B flat and C sharp are notes rather than scales. In fact, one needs to be even more specific
and pinpoint <i>which</i> C sharp you are
talking about.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">This is, in fact, what happens
in Western classical music.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">When a pianist plays, say, the
<i>Moonlight Sonata</i>, the musical score
specifies exactly which notes she needs to play.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">I will call these <u>absolute
notes</u>, as each note corresponds to a specific frequency.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Western musicians often talk
about “middle C”. This is simply C4, near the centre of the keyboard, and
having a frequency of about 262 Hz.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">The notes around C4 are most
frequently used in vocal music as they fall into the “comfort zone” of the
human voice.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span lang="EN-US" style="font-family: "Georgia","serif";">Relative
notes in Indian music<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">A Western classical musician
trying to learn Indian music starts off with complete confusion.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">To begin with, the concept of
notes seems to be absent.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Her guru will frequently
allude to the <i>Sapta Soor</i> (“Seven
Notes”) – <i>Sa Re Ga Ma Pa Dha Ni<o:p></o:p></i></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">In fact, people will often
explicitly speak out the names of these notes while singing.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">But there is no
straightforward mapping from these onto the notes on a piano.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Even worse, different singers
will sing out the same sequence of <i>Sa Re Ga Ma</i>, but the actual notes they
sing (as per piano), may be quite different !<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
</div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";"><br /></span>
<span style="font-family: Georgia, serif;">After a while, however, the
bewilderment begins to abate.</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">She observes that the <i><u>relative pitch</u></i> of the <i>Sapta Soor</i> notes remains fixed although
the absolute notes being sung may change. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">In fact, if she is familiar
with the Sound of Music movie, she quickly realizes that <i>Sa Re Ga Ma Pa Dha Ni </i>corresponds precisely to <i>Do Re Mi Fa Sol La Ti </i></span><i><span lang="EN-US" style="font-family: Wingdings; mso-ascii-font-family: Georgia; mso-char-type: symbol; mso-hansi-font-family: Georgia; mso-symbol-font-family: Wingdings;">J</span></i><i><span lang="EN-US" style="font-family: "Georgia","serif";"><o:p></o:p></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">(You see, Indian music is starting
her off at the very beginning…)<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Georgia","serif";"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-58VZdNk5XJMvtmbcYt2yMHqco0um2CzISvdp-wrqFEvWJmduqA0G8reaaUUw1MlHk6UcurcpcoLpJLzwvs0AHNijXILs_VBllfOiot4QCLY2G_cBHrEvFwOVYxg-SzhP6wItHf781fYW/s320/Swaralipi.gif" style="margin-left: auto; margin-right: auto;" /></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Swaralipi in Rabindrasangeet</td></tr>
</tbody></table>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">So, <i>Sa Re Ga Ma Pa Dha Ni </i> spans
an octave. But what if you want to go higher ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Well, just like the notes on a
piano cycling back to C, when you want to climb higher than <i>Ni</i> in Indian music, you get back to <i>Sa</i>. Only it’s a “high <i>Sa”. </i>I will call this <i>Sa<sup>*</sup>.</i><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Just like C2 and C1, the high <i>Sa</i>,
<i>Sa<sup>*</sup>, </i>corresponds to
a note with twice the frequency of the ordinary <i>Sa</i>. </span><br />
<span lang="EN-US" style="font-family: "Georgia","serif";">(<i>But which notes are they
on the piano ??</i> Patience, patience…)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Now you can continue upwards
with <i>Re<sup>*</sup>, Ga<sup>*</sup></i>
and so on.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Similarly, if you want to go
lower than ordinary <i>Sa</i>, you have <i>Ni<sub>*</sub></i> , <i>Dha<sub>*</sub></i> , (“low <i>Ni</i>”
and “ low <i>Dha</i>”) which are half the
frequency of the ordinary <i>Ni</i> and <i>Dha</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Indian vocal songs – and
especially <i>Rabindrasangeet </i>–
typically span a maximum range of two octaves, so in terms of Indian musical
notation, you rarely climb above <i>Pa<sup>*</sup></i>
(“high Pa”) or go lower than <i>Pa<sub>*</sub></i>
(“low Pa”). <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">However, in any given song,
the bulk of the notes will lie within the “ordinary notes” – i.e. between <i>Sa</i> and <i>Sa<sup>*</sup></i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Now to illustrate the crucial
difference between absolute and relative notes:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Suppose you are a western
choir singer. Your music score requires you to sing C4, D4, E4 and F4<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Instead you sing C3, D3, E3,
F3.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Your choirmaster immediately
hauls you out for singing out of tune.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Now in Indian music school,
three singers are going to sing <i>Sa Re Ga
Ma</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Singer 1 hits the notes - C4,
D4, E4, F4<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Singer 2 goes - C3, D3, E3, F3<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Singer 3 sings - A3, B3, C#4,
D4<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">To your surprise, the guru is
perfectly happy with all three!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">“They are just singing in
different scales”, he says.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Scales again! What on earth
are these pesky scales?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Well, with all this background
on absolute and relative notes, we are finally ready to answer the question.<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Georgia","serif";"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span lang="EN-US" style="font-family: "Georgia","serif";">What
scale do you sing in?<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Here’s an example of a scale
in Western music – it’s called “C Major”.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">It consists of the notes: C D
E F G A B<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Georgia","serif";">Another one is "B Minor"</span><br />
<span lang="EN-US" style="font-family: "Georgia","serif";">It consists of the notes: B C# D E F# G A</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">What the … ??!! What about
scales like B flat, G sharp and so on?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Now the Western musician gives
you a blank stare – “Those are not scales. Those are notes.”<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span lang="EN-US" style="font-family: "Georgia","serif";">A
scale in Western music is a collection of notes</span></i><span lang="EN-US" style="font-family: "Georgia","serif";">. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Vocalists and musicians
practice singing/playing these notes to perfect their technique.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Occasionally, you’ll see
things like “XYZ Symphony in C Major”.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1UCmF_7n2XqyqRU3Sthda8_UstwZ17pdBGubxPPYlC5CM01Up8AqvMgoE9-Zx1ss-EhS7AXt3qtC4-02wrriKhUOCJMxcqLPj-shfG0dmN_CHYJglxZxQyB_f9F0t1j1fbyEH5z0HB1g_/s1600/pitch+pipe.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1UCmF_7n2XqyqRU3Sthda8_UstwZ17pdBGubxPPYlC5CM01Up8AqvMgoE9-Zx1ss-EhS7AXt3qtC4-02wrriKhUOCJMxcqLPj-shfG0dmN_CHYJglxZxQyB_f9F0t1j1fbyEH5z0HB1g_/s1600/pitch+pipe.jpg" /></a><span lang="EN-US" style="font-family: "Georgia","serif";">What it means is that XYZ
Symphony only has notes from the set of notes in the scale.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">So, you may have a C5 note
somewhere in the score, but never a F#3.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">In that sense, a scale is
somewhat like a <i>raga</i> in Indian
classical music, although the rules of a raga are somewhat more elaborate.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">But then, what are Indian
musicians talking about when they mention scales ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Well, recall that the <i>Sapta Soor</i> of Indian music only defines
relative pitches.<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Georgia","serif";"><span style="font-family: Georgia, serif;">However, </span><i style="font-family: Georgia, serif;"><u>once you specify the absolute note corresponding to any one of the Sapta Soor, all the other notes become uniquely determined.</u></i></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">So, if you had to sing <i>Sa Re Ga Ma</i> and you set <i>Sa</i> to be the note C4, then you get C4,
D4, E4, F4.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">If you set <i>Sa</i> to A3 instead, you get A3, B3, C#4,
D4 instead.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">And that’s what the whole deal
is about. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<u><span lang="EN-US" style="font-family: "Georgia","serif";">Your accompanist needs to know
what absolute notes to play, and for that, all he needs is what piano note your
<i>Sa</i> (ordinary Sa, not the high Sa)
corresponds to.<o:p></o:p></span></u></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">So, here’s the dictionary
translating the most popular “scales” in<i>
Rabindrasangeet</i> to notes.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span lang="EN-US" style="font-family: "Georgia","serif";">Scale </span></b><span lang="EN-US" style="font-family: "Georgia","serif";"> <b>Piano note corresponding to <i>Sa</i> (ordinary Sa)<o:p></o:p></b></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">G Sharp G#2 or G#3<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">A A2 or A3<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">B Flat Bь2 or Bь3<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">B B2 or B3<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">C C3 or C4 (C4 = Middle C = 262 Hz)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">C Sharp C#3 or C#4<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">(Just to irritate you further,
the scale B Flat is often called “B” in Bengal and B is called “natural B”.
This is just idiotic terminology. Please, people, there are no natural versus
unnatural Bs, just good old B and B Flat.)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">But wait, hang on, why the “A2
or A3” and so on? Why not just A3, say?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">This brings us to the final question
in this post.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span lang="EN-US" style="font-family: "Georgia","serif";">Do
men really sing high?<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">“Definitely yes”, answers the <i>Rabindrasangeet </i>singer. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">In fact, it’s something he’s
known since his early days in singing school - “Men always sing the higher
notes in choruses and generally sing at higher scales.”<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">“No way”, responds the Western
choir singer. “In fact, it’s the <i>women</i>
who sing the high notes and the <i>men</i>
who sing the low notes. Ask any choirmaster.”<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">The laws of physics concur.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Humans sing by vibrating their
vocal cords. Men have longer and thicker vocal cords than women. Now, a
vibrating string which is longer and thicker generates lower frequency sounds.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Hence, men must sing at lower
frequencies – i.e. men must sing the lower notes. QED.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">So, what’s going on in Indian
music?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">As a youngster learning <i>Rabindrasangeet,</i> this question puzzled
me greatly – especially the contradiction between who sings the high notes in
Indian versus Western music.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">My music teachers in India
brushed me off with an impatient, “Oh, that’s because Indian music and Western
music are completely different.” (Nonsense !)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">And of course, nobody
understood the argument from physics.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">But, now I <b><i>do</i></b>
know the answer.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">And if you have been trained
in Indian music, you very probably don’t – so listen up!!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span lang="EN-US" style="font-family: "Georgia","serif";">In
Indian music, when men and women sing together, they are NOT singing the same
notes (in the sense of absolute notes).<o:p></o:p></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<i><span lang="EN-US" style="font-family: "Georgia","serif";">In
fact, every note the woman is singing is one octave higher – i.e. twice the
frequency – as that being sung by the man !<o:p></o:p></span></i><br />
<i><span lang="EN-US" style="font-family: "Georgia","serif";"><br /></span></i>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBgzLAqLtZiNMA9tcec0VX5G9UcDyW3DFGCvWHvRdaXnCORAfAQgahahjRGhAX8VuwPv5nlUhk6RZAI_eQdWMYrZ168avyzPr-sz54XHJ1RgUHN4c6AK_i-xI6k-xMVrKuHXbdk96xCExH/s1600/duet+singing.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="356" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBgzLAqLtZiNMA9tcec0VX5G9UcDyW3DFGCvWHvRdaXnCORAfAQgahahjRGhAX8VuwPv5nlUhk6RZAI_eQdWMYrZ168avyzPr-sz54XHJ1RgUHN4c6AK_i-xI6k-xMVrKuHXbdk96xCExH/s400/duet+singing.jpg" width="400" /></a></div>
<i><span lang="EN-US" style="font-family: "Georgia","serif";"><br /></span></i></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Knowing this immediately
explains everything.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">To begin with, the laws of
physics aren’t being violated and the choir singer is correct – it is, in fact,
women who sing at higher frequencies.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">So, why are the women in India complaining about men singing too high ?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">The reason is actually very
interesting.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">It seems that a typical
woman’s voice is pitched at about 1.5 times a typical man’s.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">In other words, if a typical
man’s singing frequency range goes from X to Y, a typical woman’s range goes
from 1.5 X to 1.5 Y. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">But, in Indian music, the woman
must sing at <i>twice</i> the frequency of a
man.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">So, if the highest frequency
the man can hit is Y, the woman must hit 2Y rather than 1.5Y, which is of
course, extremely hard on her voice, if not outright impossible.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Similarly, a man will find it
difficult to sing along when the woman sings her lowest notes. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Thus, in an amusing inversion of the Western choir, the “high parts” in Indian vocal music are left to the
men, when in fact, they are actually singing low!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">So, ladies, don’t grumble in
future about “men singing too high”. We really aren’t. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">It’s just that, apparently,
evolution didn’t intend men and women to sing in harmony. </span><span lang="EN-US" style="font-family: Wingdings; mso-ascii-font-family: Georgia; mso-char-type: symbol; mso-hansi-font-family: Georgia; mso-symbol-font-family: Wingdings;">J</span><span lang="EN-US" style="font-family: "Georgia","serif";"><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Finally, to explain the
ambiguity of “A2 or A3” in the scale-to-note translation dictionary.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">As you may have guessed, it’s
due to the fact that men and women are singing different notes.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Hence, when “setting the
scale”, the <i>Sa </i>must correspond to
different notes depending on the gender of the singer.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">Here’s the completed
dictionary which adds in that bit.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span lang="EN-US" style="font-family: "Georgia","serif";">Scale </span></b><span lang="EN-US" style="font-family: "Georgia","serif";"> <b>Piano
note corresponding to <i>Sa</i> (ordinary
Sa)<o:p></o:p></b></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">G Sharp G#2 (men) or G#3 (women)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">A A2 (men) or A3
(women)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">B Flat Bь2 (men) or Bь3
(women)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">B B2 (men) or
B3 (women)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">C C3 (men) or
C4 (women) (C4 = Middle C = 262 Hz)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">C Sharp C#3 (men) or C#4 (women)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">So, where do you go from here
?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: Georgia, serif; mso-ascii-font-family: Georgia; mso-char-type: symbol; mso-hansi-font-family: Georgia; mso-symbol-font-family: Wingdings;">Well, you can start by testing some of the things I said by connecting to this </span><a href="http://www.onlinepianist.com/virtual_piano/" style="font-family: Georgia, serif;" target="_blank"><span style="color: red;">virtual piano</span></a><br />
<span lang="EN-US" style="font-family: Wingdings; mso-ascii-font-family: Georgia; mso-char-type: symbol; mso-hansi-font-family: Georgia; mso-symbol-font-family: Wingdings;"><span style="font-family: Georgia, serif;">The key marked with a red dot is C4. The first key marked C to the left of that is C3.</span></span><br />
<span style="font-family: Georgia, serif;">Try playing the notes and matching them vocally. For best effect have a singer of the opposite gender with you. Its a real eye opener. </span><span style="font-family: Wingdings;">J</span><br />
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span lang="EN-US" style="font-family: "Georgia","serif";">PS:</span></b><span lang="EN-US" style="font-family: "Georgia","serif";"> I’ve used the words Indian
music and <i>Rabindrasangeet </i>interchangeably.
This is, of course, inaccurate. Indian music has enormously many branches with <i>Rabindrasangeet </i>being just one.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif";">However, from my experience,
the usage of “scales” as I have described and the myth that “men sing higher than women”
persists across all branches of Indian music I have encountered, including
Hindustani and Carnatic classical music.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span id="goog_312358759"></span><span id="goog_312358760"></span><br /></div>
Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com5tag:blogger.com,1999:blog-2350997094083029715.post-80271727833640714412013-01-30T02:07:00.001-08:002013-01-30T02:15:02.085-08:00Govinda - Book Review<br />
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">The
least appealing aspect of the <i>Mahabharat</i>,
I feel, is the intrusion of the divine.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">The
marvelously multifaceted narrative is, in my opinion, severely compromised by
the heavy-handed attempt to present it all as a good-versus-evil morality tale
orchestrated by God Himself in the form of Sri Krishna.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">It’s
not just that I find the idea of an <i>avatar</i>
rather ridiculous. It also makes for lousy fiction.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">An
arrogant and cocksure God-in-human-form, always slated to come out on top with
smug smile on face and pithy proverb on lip, makes for an unappealing character
at best, while ruining any element of suspense in the story.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">What
if the <i>Mahabharat </i>were presented as
historical fiction without any attempt to inject a “moral of the story”? What
if Krishna was just a man among others, struggling to shape his destiny amidst
the forces of his time, rather than achieving it all by effortless divine will?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">To
get a flavor, look no further than “<i>Govinda</i>”,
first book of the “<i>Aryavarta Chronicles</i>”
by debutante author Krishna Udayasankar.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEif5yx-51Ar0UuWmO_-smNXRwiDBCq88T8TuaOUPELH6041PWRK9yUXzroezbDr87GkeXGBG4EhR8WioY4DNafjNBB_JsyYvlQZffoNd-37TMSfQAwf-2NpjevZWB67dMuKNqS0u3RryjSI/s1600/the-aryavarta-chronicles-govinda-book-1-700x700-imadbcvmavmfczmf.jpeg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEif5yx-51Ar0UuWmO_-smNXRwiDBCq88T8TuaOUPELH6041PWRK9yUXzroezbDr87GkeXGBG4EhR8WioY4DNafjNBB_JsyYvlQZffoNd-37TMSfQAwf-2NpjevZWB67dMuKNqS0u3RryjSI/s320/the-aryavarta-chronicles-govinda-book-1-700x700-imadbcvmavmfczmf.jpeg" width="222" /></a><span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">The
geographical backdrop of the novel is Aryavarta, the northern half of the
Indian subcontinent in the second millennium BC. The historical setting is a
deadly conflict between two groups of Brahmins. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">On
one side are the Firstborn, scholar and sages, upholders of sacrificial rituals
and social norms, makers and breakers of kings. Opposed to them are the
Firewrights - master scientists, inventors, and iconoclasts.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">When
Firewright technology gone awry dries up the Saraswati river - turning a fertile
kingdom to barren desert - the Firstborn seize the opportunity to launch the
Scourge. With the active support of the ruling nobility, Firewrights across the
nation are mercilessly tortured and killed. Some survivors go into hiding;
others seek refuge with kings who are willing to surreptitiously exploit their
expertise to manufacture lethal weapons.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">One
such is Jarasandha, king of Magadha, gradually extending his sway over
Aryavarta in a bid to become Emperor. Other kingdoms, including the Kurus and
Panchalas, officially acknowledge his sovereignty, while secretly seeking to
counterbalance his power.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">Into
this landscape of intrigue steps the character of Govinda Shauri – the author’s
preferred name for Krishna. Arisen from humble origins as a <i>gwala</i> (cowherd) to become king of
Mathura, he has since abandoned his kingdom in the face of Jarasandha’s attacks
to establish an outpost at Dwarka on the southwestern edge of Aryavarta. This
act of “dishonor” has earned him the eternal contempt of the ruling Kshatriya
nobility who are already ill-disposed to accept him as an equal due to his
cowherd past.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">The
Firstborn, however, see him as a convenient pawn in their bid for supremacy. In
the past, Govinda has been instrumental in the success of the Scourge. His
enmity with Jarasandha makes him the perfect foil to the would-be emperor and
last refuge of the wrights. As Vyasa, leader of the Firstborn, tells his
disciples, Govinda is a “tiger who needs to be tamed” to their service. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">A
resourceful politician and wily warrior, Govinda is happy to aid Vyasa’s
efforts.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">But
there are hints of a deeper, darker plot in the background, a plan which makes
even his brother and friends uneasy…<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">Udayasankar
wields her prose deftly, drawing you swiftly into the story and holding your
interest thereafter. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">In
her hands, the <i>Mahabharat</i> takes on a
darker, grittier aspect. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">This
is no tale for children with the reassurance of a happy ending. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">Prisoners
are brutally tortured and executed, ruling classes treat commoners with
contempt and derision, women are marginalized and restricted – especially in
the Kuru kingdom.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">Bad
things can, and do, happen to good people.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">While
the characters are all from the <i>Mahabharat</i>,
those familiar with the epic only through <i>Amar
Chitra Katha</i> or teleserials may find themselves at a loss. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">Characters
are referred to by unfamiliar alternate names found in the epic. Yudhisthira is
Dharma, Arjuna is Partha, Duryodhana is Syodhana, Karna is Vasusena and so on<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">A
number of characters are presented in a refreshing new light.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">Dharma
(Yudhishthira) – complex and self-serving, forever trying to project an image
of righteous otherworldliness while hungering for personal glory, disguising
his bids for power as burdensome duties he must take on for the greater good.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">Syodhana
(Duryodhana) – peace-loving family man, increasingly dismayed to find himself
cast in the role of villain by Yudhsthira’s machinations.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">Shikhandin
(Shikhandi) – a guerilla warrior par excellence, maligned as a coward because
he refused to blindly condone his father’s actions.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">Panchali
(Draupadi) – Intelligent, thoughtful and every bit as fiery as her epic
counterpart, she takes a very active role in the politics of the day, far beyond
merely goading her husband/s into action.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">Finally,
there is the fascinating Govinda Shauri (Krishna).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">Charismatic,
charming, unfailingly polite and kind to everyone around him.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">Yet,
at the same time, strangely detached and dispassionate, willing to manipulate
or sacrifice loved ones and unleash great suffering to achieve a desired end.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">An
intriguing persona indeed - far more so than the annoying Bhagwan Sri Sri
Krishna we usually encounter.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Georgia","serif"; mso-bidi-font-family: "Courier New";">In
summary, I heartily recommend <i>Govinda</i>.
Can’t wait for the next book to come out.<o:p></o:p></span></div>
Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com4tag:blogger.com,1999:blog-2350997094083029715.post-72012236584669662472012-07-06T22:14:00.003-07:002012-07-08T02:34:34.008-07:00The Higgs Boson<div align="center" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: center;">
<br />
<div style="text-align: left;">
</div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<div style="text-align: left;">
<span style="font-family: Georgia, serif;">On 4<sup>th</sup> July 2012, Rolf Heuer,
director of CERN, announced that the Higgs boson had been discovered. The
packed auditorium of scientists and reporters erupted in applause, cameras
flashed and history was made. <o:p></o:p></span></div>
<div style="text-align: left;">
<span style="background-color: white; font-family: Georgia, serif;">A few hours later, Googling “Higgs boson
found” gave 262 million results. Replacing “Higgs boson” by “God particle” gave
many million more.</span><br />
<span style="background-color: white; font-family: Georgia, serif;"><br /></span></div>
</div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<div style="text-align: left;">
<span style="background-color: white; font-family: Georgia, serif;">So, what is the Higgs boson, what does it
mean for physics, and where does God come in?</span></div>
</div>
<div style="text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<b><span style="font-family: Georgia, serif;">The
Layman’s Picture </span></b></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">To start the story, let’s roll back 80
years. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">In 1932, the discovery of the neutron
completed what I call the Layman’s Picture of the universe familiar nowadays to
anyone with a high school education (if they haven’t slept through all their
science classes, that is…).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">In this picture, all the “stuff” in the
universe, aka “matter”, is made up of atoms, which are themselves composed of
protons, neutrons and electrons.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">All matter is subject to the force of
gravity which shapes the stars and galaxies.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">If, in addition, the matter has a <i>charge</i>, it also feels the forces of
electricity and magnetism.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Nearly 70 years before our story begins,
James Clerk Maxwell had shown that electricity and magnetism were two aspects
of the same force, henceforth called <i>electromagnetism</i>.
This was the first of the so called “unifications of physics” and as a stunning
consequence was that light was revealed to be an <i>electromagnetic wave</i> – a disturbance of the electromagnetic field
that propagates through space like waves through water.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgf0mZ1hjJ9tQKIDYKpkUlQmIe7eL-lgztbsPFFNbsuZCIwkX0cnoPMjJ9LDJ02YnLI2pkfTWIn0Ryg8ZIdI89sWXcgP5DCCihsBxiZW6WLLwLjV6FhnszDZioP-Z3KeuBbU4lsvoC8r8qd/s1600/images.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="226" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgf0mZ1hjJ9tQKIDYKpkUlQmIe7eL-lgztbsPFFNbsuZCIwkX0cnoPMjJ9LDJ02YnLI2pkfTWIn0Ryg8ZIdI89sWXcgP5DCCihsBxiZW6WLLwLjV6FhnszDZioP-Z3KeuBbU4lsvoC8r8qd/s400/images.jpg" width="400" /></a></div>
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<span style="font-family: Georgia, serif;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The three decades, 1900 to 1930, had
witnessed possibly the most spectacular advances ever in theoretical physics.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">On the one hand we had Einstein’s
theories of Relativity. Special Relativity reconciled Maxwell’s theory with
mechanics in a revolutionary way – among the implications was the equivalence
of mass and energy (yes, E = mc<sup>2</sup>). General Relativity subsumed
special relativity and revealed gravity an effect of the curvature of space and
time in the presence of matter and energy.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">On the other hand, the theory of Quantum
Mechanics, developed by a group of physicists in the 1920’s had incredible
success explaining the world of atoms and molecules. This theory came with its
own baggage of bizarreness – wave-particle duality, the Uncertainty Principle
and more.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Many intellectuals believed that our
picture of the world was essentially complete. All that remained was to unify
gravity with electromagnetism and Einstein was already working on it.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The truth would turn out to be far
stranger…<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<b><span style="font-family: Georgia, serif;">The
Standard Model<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Fast forward to the late 1940’s.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The terrifying potential of nuclear power
had engendered interest in peering even
deeper into the fundamental constituents of matter – after all who knew what
mightier sources of energy were waiting to be tapped ? <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">This was the age of the great particle
accelerators – huge machines that accelerate atoms to extremely high velocities
and smash them together.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The hope was to reveal even simpler
ingredients underlying protons and electrons. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">What ensued instead was complete chaos.
Instead of a few simple components, the accelerators turned out a profusion of
literally hundreds of new particles at an astonishing rate!! <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Physicists despaired of ever making sense
of the mess. Some speculated that rather than getting simpler, maybe matter
gets ever more complex as one goes further down…<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Then as the 50’s rolled into the 60’s, a
picture began to emerge, and here it is:<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjx-UOD_XLUkTC6DQrLsROn6rowWaD6bh4j5Dap_ABDRBL52APWJS7aX2gvsEPIlRHaWhir0UrjaoUqZs3TYuoCsDDHko35ijtKVdjbHuXZmZVddO8-wYm6Gu8Nb_BBYHsRcmjoMCuQ3rgp/s1600/Standard+Model.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="395" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjx-UOD_XLUkTC6DQrLsROn6rowWaD6bh4j5Dap_ABDRBL52APWJS7aX2gvsEPIlRHaWhir0UrjaoUqZs3TYuoCsDDHko35ijtKVdjbHuXZmZVddO8-wYm6Gu8Nb_BBYHsRcmjoMCuQ3rgp/s640/Standard+Model.gif" width="640" /></a></div>
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Welcome to the Standard Model. This is
the best picture we have so far of the fundamental constituents of our
universe.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">To the left of the picture are the basic
building blocks of matter.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">You will notice that protons and neutrons
are nowhere in the picture. That’s because they themselves are made of even
more basic constituents – the <i>quarks</i>.
Quarks come in 6 varieties – up, down, charmed, strange, top and bottom.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Electrons, however, <i>are</i> in the picture.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The “e” in the upper left corner of the
group called “leptons” is the electron.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Along with them in the same row are the <i>muon</i> and the <i>tau particle</i>.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The bottom row on the left shows the 3
types of <i>neutrino</i> – strange,
ghostlike particles than can pass through entire planets without leaving a
trace.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Why these twelve? Why no more or fewer?
Nobody knows.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The right side shows the “force
particles”. Come again?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Well, the theoretical framework
underlying the forces of nature is Quantum Field Theory (QFT).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">In QFT, each force is represented by a “quantum
field” pervading all of space. Associated with each such quantum field is a
characteristic particle.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">In the parlance of QFT, a matter particle
“feels” a force if it is able to “interact with” or “couple to” the particle of
the corresponding quantum field.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Just to make things fun, the force
particles can even interact with each other, or even themselves. (This comes up
later)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Clear? So now, the “γ” at bottom right is
the <i>photon</i>, which is the particle
corresponding to the electromagnetic force.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">As for the rest – two new forces were
discovered via all the particle smashing.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The <i>strong
nuclear force</i> binds quarks together into protons and neutrons. Its particle
is the <i>gluon</i>, which is the “g” on top
right.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The <i>weak
nuclear </i>force is responsible for radioactive decay and its particles are
the W and Z in middle right.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">[A diversion: You will notice that the “force particles”
are called bosons. This is because all force particles share some commonalities
that were elucidated in the early 1920’s by Bengali physicist Satyendranath
Bose. So, all Bengalis reading this – enjoy your 2 seconds of reflected glory.]<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">But talking of forces, where’s gravity?
Welcome to the biggest gap in the current foundations of theoretical physics.
Despite decades of effort, nobody has come up with a successful QFT for
gravity. Anybody who succeeds will usher in a new era of physics. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">But we digress.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Another force particle is missing in this
picture – missing because until very recently, nobody was sure it even existed.
Enter the Higgs boson.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<b><span style="font-family: Georgia, serif;">The
Higgs Field(s)<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Remember Maxwell joining electricity and
magnetism together?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The quest to unite the forces of nature
under a single description has been a prime motivator for theoretical physics
over the last century.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Starting with Paul Dirac in the 1920’s,
physicists managed to merge Maxwell’s theory with quantum mechanics to develop
the spectacularly successful theory of Quantum Electrodynamics – the QFT for
the electromagnetic field over the next several decades.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;"> (Heard of Richard Feynman? This was his
biggest work.)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Why stop with electromagnetism? From the
1960’s onwards an ongoing effort was made to incorporate both the strong and
weak nuclear force into the same framework.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">But there was a problem.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The photon and gluon have zero mass while
the W and Z particles of the weak force have mass.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Why is that a problem? <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The reasons are extremely technical
(here’s where a PhD in physics comes in handy).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Suffice it to say that the theoretical
framework constrains the particles of all three forces to have zero mass. So,
how to reconcile theory with reality?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">But what if there’s <i>another</i> “force” – another quantum field pervading the world? Maybe
the reason why the W and Z particles seem to have mass is because they feel
this strange new force.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcBKpkBnj_1v57Ymi1cmk9NcceZ6ADFdpq-udIUrz-LHX5iied1NjqSjkcj2f34cwEryxD1WaTCB_BKsMz-EnzcZRdcBJ3V6LIUXz70nwdHknY2gdyxlBSq3bf1fPX0-6LunOVzfS_CxYG/s1600/Higgs_SM.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="355" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcBKpkBnj_1v57Ymi1cmk9NcceZ6ADFdpq-udIUrz-LHX5iied1NjqSjkcj2f34cwEryxD1WaTCB_BKsMz-EnzcZRdcBJ3V6LIUXz70nwdHknY2gdyxlBSq3bf1fPX0-6LunOVzfS_CxYG/s400/Higgs_SM.jpeg" width="400" /></a></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">To make a very crude analogy, suppose you
are doing experiments with marbles on a very smooth floor. You flick all the
marbles with the same force. Some of them shoot off, while others roll away
slowly and then come to a stop. How to explain this?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">One possibility – they have different
weights and respond differently to your flicking.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">But here’s another – <i>they all weight the same</i>, <i>but
some of the marbles are made of iron, and there’s a powerful magnet behind you</i>.
<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The iron marbles feel the magnetic field
and the others don’t, hence the different reactions.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The explanation proposed for the masses
of the W and Z is similar. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The mysterious field, analogous to the
magnetic field of our example, is the <i>Higgs
field</i> – first proposed by Peter Higgs in 1964.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The idea of a Higgs field was used to
unify electromagnetism with the weak force by Sheldon Glashow, Abdus Salam and
Steven Weinberg – a feat which won them a Nobel prize in 1979.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">But now the bonus:<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">If interaction with the Higgs field can
“give mass” to the W and Z particles, <i>could
it be the case that </i>all<i> particles
derive their mass from the same mechanism</i>?!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">If true, this would mean that mass is not
a fundamental property of matter – just a consequence of some particles
“feeling the Higgs field”. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">So, does this really happen? Yes. If you believe the Standard Model<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Why do different particles have different
masses?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Nobody knows. The model says that a
particle’s mass is proportional to how strongly it interacts with the Higgs,
but that just pushes the question one step back.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Is there just one Higgs field or many?
Nobody knows.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<i><span style="font-family: Georgia, serif;">In
fact, until a week ago, nobody was sure that there was even a </span></i><span style="font-family: Georgia, serif;">single<i>
Higgs field.<o:p></o:p></i></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">A number of alternative theories had been
proposed to explain the particle masses without any need for a Higgs – although
all these theories have a bunch of side-effects.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">So, how would one know if the Higgs field
was just a figment of the imagination?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Answer: As mentioned above, every quantum
field has an associated particle. The particle associated with the Higgs field
is the <i>Higgs boson</i>. (Finally, we get
to the title of the post!)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Detect the Higgs boson and you know the
field exists.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">And <i>that’s</i>
what we did on July 4, 2012!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">(Note: Strictly speaking I should say we
detected <i>a</i> Higgs boson, and so there
is <i>at least one</i> Higgs field. For all
we know, there could be dozens of them.)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<b><span style="font-family: Georgia, serif;">Stagnation<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Okay, so if you’ve followed so far, we
just verified a theory proposed nearly 50 years ago.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">What’s the big deal? And how did we do
it?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Answering the first question requires a
bit of scientific history.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The late 70’s were a heady time for
physicists.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The past six decades had been a period of
unprecedented progress, leading us deeper than ever into the secrets of Nature.
<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The constant stream of insights and
breakthroughs, it was felt, could only have one end – the Theory of Everything,
unifying space, time, matter, energy and forces into one stupendously grand
overarching framework.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Many – including the very outspoken
Stephen Hawking – believed that this would happen by the turn of the
millennium, bringing the Century of Physics to a supremely triumphant end.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">What followed instead was three decades
of massive stagnation.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">String Theory, hailed as the most
promising candidate for unification, degenerated into a thicket of wild
speculation and unverifiable hypotheses. Currently, supporters claim that they
have made great “conceptual progress”, while detractors argue that the theory
is “not even wrong”, i.e. it can be tweaked arbitrarily to fit any observation.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">On the experimental side, things ground
to a halt with Congress refusing funding for more powerful particle accelerators.
<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Only the astronomers maintained an iota
of progress, indicating that not all was well. Evidence steadily mounted,
especially in the last decade, that the matter described by the Standard Model
only constitutes about 15% of all the matter there is. Meanwhile, cosmology
yielded a huge surprise – the expansion of the universe is accelerating and
nobody knows what’s causing it.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">But alas, no help was forthcoming from
the theorists to explain any of this, lost as they were in the wild goose chase
of strings…<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhO7mED41hteYeKWNnPGF_-PQ6LxP0K_ZkBfaU1gBAHqE1ItHFcHEV86_NKmOCAx3PitE9xcIugxvxLeRdfMytWQjCTqBKYuFlp_9zjE_0S6S1S2LcKOJ3Wc34DWLBPldbKZKj-xD_l8owL/s1600/i-ba6b0750a25c7e111fe5dcfba9c37095-lhc-sim.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhO7mED41hteYeKWNnPGF_-PQ6LxP0K_ZkBfaU1gBAHqE1ItHFcHEV86_NKmOCAx3PitE9xcIugxvxLeRdfMytWQjCTqBKYuFlp_9zjE_0S6S1S2LcKOJ3Wc34DWLBPldbKZKj-xD_l8owL/s640/i-ba6b0750a25c7e111fe5dcfba9c37095-lhc-sim.jpeg" width="640" /></a></div>
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<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">For those not enamored of string-world,
the easiest path beyond the Standard Model lay in the investigation of the
Higgs field(s). This was the only feature of the theory which remained somewhat
speculative, but concrete experimental data was needed to make headway.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Hence, all hope focused on the Large
Hadron Collider, scheduled to start operating in early 2009 at CERN in
Switzerland.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<b><span style="font-family: Georgia, serif;">Discovery<o:p></o:p></span></b></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The Large Hadron Collider (LHC) is the
most powerful particle accelerator ever built.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Magnets of tremendous power accelerate
beams of protons are along a circular tunnel 27 km long and smash them into
each other with savage force. Hundreds of detectors track the debris that
erupts and record its characterestics.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">How does this help?<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Well, quantum theory predicts that
collisions of particles don’t just give you constituents of the things
colliding. Completely new particles can arise, born out of the conversion of
energy into mass. The higher the energy of collision, the higher the mass of
the particle that may appear. (Remember, E = mc<sup>2</sup>).<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">[This, to my mind is one of the weirdest
aspects of the quantum world. Imagine a situation where you smash two stones
hard enough and a butterfly pops out and flies off. That is pretty much what
goes on in these collisions!]<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">The hope was that the collision energy of
the LHC would be sufficient to generate a Higgs boson.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">But that would just be the beginning,
because the Higgs could not be detected directly.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Instead, theory predicts that it would
quickly decay into other types of particles.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">So, the hope was to spot the <i>relics</i> of the Higgs rather than the
particle itself.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjalQylaFmx0X56i01Xh20-CaThMx-zXMvou9HZ0dV5a5R6yjSlef5lZUDJaY1RVZITu9HWjB_ugEKe6nSf2bJky4siWwpBbrIbMPVlH8m1Fjn1gaUfNIyfOni2iL8WiBKSz9axRxAPbRRJ/s1600/higgs.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="255" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjalQylaFmx0X56i01Xh20-CaThMx-zXMvou9HZ0dV5a5R6yjSlef5lZUDJaY1RVZITu9HWjB_ugEKe6nSf2bJky4siWwpBbrIbMPVlH8m1Fjn1gaUfNIyfOni2iL8WiBKSz9axRxAPbRRJ/s400/higgs.gif" width="400" /></a></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;"><br /></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">To make matters even worse, the
collisions at the LHC would generate zillions of other particles.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">Thus, a Higgs that forms and quickly
decays would only manifest as a <i>slight
excess</i> of particles spotted at certain energies. <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">And finally, the Standard Model did not
predict what the mass/energy of the Higgs boson would be. So, the experimenters
would have to comb through a vast range of energy bands, looking for tiny
excesses in the number of particles of certain types.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">See the problem? Seeking a needle in a
haystack is trivial by comparison.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">But this is what was achieved!<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">In Dec 2011, after the LHC has run for
about 2 years, researchers reported a slight excess of particles in the 120 to
130 GeV range.<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">(Gev stands for “Giga-Electron-Volts” and
is a unit of energy. The mass of a proton is about 1 GeV, which is about 1.8×10<sup>−27</sup> kg)<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: left;">
<span style="font-family: Georgia, serif;">But statistical techniques indicated this
was only a “two sigma” result – there is a 2% chance of seeing something like
this purely by chance. More work needed to be done.<o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">Over the next 6 months, a massive effort
was undertaken to scrutinize the mountains of data that had accumulated.
Towards the last weeks of June, excitement rose in the scientific blogosphere.
Rumors circulated that this was indeed the real thing. And so it was.<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjntCMf8HLB6-TUG8XFEVrIIWqrQV7SqaSDpQOlQtF-mqzwFnB9oIzL-rSDchFvsSjAtDLOipuDOm_WZLmFNMpz0Bdd7PT2Hnh8WlvsXtn0MwbxyJxhqxRj_C6cLby6Vir4mnZRFOCmgDqz/s1600/Fig3-MassFactSoBWeightedMass.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="385" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjntCMf8HLB6-TUG8XFEVrIIWqrQV7SqaSDpQOlQtF-mqzwFnB9oIzL-rSDchFvsSjAtDLOipuDOm_WZLmFNMpz0Bdd7PT2Hnh8WlvsXtn0MwbxyJxhqxRj_C6cLby6Vir4mnZRFOCmgDqz/s400/Fig3-MassFactSoBWeightedMass.png" width="400" /></a></div>
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<span style="font-family: Georgia, serif;">On July 4<sup>th</sup>, after months of
speculation, the announcement was finally made.<o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">Two separate sets of experiments had
verified the existence of a particle with a mass of about 125 GeV. <o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">In each case, the probability of a chance
occurrence was “five sigma”– less than one in a million.<o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">The Higgs boson had been found.<o:p></o:p></span></div>
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<b><span style="font-family: Georgia, serif;">Dreams
and Nightmares<o:p></o:p></span></b></div>
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<span style="font-family: Georgia, serif;">So what does this mean for the future of
physics? <o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">Very different things, depending on whom
you ask.<o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">Optimists like cosmologist Sean Carroll
believe that the Higgs only heralds the beginning of a series of discoveries by
the LHC, ushering in a new dawn for physics.<o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">But not everyone agrees.<o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">Detractors argue that all the properties
of the Higgs discovered are completely consistent with the Standard Model which
is nearly 40 years old. There is no hint yet of anything inexplicable, no trace
of any new particles or phenomena, nothing to suggest the next step forward.<o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">Maybe the only new physics occurs at much
higher energies, way beyond what the LHC can probe and way beyond our
technological capabilities.<o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">This would be a physicist’s nightmare.<o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">So, what shall it be? Dream or nightmare?
Progress at last or stagnation without end?<o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">Only time – and more data – will tell.<o:p></o:p></span></div>
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<b><span style="font-family: Georgia, serif;">Afterword
on the “God particle”:<o:p></o:p></span></b></div>
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<span style="font-family: Georgia, serif;">This nickname for the Higgs boson was
used by physicist Leon Lederman in a popular book on the subject. It remains,
to date, the most egregious example of “God-mongering” to sell a popular
science book, closely followed by Stephen Hawking’s “Brief History of Time”.<o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">So just to be clear:<o:p></o:p></span></div>
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<span style="font-family: Georgia, serif;">No, the Higgs boson has no religious
significance or divine powers, and its discovery does not prove the existence
of God.<o:p></o:p></span></div>
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</div>Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com5tag:blogger.com,1999:blog-2350997094083029715.post-67758922972843772482011-06-04T23:41:00.000-07:002011-06-05T00:19:39.644-07:00The Curious Case of Stephen Hawking<span style="font-size:180%;">C</span>ountless newspaper articles compare him to Einstein.<br />Numerous others remind us that he held the Lucasian Professorship at Cambridge, “formerly held by Isaac Newton”. In an episode in Star Trek, he appears on the spaceship’s holodeck playing cards with both Newton and Einstein – and wins.<br />Clearly, in the echelons of scientific achievement, Stephen Hawking is at the pinnacle, with the greatest of the great.<br /><br />Or is he really?<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiv874SOPND1p6zrGIblD7Oo1VngpZJUqG0pI6aBrHWhVBEVVb1qHUlZjMiUZ4KYgyiJhRSNcDEFzoFBRVO5dc9Jhlm5OyN6PhoDFBFb5qowT598d8Wv_nEY9vGe6ZLF3gU15gSqFHKgW6S/s1600/stephen_hawking.jpg"><img style="float: right; margin: 0pt 0pt 10px 10px; cursor: pointer; width: 294px; height: 400px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiv874SOPND1p6zrGIblD7Oo1VngpZJUqG0pI6aBrHWhVBEVVb1qHUlZjMiUZ4KYgyiJhRSNcDEFzoFBRVO5dc9Jhlm5OyN6PhoDFBFb5qowT598d8Wv_nEY9vGe6ZLF3gU15gSqFHKgW6S/s400/stephen_hawking.jpg" alt="" id="BLOGGER_PHOTO_ID_5614626556131567714" border="0" /></a><br />Let’s do a little comparison.<br /><br />Newton can safely be called the father of theoretical physics.<br />While scientists like Galileo and Kepler underscored the importance of experiments and observations to understand the world, Newton pioneered the use of mathematical modelling to explain observed phenomena and predict new ones.<br />His most famous achievement was formulating his laws of motion, and the inverse square law of gravity which explained both the fall of an apple and the orbit of the moon and planets with unprecedented accuracy.<br />In the process, he also invented Calculus – undoubtedly the most influential branch of mathematics we have seen in the last five centuries.<br />Newton’s book, <span style="font-style: italic;">Principia Mathematica</span>, triggered a revolution in human thought, revealing the universe as amenable to human understanding through mathematical laws.<br /><br />In 1905, Einstein unified Newtonian mechanics with Maxwell’s theory of electromagnetism through his special theory of relativity, which made highly unintuitive, but completely correct predictions.<br />A decade later, he went even further, to create the general theory of relativity – a truly revolutionary theory that unified Newton’s law of gravity with special relativity by showing that gravity was the result of the curvature of space-time due to matter and energy.<br />General Relativity is one of the great pillars of modern physics, subsuming Newton’s laws of gravity and predicting a plethora of new phenomena – from black holes to the Big Bang.<br />Einstein also made very significant contributions to Quantum Mechanics, the other great pillar of modern physics. Most notably, he was the first to analyze light as a stream of particles rather than a wave.<br /><br />Now for Stephen Hawking.<br />Fundamental areas of science established? None.<br />Influential new branches of math invented? None.<br />Any new fundamental particles predicted, as in the case of Paul Dirac? Nope.<br />How about a revolutionizing to an existing theory, like Richard Feynman? Not at all.<br />Don’t mean to rude or anything, but what has he done, then?<br /><br />Two things, mainly.<br />In the late 1960’s, Roger Penrose proved a theorem to the effect that under certain conditions, general relativity predicts the occurrence of “singularities” – points in space where the laws of physics break down.<br />Hawking applied Penrose’s results to cosmological models, showing that under generic assumptions the point of origin of the Universe – the “moment of creation”, so to speak – is a singularity. Note that this is not a new theory, but a mathematical theorem derived from an existing theory (general relativity).<br /><br />In 1974, Hawking proved the result which made him famous.<br />Using results from quantum field theory, he showed that black holes radiate energy and eventually evaporate. This created a huge stir, because black holes were by definition, objects from which nothing could escape, not even light. This was a strong indication that combining general relativity with quantum field theory could lead to unexpected results.<br />Hawking’s result has never been experimentally verified, but is accepted as true. Other scientists arrived at the same conclusion from a several different approaches, and the result explains a number of other theoretical issues in the field.<br /><br />Impressive contributions? Certainly.<br />Worthy of a Nobel prize? Possibly.<br />Comparable to Newton and Einstein? Not by a very long run.<br />Then why is Hawking thus compared in the media? How has he gained a level of public prominence completely disproportionate to his actual achievements?<br />Why did his popular book, <span style="font-style: italic;">A Brief History of Time</span>, top the New York Times bestseller list for three months straight, despite the fact that most readers claimed not to understand it?<br /><br />Many think it is due to his physical condition which gets media attention– Hawking has been paralyzed by motor neuron disease since his early twenties.<br />In my opinion, the real answer lies elsewhere.<br /><br /><span style="font-weight: bold;">Grandiose Claims<br /><br /></span>Most great scientists are noted for their profound humility in the face of the universe.<br />The famous quote of Newton, comparing himself to a little boy playing on the shore of the ocean of knowledge comes to mind.<br />Now let’s hear Hawking:<br /><span style="font-style: italic;">“My goal is simple. It is a complete understanding of the universe, why it is as it is and why it exists at all.”</span><br /><br />A simple goal indeed!! This is not an isolated statement.<br />In my view, audaciously grandiose claims like these play a key role in explaining Hawking’s arc to scientific superstardom.<br />While most popular science books are content to explore the wonders of the Universe and explain some of their workings, <span style="font-style: italic;">A Brief History of Time</span> promises nothing less than an Ultimate Understanding – an answer to all the biggest questions ever asked my mankind.<br /><br />In the book, Hawking claims that we are on the verge of a revolution in our understanding of the cosmos – not just any revolution, but the one to end them all.<br />We are on the verge of completing our quest to understand the ultimate laws of the universe, says Hawking. We will soon be in possession of a Theory of Everything – a grand unifying principle that subsumes all of physics and explains all features of the cosmos hitherto unknown, including the physical properties of all the fundamental particles, the forces and interactions between them and even the underlying structure of space and time.<br /><br />What will follow thereafter is nothing short of intellectual Nirvana:<br /><span style="font-style: italic;">“If we do discover a complete theory.... we shall all be able to take part in the discussion of the question of why it is that we and the universe exist. If we find the answer to that, it would be the ultimate triumph of human reason — for then we would know the mind of God.”</span><br />A parenthetic claim – never directly made by Hawking, but widely inferred by the media and lay public – is that he would be the one to achieve this ultimate triumph of reason.<br /><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPpLTF_GZVQX6CINquxQdDgUqS0hEGTQZvD8xrtu8zTpqg_a53yWE5PPFD6eIdR_HZIr-SBwSOQ3NWkWpOyq6LD3H4jo1XIn8-uUtezacAPbAXOpeFLcUnvHm4_RjxyLlhduv9kWCUsHDd/s1600/stephen_hawking.png"><img style="display: block; margin: 0px auto 10px; text-align: center; cursor: pointer; width: 400px; height: 119px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPpLTF_GZVQX6CINquxQdDgUqS0hEGTQZvD8xrtu8zTpqg_a53yWE5PPFD6eIdR_HZIr-SBwSOQ3NWkWpOyq6LD3H4jo1XIn8-uUtezacAPbAXOpeFLcUnvHm4_RjxyLlhduv9kWCUsHDd/s400/stephen_hawking.png" alt="" id="BLOGGER_PHOTO_ID_5614627724821751410" border="0" /></a><br />The grand vision promulgated by Hawking is, of course, vastly misleading.<br />Even if a unified theory of physics is found, its real applications will be primarily in esoteric realms far beyond the reach of experiment or observation, such the centres of black holes and the moment of the Big Bang. While this would certainly be impressive, the discussion of "why it is that we and the universe exist” has proceeded quite far without it.<br />Our current theories of physics, while incomplete and partial, have nevertheless been very successful in explaining how the world around us emerged from the blazing radiation of the Big Bang. Most areas of physics itself, let alone the other sciences, would not be impacted at all by the discovery of a “complete theory”.<br /><br />Furthermore, Hawking’s own role in unifying physics has been minimal.<br />Currently, the most popular approach for unification is String Theory. If it succeeds in producing a Theory of Everything – a very big ‘If’ – the mantle of Einstein would pass to Edward Witten. Vastly respected within the theoretical physics community for his path-breaking contributions and insights, Dr. Witten is virtually unknown to the common man due to his preference for sticking to science rather than airing extravagant claims.<br /><br />Hawking has never been at the forefront of the unification program, or even a significant contributor. But his starry eyed vision of the imminent arrival of the unified theory, his self declared goal of “complete understanding of the universe”, his constant references to God in his popular books and public lectures, has given the quest of unify physics an almost religious significance in the public psyche – with the Theory of Everything as God and Stephen Hawking as It’s soon-to-be prophet.<br /><br />Unfortunately, scientific gurus, unlike religious ones, must test their claims against reality.<br />So how has Hawking fared?<br /><br /><span style="font-weight: bold;">The Rather Pathetic Design<br /><br /></span>It is now past 22 years since <span style="font-style: italic;">A Brief History of Time</span> was written and over three decades since Hawking’s first public pronouncement that the end of physics was imminent.<br />The unified theory is nowhere in sight, and while string theorists keep ploughing on and claiming progress, their best efforts are yet to yield a single experimental prediction.<br /><br />It is also becoming abundantly clear that even if a unified theory is eventually discovered, Stephen Hawking’s name will not figure on the credits list.<br />Hawking seems to be having trouble digesting this fact – his strategy is to alternate between suggesting that no unified theory is possible (sour grapes, anyone?) and declaring that it has already been found (no way).<br /><br />His latest popular book, <span style="font-style: italic;">The Grand Design</span>, is a good example of this.<br />He starts by saying that there may be no unified theory of physics, but just a bunch of “observer dependent theories”, whatever that means. The next moment, he is spinning on a dime to declare that “M-theory” is the unified theory.<br />Furthermore, apparently, M-theory predicts that the universe can spontaneously originate from absolute nothingness, so God is not necessary (once again God appears to help Hawking make headlines and sell copies).<br />So there! Dr. Hawking has Explained It All. Hip, hip, hooray!<br /><br />Nobody is convinced.<br />Because M-theory makes no predictions. The reason being that nobody even knows what M-theory is.<br />Let me explain.<br /><br />In 1995, Dr. Edward Witten whom we saw earlier, demonstrated the presence of a number of “dualities” between various versions of string theory – roughly speaking, a difficult problem in one version could be translated into an easy one in another version.<br />This led to the hope that all the string theories were special aspects of a greater theory subsuming them all, which was termed “M-theory”.<br />In Witten’s own words: “<span style="font-style: italic;">The M stands for magic, mystery or matrix according to taste</span>”.<br />Thus M-theory is a <span style="font-style: italic;">hypothetical </span>theory which <span style="font-style: italic;">may</span> exist – not an existing theory with concrete predictions.<br /><br />Similarly, the spontaneous origin of the universe is not a prediction of any theory we have – it is merely <span style="font-style: italic;">speculated</span> that a unified theory of physics which merges general relativity and quantum mechanics <span style="font-style: italic;">might </span>allow something like this to happen.<br />The “grand design” revealed by Hawking is, thus, misleading on many levels.<br />Much like religious gurus who refuse to admit a mistake, Hawking attempts to “explain it all” through obfuscation and incorrect statements.<br /><br />So, where does this leave us?<br />In my view, with a lesson that we keep forgetting despite endless reminders.<br />We humans are very small creatures in an incredibly vast Universe and our attempts to unravel its mysteries work best when we work sincerely to answer small questions.<br />It is only by carefully knitting together the answers to those little questions that the big picture slowly emerges.<br />By contrast, whenever an individual pompously proclaims an “Answer to Life, The Universe and Everything”, it inevitably turns out to be incorrect or a meaningless 42.<br /><br />Serious students and followers of science would do well to ignore the self-aggrandizing hype of Stephen Hawking, our media-made “Einstein”, and heed the words of the real one:<br /><span style="font-style: italic;">“Enough for me an inkling of the marvellous structure of Reality, the endeavour to comprehend a portion, be it ever so tiny, of the Reason that manifests itself in Nature.”</span>Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com5tag:blogger.com,1999:blog-2350997094083029715.post-62447952497526366132011-02-24T03:19:00.000-08:002011-02-24T05:12:51.902-08:00AI Is Creeping Up On You<span style=";font-family:";font-size:100%;" lang="EN-US" ><span style="font-size:180%;">I</span><span style="font-family:georgia;">n movies, Artificial Intelligence, aka AI, always arrives with a bang.</span></span><span lang="EN-US" style="font-size:100%;"><br />The machines wake up, realize their power and immediately launch a nuclear holocaust or trap us in the Matrix or something similarly</span><span lang="EN-US" style="font-size:100%;"> unpleasant.<span style=""> </span>I strongly suspect this will never happen. Instead, as the decades go by, we will increasingly be surrounded by AI at many levels – while vigorously insisting all through that it’s “no big deal”.</span><span lang="EN-US" style="font-size:100%;"><br /><br />A milestone for artificial intelligence was achieved last week in a three-day <i style="">Jeopardy</i> contest held from February 14 – 16. For those unfamiliar with Jeo</span><span lang="EN-US" style="font-size:100%;">pardy, it is a version of our beloved Quiz contests, wit</span><span lang="EN-US" style="font-size:100%;">h some differences.</span><span lang="EN-US" style="font-size:100%;"><br />For one, the clues are often presented in deliberately convoluted language, often with more than one meaning. As a further twist, the quizmaster presents the question as an “answ</span><span lang="EN-US" style="font-size:100%;">er”, and the contestant must present the answer as a “question”.</span><span lang="EN-US" style="font-size:100%;"><br />For instance, rather than asking “Who wrote Hamlet and Macbeth?” the host will say, “This is the author of Hamlet and Macbeth” and the contestant will answer “Who is Shakespeare?”<br />Rather than the straightforward scoring system of quizzes, each clue comes with a “d</span><span lang="EN-US" style="font-size:100%;">ollar value”, which is added or deducted to the contestant’s total depending on their answer.</span><span lang="EN-US" style="font-size:100%;"><br />There are also several “Daily Double” clues, where the contestant can wager a sum of money all the way up to their total “earnings” till that point.</span><span lang="EN-US" style="font-size:100%;"><br />The score of the contestant is the total amount of “money accumulated”.<br /><br /></span><p class="MsoNormal" style="margin-bottom: 0.0001pt;font-family:georgia;"><span style="font-size:100%;"><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIQI8tE-BIihwJoZwwOldIA6ga00nxJDQUL569xmRFN-DSYa6QauNoidamqXTMpsNYl236V2H_72Lxg779HQHOGNvQbcfZf9hpg6IozLd40FIJaXDPNbjnss6yFGLx95FQuBeVhbnnKZBU/s1600/Watson_Jeopardy.jpg"><img style="display: block; margin: 0px auto 10px; text-align: center; cursor: pointer; width: 400px; height: 225px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIQI8tE-BIihwJoZwwOldIA6ga00nxJDQUL569xmRFN-DSYa6QauNoidamqXTMpsNYl236V2H_72Lxg779HQHOGNvQbcfZf9hpg6IozLd40FIJaXDPNbjnss6yFGLx95FQuBeVhbnnKZBU/s400/Watson_Jeopardy.jpg" alt="" id="BLOGGER_PHOTO_ID_5577215184897799746" border="0" /></a></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;font-family:georgia;"><span lang="EN-US" style="font-size:100%;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;font-family:georgia;"><span lang="EN-US" style="font-size:100%;"><br /></span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;font-family:georgia;"><span lang="EN-US" style="font-size:100%;">The tournament last week featured two superstars of the Jeopardy world – Brad Rutter, the biggest all-time money winner on the show, and Ken Jennings, record holder for the longest championship streak.</span></p><span lang="EN-US" style="font-size:100%;">But the spotlight was on the non-human entrant, Watson – a supercomputer designed by IBM running natural language processing software.</span><span lang="EN-US" style="font-size:100%;"><br />The clues were sent to Watson as a text message at exactly the same time they were made visible to the other contestants. Watson would have to unravel the language in the clue, find the answer, and press the buzzer before the other contestants did to have a chance at scoring.</span><span lang="EN-US" style="font-size:100%;"><br /><br />The first day of the match on Feb 14 ended with Watson and Rutter tied at $50</span><span lang="EN-US" style="font-size:100%;">00 with Jennings trailing at $2000. The internet was abuzz with theories about how the champions were merely “warming up” before trouncing the machine over the next two days.</span><span lang="EN-US" style="font-size:100%;"><br />All such speculations were crushed on Day Two, which ended with Jennings at $4,800, Rutter at $10,400… and Watson massively ahead with $35,734 !</span><span lang="EN-US" style="font-size:100%;"><br />The final day ended with Jennings at $24,000,<span style=""> </span>Rutter at $21,600 and Watson at $77,147 – a thoroughly convincing victory.</span><span lang="EN-US" style="font-size:100%;"><br />The answer to “Jeopardy world champion” is now “Who is Watson?”</span><span lang="EN-US" style="font-size:100%;"><br /><br />Apart from the immense entertainment, a pleasant aspect of the program was the graceful acceptance of defeat by the humans. The affable Ken Jennings even quipped, “I, for one, welcome our computer overlords.”</span><span lang="EN-US" style="font-size:100%;"><br /><br />This was a marked contrast to the acrimonious ending of a similar Man vs. Machine event fourteen years earlier, when IBM supercomputer Deep Blue defeated world chess champion Garry Kasparov in a 6 game match played in 1997.</span><span lang="EN-US" style="font-size:100%;"><br />Kasparov proved to be a very poor loser – storming away after the last game, b</span><span style="font-size:100%;"><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbJEJyVckFZaqCYUXBnGno88wul8_x1R4ZiRcnC8xwwpii4BidpMsZSwad_Ird5q-3uGvd0N3xB8tEP8wGrFNDc9SsLzgUcg_yzcTFK9WsXM9BkjT8z3nPy7f-6eHFdm7psLtyUJtp3KKq/s1600/kasparov-deep-blue.jpg"><img style="float: right; margin: 0pt 0pt 10px 10px; cursor: pointer; width: 400px; height: 271px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbJEJyVckFZaqCYUXBnGno88wul8_x1R4ZiRcnC8xwwpii4BidpMsZSwad_Ird5q-3uGvd0N3xB8tEP8wGrFNDc9SsLzgUcg_yzcTFK9WsXM9BkjT8z3nPy7f-6eHFdm7psLtyUJtp3KKq/s400/kasparov-deep-blue.jpg" alt="" id="BLOGGER_PHOTO_ID_5577215681391379314" border="0" /></a></span><span lang="EN-US" style="font-size:100%;">eing conspicuously absent at the prize distribution ceremony and accusing the IBM team of cheating.</span><span lang="EN-US" style="font-size:100%;"><br />IBM retaliated by refusing a re-match and decommissioning Deep Blue.</span><span lang="EN-US" style="font-size:100%;"><br />The whole episode remains mired in controversy and bad feeling.</span><span lang="EN-US" style="font-size:100%;"><br /><br />Computer chess advanced considerably over the next decade.</span><span lang="EN-US" style="font-size:100%;"><br />In November 2006, the reigning world champion Vladimir Kramnik played Deep Fritz.</span><span lang="EN-US" style="font-size:100%;"><br />In contrast to Deep Blue which was specially designed software running on a customized supercomputer, Fritz was a commercially available program running on a high-end laptop.</span><span lang="EN-US" style="font-size:100%;"><br />Nevertheless, the computer won the 6 game match with 2 wins and 4 draws.</span><span lang="EN-US" style="font-size:100%;"><br /><br />Since then, interest in human-computer chess matches has waned. Though not proved by actual play, it is quietly acknowledged that today’s best chess programs like Rybka running on a supercomputer would trounce any human chess player.</span><span style="font-size:100%;"><b style=""><span style="" lang="EN-US"><br /><br />Lame Excuses</span></b></span><span lang="EN-US" style="font-size:100%;"><span style="font-weight: bold;"><br /></span>What amuses me about both the Watson and Deep Blue incidents is the subsequent proliferation of excuses from the human side for why these incidents were “nothing special” and “not really artificial intelligence”. The excuses fall into roughly four categories, which I list below in decreasing order of silliness, along with my responses.</span><span style="font-size:100%;"><b style=""><span style="" lang="EN-US"><br /><br />Excuse 1:</span></b></span><span lang="EN-US" style="font-size:100%;"> “Deep Blue and Watson were both supercomputers with top end hardware. So it’s no big deal that they could do what they did”</span><span style="font-size:100%;"><b style=""><span style="" lang="EN-US"><br />Response:</span></b></span><span lang="EN-US" style="font-size:100%;"> And your point is? I can similarly imagine a rabbit saying, “It’s no big deal that humans are so intelligent, given their big brains and all.”</span><span lang="EN-US" style="font-size:100%;"><br />The power of the hardware is <i style="">part of</i> what makes the system impressive. I agree that Watson wouldn’t have won if it was running on a laptop, but I can bet you that Jennings wouldn’t do too well after a frontal lobotomy either.</span><span lang="EN-US" style="font-size:100%;"><br />Also note how quickly we jump from “A computer can never do X” to “It’s no big deal that a computer can do X”!!</span><span style="font-size:100%;"><b style=""><span style="" lang="EN-US"><br /><br />Excuse 2:</span></b></span><span lang="EN-US" style="font-size:100%;"> “The computer isn’t really thinking. It is only doing what its program tells it to do”</span><span style="font-size:100%;"><b style=""><span style="" lang="EN-US"><br />Response:</span></b></span><span lang="EN-US" style="font-size:100%;"> This is in strong competition for the silliness top spot.</span><span lang="EN-US" style="font-size:100%;"><br />If a human had beaten the world chess champion, would you have agreed that he or she was thinking?</span><span lang="EN-US" style="font-size:100%;"><br />Conversely, why not argue that when Kasparov plays “he isn’t really thinking. He is only doing what the firing of neurons in his brain tells him to do”?</span><span style="font-size:100%;"><b style=""><span style="" lang="EN-US"><br /><br />Excuse 3:</span></b></span><span lang="EN-US" style="font-size:100%;"> “The computer has no credit in this. The credit belongs entirely to the humans who programmed it”</span><span style="font-size:100%;"><b style=""><span style="" lang="EN-US"><br />Response:</span></b></span><span lang="EN-US" style="font-size:100%;"> No wait, it’s not the credit of the programmers at all, but of the genes and environment that shaped their brains. No wait, actually all credit is due to the process of evolution which shaped those genes. No wait…</span><span lang="EN-US" style="font-size:100%;"><br />See how this goes?</span><span lang="EN-US" style="font-size:100%;"><br />My point is, if we follow any consistent standard for giving credit, we should certainly congratulate the programmers who for <i style="">designing</i> Watson or Deep Blue, but after that we must credit the systems for their subsequent performance.</span><span style="font-size:100%;"><b style=""><span style="" lang="EN-US"><br /><br />Excuse 4:</span></b></span><span lang="EN-US" style="font-size:100%;"> “Computers may be able to play chess and win Jeopardy, but they cannot invent new technology or compose music or *fill in the blanks*”</span><span style="font-size:100%;"><b style=""><span style="" lang="EN-US"><br />Response:</span></b></span><span lang="EN-US" style="font-size:100%;"> The sentence above is missing a “Yet” at the end.</span><span lang="EN-US" style="font-size:100%;"><br /><br />One must remember that the first ‘computers’ in society were not machines, but a group of people, mostly women, working in science laboratories. They were so called because of their ability to perform complex arithmetic accurately and repeatedly – an ability much valued and taken to indicate great mental stamina.</span><span lang="EN-US" style="font-size:100%;"><br />Fifty years ago, anyone would have agreed that playing chess well required intelligence, and a high degree of intelligence, at that.</span><span lang="EN-US" style="font-size:100%;"><br />Talking computers which understand language have traditionally been science-fiction territory – a hallmark of intelligent machines and droids of the far future.</span><span lang="EN-US" style="font-size:100%;"><br /><br />But every time real computers reach one of these milestones, the significance of the event is denied and the bar of “true intelligence” reset several notches higher.</span><span lang="EN-US" style="font-size:100%;"><br />The current list of “what computers can never do” includes “appreciating poetry” and “falling in love”. True, perhaps, but the question is, do they need to?</span><span lang="EN-US" style="font-size:100%;"><br />The goal of AI is not to create artificial humans, any more than the goal of aircraft designers is to create a machine which flaps its wings and lays eggs.</span><span lang="EN-US" style="font-size:100%;"><br /><br />I personally believe that Artificial Intelligence will not take the form of an all-encompassing, godlike Supermind, so beloved of science fiction authors and fans. <br />Instead, as the centuries roll on, we will see a proliferation of specialized applications tailored to specific tasks, that we would definitely call intelligent, but our descendants may not. <br />Ultimately, the only remaining special feature of human intelligence may be the ability to invent excuses for why we are special!</span> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style=";font-family:";" lang="EN-US"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style=";font-family:";" lang="EN-US"> </span></p>Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com11tag:blogger.com,1999:blog-2350997094083029715.post-58236231440492425392011-01-30T22:05:00.000-08:002011-01-30T23:55:20.146-08:00Male and Female<p class="MsoNormal" style="margin-bottom: 0.0001pt;"><b style=""><span style="line-height: 115%;font-family:";font-size:16pt;" > </span></b><span style=";font-family:";" ><span style="font-size:180%;">W</span><span style="font-family:georgia;">e are all half-human. Think about it.</span></span></p><span style="">Imagine an alien zoologist who abducts one of us to conduct a biological study of the human species. However judiciously chosen, the sample will lack one major organ system and some attendant secondary characteristics. Furthermore, our inquisitive alien will be at a loss trying to figure out how the species propagates.</span><br /><span style=""><br />Much ado about little has been made in human society over the “inexorable differences between male and female” – from different dressing conventions to different </span><span style="">expectations about career and lifestyle.</span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtS9Gv4BbN-wF_ZIa6ylXBLSuq_cshM01mrCHgfLAvfzIYkAduHcJXdRwk9EfIAgpVanlT7E-_gwDzpbCCvcDrjizWSEr7v690I7hIT1Y2qhCc5RimKh7ZUgHHWMTmzcCn0r0lgRjy4J8A/s1600/yin-yang-cracked.jpg"><img style="float: left; margin: 0pt 10px 10px 0pt; cursor: pointer; width: 400px; height: 300px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtS9Gv4BbN-wF_ZIa6ylXBLSuq_cshM01mrCHgfLAvfzIYkAduHcJXdRwk9EfIAgpVanlT7E-_gwDzpbCCvcDrjizWSEr7v690I7hIT1Y2qhCc5RimKh7ZUgHHWMTmzcCn0r0lgRjy4J8A/s400/yin-yang-cracked.jpg" alt="" id="BLOGGER_PHOTO_ID_5568240121943390834" border="0" /></a><br /><span style="">In fact, so</span><span style="">me philosophers have gone so far as to</span><span style=""> explain the entire Universe as the interaction of “male and female principles” – Shiva and Sha</span><span style="">kti, Yin and Yang.</span><span style="">This is, of course, one of the many examples where over-extrapola</span><span style="">ting from the human condition leads to complete</span><span style=""> </span><span style="">nonsense. The “principles” governing the Universe are impersonal mathematical laws which know nothing of male, female or transg</span><span style="">ender.</span><span style=""><br /></span><br /><span style="">All right then, what about the biosphere?</span><span style=""><br />Aren’t male-female distinctions fundam</span><span style="">ental to the living world? Hardly.</span><span style=""><br /><br />Let’s start at the very beginning – with microorganisms.</span><br />In case that’s too humble a beginning for you, consider the fact that at least 50% of the Earth’s biomass consists of bacteria alone, and for 85% of its history, life on Earth was nothing <i style="">but</i><span style=""> microorganisms.</span><span style=""><br />There are no genders in the unicellular world.</span><span style=""><br />Virtually all microorganisms reproduce asexually – splitting into clones which mutate slowly over the </span><span style="">ages. Among bacteria, a pair of creatures will sometimes come together to exchange genetic material, but this is not connected to reproduction.</span><span style=""><br /><br />Still too microscopic for you? Then let’s consider the green world of autotrophs.</span><span style=""><br />Plants are the basis of the biosphere, converting the energy of sunlight into useful forms which all other creatures can process.</span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvGbZHmukPECa9_JCn_Is_C5imMyMfFu-b4wK-FFeu3XcGlz_JihqCicPtjSflXcS3a2WHA9xGS4yITA18Jw7JocVgmpCw883h-T-4BSjaIz14zsLwlMJIUV1RbC6N5mDPnZBIahlg17lO/s1600/flower-cross-section.jpeg"><img style="float: right; margin: 0pt 0pt 10px 10px; cursor: pointer; width: 386px; height: 290px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvGbZHmukPECa9_JCn_Is_C5imMyMfFu-b4wK-FFeu3XcGlz_JihqCicPtjSflXcS3a2WHA9xGS4yITA18Jw7JocVgmpCw883h-T-4BSjaIz14zsLwlMJIUV1RbC6N5mDPnZBIahlg17lO/s400/flower-cross-section.jpeg" alt="" id="BLOGGER_PHOTO_ID_5568238562219541378" border="0" /></a><br /><span style="">An alien botanist would face none of the “incomplete sample” problems plaguing our hapless zoologist. The vast majority of plant species are hermaphrodites – a single sample gives you the entire picture. The situation is worth a deeper look.</span><span style=""><br /><br />Each plant has many reproductive org</span><span style="">ans. We call them flowers.</span><br /><span style="">In the vast majority of cases, <i style="">each </i>flower has bo</span><span style="">th male and female parts – an a</span><span style="">ndroecium that produces pollen, and a gynaecium that contains ova. In botanist’s jargon, these are “complete flowers”.</span><br /><span style="">These include roses, lilies and the hibiscus, much beloved of authors of sixth-grade biology texts. In fact, virtually all the bright and co</span><span style="">lourful flowers that gardeners cherish are complete flowers, and the corresponding plants are called “hermaphrodite”.</span><span style=""><br /><br />Now some plants produce flowers that only make pollen or only have ova – male and female flowers so to speak. But each <i style="">plant</i> has both male and fem</span><span style="">ale flowers.</span><span style=""><br />Only a paltry 6% or so of plant species have exclusively male or female plants, with flowers of only one type. These tend to be rather small and unattractive, usually pollinated by wind.</span><span style=""><br />Clearly, plants think gender is a bad idea.</span><span style=""><br /><br />So, how about things that move? If you take a walk in your garden, the earthworms and snails you come across are all hermaphrodites. During mating time, two indiv</span><span style="">iduals will line up and simultaneously fertilize each other!</span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_okXQHdRomRsP47OBUrnumM4HgVRGOmoGM1K1792pXSWPClgEVggRmlIjiZxUVzBe6bBtPHbNpmWwQ8X0xE25EoQKU-itD0u8CkDn_W2ue2vUfSskQ3bFJKcVcle9GMpnwNTpwPTShWx-/s1600/clownfish.jpg"><img style="float: left; margin: 0pt 10px 10px 0pt; cursor: pointer; width: 400px; height: 300px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_okXQHdRomRsP47OBUrnumM4HgVRGOmoGM1K1792pXSWPClgEVggRmlIjiZxUVzBe6bBtPHbNpmWwQ8X0xE25EoQKU-itD0u8CkDn_W2ue2vUfSskQ3bFJKcVcle9GMpnwNTpwPTShWx-/s400/clownfish.jpg" alt="" id="BLOGGER_PHOTO_ID_5568238822608775282" border="0" /></a><br /><span style="">Looking for something less slimy? Take a crui</span><span style="">se down to the Great Barrier Reef. Among the colourful corals and sea anemones, you will see sc</span><span style="">hoo</span><span style="">ls of bright orange and white fish, which </span><span style="">may remind you of the movie <i style="">Finding Nemo</i>.</span><br /><span style="">These are clownfish, and their gender behaviour is</span><span style=""> most interesting. Each school consists of many males and one dominant female. But if the fem</span><span style="">ale dies or goes missing, <i style="">one of the males changes sex to become female</i>. How extremely convenient!</span><span style=""><br />In fact, clownfish will sometimes pair up and inhabit a sea anemone. If the pair consists of two males, no problem at all. 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name="Subtle Reference"> <w:lsdexception locked="false" priority="32" semihidden="false" unhidewhenused="false" qformat="true" name="Intense Reference"> <w:lsdexception locked="false" priority="33" semihidden="false" unhidewhenused="false" qformat="true" name="Book Title"> <w:lsdexception locked="false" priority="37" name="Bibliography"> <w:lsdexception locked="false" priority="39" qformat="true" name="TOC Heading"> </w:LatentStyles> </xml><![endif]--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin-top:0cm; mso-para-margin-right:0cm; mso-para-margin-bottom:10.0pt; mso-para-margin-left:0cm; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-fareast-language:EN-US;} </style> <![endif]--><span style="line-height: 115%;font-family:";font-size:11pt;" ><br /><br /><span style="font-family:georgia;">Such sex change behaviour, where males become females or vice versa in response to the gender ratio in the population is in fact quite common among fish in the coral reefs and several species of amphibians as well (some of you may recall that this was a plot element in </span><i style="font-family: georgia;">Jurassic Park</i><span style="font-family:georgia;">). Imagine the wonders such a system would work in your average engineering college. </span></span><span style=""><br /><br />So then, <i style="">what happened to us</i>?</span><span style=""><br />Why are we stuck with the most boring of alternatives, where each individual is doomed from birth to see only “one half of life”, as it were, with much inane speculation about what the other half “really want</span><span style="">s”? Why aren’t we hermaphroditic “complete </span><span style="">individuals” like the plants, or conveniently mutable like the clownfish?</span><span style=""><br />Turns out, this is quite a puzzle.</span><span style=""><br /><br />The only major classes of animal where all individuals are exclusively male or female are reptiles, birds and mammals (and dinosaurs too, from fossil evidence, but then birds are just living dinosaurs.) But they are produced rather differently in each case.</span><span style=""><br />If you think male versus female is a matter of different versus similar sex chromosomes (XY vs XX) think again. That’s only for mammals.</span><span style=""><br />Among reptiles, the sex of the individual depends on various external factors like the temperature at which the egg was incubated. For birds, it’s the <i style="">males</i> that have the same sex chromosomes (ZZ, as opposed to ZW for females).</span><span style=""><br />And even for humans, chromosomes aren’t destiny. It turns out that in addition to XY chromosomes, you need functioning testosterone receptors in the foetus, otherwise it just develops into a female.</span><span style=""><br />Nature has to work quite hard to produce separate-sex individuals, apparently.</span><span style=""><br /><br />But why? <span style=""> </span>What’s wrong with good old hermaphrodites?</span><span style=""><br />The possible diversity in the gene pool is reduced to half in ev</span><span style="">ery generation, thanks to the exclusive single-sex system, so what are the compensating features for evolution?</span><span style=""><br />Is some kind of fitness parameter being optimized by having separate males and females?</span><span style=""><br />Or is it just a “frozen accident” of biological history – the comm</span><span style="">on ancestor of mammals, birds and reptiles had males and females, so that’s how it remained?</span><span style=""><br /><span style=""><span style=""><span style=""><span style=""><span style=""><span style=""><span style=";font-family:";" ><span style="font-family:georgia;"><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0veAC0bmsOEGt0V_tJw46NfpVHf1PfsgYNfJY3M0ejfsPyMIjNKs0I-lSXwsnU-T_8ZY8Be3dxz49cf6m5wsTw7hcDRvQr9VXT44JDVuuAPFcK7_bASHBD5leelQDZ3vEgIKpkx7bmqC5/s1600/Copy+of+DSC_6755.JPG"><img style="display: block; margin: 0px auto 10px; text-align: center; cursor: pointer; width: 320px; height: 184px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0veAC0bmsOEGt0V_tJw46NfpVHf1PfsgYNfJY3M0ejfsPyMIjNKs0I-lSXwsnU-T_8ZY8Be3dxz49cf6m5wsTw7hcDRvQr9VXT44JDVuuAPFcK7_bASHBD5leelQDZ3vEgIKpkx7bmqC5/s320/Copy+of+DSC_6755.JPG" alt="" id="BLOGGER_PHOTO_ID_5568246806595502178" border="0" /></a></span></span></span></span></span></span></span></span><br />The answers lie hidden somewhere in the genetic code. But for now, we half-humans can only wonder and our alien zoologist needs a sample of at least two....</span> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style=";font-family:";" > </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style=";font-family:";" > </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span class="apple-style-span"><span style=";font-family:";color:black;" > </span></span></p> <p class="MsoNormal"><span class="apple-style-span"><span style=";font-family:";color:black;" > </span></span></p> <p class="MsoNormal"><span class="apple-style-span"><span style=";font-family:";color:black;" > </span></span></p> <p class="MsoNormal"><span class="apple-style-span"><span style=";font-family:";color:black;" > </span></span></p> <p class="MsoNormal"><span class="apple-style-span"><span style=";font-family:";color:black;" > </span></span></p> <p class="MsoNormal"><span style=";font-family:";" > </span></p> <p class="MsoNormal"><b style=""><span style=";font-family:";" > </span></b></p>Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com7tag:blogger.com,1999:blog-2350997094083029715.post-27460048070325966072010-10-24T08:56:00.000-07:002010-10-26T08:52:50.172-07:00Aaji Joto Taara Tobo Akashe<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWpTB8SzSBaaVsc49ggD3Y4lMUDlIPgvhbolBtB6OmS3ZICBb1U4G02laWujdh6K5PONfugFkF1liyagZW8x7-C9rQMg7HJv8rlyRMJcXef_XCWHzJi7t_O47wbQ3C1aJ6ZOVZa3hJj-rn/s1600/Carina+Nebula.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 397px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWpTB8SzSBaaVsc49ggD3Y4lMUDlIPgvhbolBtB6OmS3ZICBb1U4G02laWujdh6K5PONfugFkF1liyagZW8x7-C9rQMg7HJv8rlyRMJcXef_XCWHzJi7t_O47wbQ3C1aJ6ZOVZa3hJj-rn/s400/Carina+Nebula.jpg" alt="" id="BLOGGER_PHOTO_ID_5532007586958549026" border="0" /></a><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:punctuationkerning/> <w:validateagainstschemas/> <w:saveifxmlinvalid>false</w:SaveIfXMLInvalid> <w:ignoremixedcontent>false</w:IgnoreMixedContent> <w:alwaysshowplaceholdertext>false</w:AlwaysShowPlaceholderText> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> <w:dontgrowautofit/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:latentstyles deflockedstate="false" latentstylecount="156"> </w:LatentStyles> </xml><![endif]--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style> <![endif]--> <p class="MsoNormal" style="margin-left: 1.5in;"><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:punctuationkerning/> <w:validateagainstschemas/> <w:saveifxmlinvalid>false</w:SaveIfXMLInvalid> <w:ignoremixedcontent>false</w:IgnoreMixedContent> <w:alwaysshowplaceholdertext>false</w:AlwaysShowPlaceholderText> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> <w:dontgrowautofit/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:latentstyles deflockedstate="false" latentstylecount="156"> </w:LatentStyles> </xml><![endif]--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style> <![endif]--> </p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" ><span style="font-size:180%;"><br /></span></span></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" ><span style="font-size:180%;">A</span>ll the stars that grace Thy skies</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >Shine within my soul tonight</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >Thy Cosmos surges in from every side</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >Pouring into my depths.</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >Every blossom in all Thy groves</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >Blooms within my being.</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" > </span></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" ><br /></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >Unbounded joy from horizons afar</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >Flows through my soul </span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >A fragrance deep</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >Mingling, merging, gushing forth again</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >In ecstatic praise of Thee.</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >This day I heed not any one</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >I can hear the words of none</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >The Cosmic Breath</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >Sings in my heart</span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" lang="EN-GB" >Entrancing as the sweetest flute.</span></p><br /><br /><object width="335" height="28"><param value="http://www.divshare.com/flash/audio_embed?data=YTo2OntzOjU6ImFwaUlkIjtpOjQ7czo2OiJmaWxlSWQiO2k6MTI5NTcwMTc7czo0OiJjb2RlIjtzOjEyOiIxMjk1NzAxNy01MjkiO3M6NjoidXNlcklkIjtpOjE1OTU1MzA7czoxMjoiZXh0ZXJuYWxDYWxsIjtpOjE7czo0OiJ0aW1lIjtpOjEyODgwMjA5NDY7fQ==&autoplay=" name="movie"><param name="allowFullScreen" value="true"><param name="allowscriptaccess" value="always"><param name="wmode" value="transparent"><embed wmode="transparent" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" src="http://www.divshare.com/flash/audio_embed?data=YTo2OntzOjU6ImFwaUlkIjtpOjQ7czo2OiJmaWxlSWQiO2k6MTI5NTcwMTc7czo0OiJjb2RlIjtzOjEyOiIxMjk1NzAxNy01MjkiO3M6NjoidXNlcklkIjtpOjE1OTU1MzA7czoxMjoiZXh0ZXJuYWxDYWxsIjtpOjE7czo0OiJ0aW1lIjtpOjEyODgwMjA5NDY7fQ==&autoplay=" width="335" height="28"></embed></object><br /><br />Click the button to play the song.<br />Translation and song are both mine. The photo is not, but click on it for the full impact.Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com15tag:blogger.com,1999:blog-2350997094083029715.post-67054641314476932432010-10-17T05:00:00.000-07:002010-10-17T05:25:07.589-07:00The Official Phoren Glamour Ranking List<!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:punctuationkerning/> <w:validateagainstschemas/> <w:saveifxmlinvalid>false</w:SaveIfXMLInvalid> <w:ignoremixedcontent>false</w:IgnoreMixedContent> <w:alwaysshowplaceholdertext>false</w:AlwaysShowPlaceholderText> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> <w:dontgrowautofit/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:latentstyles deflockedstate="false" latentstylecount="156"> </w:LatentStyles> </xml><![endif]--><!--[if !mso]><object classid="clsid:38481807-CA0E-42D2-BF39-B33AF135CC4D" id="ieooui"></object> <style> st1\:*{behavior:url(#ieooui) } </style> <![endif]--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style> <![endif]--><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:punctuationkerning/> <w:validateagainstschemas/> <w:saveifxmlinvalid>false</w:SaveIfXMLInvalid> <w:ignoremixedcontent>false</w:IgnoreMixedContent> <w:alwaysshowplaceholdertext>false</w:AlwaysShowPlaceholderText> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> <w:dontgrowautofit/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:latentstyles deflockedstate="false" latentstylecount="156"> </w:LatentStyles> </xml><![endif]--><!--[if !mso]><object classid="clsid:38481807-CA0E-42D2-BF39-B33AF135CC4D" id="ieooui"></object> <style> st1\:*{behavior:url(#ieooui) } </style> <![endif]--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style> <![endif]--> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><span style="font-size:180%;">F</span>inally! After months of painstaking research and much conversation with non-resident Indians (henceforth, NRI’s) and resident ones, I have successfully compiled the definitive ranking of the various foreign (henceforth, phoren) countries where Indian diaspora are settled.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">To provide the necessary background, one must realize that beloved though our country is, it is much lacking in glamour and glitz. Resident Indians who have never left the country sadly shake their heads at the materialistically fulfilling, but spiritually devoid lifestyle of their NRI brethren. Interestingly though, they tend to make a rapid beeline for aforesaid materialistic lifestyle if given the opportunity.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">But having successfully transitioned to NRI status, it is considered extremely non-kosher to declare that you want to stay on in the country you spent so much effort moving to. Instead you must always say that you intend to “go back after some years”, where the “some” is very nebulously defined.<br /></span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">On the other hand, <i style="">actually</i> going back to India is considered even more non-kosher, and causes much shock and awe – like saying you want to drop out of college and then really doing it.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Now, having lived in two phoren countries and then come back to India (shock! awe!), I have also observed that all phorens are not equal in the Indian psyche and some have decidedly more glamour than others. Hence, here I compile a carefully researched ranking for my readership, so that they may choose the level of glamour that best suits them.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">So, firstly, the three major criteria that glamorize your phoren land:</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><b style=""><span style="font-family:Georgia;">1) Wealth and shine:</span></b><span style="font-family:Georgia;"> No country is truly phoren unless it obviously has mucho moolah with shiny-shiny buildings, fast-fast cars and big-big highways.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">As the brother of my Philadelphia resident friend keeps getting asked, <i style="">“Aap ka behen phoren mein rahti hai, toh aap </i>Nigeria<i style=""> mein kyon rahte hain?”</i> Clearly, Nigeria is not phoren.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Ditto for Russia, Iran, Brazil, and Thailand (whatever other appeal it may have.)</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><b style=""><span style="font-family:Georgia;">2) White people:</span></b><span style="font-family:Georgia;"> The race of our former colonial overlords must be present in copious quantities to legitimize phoren glamour. Don’t ask why. It’s an axiom.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><b style=""><span style="font-family:Georgia;">3) English:</span></b><span style="font-family:Georgia;"> More a matter of convenience, but helps fine-tune the rankings as we shall see.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">So noooow…without further delay…. we give you the rankings.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><b style=""><span style="font-family:Georgia;">Rank 1: Take a guess, anyone!</span></b></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Numero Uno! The Best of the Best!! Undisputed All-Time Champ!!!!</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Introduuuucing…. THE <span style=""> </span>U…..S……OF…..A!!!!!! (Drum roll and thunderous applause)</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Nothing, and I mean nothing, remotely compares to the glamour of living in the land of the brave, the free and the morbidly obese.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Even if you live in a nondescript small town of Idaho, the aura of the USA surrounding you will dazzle all onlookers, and allow you to sneer with impunity at Bangaloreans, Bombayites and maybe even Londoners.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><b style=""><span style="font-family:Georgia;">Rank 2: UK, Canada and Australia</span></b></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Presented in no particular order, these are the Salieris to USA’s Mozart – all the right attributes, but not quite there. The NRIs living in these countries also exhibit the mentality of those who have almost reached heaven’s door, but not quite.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">So, the NRIs in the US keep talking of “going back home” while applying for a Green Card.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">The UK ones talk of going back home while applying to jobs in the US.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Similar behaviour is reported for those in Canada and Australia.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><b style=""><span style="font-family:Georgia;">Rank 3: Non-UK Western Europe</span></b></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">So, that means Germany, Switzerland, France, etc. Not Hungary, Poland and all <i style="">those</i> places.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Shiny-shiny – tick. White people – definitely.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">But language. <i style="">Such</i> a problem, the language. Can’t you possibly move to the UK? Or maybe even, nudge, nudge, the US? </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Amsterdam</span><span style="font-family:Georgia;"> gets a partial exemption since the language of cannabis and red light districts is universal.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><b style=""><span style="font-family:Georgia;">Rank 4: Singapore and Hong Kong</span></b></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Aaaah, we can speak English again. But what’s with all these <i style="">Chinese</i> people everywhere?</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">I mean, can you even make out what they are <i style="">thinking</i>? No, no, definitely not done.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Hong Kong</span><span style="font-family:Georgia;"> is sinking in status as Mandarin is increasingly becoming the spoken language.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Meanwhile, Singapore has almost risen to rank 3, because of proximity to India and relaxed visa procedures. But the Chinese itch refuses to go away, it seems.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><b style=""><span style="font-family:Georgia;">Rank 5: Dubai and the UAE</span></b></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Immense flash and glitter, no doubt. English speaking - good. Also, Muslim. Long pause.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">We are in serious bad news territory here, folks. Whispers of brutal punishments, second class citizenship, Indian labourers treated as virtual slaves. Plus Dawood Ibrahim and links with Pakistan (oh no, the Muslim connection again!). </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Some phoren glamour remains here, but tinged with darker hues.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><b style=""><span style="font-family:Georgia;">Rank 6: Japan</span></b></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Once a favourite of the older generation, only a lot of wealth and tremendous shine keep it on the glamour list at all. Severe language problems and a worldwide reputation for xenophobia.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Plus, what’s with all the raw fish and cut throat prices, eh?</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">This is definitely a place where you can tell the visa official, “I intend to stay here a few years and then leave”, without having your fingers crossed behind your back.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">So, what do you think? Feel free to suggest fine tunings and alternative rankings, or make indignant comments.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> </span></p>Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com18tag:blogger.com,1999:blog-2350997094083029715.post-33299939783539846712010-09-09T11:43:00.000-07:002010-09-19T05:37:37.151-07:00Much Ado About Nothing<div style="text-align: justify;"><span style="font-size:180%;">A</span>nybody seeking a perfect example of making a mountain out of a very harmless molehill need look no further than the supremely silly "Park 51 Mosque Controversy" dominating the US media nowadays.<br />Rumours of a vague plan to erect a mosque on Ground Zero - the site of the World Trade Center buildings - have even reached India. It sounded rather unlikely, and I was disinclined to pay much attention to the matter. After all, the US media is always roiled by some storm in a teacup or another, unlike India where we only discuss issues of the greatest import. :)<br /><br />All this changed when I ran into an old friend, Ameena, on Facebook, after several years.<br /><br />What follows is a guest post by my sister, Bisakha Sen, written after I reported my e-conversation with Ameena. Read on, and contemplate the idiocy of the human race.<br /><br />---------------------------------------------------------------------------------<br /><br /><span style="font-weight: bold; color: rgb(204, 0, 0);font-size:130%;" >Musings on Park51</span><br /><br />Maybe I should have titled it "Banning Unitarian Universalist churches because you hate Pat Robertson"?<br />If there is a God that specializes in irony.....this whole mess must surely be laid at His altar !<br />Till recently, the whole Park51 'Mosque at WTC' was a matter of principle to me, not much more. I have no particular fondness for ANY organized religion, including Islam. However, there are certain religious freedoms in USA. Those religious freedoms cannot be denied to Muslims. And anyway, the site is a few blocks AWAY from WTC, in an area where strip clubs and bars abound -- so its not exactly 'desecrating hallowed ground' to begin with. Plus there are mosques in that vicinity already.What changed things was an email from my brother, a few days ago with a subject line screaming 'UNBELIEVABLE'."Can you believe<br />it, this is AMEENA'S community mosque that all these (unprintables) are going loony over !"<br /><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbuAG8SxYJ8IassElf89-T68QxAwxiMadHvBcp2UDeMxL_svQddxmwlH7FldMqjSSsjxY29JRTlRbPyRKZNT2mEVwFguawNWgDyE0O5MFxmny6rw5UTS4M2Tlev_3jycMsfJ2Nyb3dqwrP/s1600/Ground+Zero+Mosque.jpg"><img style="display: block; margin: 0px auto 10px; text-align: center; cursor: pointer; width: 400px; height: 266px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbuAG8SxYJ8IassElf89-T68QxAwxiMadHvBcp2UDeMxL_svQddxmwlH7FldMqjSSsjxY29JRTlRbPyRKZNT2mEVwFguawNWgDyE0O5MFxmny6rw5UTS4M2Tlev_3jycMsfJ2Nyb3dqwrP/s400/Ground+Zero+Mosque.jpg" alt="" id="BLOGGER_PHOTO_ID_5515259492309796610" border="0" /></a><br /><br />Ameena. My lovely friend. Looks like a sylph and has a spine of steel. Writer, fashionasta, feminist, passionately progressive, gay-rights advocate, generous-to-a-fault. Who has been known to march out of 'traditional' mosques with her three daughters in tow because the imam had spouted some typical nonsense of 'good women should be meek and subservient' during service! Who can destroy every stereotype you held about Muslims in about 15 minutes of conversation. Who is a dedicated Sufi Muslim herself. Part of a Interfaith community full of free-thinkers of every stripe. My brother visited Friday services at this mosque when he first visited Ameena. He was absolutely enchanted -- "Sis, we had the BEST dancing with chants......sooo much fun". He assured me he had greatly enjoyed conversations with New-age artistic free-thinking types from every background as well, but clearly it was the dancing that truly impressed him ! (side note, this is a community mosque where men and women not only pray together, they dance together !!!).<br /><br />When we last visited Ameena in New York, we met El-Farouk Khaki at her place. Brilliant, impassioned, extravagantly tattooed and as flamboyant as they come -- El Farouk Khaki is a Canadian activist for Muslim LGBT rights and founder of the 'Salaam', one of the biggest suport groups for gay Muslims in the west. He'd been a guest at the community mosque earlier. We all chatted deep into the night over an endless supply of kabobs -- starry-eyed social liberals imagining days when ALL organized religious groups would be welcoming to all, when imams and priests presiding over exchange of vows of gay/lesbian couples would be a norm, not an exception.<br />As I now hear the cacophony about this community's mosque, I can't help but wonder -- is THIS why Newt Gingrich, Sarah Palin and the rest are up in arms against this community and their mosque ? Are they worried that a progressive interfaith Muslim group that is pro-gay rights will dent their cherished construct of Muslims uniformly being more 'socially backwards' and 'anti human-rights' than Christians of their ilk ?<br /><br />Ameena wrote to us saying: "Traditional Muslims don't like us because we're hippy New Age types, and now Fox News says we're jihadists dancing on graves of 9/11 victims !" Frankly, I'm struggling to get my head around the sheer scale of the irony ! That opposition to Al Qaeda and Islamist fundamentalism is now taking the form of opposition to a highly progressive Sufi group! What could be a possible parallel ? Opposition to Pat Robertson's followers resulting in demonizing Unitarian Universal churches might come close!<br /><br />I fear that the average American protesting this community center is getting played for a fool, and Al Qaeda members are hooting with laughter! An article in this week's Newsweek confirmed today that the Taliban are DEFINITELY hooting with laughter (<a style="color: rgb(51, 102, 255);" href="http://www.newsweek.com/2010/08/30/taliban-using-mosque-controversy-to-recruit.html">http://www.newsweek.com/2010/08/30/taliban-using-mosque-controversy-to-recruit.html</a>) -- it doesn't get better than this for them! Americans turn on Sufi Muslims --- whom both Al Qaeda and Taliban hate with a passion -- and in process of doing so, America projects an 'anti-Muslim' image around the world that helps Al Qaeda and Taliban to recruit !!!!!!!<br /><br />I keep hoping for a happy resolution. Maybe the cacophony will die down a bit, and people will set aside those "9-11 taught me ALL I need to know about Islam" posters, and just be a LITTLE more curious ? Even asking "so who are the Sufis anyway" -- would be a good start! Now that its been revealed that the so-called 'shady Saudi Prince' who MIGHT have been helping with the mosque funding is actually Fox News's second largest shareholder (and among other things, has funded the George HW Bush scholarship at Phillips Academy, MA), maybe some of the anti-mosque protestors will take a small step back and reconsider the possibility that maybe they haven't *quite* been given the full picture ?<br /><br />If nothing else -- maybe some people could at least start railing against this particular group of Muslims for more APPROPRIATE reasons, like they are "TOO darned" socially liberal, and therefore un-Real-American ?<br /><br />I can think of no better way to finish this note than with a link from Ameena's own blog, where she talks about her own hope for Park51 leading to awareness and dialogue. Amen to that!<br /><a style="color: rgb(51, 51, 255);" href="http://ameenameer.blogspot.com/2010/08/why-park51-needs-ayaan-hirsi-ali-and.html">http://ameenameer.blogspot.com/2010/08/why-park51-needs-ayaan-hirsi-ali-and.html</a><br /></div>Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com1tag:blogger.com,1999:blog-2350997094083029715.post-57359303024844898822010-07-22T18:16:00.000-07:002010-07-22T18:22:42.967-07:00Of Prisoners and Hats<span style="font-size:180%;">A</span> bored prison warden calls together 100 very smart prisoners and announces that they will be playing the following Sadistic Game.<br /><br />Each prisoner will be made to wear a hat which they cannot see. The colour of the hat may be black or white. After this they will be lined up in positions marked 1 through 100.<br />Each prisoner gets to see the hats of everybody in a position higher than his. (So prisoner 1 gets to see all the hats but his own, prisoner 2 sees all hats but his own and prisoner 1’s, etc)<br /><br />Now starting with prisoner 1, the warden will ask each man the colour of his hat.<br />He can only answer “Black” or “White”. The answer will be announced on the public address system so that all can hear. The warden, in turn, will announce whether the answer is right or wrong. This, too, is announced on the PA system.<br />And yeah, by the way, anyone who gets their colour wrong will be executed immediately.<br />(That’s where the sadism comes in!)<br />The prisoners are given a day to come up with any scheme they can think of to save themselves.<br /><span style="font-style: italic;">Question</span>: How many prisoners can be saved for certain?<br /><br />So, as a first shot, suppose each man simply makes a guess. There’s a 50-50 chance of being right, so on average about half will survive.<br />But this is not guaranteed. There’s a very small, but non-zero, probability that everyone will be wrong! Guesswork gives zero certainty.<br /><br />To <span style="font-style: italic;">guarantee</span> that at least 50 people survive, here’s a scheme.<br />Every odd numbered person calls out the hat colour of the man in front of him. The even numbered people just repeat what they heard from the man behind them.<br />But can we do better? Yes.<br /><br /><span style="font-style: italic;">In fact, there’s a scheme which ensures that all but the first prisoner survives for sure!</span><br />This is an extremely neat problem, so you might want to think about it yourself.<br /><br />…………<br />…………<br /><br />Still thinking?<br /><br />…………<br />…………<br /><br />Had enough? Giving up??<br /><br />Ok here goes.<br />Prisoner 1 looks at the hats of all in front of him. If among them, there are an odd number of white hats, he says “White” else he says “Black”.<br />Knowing this answer and comparing with how many white hats he can see, prisoner 2 can deduce the colour of his own hat. (For eg: If prisoner 1 says “White” and prisoner 2 sees only an even number of white hats, he can deduce that his own hat is white).<br />Now knowing prisoner 2’s answer, prisoner 3 can similarly deduce his own hat colour and so on.<br />Done!<br />Only prisoner 1 has a 50-50 chance of being executed, but every noble cause needs a martyr.<br /><br /><span style="font-weight: bold;">Let’s Have More Colour</span>:<br /><br />Flabbergasted by the amazing escape rate, the warden tries another ploy. Maybe adding another hat colour, say, Red will improve the odds (his, not the prisoners’). Of course, he now has to give them the option of saying Black, White or Red in answer to his question, but he reckons that more prisoners will be wrong this time.<br />Sorry, Mr. Warden, you are still out of luck! Once again, all but the first prisoner get their colour right – this time using some “mod 3 arithmetic”.<br /><br />The idea is, replace every integer, positive or negative, with its remainder when divided by 3.<br />So, for example, 9 is 0, 19 is 1 and -1 is 2.<br />For brevity, for any integer N, we say “N mod 3” to mean “the remainder left when N is divided by 3”.<br /><br />Now the thing is, you can add numbers “mod 3” – which means simply add the numbers and take the remainder mod 3.<br />So, (46 + 21) mod 3 = 67 mod 3 = 1<br /><br />It doesn’t matter if you first add the numbers and go mod 3, or take each number mod 3 and then add. For example:<br />(46 + 21) mod 3 = (46 mod 3 + 21) mod 3 = (1 + 21) mod 3 = (1 + 21 mod 3) mod 3 = 1<br /><br />Similarly, one can subtract and multiply mod 3. Division is trickier and we don’t need it here.<br /><br />So, back to the prisoners. Once again, the plan is ingenious, but elegant.<br />Black hats are given a “value” of 0, white hats get value 1, and red hats get the value 2.<br />Prisoner 1 simply calculates the total value of all the hats he sees mod 3.<br />He calls out Black, White or Red, if his answer is 0, 1 0r 2 respectively.<br />Using this, the other prisoners can work out the colour of their hat.<br /><br />To illustrate, suppose prisoner 1 sees 45 black, 39 white and 15 red hats.<br />The total mod 3 is: (45*2 + 39*1 + 15* 2) mod 3 = 0<br />So, he calls out “Black”.<br />Now suppose prisoner 2 is wearing a red hat.<br />Then he will see, in front of him, 45 black, 39 white and 14 red hats.<br />His total mod 3 is: (45*2 + 39*1 + 14* 2) mod 3 = 28 mod 3 = 1<br />Having heard prisoner 1’s answer on the PA system, he simply subtracts his total from prisoner 1’s total mod 3.<br />This gives: (0 – 1) mod 3 = -1 mod 3 = 2<br />2 is, of course, the value for a red hat.<br />Similarly, prisoner 3 can get his hat colour using the answers of prisoner 1 and 2, plus the hats he can see.<br />Once again, everyone but the first prisoner escapes with certainty.<br /><br />Our warden is apoplectic!<br />To his growing frustration, he finds that changing the number of prisoners makes absolutely no difference. A hundred prisoners, or a billion, everyone escapes for sure, except the first (and even he might get lucky sometimes).<br />Increasing the number of hat colours is similarly futile.<br />Given a million colours, the prisoners simply use “mod 1 million arithmetic” (plus some very rapid mental calculation), to get the correct answer every time.<br />With more colours, the likelihood of the first prisoner being executed increases, but that is small consolation for the jailor who was hoping for some serious body count.<br /><br /><span style="font-weight: bold;">The Seriously Sadistic Game:</span><br /><br />Despondent at the high prisoner survival rate, which has made him the butt of prison jokes, our warden e-mails his odious cousin in Tyrannistan, detailing his woes.<br />The reply is swift: “Turn off the public address system. That’s how we play the game in T-stan.”<br /><br />Success at last!<br />Without the PA system, the hapless prisoners have no information other than the colours of the hats in front of them. Every one of them faces the same position as the first prisoner in the earlier game. The only option available is to simply guess.<br />With N colours, this gives a mere 1/N chance of success. If N is significantly bigger than the total number of prisoners, there is a decent chance that everybody dies.<br />A dismal end indeed for our highly ingenious prisoners.<br /><br />The story should end here with, perhaps, a denunciation of capital punishment and wanton wardens, but the mathematician in me has one last question.<br /><br />What if we play this game with<span style="font-style: italic;"> infinitely many</span> prisoners?<br /><br />You may wonder how that could possibly help – after all, the situation is still the same, isn’t it?<br />Without any information on what others can see, how could anybody do better than just guess?<br />All I can say is “Have faith in Infinity and read on....”Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com4tag:blogger.com,1999:blog-2350997094083029715.post-9693728050830903012010-03-23T13:23:00.000-07:002010-03-28T19:40:43.041-07:00Science and Math<meta equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 10"><meta name="Originator" content="Microsoft Word 10"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CBAISHA%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style> <!-- /* Font Definitions */ @font-face {font-family:Calibri; panose-1:2 15 5 2 2 2 4 3 2 4; mso-font-charset:0; mso-generic-font-family:swiss; mso-font-pitch:variable; mso-font-signature:-1610611985 1073750139 0 0 159 0;} @font-face {font-family:Georgia; panose-1:2 4 5 2 5 4 5 2 3 3; mso-font-charset:0; mso-generic-font-family:roman; mso-font-pitch:variable; mso-font-signature:647 0 0 0 159 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin-top:0cm; margin-right:0cm; margin-bottom:10.0pt; margin-left:0cm; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:Calibri; mso-fareast-font-family:"Times New Roman"; mso-bidi-font-family:"Times New Roman"; mso-fareast-language:EN-US;} @page Section1 {size:612.0pt 792.0pt; margin:72.0pt 90.0pt 72.0pt 90.0pt; mso-header-margin:36.0pt; mso-footer-margin:36.0pt; mso-paper-source:0;} div.Section1 {page:Section1;} --> </style><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style> <![endif]--> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><span style="font-size:180%;">M</span>y last post was a rather bare bones analysis of mathematics which separated it from the arts and placed it close to, but not within, the circle of</span><span style="font-family:Georgia;"> the sciences. Given my emphasis on the top-down approach of science versus the bottom up style of math in that post, I thought it might be worth exploring both the similarities and differences with an (admittedly artificial) example.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">The scientist does not forever create new theorie</span><span style="font-family:Georgia;">s about the universe. Nor does the mathematician relentlessly grind out new mathematical properties ad infinitum. In both cases, discovery usually starts with observing something new.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">For instance, the astronomer looks up on a clear n</span><span style="font-family:Georgia;">ight, a</span><span style="font-family:Georgia;">nd whoa! There’s a new star in the sky! (I must confess I am very p</span><span style="font-family:Georgia;">artial to astronomy, far and away my favourite science. Aficionados of other sciences can undoubtedly find analogous examples.)</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">
<br /></span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCcgsUFcPoF-Fr6EmEX8XJyGKHwsSCG5MFOYctSJEkGDjVBR6wEzdkWXJHp5o-sQWQDqfGiCQlg2zhhaovxTV0doR_ZDIb5-J5y8wnzIUwsx8Hg1cDb6zS8z2N15DQluWyYJ4uzcGQCAqM/s1600-h/full_jpg.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 391px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCcgsUFcPoF-Fr6EmEX8XJyGKHwsSCG5MFOYctSJEkGDjVBR6wEzdkWXJHp5o-sQWQDqfGiCQlg2zhhaovxTV0doR_ZDIb5-J5y8wnzIUwsx8Hg1cDb6zS8z2N15DQluWyYJ4uzcGQCAqM/s400/full_jpg.jpg" alt="" id="BLOGGER_PHOTO_ID_5451963795396626882" border="0" /></a></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Similarly, the mathematician draws a triangle and all its medians, and hey, look ! They all pass through the same point!</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">At this point one usually makes sure that the observation is really correct. The astronomer needs to ensure that what she saw wasn’t just a plane, or a </span><span style="font-family:Georgia;">comet or something nearby. <o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">The mathematician is also making sure of his facts. Maybe the medians just pass very close by, and it only looks like they go through the same point beca</span><span style="font-family:Georgia;">use his pencil is blunt.
<br /></span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">But no, it’s authentic. The astronomer focuses a telescope, calls her colleagues, takes parallax measurements and everything confirms that what is b</span><span style="font-family:Georgia;">eing seen is indeed an object many thousand light years from earth. In fact, a search through the image archives reveals a very faint star, invisible to the n<span class="Apple-style-span" style="font-family: Calibri; "><span style="font-family:Georgia;">aked eye, at that </span><span style="font-family:Georgia;">exa</span><span style="font-family:Georgia;">ct location in the sky.</span></span></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Meanwhile the mathematician has sharpened his pencils, and redrawn the diagram. Nope, no mistake.
<br />
<br /><span style="font-family:Georgia;"><o:p></o:p></span></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Now at this point, the astronomer will not </span><span style="font-family:Georgia;">try to pull a brilliant <span class="Apple-style-span" style="font-family: Calibri; "><span style="font-family:Georgia;">new theory out of her hat. The first several steps are always to see if what was observed, i.e., the “new star”, can be explained within the framework of existin</span><span style="font-family:Georgia;">g theory.</span></span></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">For starters, it looks like this wasn’t a new star</span><span style="font-family:Georgia;"> really, but an existing one which has suddenly brightened drasti<span class="Apple-style-span" style="font-family: Calibri; "><span style="font-family:Georgia;">cally. </span><span style="font-family:Georgia;">Can existing theories of stellar structure and </span><span style="font-family:Georgia;">evolution explain how this may h</span><span style="font-family:Georgia;">appen? Time to solve some equations and run</span><span style="font-family:Georgia;"> simulations...</span></span></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">The mathematician, meanwhile, d</span><span style="font-family:Georgia;">oesn’<span class="Apple-style-span" style="font-family: Calibri; "><span style="font-family:Georgia;">t have the luxury of inventing new theories. Math starts from the bedrock, remem</span><span style="font-family:Georgia;">ber? Triangles are triangles, medians are medians. Their properties are clearly given. All one can do is try to see what consequences follow.</span></span></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Maybe he can wor</span><span style="font-family:Georgia;">k this out just by calculating side lengths and angles. Maybe<span style=""> </span>not...</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Success!! The astr</span><span style="font-family:Georgia;">onomer discovers that it is possible for massive stars to explode cataclysmically at the end of </span><span style="font-family:Georgia;">their lives. An exploding star might brighten by a factor of 100 billion or more - definitely enough for a hitherto invisible star to dazzle into prominence.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">A new observation has been given a satisfying theoretical explanation. She decides to call this phenomenon a supernova.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Our mathematician is also very pleased. It wasn’t trivial. He had to make some constructions, which weren’t at all apparent at t</span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8aCDxjaK8SmNZ_0th3OIMO-KWxAMtzw4u7mSVcTpomClqfm58lmTn7tu-xJ9f77X-RuqeWn1UgK6h426klleb_kCpF1wWwTLgM8dsvEUlp04eW4edr2u1rk6PSDXvv5bQII8gO_WAXuka/s1600-h/triangle.centroid.median.png"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 300px; height: 259px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8aCDxjaK8SmNZ_0th3OIMO-KWxAMtzw4u7mSVcTpomClqfm58lmTn7tu-xJ9f77X-RuqeWn1UgK6h426klleb_kCpF1wWwTLgM8dsvEUlp04eW4edr2u1rk6PSDXvv5bQII8gO_WAXuka/s400/triangle.centroid.median.png" alt="" id="BLOGGER_PHOTO_ID_5451974333280882418" border="0" /></a><span style="font-family:Georgia;">he start, and think quite hard. But it’s done and he has a new theorem: The medians of any triangle always intersect at a point.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">At this point, our stories diverge. The mathematician’s colleagues have all verified his proof, and he can forever rest easy.</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">
<br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">For the astronomer, there’s always a smidgen of doubt. It looks like the theory fits, and all observations match. But it could always be the case that this was something quite different, some hitherto unknown astrophysical phenomenon with the same luminosity pattern. Maybe it was something really wild, like aliens conducting an inter-dimensional experiment.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Nobody knows what the Universe really is and the possibilities are endless.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">All she can do is use a principle called Occam’s razor – go for the simplest explanation which fits the facts.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Now what if we don’t have the happy ending?</span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"> What if the new star defies explanation? What if the result can’t be proved?</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">
<br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">At this point, things get interesting for the astronomer. The first step is to try very hard to see if the existing theories genuinely can’t explain what is being seen. Maybe something obvious is being overlooked, maybe she just has to think harder. <o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">But if the evidence keeps piling up, it’s time to look for a new theory!! <o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">These are the situations that scientists really love – when the boundaries of current knowledge break down, and we begin to catch glimpses of the vast unknown beyond...</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">The mathematician’s choices are a bit more limited. No possibility of “new theories”. Triangles are still triangles and will stubbornly continue to be so. What he can do is start looking for a <i style="">counterexample</i> – a triangle whose medians don’t intersect. In this case, the search will be futile because the result is true.
<br /></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">But there are examples of mathematical conjectures which were verified in thousands of cases, but no proof was forthcoming. And then a counterexample was discovered, showing that the search for proof had been misguided all along.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">If neither proof nor counterexample can be found, despite the best efforts of mathematicians over many, many years, a conjecture tends to gather fame and notoriety, and sometimes, money. One such example is the Riemann hypothesis, proposed in 1859 and still unresolved.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Currently, there is a one million dollar prize offered for a correct proof or counterexample.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">I might have given the impression that the astronomer’s task is simple. Just pull a theory out of thin air and voila!<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">In reality, it is extremely difficult to come up with a successful new theory – particularly if the current theory is well-established. <i style="">Not only must a new theory explain the new observations</i>, <i style="">it must also be consistent with everything that is already known and explained by current theory</i>.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Theoretical physicists know this very well.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">The two main pillars of modern physics – general relativity and quantum field theory – are mutually incompatible. If you try to use them together, they give nonsense results.
<br /></span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Over the last two decades, observations have revealed that 96% of the universe is composed of substances whose nature is not described by any existing theory. Observations are also making it clear that the universe began with a huge burst of hyper-accelerated expansion, but once again, our known theories of physics can’t suggest anything which could cause this.</span></p><p class="MsoNormal" style="margin-bottom: 0.0001pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">For the past three decades, theoretical physics has been stuck in an impasse with virtually no progress and no successful new theories. String theory has been much hyped as a candidate to take us beyond the boundaries, but it has yet to make a single correct prediction.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Meanwhile, in 2004, mathematics saw the solution of one of the most famous unsolved math problems of the past century – the Poincare conjecture.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">Life goes on for science and math – similar but different.<o:p></o:p></span></p> Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com2tag:blogger.com,1999:blog-2350997094083029715.post-2695130022447649812010-03-23T08:09:00.000-07:002010-03-24T16:57:14.000-07:00What is Mathematics<span style="font-size:180%;">I</span>s it a science, an art, or something else altogether?<br />Most universities give math majors a Bachelor of Science degree. And I am sure anybody who has studied or worked with math would agree that it feels far more like doing science than painting a picture or composing a poem.<br /><br />Yet a number of mathematicians ardently believe that mathematics is an art. As a friend of mine from grad school used to say, “Science is about the Why and How, but math is about the Wow.” He is echoing the great mathematician G.H. Hardy, (of Hardy-Ramanujan fame) who wrote, “A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”<br /><br />The way I see it, the crucial difference between the arts and sciences is the primacy of Truth. One may argue endlessly about whether Tagore’s writings are more beautiful or appealing or resonant than Shakespeare’s, but there’s no meaningful sense in which either can be said to be true or false.<br />But in science, truth is all. A scientist may use aesthetic appeal as a guide to formulating her theories, but “even the most beautiful theory may be slain by just one ugly fact”. And once that happens, it’s just another discarded<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOgsZ11duVF0Nw06Q88qSzUgD7hmC3jVmQFmUdBJj42vYfpfbhQyPv8ambzCxEx-bCTFDPYiYI6oJW7trR0x3Geq9CxmNLEsaZuYOgbIfzxhxWI-7iDr_k5toBeHESvbUdY6W-uiVkFqNr/s1600-h/fractal-art-alfred-laing-spiral-fantasy.jpg"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 400px; height: 319px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOgsZ11duVF0Nw06Q88qSzUgD7hmC3jVmQFmUdBJj42vYfpfbhQyPv8ambzCxEx-bCTFDPYiYI6oJW7trR0x3Geq9CxmNLEsaZuYOgbIfzxhxWI-7iDr_k5toBeHESvbUdY6W-uiVkFqNr/s400/fractal-art-alfred-laing-spiral-fantasy.jpg" alt="" id="BLOGGER_PHOTO_ID_5451977510547372642" border="0" /></a> theory, of interest only to historians of science.<br /><br />Truth is central to mathematics as well. A mathematician may formulate the most elegant conjectures, but if proved false, he must discard them. To paraphrase Hardy, a mathematician may be a maker of patterns, but the only worthwhile patterns are the ones which are true. So, I’d say, math is certainly much more a science than an art in terms of its primary objectives.<br /><br /><span style="font-weight: bold;">Theories versus Theorems</span><br /><br />So does that end the argument? Not quite.<br />While truth may be central to both science and math, a vital difference exists regarding how truths are established.<br /><br />To illustrate, consider the General Theory of Relativity. It is the theory that gravity is caused by the curvature of space and time in the presence of matter and energy. Over the past century, the predictions of the theory have been confirmed by dozens of observations and tests – it forms one of the main pillars of our current understanding of the universe.<br /><br />In contrast, consider the Riemann Hypothesis. It is the statement that all nontrivial zeroes of the Riemann Zeta function lie on the straight line with real part equal to half (<a href="http://en.wikipedia.org/wiki/Riemann_hypothesis"><span style="color: rgb(51, 51, 255);">http://en.wikipedia.org/wiki/Riemann_hypothesis</span></a>). Over the past century, about 10 trillion zeroes of the function have indeed been found to lie on this line. But all this “observational support” makes no difference – the Riemann hypothesis remains a conjecture, albeit probably the most famous conjecture in mathematics.<br /><br />On the flip side,<span style="font-style: italic;"> none of the confirmations establishes general relativity beyond doubt</span>.<br />The theory is extremely elegant and has withstood the test of time really well. But the proverbial ugly fact may always show up. In fact, general relativity itself replaced Newton’s theory of gravitation which was considered unassailable for over two centuries before it!<br /><br />In contrast, if a correct proof is given for the Riemann hypothesis, the hypothesis would become a theorem. One would never have to worry about an unruly zero showing up to topple the edifice, any more than one worries about someone drawing a triangle whose angles add up to more than 180 degrees.<br /><br />So, whence this difference? Why isn’t it considered adequate to justify a mathematical conjecture by providing a bunch of observations and tests? Conversely, why can’t one prove a scientific theory once and for all?<br />I will argue that the reason is the essence of the difference between science and mathematics.<br /><br /><span style="font-weight: bold;">Top-down versus Bottom-up</span><br /><br />Nobody knows what the Universe really is.<br />The abstract structure of the universe is hidden from us and we only have access to sensory data about the objects within it – either directly or via our instruments. Trying to figure out this structure is the work of science.<br /><br /><span style="font-style: italic;">A scientific theory is a statement about fundamental properties of the universe, inferred from observed properties of objects within it.</span><br />Of course, not all theories are quite so grand. Most are merely attempt to explain features of a particular class of objects, - say the luminosity of stars. These theories may themselves be based on “deeper” theories – the luminosity of stars, may be explained in terms of nuclear physics, which in turn is based on quantum mechanics.<br />But ultimately, we want to “get to the bottom” - to work out the ultimate structure of the universe from our observed data.<br />Science works from the top down.<br /><br />And this is why a scientific theory can never be proved beyond doubt.<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-aOfIRYP2btW8JAIe5QprsA4xli7gtcnr9uH05HQiJ3u-vbTWuFlqNIpP8loUCUauEl-O7cxkQJMO8pLbCcnNRCTa6hsUBipALy7qszJ6wSTlBOsEYncMVjgN7HOTef9OWXFhJqKPJnc7/s1600-h/blackboard.jpg"><img style="margin: 0pt 0pt 10px 10px; float: right; cursor: pointer; width: 320px; height: 320px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-aOfIRYP2btW8JAIe5QprsA4xli7gtcnr9uH05HQiJ3u-vbTWuFlqNIpP8loUCUauEl-O7cxkQJMO8pLbCcnNRCTa6hsUBipALy7qszJ6wSTlBOsEYncMVjgN7HOTef9OWXFhJqKPJnc7/s320/blackboard.jpg" alt="" id="BLOGGER_PHOTO_ID_5451976534495151826" border="0" /></a><br />It is ultimately a guess.<br />We see the surface of the ocean of reality, and try to guess where the bottom is. Our guesses may be very educated indeed, but it is always possible that the bottom is further down, since we can never see it directly. It is even possible that there is no bottom, no ultimate Theory to explain it all.<br /><br />We can never know with absolute certainty, only with provisional degrees of confidence, always tinged with doubt. We can support our best guesses by gathering additional data, but one never knows whether the next data point will confound us all.<br /><br />Mathematics, by contrast, works from the bottom up.<br />There is no doubt at all about what the mathematical entities being studied “really are” – in fact, they are very clearly defined right at the outset. And given the basic properties and definitions, a mathematician tries to work out further properties of the objects of interest.<br /><br /><span style="font-style: italic;">A mathematical theorem is a statement about properties of mathematical objects, deduced from their fundamental properties.</span><br />This is why a mathematical theorem can be proved. It is not a guess about fundamental properties of objects – those are laid out right at the beginning. We already know where the bottom is. The rest is just (?!) deduction and logic.<br /><br />So where does this leave us?<br />Given the centrality of “true propositions”, I would place math much closer to the sciences than the arts. However, because of the opposite directions of inference, and the consequent difference in how propositions are verified/falsified, I’d say it’s not entirely accurate to call mathematics a science. But they are so similar, that it makes sense to say “science and math” in the same breath, as people often do.Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com15tag:blogger.com,1999:blog-2350997094083029715.post-3645177720699028082010-01-26T00:34:00.000-08:002010-02-02T00:51:56.525-08:00National Anthem<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgA9EbIwwJFcTJJhrJzkGVIxGB88yYkMfCGMJjT44td_WZc93aDoOkw11V_Pd-wemrV7WgbudyvHaEDLjm7QUiGjeF8pmidfEtg8Khp5_UpjdM4bVrT6uYYk0PyI9opIYUEtZerBv0gIsM/s1600-h/0.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 300px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgA9EbIwwJFcTJJhrJzkGVIxGB88yYkMfCGMJjT44td_WZc93aDoOkw11V_Pd-wemrV7WgbudyvHaEDLjm7QUiGjeF8pmidfEtg8Khp5_UpjdM4bVrT6uYYk0PyI9opIYUEtZerBv0gIsM/s400/0.jpg" alt="" id="BLOGGER_PHOTO_ID_5431136614514939522" border="0" /></a><br /><span class="Apple-style-span" style="font-size:x-large;">I</span>t is India’s 60th Republic Day today. Even as I write, flags are being hoisted across the country and the national anthem is being sung.<div><br />This is, therefore, an opportune moment to put to rest an extremely silly, but annoyingly persistent, ‘controversy’ about our national anthem, “Jana Gana Mana”, composed by Indian poet laureate Rabindranath Tagore. The song was written, so the accusation goes, to felicitate the coronation of British emperor George V. Hence, it should be replaced by ‘something patriotic’.</div><div>So, what do I make of this?</div><div><br />My shortest answer is unprintable. A slightly longer answer is, “No, the song was not addressed to George V and ‘<i>Bharatabhagyavidhata</i>’ in the song does not refer to the British emperor, except in the imaginations of the ignorant and/or deliberately obtuse.”</div><div>The details of the song’s composition can be found in a delightful little book by Dr. Prabodh Chandra Sen – literary scholar, historian and a contemporary of Tagore.</div><div>To summarize, Tagore was asked to compose a song of felicitation at the coronation of the British emperor by a high ranking acquaintance in the Indian Civil Service.</div><div>Incensed by this request, the poet stayed awake all night, penning this masterpiece of a song.</div><div>It was first sung on 27th December, 1911 at the proceedings on the Indian National Congress.<br />The British press dutifully reported that “the Bengali poet Babu Rabindranath Tagore sang a song specially composed by him to welcome the emperor”.</div><div><br />Little did they know that what they had heard was an eulogy to the eternal spirit of India, referring to the geographic and religious unity of the country, the oppression of foreign rule and presaging an independence to come! </div><div>If you are frowning in puzzlement at this, you very probably don’t know that the song has five verses. What is sung at the flag hoisting ceremonies is the first verse – merely the ‘geographical introduction’, so to speak.</div><div><br />So, here I give you all five verses, sung and translated.</div><div><br />The singing is mine. </div><div><br />The translation is a bit of a ‘family project’. My parents remembered that my sister had done a translation way back. So, late last night, Dad emails me saying, “Here it is.”</div><div>Both sis and I take a look at it, and our conclusion was, “Hmm, not that great, needs some serious changing.” In fact, we are both secretly convinced that this might have been mailed to Dad by somebody else, though parents insist it wasn’t.</div><div>Anyway, I heavily modify the second and fifth verses, and sis does a complete rewrite of the third and fourth. And after some further fine tuning, we think, “Now it looks decent.”</div><div><br />After all this, I get another email from parents. Turns out that ‘sister’s original attempt’, which we pretty much rewrote, was actually Tagore’s own translation. Aaaah, well !! :) :)</div><div><br />However, what you see below, is our translation of the song.</div><div><br />Listen, read and judge for yourself.<br /><br /><object classid="clsid:d27cdb6e-ae6d-11cf-96b8-444553540000" codebase="http://fpdownload.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=8,0,18,0" width="325" height="28" id="divmp3"><param name="movie" value="http://www.divshare.com/flash/playlist?myId=10352299-98e"><embed src="http://www.divshare.com/flash/playlist?myId=10352299-98e" width="325" height="28" name="divmp3" type="application/x-shockwave-flash" pluginspage="http://www.macromedia.com/go/getflashplayer"></embed></object><br /><br /><br /><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">Thou art the ruler of the minds of the people,<br />O Creator of India’s destiny.<br />Thy name rouses the hearts of Punjab, Sind,<br />Gujarat and Maratha, of Dravida, Orissa and Bengal.<br />It echoes in the hills of the Vindhyas and Himalayas,</span></i></div><div><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">Mingles in the music of Jamuna and Ganges,<br />In the surging of the Indian Seas.<br />They pray for thy blessing and sing thy praise,<br />O Creator of India’s destiny,<br />Victory, Victory, Victory to thee.</span></i><div><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><br />Day and night thy call is heard across the lands,<br />Drawing Hindus, Buddhists, Jains and Sikhs to thy throne<br />And Parsees, Musalmans and Christians.<br />East and West bring offerings to thy shrine,<br />A garland of Love is woven.<br />O Uniter of the hearts of all,<br />Creator of India’s destiny,<br />Victory, Victory, Victory to thee.</span></i></div><div><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><br />Along that road rugged with the rise and falls of Nations<br />Which Mankind travels down the ages.<br />Eternal Charioteer, the thunder of thy Chariot's wheels echo.<br />Amidst the revolutions of history,<br />Thy trumpet sounds its clarion call,<br />Thou guide to all travelers in their paths of peril,<br />Creator of India’s destiny,<br />Victory, Victory, Victory to thee.<br /><br />When the long dreary night was dense with gloom<br />And the Country lay in a fevered stupor,<br />Thy sheltering arms held her,<br />Thy watchful, compassionate eyes bent upon her face.<br />Against the dark evil dreams of oppression<br />Did thou protect her, most Compassionate Mother.<br />Creator of India’s destiny,<br />Victory, Victory, Victory to thee.</span></i></div><div><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><br />The night dawns, the Sun rises<br />Over the mountains of the East<br />The birds sing, life stirs in the morning breeze.<br />Blessed by the golden rays of thy love,<br />India awakes again from sleep<br />And bends her head at thy feet.<br />Victory, victory, victory to thee,<br />O Lord of all kings,<br />Creator of India’s destiny,<br />Victory, Victory, Victory to thee.<br /></span></i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><br /></span><br /><br /></div></div>Baishalihttp://www.blogger.com/profile/02949778612648204281noreply@blogger.com124tag:blogger.com,1999:blog-2350997094083029715.post-50124423348488759942010-01-22T14:32:00.000-08:002010-01-23T07:23:47.312-08:00God and Aliens<span style="font-size:180%;">A</span> very belated Happy New Year to all.<br />I’m back after a long hiatus, so let’s start on a high note.<br /><br />Over the past few months, I’ve often been asked “Do you believe in God ?”.<br />The problem is, of course, that everyone seems to have a different definition of God. However, if you statistically accumulate the various views, (and ignore the obscure or tautological ones), God is generally defined as:<br />“An all-powerful intelligent Being that created the universe, loves and protects us, and watches over our lives”.<br /><br />Hmm, still problematic.<br />To illustrate, suppose you are asked, “Do you believe in aliens?” (being an avid sci-fi fan, I get asked that a lot, too) and aliens are defined as :<br />“Intelligent beings from elsewhere in the universe who are visiting us in their spaceships.”<br />See the issue ? Both definitions are conflations of two concepts which are not necessarily linked.<br />To resolve the problem, let’s split the Alien definition into:<br /><span style="font-weight: bold;">ET’s</span>: “<span style="font-style: italic;">Intelligent creatures elsewhere in the u</span><span style="font-style: italic;">niverse</span>”.<br /><span style="font-weight: bold;">UFO’s</span>: “<span style="font-style: italic;">Intelligent creatures who are visiting us in their spaceships</span>”.<br />Realize that it is perfectly possible to believe in ET without believing in UFO’s. The Alien question is actually two questions in one.<br />Similarly, God splits into:<br /><span style="font-weight: bold;">Alpha</span>: “<span style="font-style: italic;">An all-powerful intelligence that </span><span style="font-style: italic;">created the universe</span>.”<br /><span style="font-weight: bold;">Omega</span>: “<span style="font-style: italic;">An all-powerful, loving being w</span><span style="font-style: italic;">ho ca</span><span style="font-style: italic;">res about and protects us</span>.”<br /><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfknQ6d810P1yNMz2gzeLKuAwksFes55tP9uv753R5Oyl70RQDXRrOvgUr__aQxQHN8P_TliJkWo6QDB36ljt5XxSlBvF5gKJapcydumVNo67MnDIfGVm5sbEFlc2VEv9ugg9WfykwV6yF/s1600-h/n3463.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 307px; height: 333px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfknQ6d810P1yNMz2gzeLKuAwksFes55tP9uv753R5Oyl70RQDXRrOvgUr__aQxQHN8P_TliJkWo6QDB36ljt5XxSlBvF5gKJapcydumVNo67MnDIfGVm5sbEFlc2VEv9ugg9WfykwV6yF/s400/n3463.jpg" alt="" id="BLOGGER_PHOTO_ID_5429956220834957266" border="0" /></a><br /><br />Observe that when people talk about God, they mostly mean Omega.<br />It is assumed that God created the universe as well, but the primary attribute is that God cares about us as individuals and a species. All the paraphernalia of religion – prayers, rituals, efforts to live according to specified ‘God given’ norms – are useless if the deity does not care. Additionally, Omega is typically imagined as an enhanced human, complete with very human-like emotions, though one is occasionally admonished that “God cannot be understood in human terms”.<br /><br />Alpha is a different entity altogether. An intelligence that created the universe could be infinitely mysterious, utterly alien. Such a Being may be totally indifferent to us carbon based life-forms in our obscure corner of a typical galaxy. It may not even notice our existence. Religious literature, being overly anthropocentric, provides few examples, but think of the Overmind in ‘Childhood’s End’, or Olaf Stapledon’s ‘Star Maker’.<br /><br />Distinctions similar to Alpha and Omega have been made in the past. Religious philosophies, especially in Hinduism, make a distinction between an ‘impersonal’ and a ‘personal’ God.<br />Alpha would roughly correspond to Brahman, and Omega to Ishwara.<br /><br />Ok, then, what do I believe? First, let’s face the facts.<br /><span style="font-style: italic;">There is no concrete evidence for Alph</span><span style="font-style: italic;">a, Omega, ET’s or UFO’s.</span><br />Anyone claiming to “know that God exists” is either using ‘know’ as a synonym for ‘strongly believe’, or confusing belief and fact.<br />I’m not about to provide any such evidence either, I’ll just say what I believe and a bit of why. So, here goes.<br /><br /><span style="font-weight: bold;">I don’t believe in UFO’s</span>.<br />The idea that intelligent extraterrestrials are flying all the way to Earth, only to remain in hiding and get occasionally spotted by unreliable witnesses, strains my credulity to breaking point. UFO-believers typically offer arguments like “maybe they are trying to stay hidden” Sounds more like an excuse than a reason – why would they, after coming so far?<br />Usually this degenerates into “Who can fathom their purpose?”. Still an excuse (and one which we shall see again).<br />I can’t prove with 100% certainty that UFO’s don’t exist, but I can’t do that for the Lochness Monster either. But it’s clear to me that the evidence is strongly against them.<br /><br /><span style="font-weight: bold;">I don’t believe in Omega</span>.<br />There is just way too much unfair misery in the world. Start with the recent disaster in Haiti and continue with all the natural disasters that have struck in this century alone. Add all the horrors of history – the Holocaust, the Khmer Rouge, the Partition riots.... Now continue with the countless cases of individual suffering from birth defects to cancer to unfortunate accidents. This list goes on <span style="font-style: italic;">ad nauseam</span>.<br />Just does not make sense in a world where Omega is out there to love and protect us.<br />Once again, excuses abound. “God is testing us”, “Bad karma from previous births”, “All works out for good in the end”, and finally, “Who can fathom the ways of God?”.<br />We saw this before and I am not convinced. It really looks like people believe in Omega despite the evidence rather than because, just like the UFO-maniacs.<br /><br /><span style="font-weight: bold;">I believe in ET’s</span>.<br />Our galaxy alone has 400 billion stars. There are about a 100 billion galaxies in the observable universe. It is becoming clear that most stars have planetary systems orbiting them. Molecules of life are abundant in interstellar clouds. Plus, it is entirely possibly that life may be based on completely different chemicals from terrestrial life.<br />None of this proves that ET’s exist. Maybe the odds of life forming are incredibly low, low enough to make it improbable despite all the stars out there. Maybe life happens, but it remains at the level of bacteria with overwhelming odds. But the existence of ET’s is at least <span style="font-style: italic;">consistent with the evidence</span>.<br />Still, why believe without proof ? <span style="font-style: italic;">Because I like the idea!</span><br />I find it incredibly depressing to think that Earth is the only place in this incredibly vast universe to have complex life-forms and intelligence. If the evidence points overwhelmingly in that direction, I will have to accept it.<br />But all the evidence so far is consistent with the vastly more appealing picture (to me), of a universe where life and intelligence are commonplace, where we are but one voice in a cosmic symphony. So, I’ll stick with it until forced otherwise.<br /><br /><span style="font-weight: bold;">I believe in Alpha</span>.<br />We live in an amazing universe “rich beyond measure – in elegant facts, in exquisite interrelationships, in the subtle machinery of awe”. Our investigations of the universe reveal a profound and elegant underlying order that continually challenges the imagination and ingenuity of our brightest minds. The depth and intricacy of the cosmic order frequently leaves our best scientists with a feeling of “rapturous amazement at the harmony of natural law, which reveals an intelligence of such superiority that, compared with it, all the systematic thinking and acting of human beings is an utterly insignificant reflection”.<br />To me, it seems entirely plausible that Alpha “breathes fire into the equations, creating an universe for them to describe”.<br />Once again, this is not the only possibility. The mathematical order of the universe may just be a ‘brute fact’, admitting of no further explanation. Alternatively, our universe might be a tiny part of a far bigger Cosmos, where natural laws self-organize and emerge like galaxies and ecosystems.<br />But yes, I personally find that elegant though the laws are, I am disappointed by the idea of “All that is or was or ever will be” being merely the working out of an algorithm that could perhaps be written down on a T-shirt.<br />Much more appealing to me is the concept of our universe as just one creation or manifestation of a numinous and ineffable Alpha, a <span style="font-style: italic;">mysterium tremendum</span> without beginning or end.<br />Nebulous, imprecise, unverifiable – I know. But, hey, this is a belief, not a scientific hypothesis. And it is consistent with all the facts so far (though not implied by them).<br /><br />Finally, a related question – Am I religious ? Short answer: No.<br />While religions pay lip-service to Alpha and occasionally come up with evocative poetry<br />(eg: The “<span style="font-style: italic;">Om Purnamadah Purnamidam</span>” shloka of the Upanishads), their primary concern is Omega and how we can relate to Him/Her (never “It”). Doesn’t work for me.<br /><br /><span style="font-weight: bold;">Addendum</span>: Throughout this piece I’ve quoted liberally from Carl Sagan, Stephen Hawking and Einstein. Any reader of popular science will recognise them at once. If you don’t, what are you waiting for? Get reading! :)Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com8tag:blogger.com,1999:blog-2350997094083029715.post-67065565909254901472009-11-16T21:18:00.000-08:002009-12-05T07:28:16.077-08:00Mahabishwe Mahakashe<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrzRj0FyeRBmn7W_0TNxneoSFCDO8W_zPvP0Bm2_XKiXu3Rkb73KX7uPg_NH3rTl5T349IK5epiUNawx20NG2rsFvMPoiPEDFDcV54Xy8ExcHGRnaYRBMmn-qK2hbwr2LRl5QPVJw99sI3/s1600/Gravitational+Lens.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 385px; height: 400px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgrzRj0FyeRBmn7W_0TNxneoSFCDO8W_zPvP0Bm2_XKiXu3Rkb73KX7uPg_NH3rTl5T349IK5epiUNawx20NG2rsFvMPoiPEDFDcV54Xy8ExcHGRnaYRBMmn-qK2hbwr2LRl5QPVJw99sI3/s400/Gravitational+Lens.jpg" alt="" id="BLOGGER_PHOTO_ID_5404939071696169154" border="0" /></a><br /><p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><span class="Apple-style-span" style="font-size:x-large;">A</span></span></i></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">midst this vast cosmos</span></i></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><o:p></o:p></span></i></span></span></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">Through deep space and time</span></i></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><o:p></o:p></span></i></span></span></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">Alone I walk ever, in awe.</span></i></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><o:p></o:p></span></i></span></span></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">Amidst infinite mystery</span></i></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><o:p></o:p></span></i></span></span></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">Art Thou, Lord of All,</span></i></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><o:p></o:p></span></i></span></span></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">Alone in Thy glory, ever silent.</span></i></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><o:p></o:p></span></i></span></span></span></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><br /></span></i></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><o:p><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"> </span></i></span></span></o:p></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">Across endless space, </span></i></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><o:p></o:p></span></i></span></span></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">From the stars beyond count</span></i></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><o:p></o:p></span></i></span></span></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">Thy gaze falls on me,</span></i></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><o:p></o:p></span></i></span></span></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">I gaze back at Thee.</span></i></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><o:p></o:p></span></i></span></span></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">Silence descends,</span></i></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><o:p></o:p></span></i></span></span></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">The world rests in peace.</span></i></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><o:p></o:p></span></i></span></span></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">Eternal One, amidst Thy Oneness</span></i></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);"><o:p></o:p></span></i></span></span></span></p> <p class="MsoNormal" style="margin: 0in 0in 0.0001pt 2in; text-align: justify; line-height: 150%;"><span lang="EN-GB"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="font-family:georgia;"><i><span class="Apple-style-span" style="color: rgb(204, 0, 0);">Alone I dwell, ever fearless.</span></i></span></span></span></p><p></p><br /><br /><object classid="clsid:d27cdb6e-ae6d-11cf-96b8-444553540000" codebase="http://fpdownload.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=8,0,18,0" id="divmp3" width="325" height="28"><param name="movie" value="http://www.divshare.com/flash/playlist?myId=9383722-044"><embed src="http://www.divshare.com/flash/playlist?myId=9383722-044" name="divmp3" type="application/x-shockwave-flash" pluginspage="http://www.macromedia.com/go/getflashplayer" width="325" height="28"></embed></object><div><br /></div><div><span class="Apple-style-span" style="line-height: 22px;font-size:14px;" >Click the button to play the song.<br />Translation and song are both mine. The photo is not, but click on it for the full impact.</span></div><div><br /></div>Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com6tag:blogger.com,1999:blog-2350997094083029715.post-91421440841203908692009-10-18T02:52:00.000-07:002009-10-27T17:25:09.367-07:00Prothomo Aadi Tobo Shakti<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0abcU0qgFGKu7RMPdqSYzcEihrHRJFmv2CsUa3f8nZKCybIJemWhGRq6wNqSup_3iP0r-cuq_TwofYKJi_hy1MYbhzPd8SoVgQiIYvmj7CN70_wKpfGOSsWIDxJq-0OOJ3fNPIMuHyDIw/s1600-h/M82+Starburst.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 324px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0abcU0qgFGKu7RMPdqSYzcEihrHRJFmv2CsUa3f8nZKCybIJemWhGRq6wNqSup_3iP0r-cuq_TwofYKJi_hy1MYbhzPd8SoVgQiIYvmj7CN70_wKpfGOSsWIDxJq-0OOJ3fNPIMuHyDIw/s400/M82+Starburst.jpg" alt="" id="BLOGGER_PHOTO_ID_5393879395203134130" border="0" /></a><br /><p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 108pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" ><span style="font-size:180%;">T</span>hine is the power primordial,<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 108pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" >Thy dazzling radiance fills the skies.<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 108pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" >Thy ancient Word, bearing Thy eternal bliss,<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 108pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" >Arises anew in endless forms.<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 108pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" >In the heavens of Thy mind,<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 108pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" >Shine the sun, moon and stars,<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 108pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" >Life ebbs and flows with Thy breath.</span></p><p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 108pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"></p><p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 108pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" >Thou art the primal poet,<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 108pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" >O guru of poets,<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 108pt; line-height: 150%;"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;" ><span style="font-style: italic; color: rgb(204, 0, 0);">The cosmos resounds to thy chant.</span></span></p><span dir="ltr" id=":a6"><br /></span><p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 108pt; line-height: 150%;"><br /></p><object classid="clsid:d27cdb6e-ae6d-11cf-96b8-444553540000" codebase="http://fpdownload.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=8,0,18,0" id="divmp3" width="325" height="28"><param name="movie" value="http://www.divshare.com/flash/playlist?myId=8934478-478"><embed src="http://www.divshare.com/flash/playlist?myId=8934478-478" name="divmp3" type="application/x-shockwave-flash" pluginspage="http://www.macromedia.com/go/getflashplayer" width="325" height="28"></embed></object><br /><br />Click the button to play the song.<br />Translation and song are both mine. The photo is not, but click on it for the full impact.Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com6tag:blogger.com,1999:blog-2350997094083029715.post-62017653308553610752009-10-18T02:48:00.000-07:002010-02-09T02:13:47.253-08:00Akash Bhora Shurjo Taara<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2OrOtJCDojuHj9PA_Pe0b-ztzxzamue9T9BHMENcmqCxEY-9MDZlimkKQqviTRZzFuU1P4LxxurrWxswM8xTj6zwOcZE2Xk1oLNYXWnmtAFJtP48roIiQWKt9TyenySLcxMra69RpD_tw/s1600-h/Milky+Way_desert.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 395px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2OrOtJCDojuHj9PA_Pe0b-ztzxzamue9T9BHMENcmqCxEY-9MDZlimkKQqviTRZzFuU1P4LxxurrWxswM8xTj6zwOcZE2Xk1oLNYXWnmtAFJtP48roIiQWKt9TyenySLcxMra69RpD_tw/s400/Milky+Way_desert.jpg" alt="" id="BLOGGER_PHOTO_ID_5393875304148224834" border="0" /></a>
<br /><meta equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 10"><meta name="Originator" content="Microsoft Word 10"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CBAISHA%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style> <!-- /* Font Definitions */ @font-face {font-family:Georgia; panose-1:2 4 5 2 5 4 5 2 3 3; mso-font-charset:0; mso-generic-font-family:roman; mso-font-pitch:variable; mso-font-signature:647 0 0 0 159 0;} @font-face {font-family:Calibri; panose-1:2 15 5 2 2 2 4 3 2 4; mso-font-charset:0; mso-generic-font-family:swiss; mso-font-pitch:variable; mso-font-signature:-1610611985 1073750139 0 0 159 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin-top:0cm; margin-right:0cm; margin-bottom:10.0pt; margin-left:0cm; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:Calibri; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman"; mso-fareast-language:EN-US;} @page Section1 {size:612.0pt 792.0pt; margin:72.0pt 90.0pt 72.0pt 90.0pt; mso-header-margin:36.0pt; mso-footer-margin:36.0pt; mso-paper-source:0;} div.Section1 {page:Section1;} --> </style><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style> <![endif]--> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;"><span style="font-size:180%;">A</span>midst skies full of stars, a world full of life,<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;">Have I found my abode.<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;">My song awakes in wonder.<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;">The vast surge of Endless Time<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;">That ebbs and flows through all Creation<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;">Ripples through the blood in my veins today.<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;">My song awakes in wonder.</span></p><p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);">
<br /><span style="line-height: 150%;font-family:Georgia;font-size:12pt;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;">I have tread the soft grass on forest paths,<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;">Delighting in the scent of woodland flowers,<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;">I have seen the gifts of joy strewn all around.<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;">My song awakes in wonder.<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;">I have opened my eyes, aroused my senses,<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;">Poured myself on the bosom of this world,<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%; font-style: italic; color: rgb(204, 0, 0);"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;">Ever amidst the known have I sought the Unknown.<o:p></o:p></span></p> <p class="MsoNormal" style="margin: 0cm 0cm 0.0001pt 117pt; line-height: 150%;"><span style="line-height: 150%;font-family:Georgia;font-size:12pt;"><span style="font-style: italic; color: rgb(204, 0, 0);">My song awakes in wonder.</span></span></p><span dir="ltr" id=":a6">
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<br /></p><object classid="clsid:d27cdb6e-ae6d-11cf-96b8-444553540000" codebase="http://fpdownload.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=8,0,18,0" id="divmp3" width="325" height="28"><param name="movie" value="http://www.divshare.com/flash/playlist?myId=8941200-912"><embed src="http://www.divshare.com/flash/playlist?myId=8941200-912" name="divmp3" type="application/x-shockwave-flash" pluginspage="http://www.macromedia.com/go/getflashplayer" width="325" height="28"></embed></object>
<br />
<br />Click the button to play the song.
<br />Translation and song are both mine. The photo is not, but click on it for the full impact.
<br />Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com2tag:blogger.com,1999:blog-2350997094083029715.post-69614023045195888682009-10-15T10:48:00.000-07:002010-01-22T14:36:45.862-08:00The Bhagavad Gita In Brief<p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><span style="font-size:180%;">
<br /></span></span></p><p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><span style="font-size:180%;">I</span> am guessing many of you have never read the Gita, but at least some have felt <span style=""> </span>they should have made an attempt. I’m in the category that trie</span><span style="font-family:Georgia;">d, made some progress (making notes and all), then quit. Until recently, that is.<span style=""> </span><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">A late evening attempt to decipher the Sanskrit text with a friend resulted in a renewed surge of interest, and I decided to just go ahead and finish the project. It’s done now and what follows is my roadmap of the 18 chapters of the Gita – with some shl</span><span style="font-family:Georgia;">okas to highlight the path - for those wishing to de</span><span style="font-family:Georgia;">lve in and explore further.</span></p><p class="MsoNormal" style="margin-bottom: 6pt;"><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfZgS1ZKslkog8haJAIFLH7LaWyEaAgbx6oqN5XGZB6RimeuAErxvkyc2chxtFA3E3PoKdyZGX3bkn5J-cvLHIdtrWZxG2ERK_RYUmNRJgU06w_OHR2ZVNyqr420gQU6n5EtknyRPEeuig/s1600-h/gitapotrait.jpg"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 292px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfZgS1ZKslkog8haJAIFLH7LaWyEaAgbx6oqN5XGZB6RimeuAErxvkyc2chxtFA3E3PoKdyZGX3bkn5J-cvLHIdtrWZxG2ERK_RYUmNRJgU06w_OHR2ZVNyqr420gQU6n5EtknyRPEeuig/s400/gitapotrait.jpg" alt="" id="BLOGGER_PHOTO_ID_5393163621535284354" border="0" /></a></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 1:</span></b><span style="font-family:Georgia;"> Setting the stage. Surveying the armies at Kurukshetra, Arjuna is overcome by grief when he realizes what the war will entail. Good for him. <o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">Standard interpretations speak of this as ‘weakness’ on Arjuna’s part. <i style="">Au contraire</i>, I find the very human anguish, the questioning of a war which entails t</span><span style="font-family:Georgia;">he mass slaughter of kinsmen and loved ones for the sake of a kingdom, coming from a man who is expected to be a relentless killing machine, to be one of the brilliant character touches that makes the Mahabharat great. </span><st1:place><span style="font-family:Georgia;">Krishna</span></st1:place><span style="font-family:Georgia;"> responds, and the main Gita begins.</span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 2:</span></b><span style="font-family:Georgia;"> The soul</span><span style="font-family:Georgia;"> stuff. “As a person casts off old garments and puts on new ones, so the soul leaves one body for another” and “Weapons do not cleave the soul, fire does not burn it, water does not wet it, nor wind make it dry” (Shlokas 22 and 23)</span><span style="font-family:Georgia;">. </span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">Get the point? </span><st1:place><span style="font-family:Georgia;">Krishna</span></st1:place><span style="font-family:Georgia;"> is saying that Arjuna won’t <i style="">really</i> be killing his loved ones, just destroying their </span><span style="font-family:Georgia;">bodies. The soul is the real thing and that’s safe. Of course, if one doesn’t buy the soul stuff, this argument goes for a toss, along with much of the Gita.
<br /></span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 3:</span></b><span style="font-family:Georgia;"> All work and no attachment makes Jack a Karmayogi.<span style=""> </span>To be accurate, this starts off in chapter 2 with the famous<span style=""> </span>“Karmanyevadhikaraste, ma phaleshu kadachana” (Chap 2, Sh 47). Acting without attachment – one of the central principles</span><span style="font-family:Georgia;"> of the Gita. <o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">To be honest, I find Karmayoga unconvincing. Why would one do anything if one has no interest in the results? </span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">Furthermore, <i style="">how</i> is one supposed to act? The answer, as per Gita, is to follow ‘<i style="">swadharma</i>’ – nebulously interpreted as ‘one’s innate nature’ or ‘duty’, it’s never quite clear. “Better is swadharma, though carried out imperfectly, than the dharma of another carried out perfectly”(Sh 35). <o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">Highly unsatisfactory. This suggests that people are born with some kind of innate duty (who dictates what that is?) and one should ‘just stick to it’. More problematically, if you interpret swadharma as ‘duty’, what if that duty conflicts with basic human values like compassion, love, loyalty and so on? This is really Arjuna’s question </span><span style="font-family:Georgia;">and </span><st1:place><span style="font-family:Georgia;">Krishna</span></st1:place><span style="font-family:Georgia;"> ne</span><span style="font-family:Georgia;">ver quite answers it, IMO.</span></p> <p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 4:</span></b><span style="font-family:Georgia;"> </span><st1:place><span style="font-family:Georgia;">Krishna</span></st1:place><span style="font-family:Georgia;"> to the rescue !! “Yada yada hi dharmasya...” is right here ! (Sh 7 & 8)<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">The rest is largely a rehash of Karmayoga. Apparently, Arjuna is not convinced first time, and I don’t blame him. Apart from all the swadharma issues there’s the growing problem of “Who should take responsibility for one’s actions?” - very relevant when b</span><span style="font-family:Georgia;">attle is about to commence. Well, over the past two chapters, there have been growing hints that humans are mere intermediaries and God is the ultimate source of all actions, so the answer is.....</span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;"><o:p> </o:p></span></b></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 5:</span></b><span style="font-family:Georgia;"> Blame it all on God!! Or at least, if you do, the responsibility won’t stick to you. “He who works, resigning his actions to God, is not touched by sin, even as a lotus leaf is untouched by water” (Sh 10).</span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">
<br /></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 6:</span></b><span style="font-family:Georgia;"> Introducing <span style=""> </span>Mr. Detachee, aka, the Yogi, “to whom a lump of mud, a stone and a piece of gold are the same”, “who is equal minded among friends and foes” (Sh 8 & 9), “who does not rejoice on obtaining what is pleasant nor sorrow on obtaining what is unpleasant”. Personally speaking, I am more for rejoicing on getting the pleasant and staying calm in the </span><span style="font-family:Georgia;">face of the unpleasant, rather than zoning out equally on both. But this picture of the Yogi, devoid of worldly attachments, is a familiar presence in our culture. </span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">The rest of the chapter is about what the Yogi should do – <i style="">Dhyanayoga</i>, meditation in the grand old style. “Holding the body erect and still, looking fixedly at the point of his nose, serene and fearless, firm in the vow of celibacy, let him sit, his mind turned to Me alone”<span style=""> </span>(Sh 13 & 14). The ultimate goal of all this –“Thus making the sel</span><span style="font-family:Georgia;">f ever harmonized, the yogi experiences the infinite bliss of contact with the Eternal. He w</span><span style="font-family:Georgia;">ho sees Me everywhere and sees all in Me: I am never lost to him nor he lost to Me” (Sh 29 & 30).</span></p> <p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><span style=""> </span><b style="">The story so far:</b> Act without attachment, remember that all actions are rooted in God, then focus, focus, <i style="">focus</i> on God. At this point, the reader may well ask, “So who is this God person anyway?” We’ll see.</span></p> <p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 7:</span></b><span style="font-family:Georgia;"> Ok, so let’s talk God.<span style=""> </span>Here goes.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">“I am the origin of the universe and its dissolution as well. There is nothing whatsoever higher than I. All that exists is strung on me like gems on a string.” (Sh</span><span style="font-family:Georgia;"> 6 & 7).<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">Straight to the point an</span><span style="font-family:Georgia;">d no mincing words either. That’s pretty much the message in this chapter. But some interesting asides exist. <o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">Firstly, some glimmerings of the concept of Maya which hides the true form of God from the world.<span style=""> </span>“Deluded by the three <i style="">gu</i></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhk-cbmHoTKaF6AyADpPdIamJkGbPu55au9GMeO-clbnKV1DbXj4XgmXWmaMzmjP_zjmjirzLmKp3ILsDesj-Fz-XplNTkXJvORoxXkY1vgRnu_TcyUFW8PcnYvn7I0eKMHBN0MUnKY0WXi/s1600-h/gita-005b.jpg"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 297px; height: 400px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhk-cbmHoTKaF6AyADpPdIamJkGbPu55au9GMeO-clbnKV1DbXj4XgmXWmaMzmjP_zjmjirzLmKp3ILsDesj-Fz-XplNTkXJvORoxXkY1vgRnu_TcyUFW8PcnYvn7I0eKMHBN0MUnKY0WXi/s400/gita-005b.jpg" alt="" id="BLOGGER_PHOTO_ID_5393163989791448962" border="0" /></a><span style="font-family:Georgia;"><i style="">nas</i>, the world does not recognise Me who am above them. This divine Maya of mine is hard to overcome.” (Sh 13 & 14)<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">Secondly, an interesting piece about tolerance of worship for other gods, but emphasizing that they are inferior. This is clearly the period when the existing Vishnu and Shiva cults were taking over from the old Vedic deities. </span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">“Those whose minds are distorted by desires resort to other gods, observing various rites. Whatever form a faithful devotee wishes to worship, I make his faith firm. But temporary are the results of their worship. Those who worship the gods go to them, but My devotees come to Me.” (Sh 20 to 23)</span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 8:</span></b><span style="font-family:Georgia;"> Not too much going on here, frankly.<span style=""> </span>The main point seems to be “He who utters Aum at the moment of death come to me” (Sh 13) and </span><span style="font-family:Georgia;">“having come to Me, they do not get back to rebirth” (Sh 15). The rest is a rehash of chapter 7.</span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 9:</span></b><span style="font-family:Georgia;"> Panentheism. Distinct from pantheism (no ‘en’), which identifies God and the universe, panentheism is the belief that God pervades the universe, but also transcends it. Here it is, loud and clear.</span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">“By Me is all the universe pervaded in My unmanifested form. All beings abide in Me, but I do not abide in them. My spirit which is the source all beings, sustains them, but does not abide in them. All beings pass into Me at the End, and I send them f</span><span style="font-family:Georgia;">orth again at the Beginning” (Sh 4 to 7).</span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">The superiority of bhakti to God over Vedic rituals is re-emphasized very explicitly.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">“The knowers of the Vedas who drink the soma, pray for the way to heaven. They reach the holy world of Indra and enjoy the pleasures of the gods. Having enjoyed paradise, they return to the world of mortals again. But to those who worship Me alone, I bring attainment of what they have not.” (Sh 20 to 22).</span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 10:</span></b><span style="font-family:Georgia;"> This is just God going ‘I am the Greatest”, over and over. “Of the Adityas I am Vishnu, of the heavenly lights I am the Sun, .....of the gods I am Indra....... of weapons I am the thunderbolt” and so on and on and on. Finally ends with, “I support this entire universe, pervading it with but a fraction of Myself”. E</span><span style="font-family:Georgia;">nuff said.</span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 11:</span></b><span style="font-family:Georgia;"> Vishwarupa!! The grand climax of the Gita. Arjuna gets to see the Cosmic Form of God.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">Imagine that! A mind-blowing, soul-searing, all-encompassing vision of Ultimate Reality, of All That Is or Was or Ever Will</span><span style="font-family:Georgia;"> Be!!! What could ever be more awesome and humbling, more ecstatic and terrifying, more devastating and transformative than that?! What an absolutely brilliant concept.</span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">Unfortunately, followed up by poor execution. It starts off well enough with, “If the radiance of a thousand suns were to blaze forth at once in the sky, that might resemble the splendour of the Exalted one” (Sh 12), but<span style=""> </span>is followed by a huge overload of anthropomorphic imagery – eyes, hands, stomach, mouths, teeth, garlands and such. Just doesn’t do it for me.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">IMO, contemplating the Hubble Ultra Deep Field or zooming into the Mandelbrot set can convey a far better picture of what Vishwarupa mi</span><span style="font-family:Georgia;">ght have been like. :-)
<br /></span></p><meta equiv="Content-Type" content="text/html; charset=utf-8"><meta name="ProgId" content="Word.Document"><meta name="Generator" content="Microsoft Word 10"><meta name="Originator" content="Microsoft Word 10"><link rel="File-List" href="file:///C:%5CDOCUME%7E1%5CBAISHA%7E1%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml"><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style> <!-- /* Font Definitions */ @font-face {font-family:Georgia; panose-1:2 4 5 2 5 4 5 2 3 3; mso-font-charset:0; mso-generic-font-family:roman; mso-font-pitch:variable; mso-font-signature:647 0 0 0 159 0;} @font-face {font-family:Calibri; panose-1:2 15 5 2 2 2 4 3 2 4; mso-font-charset:0; mso-generic-font-family:swiss; mso-font-pitch:variable; mso-font-signature:-1610611985 1073750139 0 0 159 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin-top:0cm; margin-right:0cm; margin-bottom:10.0pt; margin-left:0cm; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:Calibri; mso-fareast-font-family:Calibri; mso-bidi-font-family:"Times New Roman"; mso-fareast-language:EN-US;} a:link, span.MsoHyperlink {mso-style-noshow:yes; color:blue; text-decoration:underline; text-underline:single;} a:visited, span.MsoHyperlinkFollowed {color:purple; text-decoration:underline; text-underline:single;} @page Section1 {size:612.0pt 792.0pt; margin:72.0pt 90.0pt 72.0pt 90.0pt; mso-header-margin:36.0pt; mso-footer-margin:36.0pt; mso-paper-source:0;} div.Section1 {page:Section1;} --> </style><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style> <![endif]--> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><span style="font-family:Georgia;">(Deep field: <a href="http://www.spacetelescope.org/images/wallpaper3/opo0428b.jpg">http://www.spacetelescope.org/images/wallpaper3/opo0428b.jpg</a><span style=""> </span>Every little speck in the image is a galaxy!<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><span style=""> </span>Mandelbrot set: <a href="http://www.youtube.com/watch?v=WAJE35wX1nQ">http://www.youtube.com/watch?v=WAJE35wX1nQ</a></span> <span style="font-family:Georgia;">)<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><span style=""> </span>
<br /></span><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Brief Interlude:</span></b><span style="font-family:Georgia;"> So, now with the Cosmic Form shown and described, does the Gita end on a high note? Alas, no. Seven more chapters follow. Many of them are just remixes of the old ones, and give the impression of being added on to make the magic number, 18.
<br /></span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 12:</span></b><span style="font-family:Georgia;"> Strong advertisement for bhakti. Devotion to a personal God (Ishwara) is declared at the outset to be superior to contemplation of the impersonal Brahman. And what is the ideal bhakta like ? “He who behaves alike to friend and foe, who is alike in heat and cold, pleasure and pain” (Sh 18)... sounds familiar? Yes ! Mr. Detachee from chapter 6 is back.</span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 13:</span></b><span style="font-family:Georgia;"> Just Sankhya philosophy – Purush and Prakriti. Prakriti is dynamic and everchanging, ceaselessly bringing forth myriad different forms. Purush just sits around, observing and unobserved, unaffected by anything that happens. Yet, somehow, Purush is supposed to be the better of the two. Possibly because He is the </span><st1:state><st1:place><span style="font-family:Georgia;">Man.</span></st1:place></st1:state></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><st1:state><st1:place></st1:place></st1:state><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 14:</span></b><span style="font-family:Georgia;"> More on the three <i style="">gunas – sattva, rajas, tamas</i> – first seen in chapter 7. Sattva is goodness, rajas is passion, tamas is darkness/ignorance. Most of the chapter is an elaboration on what qualities are associated with each. But towards the end, Arjuna asks, “What are the qualities of one who has risen above the three gunas?” Turns out, this is someone who “regards pain and pleasure alike, treats alike a lump of<span style=""> </span>mud, a stone and a piece of gold, who...” ok, ok, we got it, we <i style="">got</i> it. It’s Mr D. again.</span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 15:</span></b><span style="font-family:Georgia;"><span style=""> </span>Remember the cosmic tree? It’s this very interesting image of the reality as an inverted tree, with its roots originating in Brahman, its leaves and branches constituting the world. Quite a unique picture, really. (Ok, there’s Yggdrasil of Norse myth, but it’s the right side up). <span style=""> </span>I had no idea the image was in the Gita, so that was interesting.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">But what are we told to do? “Cut off this firm rooted tree with the sword of non-attachment”. Uh-oh, it’s the D-word again. The rest of the chapter is about how the Lord dwells in all things and suchlike, but we’ve seen this all before in chapters 7 to 9.</span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 16:</span></b><span style="font-family:Georgia;"> All about the daivic (divine) and asuric (demonic) natures intrinsic to human beings. The message: Daivic nature, very good, very good, Asuric nature, very bad, very bad.</span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 17:</span></b><span style="font-family:Georgia;"> Rather artificial classification of diets, forms of worship, types of penance etc as sattvic, rajasic and tamasic. Don’t know why this wasn’t just attached to chapter 14. Sudden digression into the meaning and significance of “Aum Tat Sat” at the end.</span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><b style=""><span style="font-family:Georgia;">Chapter 18:</span></b><span style="font-family:Georgia;"> The finale, and a fairly eclectic mix of previous material. The three-way classification of the last chapter goes on for a bit. <span style=""> </span>We have the three kinds of knowledge, three kinds of work, three kinds of understanding and so forth.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">Karmayoga and swadharma re-appear, but this time, the dharma is laid out by caste in shlokas 42 – 44. (I <i style="">really </i>don’t buy that stuff.) A bit of Dhyanayoga pops in. Then, a particularly strong form of ‘God is responsible for all action’ – “The Lord abides in all hearts, driving them as machines” (Sh 61). The implication is that if Arjuna refuses to fight, the will of God will compel him.<o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">And finally, “Fix thy mind on Me, be devoted to Me, prostrate thyself before Me. Abandoning all duties, come to Me alone for shelter. I shall deliver thee from all evils” (Sh 65 & 66).</span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 0.0001pt;"><b style=""><span style="font-family:Georgia;">In conclusion:</span></b><span style="font-family:Georgia;"> Thus, ultimately, it is ‘Surrender to Me and do My will’.<span style=""> </span><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;">How disappointing! So much better would have been, “Use your new knowledge, think carefully and make up your own mind.” Don’t you think so?</span></p><p class="MsoNormal" style="margin-bottom: 6pt;">
<br /><span style="font-family:Georgia;"><o:p></o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <p class="MsoNormal" style="margin-bottom: 6pt;"><span style="font-family:Georgia;"><o:p> </o:p></span></p> <span style="font-family:Georgia;"><o:p></o:p></span>Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com16tag:blogger.com,1999:blog-2350997094083029715.post-87752964242944863922009-09-10T05:18:00.000-07:002009-10-27T17:26:19.384-07:00Large Numbers<span style="font-size:180%;"><br />L</span>et’s talk about large numbers. Not your humdrum, everyday sort of large number, like the number of stars in the galaxy (about 10^11 or 100,000,000,000) or drops of water in the ocean (about 10^25 or 10,000,000,000,000,000,000,000,000). Not even the somewhat larger numbers like atoms in the visible universe (10^80, I won’t bother to write it out) or the Googol (10^100, not to be confused with Google).<br /><br />No, today I want to talk about seriously large numbers. A good first try is the Googolplex, or 10^(10^100), that is 1 followed by a <span style="font-style: italic;">googol</span> zeroes. While a googol vastly exceeds the number of atoms in the known universe, a googolplex can’t even be written out in full within the universe, even assuming you could write each zero using only one atom!<br /><br />Ok, now we’re talking, but this is just the beginning. To represent the sort of numbers I have in mind, we need a new type of notation, called Arrow Notation. Here goes.<br /><br /><span style="font-weight: bold;">Arrow Notation:</span><br /><br />If a and b are positive integers, a^b is just defined as ab.<br /><br />The general definition is recursive:<br /><br />a^...n arrows...^b = a^..(n-1) arrows..^a^..(n-1) arrows..^a^.......^a<br /><br />where the (n-1) arrows operation is executed (b-1) times. The arrow operations are executed from right to left.<br /><br />Let’s illustrate by example. It’s easy to check that a^^..n arrows ..^^ 2 is just a^2 for any n and any a. The smallest value of b which gives us something interesting is 3.<br /><br />So, to start off:<br /><br />3^3 = 33 = 27<br /><br />3^^3 = 3^3^3 = 327 = 7,625,597, 484, 987<br /><br />Ok, so that’s one of our garden variety large numbers. The ^^ operation, known as the ‘tower’ operation quickly gives us much bigger numbers.<br /><br />For example, 3^^4 = 3^3^3^3 = 3^7,625,597,484,987, which is a number with about 3.5 trillion digits. 3^^5 would be 3^(3^7,625,597,484,987), which means if you wrote it out in base 3, the <span style="font-style: italic;">number of digits</span> would be 3^7,625,597,484,987 !! And so on...<br /><br />But now let’s get serious. How about adding yet another arrow?<br /><br />3^^^3 = 3^^(3^^3) = 3^^7,625,597, 484, 987<br /><br />How big is this? Well, when we look at 3^^3, 3^^4 and 3^^5, we see the incredible impact of increasing the number to the right of the ^^ by 1. Well, we’ve just increased it by about 7.6 trillion, so it’s impossible to imagine not only the number itself, but even the number of digits in the number, or even the number of digits of the number of digits of the number , or even....hmmm, running into some serious linguistic limitations here, but you get the idea.<br /><br />But ok, let’s quit trying to imagine and just add one more arrow.<br />Let’s look at 3^^^^3<br /><br />3^^^^3 = 3^^^3^^^3 = 3^^3^^......3^^3<br /><br />where the ^^ operation is done 3^^^3 times.<br /><br />Take another quick read through the part where I describe 3^^^3. Now take a very deep breath.<br /><br />Imagine you are doing the evaluation of the right hand side in the expression above. Remember it’s done from right to left.<br /><br />So, at step 1, you get 3^^3 which is kid-stuff.<br /><br />But at step 2, you already have 3^^(3^^3) which is our mind-cracking 3^^^3 !!! Now, you just have to continue for another (3^^^3 – 3) steps....<br /><br />If you’re really feeling masochistic, you can try working out 3^^^^^3, but by now I hope you’ve realized the effect of adding just one extra arrow. So, I’ll go ahead instead and mention the biggest number ever used in a mathematical proof.<br /><br /><span style="font-weight: bold;">Graham’s Number</span><br /><br />First brought to attention in 1977, the number was used by the mathematician Ronald Graham working in a field called Ramsey theory. Ramsey theory deals with problems of the form, “How many elements must a set have for a certain property to occur.”<br /><br />So, for example, suppose you have a gathering where any two people either know each other or don’t. How many people must there be, so that you always have either three people who all know each other or three people who all don’t know each other ? The answer in this case is 6. (Prove it!).<br /><br />If you make the property more complex, the size of the set increases correspondingly. Graham showed that for his problem, the desired property is always satisfied if the set has at least Graham’s number of elements.<br /><br />So, what is this number? Let’s define a sequence as follows.<br /><br />G1 = 3^^^^3, our humongous old friend.<br /><br />Now let G2 = 3^^...^^3<br /><br />where - and read this bit very carefully – <span style="font-style: italic;">the number of arrows is G1</span> !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (What I really feel like doing is adding at least a googol exclamation marks here, or maybe a googolplex.)<br /><br />G3 = 3^^...^^3 where the number of arrows is G2.<br /><br />Still hanging on? Ok then, so we define G3, G4, G5 etc in the same style.<br />Graham’s number is G64.<br /><br /><br /><span style="font-weight: bold;">Infinity</span><br /><br />So, finally, we come to the biggest of them all. “Nonsense!”, interjects the mathematician, “Infinity is not a number at all. It’s an abstract concept and a pretty tricky one at that.”<br /><br />Of course, of course. But when we mere mortals think of Infinity, we do tend to think of something very, very big which goes on and on and on and on.<br /><br />How big? Well, the ancients used to mention things like “stars in the heavens”, “drops of water in the ocean”, “grains of sand in the desert” etc to convey a sense of infinity. But as we saw right at the start, these concepts are easily tamed with standard mathematical notation, and turn out to be all too finite and very manageable.<br /><br />If we go beyond and introduce arrow notation, we can quickly write down numbers which completely drown the imagination all the way up to Graham’s number (and way beyond of course.)<br /><br /><span style="font-style: italic;">However, compared to Infinity, there is no difference between Graham’s number and zero. </span><br /><br />In fact, think about the sequence G1, G2, etc, where Graham’s number if G64. If we take the “Graham’s number”-th number in this sequence (!!!) and subtract it from Infinity, it makes, not a small difference, not a tiny, puny, minute difference, but absolutely no difference whatsoever.<br /><br />That’s what Infinity is. Treat it with respect.Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com4tag:blogger.com,1999:blog-2350997094083029715.post-11098791205668097212009-08-23T00:15:00.000-07:002010-01-22T14:36:45.862-08:00Project Mahabharata<span style="font-size:180%;"><br />I</span> am planning to read the Mahabharat. All of it. You can, too. A complete and unabridged translation of the Sanskrit original is available at <span style="font-weight: bold;"> http://www.sacred-texts.com/hin/maha/index.htm</span><br /><br />In addition, I have a copy of Kaliprasanna Singha’s translation in Bengali. I remember seeing it first in my grandfather’s house when I was about 9. It seemed inconceivable that anybody could finish reading even a chapter of that humongous tome, let alone the entirety. My grandfather, it seemed, had done this, proving once again that he was the smartest man in the world, ever.<br /><br />Now, over two decades later I have a copy of my own. I estimate that if it was written out in the same style as, say, the US hardbound edition of the Harry Potter series, it would be about 5000 pages. Now that’s big, but nowhere near infinite. The HP series itself is a total of about 2500 pages, and children routinely read it all. So, it’s eminently possible to read the entire Mahabharat from start to finish! How cool is that?<br /><br />So, I’m reading the Mahabharat and I must admit it’s a lot tougher going than Rowling’s work or Tolkien’s, because, boy, this book needed an editor!!!<br /><br />To begin with there’s an incredible amount of fluff, and descriptions tend to be overly detailed and highly formulaic. If I have to read yet another description of warriors fighting ‘like passionate bull elephants during mating season’, I swear I’m going to throw the book at somebody’s head (and severely sprain a muscle in the process). Every time a deity or rishi or even a minor dignitary enters the scene, be prepared for a mile-long stava (‘oratory of praise’) elevating him all the way to Param-Brahma. And if anybody starts lamenting, especially Draupadi, you might as well head for the hills, or at least skip the next several pages.<br /><br />Secondly, the narrative is highly nonlinear. There’s no “Once upon a time there were five brothers.....”. The Mahabharata is more like the Arabian nights. People start telling a story, and the characters in the story start telling each other another story, and on it goes. Then, all of a sudden, somebody gets beheaded or something and you're like, "Wait, wait, wait, what?? Which level of the loop was I on???"<br /><br />Then there’s the whole blasted habit of giving away the ending. Happens all the time in the war chapters. For example, right at the outset of Karnaparva, Sanjay goes, "And so after two days of valiant fighting, the mighty Karna was slain by Arjuna..." . Dhritarashtra promptly faints away in shock, and I'm going, "Damn you, that was supposed to come at the END." Then they revive the old King, and Sanjay goes, "So, as I was saying, after Drona died, Duryodhana made Karna the new senapati...." and so on.<br />Honestly, those ancients had NO concept of slowly building up the tension to a big climax!!<br /><br />It makes more sense if you remember that the epic was designed for listening in real time. So, during an endless stava or lamentation sequence, you could let your mind wander or run a small errand. Whereas maybe an ending-giveaway was a way of saying, “Hey, don’t go to the bathroom now. Big action scene coming up!” But for a modern reader, it can be rather frustrating.<br /><br />So, then why bother, you may ask? Why not just read one of the dozens of abridged versions available in the market?<br /><br />Well, I already have. In fact, if you can read Bengali, I will highly, highly recommend Upendrakishore Raychaudhuri’s (he was Satyajit Ray’s grandfather) version for children. I knew it as ‘<span style="font-style: italic;">CCheleder Mahabharat’ (Mahabharat for Boys)</span>, but apparently its been PC-ly renamed ‘<span style="font-style: italic;">CChotoder Mahabharat” (Mahabharat for Youngsters)</span>. The book is a gem – very entertainingly written and very faithful to the original as I am finding.<br /><br />Part of my motivation is a reality check – one constantly hears that something or the other is ‘written in the Mahabharat’, I want to see if that’s really so.<br /><br />But what really drives me is the sheer feel of reading the massive old epic. It’s not like reading a novel at all. Apart from the main plotline, the Mahabharat contains tons of other stories. Virtually every ancient Indian story you’ve heard of is present somewhere or the other – ‘Shakuntala-Dushyanta’, ‘Nala-Damayanti’, ‘Kacha-Devayani’, a condensed version of the Ramayan and much, much more. And often in the most unexpected of places – for example, the story of Shiva, Parvati and the birth of Kartika suddenly shows up before the final battle between Bhima and Duryodhana. Add to all this a huge dollop of philosophical discourse, political advice, social directives and whimsical folk tales thrown in almost at random and you have the big, bewildering mishmash that is the Mahabharat. Reading the great epic is a lot like walking the ways of an ancient city – you wander around pondering the well-known sights, and all of a sudden you are in a dingy little alleyway which leads to an exquisite garden, tucked away where you least expect it. It’s an experience you shouldn’t miss.<br /><br />P.S: If you want to try the Mahabharat project yourself and want some company, some of us have set up a discussion blog at <span style="font-weight: bold;">http://groups.google.com/group/mahabharata-discussion<br /></span>All are welcome.Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com5tag:blogger.com,1999:blog-2350997094083029715.post-81605858604500250562009-08-08T13:57:00.000-07:002009-10-27T17:26:19.384-07:00BANG!!!<span style="color: rgb(102, 0, 0);"></span><span style="color: rgb(102, 0, 0);"></span><div><br /><span style="color: rgb(0, 0, 0);"><span style="font-size:180%;">“</span><span style="font-size:180%;">T</span>en to twenty billion years ago, something happened. The Big Bang, the event that started our Universe. Why it happened is the biggest mystery we know. <em>That</em> it happened is reasonably clear”. Thus wrote Carl Sagan in 1980.<br /><br />Now nearly three decades later, we know that it happened not “ten to twenty billion years ago”, but more like 13.7 billion years give or take 200 million. And we’re beginning to home in on why it happened.<br /><br />For the cosmologically uninitiated, the Big Bang is the colossal explosion which started our Universe. It’s not really your garden variety explosion - it happened everywhere at the same time and space itself began to expand and has been expanding ever since. (Expanding into what ? Wrong question! Space just expands. There need not be anything outside for it to expand into.). As a bonus, all the matter and energy in the Universe came into existence at the Big Bang. Must have been really something to see (for the picosecond before your retina evaporated.) </span><br /><span style="color: rgb(0, 0, 0);"><br />Ok then, so what caused the Big Bang? Answer: Gravity. </span></div><div style="color: rgb(0, 0, 0);"></div><div style="color: rgb(0, 0, 0);"><span style="color: rgb(51, 0, 0);"><br /></span></div><img id="BLOGGER_PHOTO_ID_5367899215991225378" style="margin: 0px auto 10px; display: block; width: 301px; height: 400px; text-align: center; color: rgb(0, 0, 0);" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXCX5tz_d2Q39nkF3Z4YreOm4jgNQ9GrqCDSggLL2ELBisLI8erWaAVakOMtUWpnMzsZbd30gsKbXrRGkhbYQ8xtG6ysIxMKDlmEqiVbrY0p2u0iR_ls4dVPkpxj-DYg7LLox30O8Ds_IS/s400/bigbang_multiverse.jpg" border="0" /><br /><span style="color: rgb(0, 0, 0);">“Now hang on”, you say, “gravity is an </span><em style="color: rgb(0, 0, 0);">attractive</em><span style="color: rgb(0, 0, 0);"> force. It makes everything collapse and clump together. The Big Bang was an </span><em style="color: rgb(0, 0, 0);">explosion</em><span style="color: rgb(0, 0, 0);">. Am I missing something here?”</span><br /><br /><span style="color: rgb(0, 0, 0);">We-e-ell, you see, there’s gravity and gravity. Our usual type of gravity, the type that foils your attempts at spontaneous levitation, is caused by our everyday type of matter. But make that matter sufficiently weird, and gravity can push things apart with enough OOOMPH to satisfy the most ardent anti-gravity aficionado.</span><br /><br /><span style="color: rgb(0, 0, 0);">Let me elaborate. Our best model of gravity today is Einstein’s General Theory of Relativity. What GTR says is that the gravitational force generated by an object is proportional to its mass/energy. This is pretty much what we know, the heavier the body, the more the gravity. But there’s also something else – the pressure of the system also contributes.</span><br /><br /><span style="color: rgb(0, 0, 0);">To illustrate, if you have a spring, then its mass will create a gravitational attraction. If you squeeze the spring, its pressure increases and so does the pull of its gravity. Now here’s the crux – if you </span><em style="color: rgb(0, 0, 0);">stretch</em><span style="color: rgb(0, 0, 0);"> the spring, the pressure term becomes negative and the gravitational pull </span><em style="color: rgb(0, 0, 0);">decreases</em><span style="color: rgb(0, 0, 0);">. </span><em style="color: rgb(0, 0, 0);">And if you stretch it enough, the negative pressure term can totally overwhelm the effect of the spring’s mass, and the gravitational field will repel rather than attract</em><span style="color: rgb(0, 0, 0);">. A sufficiently stretched spring is an anti-gravity machine!</span><br /><br /><span style="color: rgb(0, 0, 0);">Unfortunately, any real spring will rip into pieces long before anti-gravity sets in. But there does exist a substance which can act like our hyper-stretched theoretical spring. What is it? Nothingness.</span><br /><br /><span style="color: rgb(0, 0, 0);">No, I haven’t suddenly gone Zen on you. If you remove all our usual matter and energy, the empty space that remains (which would qualify as nothing for most people) might still have an ‘energy’. The details are technical, but the gist is, this weird ‘vacuum energy’ generates a ferocious burst of antigravity, the Bang of the Big Bang. In the space of 10^(-35) seconds the nascent universe expands by a factor of at least 10^30, and the rest is history.</span><br /><br /><span style="color: rgb(0, 0, 0);">But wait, first let’s give a sense of those numbers. 10^30 is one followed by thirty zeroes. That’s like blowing up a virus to the size of our galaxy! And how fast did this happen? Well, if a blink of an eye was scaled down to 10^(-35) seconds, then the lifetime of our universe would be one hundred millionth of a nanosecond. Unimaginable doesn’t even begin to describe it.</span><br /><br /><span style="color: rgb(0, 0, 0);">As if this wasn’t enough, we have every reason to believe that the expansion went on way beyond this. A consequence is, our </span><em style="color: rgb(0, 0, 0);">observable</em><span style="color: rgb(0, 0, 0);"> universe is miniscule compared to the actual universe. To picture this, imagine the entire Universe to be the surface of a giant balloon studded with galaxies. The observable universe, which is what we can see, from the Earth to the most distant galaxies spotted by our telescopes, would be like a patch drawn on the balloon. The diameter of that patch would be about 90 billion light years, which is about 10^27 metres. Now the question is, “What is the circumference of the entire balloon?” A possible estimate is, 10^(10^12) metres. Yes, you read that right. One followed by a </span><em style="color: rgb(0, 0, 0);">trillion </em><span style="color: rgb(0, 0, 0);">zeroes!!!</span><br /><br /><span style="color: rgb(0, 0, 0);">Usually astronomers like to convey the scale of things with analogies like, “Suppose the earth was the size of an orange, then the Sun would be...” etc. So, let me try to convey the size of the whole Universe. Shrink the Universe by a factor of 10^40. The observable universe, with all its hundreds of billions of galaxies is now the size of an atomic nucleus. Now repeat the procedure </span><em style="color: rgb(0, 0, 0);">25 billion</em><span style="color: rgb(0, 0, 0);"> times. At this point, the Universe has the same size as our observable universe... or an atomic nucleus if we’ve scaled down one too many times by accident! Does it really matter?</span><br /><span style="color: rgb(0, 0, 0);"><br />Now let’s go back to the beginning, before that miniscule speck of vacuum energy expanded to incomprehensible size. On the smallest of scales, quantum physics rules, which implies roughly that “nothing ever sits still”. So, even on the tiniest scales, the vacuum energy cannot be uniform- it is perpetually roiled by quantum fluctuations which ensure that its values are never quite the same everywhere. What do these elusive quantum fluctuations look like? Take a look below.<br /></span><span style="color: rgb(0, 0, 0);"><br /><p><img id="BLOGGER_PHOTO_ID_5367703718756503490" style="margin: 0px auto 10px; display: block; width: 400px; height: 400px; text-align: center;" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3rnJeeJrnBxyLfnTGV3xqNm9sVRfnRqT6X8yVSLvqQ-iW_6bkzMwDWnA2P2QTm5V06XCoOj1wymHDhQf24qVrWfryuZSPz_oV2quXeYeM0Yu-49tRRxJd27u4XJwKQlnCv608A_Pd-SH3/s400/Galaxy+Clusters.jpg" border="0" /><br />What you saw was a map of our universe on the largest scales we can see. The reddish foam-like structure represents clusters of galaxies strewn across space with vast voids in between. The gigantic cosmic expansion stretches the quantum fluctuations out from submicroscopic scales to cosmic ones - resulting in a froth-like clustering of galaxies as far as we can see. The largest structures we see in the visible universe are quantum fluctuations stretched across the heavens.</p><p><br />And finally the grand slam. The universe contains an enormous amount of matter and energy. At least a hundred billion galaxies with a hundred billion stars each. Where did all this come from? The traditional answer is, “From the Infinite and Eternal Mind of God”.<br /><br />But we’ll try for something much humbler – by starting with a glass falling from your hand. The glass gains kinetic energy from the earth’s gravitational field, which then gets converted to sound, heat and a mess on the floor. A rather more spectacular example is a supernova. A giant star at the end of its life collapses under its own gravity. The energy gained from the gravitational field converts into a cataclysmic explosion which blows the star apart and temporarily outshines an entire galaxy. This is all with our familiar matter and energy. </p><p>A speck of vacuum will also gain a gigantic amount of energy from its own gravitational field. But instead of collapsing while doing so, it blows apart in a Big Bang. At the end of this blast of expansion, the energy gained at gravity’s expense is let loose in a flood of radiation and matter that makes up everything you see and much more. So, there we go - you don’t need a hundred billion galaxies worth of material to start off the universe. Just a tiny seed of vacuum energy and the rest will follow.<br /><br />At this point, it’s a case of ‘almost, but not quite’. After all, where did that seed come from? Nobody knows, but here’s a possibility. Start off with truly empty space – so empty, it’s even devoid of vacuum energy. Now, remember those pesky quantum fluctuations? They’re always around, wriggling, wiggling, flickering and jittering. So, even if you assume that the vacuum energy of empty space is zero, it’s never <em>quite </em>so. The value keeps fluctuating up and down, here and there. Given enough time, somewhere, on some miniscule, submicroscopic domain, a fleeting fluctuation will exceed a critical threshold, and.... BANG!!!<br /><br />So, at the end of things (or is it the beginning?), think of Space. From the endless voids between the galaxies and the yawning gulfs separating the stars therein to the emptiness within an atom, space pervades everything. Think of the fact that every bit of that space, every minute, infinitesimal bit – so small that an atom looks is a galaxy in comparison – has the potential to blossom into an entire Universe of stars and galaxies, life and mind. Maybe it is happening somewhere right now. Maybe, even as you read this sentence, a billion, trillion, zillion Universes are exploding into existence....<br /></p></span><p style="color: rgb(0, 0, 0);"></p>Anindyahttp://www.blogger.com/profile/17094522198568878911noreply@blogger.com6