National Anthem

It 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.

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’.

So, what do I make of this?

My shortest answer is unprintable. A slightly longer answer is, “No, the song was not addressed to George V and ‘Bharatabhagyavidhata’ in the song does not refer to the British emperor, except in the imaginations of the ignorant and/or deliberately obtuse.”

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.

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.

Incensed by this request, the poet stayed awake all night, penning this masterpiece of a song.

It was first sung on 27th December, 1911 at the proceedings on the Indian National Congress.
The British press dutifully reported that “the Bengali poet Babu Rabindranath Tagore sang a song specially composed by him to welcome the emperor”.

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!

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.

So, here I give you all five verses, sung and translated.

The singing is mine.

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.”

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.

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.”

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 !! :) :)

However, what you see below, is our translation of the song.

Listen, read and judge for yourself.




Thou art the ruler of the minds of the people,
O Creator of India’s destiny.
Thy name rouses the hearts of Punjab, Sind,
Gujarat and Maratha, of Dravida, Orissa and Bengal.
It echoes in the hills of the Vindhyas and Himalayas,

Mingles in the music of Jamuna and Ganges,
In the surging of the Indian Seas.
They pray for thy blessing and sing thy praise,
O Creator of India’s destiny,
Victory, Victory, Victory to thee.

Day and night thy call is heard across the lands,
Drawing Hindus, Buddhists, Jains and Sikhs to thy throne
And Parsees, Musalmans and Christians.
East and West bring offerings to thy shrine,
A garland of Love is woven.
O Uniter of the hearts of all,
Creator of India’s destiny,
Victory, Victory, Victory to thee.

Along that road rugged with the rise and falls of Nations
Which Mankind travels down the ages.
Eternal Charioteer, the thunder of thy Chariot's wheels echo.
Amidst the revolutions of history,
Thy trumpet sounds its clarion call,
Thou guide to all travelers in their paths of peril,
Creator of India’s destiny,
Victory, Victory, Victory to thee.

When the long dreary night was dense with gloom
And the Country lay in a fevered stupor,
Thy sheltering arms held her,
Thy watchful, compassionate eyes bent upon her face.
Against the dark evil dreams of oppression
Did thou protect her, most Compassionate Mother.
Creator of India’s destiny,
Victory, Victory, Victory to thee.

The night dawns, the Sun rises
Over the mountains of the East
The birds sing, life stirs in the morning breeze.
Blessed by the golden rays of thy love,
India awakes again from sleep
And bends her head at thy feet.
Victory, victory, victory to thee,
O Lord of all kings,
Creator of India’s destiny,
Victory, Victory, Victory to thee.

God and Aliens

A very belated Happy New Year to all.
I’m back after a long hiatus, so let’s start on a high note.

Over the past few months, I’ve often been asked “Do you believe in God ?”.
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:
“An all-powerful intelligent Being that created the universe, loves and protects us, and watches over our lives”.

Hmm, still problematic.
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 :
“Intelligent beings from elsewhere in the universe who are visiting us in their spaceships.”
See the issue ? Both definitions are conflations of two concepts which are not necessarily linked.
To resolve the problem, let’s split the Alien definition into:
ET’s: “Intelligent creatures elsewhere in the universe”.
UFO’s: “Intelligent creatures who are visiting us in their spaceships”.
Realize that it is perfectly possible to believe in ET without believing in UFO’s. The Alien question is actually two questions in one.
Similarly, God splits into:
Alpha: “An all-powerful intelligence that created the universe.”
Omega: “An all-powerful, loving being who cares about and protects us.”




Observe that when people talk about God, they mostly mean Omega.
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”.

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’.

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.
Alpha would roughly correspond to Brahman, and Omega to Ishwara.

Ok, then, what do I believe? First, let’s face the facts.
There is no concrete evidence for Alpha, Omega, ET’s or UFO’s.
Anyone claiming to “know that God exists” is either using ‘know’ as a synonym for ‘strongly believe’, or confusing belief and fact.
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.

I don’t believe in UFO’s.
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?
Usually this degenerates into “Who can fathom their purpose?”. Still an excuse (and one which we shall see again).
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.

I don’t believe in Omega.
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 ad nauseam.
Just does not make sense in a world where Omega is out there to love and protect us.
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?”.
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.

I believe in ET’s.
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.
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 consistent with the evidence.
Still, why believe without proof ? Because I like the idea!
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.
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.

I believe in Alpha.
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”.
To me, it seems entirely plausible that Alpha “breathes fire into the equations, creating an universe for them to describe”.
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.
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.
Much more appealing to me is the concept of our universe as just one creation or manifestation of a numinous and ineffable Alpha, a mysterium tremendum without beginning or end.
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).

Finally, a related question – Am I religious ? Short answer: No.
While religions pay lip-service to Alpha and occasionally come up with evocative poetry
(eg: The “Om Purnamadah Purnamidam” shloka of the Upanishads), their primary concern is Omega and how we can relate to Him/Her (never “It”). Doesn’t work for me.

Addendum: 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! :)

Mahabishwe Mahakashe

Amidst this vast cosmos

Through deep space and time

Alone I walk ever, in awe.

Amidst infinite mystery

Art Thou, Lord of All,

Alone in Thy glory, ever silent.

Across endless space,

From the stars beyond count

Thy gaze falls on me,

I gaze back at Thee.

Silence descends,

The world rests in peace.

Eternal One, amidst Thy Oneness

Alone I dwell, ever fearless.

Click the button to play the song.
Translation and song are both mine. The photo is not, but click on it for the full impact.

Prothomo Aadi Tobo Shakti

Thine is the power primordial,

Thy dazzling radiance fills the skies.

Thy ancient Word, bearing Thy eternal bliss,

Arises anew in endless forms.

In the heavens of Thy mind,

Shine the sun, moon and stars,

Life ebbs and flows with Thy breath.

Thou art the primal poet,

O guru of poets,

The cosmos resounds to thy chant.

Click the button to play the song.
Translation and song are both mine. The photo is not, but click on it for the full impact.

Akash Bhora Shurjo Taara

Amidst skies full of stars, a world full of life,

Have I found my abode.

My song awakes in wonder.

The vast surge of Endless Time

That ebbs and flows through all Creation

Ripples through the blood in my veins today.

My song awakes in wonder.

I have tread the soft grass on forest paths,

Delighting in the scent of woodland flowers,

I have seen the gifts of joy strewn all around.

My song awakes in wonder.

I have opened my eyes, aroused my senses,

Poured myself on the bosom of this world,

Ever amidst the known have I sought the Unknown.

My song awakes in wonder.

Click the button to play the song.
Translation and song are both mine. The photo is not, but click on it for the full impact.

The Bhagavad Gita In Brief

I am guessing many of you have never read the Gita, but at least some have felt they should have made an attempt. I’m in the category that tried, made some progress (making notes and all), then quit. Until recently, that is.

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 shlokas to highlight the path - for those wishing to delve in and explore further.

Chapter 1: Setting the stage. Surveying the armies at Kurukshetra, Arjuna is overcome by grief when he realizes what the war will entail. Good for him.

Standard interpretations speak of this as ‘weakness’ on Arjuna’s part. Au contraire, I find the very human anguish, the questioning of a war which entails the 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. Krishna responds, and the main Gita begins.

Chapter 2: The soul 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).

Get the point? Krishna is saying that Arjuna won’t really be killing his loved ones, just destroying their 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.

Chapter 3: All work and no attachment makes Jack a Karmayogi. To be accurate, this starts off in chapter 2 with the famous “Karmanyevadhikaraste, ma phaleshu kadachana” (Chap 2, Sh 47). Acting without attachment – one of the central principles of the Gita.

To be honest, I find Karmayoga unconvincing. Why would one do anything if one has no interest in the results?

Furthermore, how is one supposed to act? The answer, as per Gita, is to follow ‘swadharma’ – 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).

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 and Krishna never quite answers it, IMO.

Chapter 4: Krishna to the rescue !! “Yada yada hi dharmasya...” is right here ! (Sh 7 & 8)

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 battle 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.....

Chapter 5: 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).

Chapter 6: Introducing 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 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.

The rest of the chapter is about what the Yogi should do – Dhyanayoga, 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” (Sh 13 & 14). The ultimate goal of all this –“Thus making the self ever harmonized, the yogi experiences the infinite bliss of contact with the Eternal. He who sees Me everywhere and sees all in Me: I am never lost to him nor he lost to Me” (Sh 29 & 30).

The story so far: Act without attachment, remember that all actions are rooted in God, then focus, focus, focus on God. At this point, the reader may well ask, “So who is this God person anyway?” We’ll see.

Chapter 7: Ok, so let’s talk God. Here goes.

“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 6 & 7).

Straight to the point and no mincing words either. That’s pretty much the message in this chapter. But some interesting asides exist.

Firstly, some glimmerings of the concept of Maya which hides the true form of God from the world. “Deluded by the three gunas, the world does not recognise Me who am above them. This divine Maya of mine is hard to overcome.” (Sh 13 & 14)

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.

“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)

Chapter 8: Not too much going on here, frankly. The main point seems to be “He who utters Aum at the moment of death come to me” (Sh 13) and “having come to Me, they do not get back to rebirth” (Sh 15). The rest is a rehash of chapter 7.

Chapter 9: 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.

“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 forth again at the Beginning” (Sh 4 to 7).

The superiority of bhakti to God over Vedic rituals is re-emphasized very explicitly.

“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).

Chapter 10: 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”. Enuff said.

Chapter 11: Vishwarupa!! The grand climax of the Gita. Arjuna gets to see the Cosmic Form of God.

Imagine that! A mind-blowing, soul-searing, all-encompassing vision of Ultimate Reality, of All That Is or Was or Ever Will Be!!! What could ever be more awesome and humbling, more ecstatic and terrifying, more devastating and transformative than that?! What an absolutely brilliant concept.

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 is followed by a huge overload of anthropomorphic imagery – eyes, hands, stomach, mouths, teeth, garlands and such. Just doesn’t do it for me.

IMO, contemplating the Hubble Ultra Deep Field or zooming into the Mandelbrot set can convey a far better picture of what Vishwarupa might have been like. :-)

(Deep field: http://www.spacetelescope.org/images/wallpaper3/opo0428b.jpg Every little speck in the image is a galaxy!

Mandelbrot set: http://www.youtube.com/watch?v=WAJE35wX1nQ )

Brief Interlude: 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.

Chapter 12: 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.

Chapter 13: 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 Man.

Chapter 14: More on the three gunas – sattva, rajas, tamas – 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 mud, a stone and a piece of gold, who...” ok, ok, we got it, we got it. It’s Mr D. again.

Chapter 15: 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). I had no idea the image was in the Gita, so that was interesting.

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.

Chapter 16: 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.

Chapter 17: 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.

Chapter 18: The finale, and a fairly eclectic mix of previous material. The three-way classification of the last chapter goes on for a bit. We have the three kinds of knowledge, three kinds of work, three kinds of understanding and so forth.

Karmayoga and swadharma re-appear, but this time, the dharma is laid out by caste in shlokas 42 – 44. (I really 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.

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).

In conclusion: Thus, ultimately, it is ‘Surrender to Me and do My will’.

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?

Large Numbers

Let’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).

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 googol 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!

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.

Arrow Notation:

If a and b are positive integers, a^b is just defined as ab.

The general definition is recursive:

a^...n arrows...^b = a^..(n-1) arrows..^a^..(n-1) arrows..^a^.......^a

where the (n-1) arrows operation is executed (b-1) times. The arrow operations are executed from right to left.

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.

So, to start off:

3^3 = 33 = 27

3^^3 = 3^3^3 = 327 = 7,625,597, 484, 987

Ok, so that’s one of our garden variety large numbers. The ^^ operation, known as the ‘tower’ operation quickly gives us much bigger numbers.

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 number of digits would be 3^7,625,597,484,987 !! And so on...

But now let’s get serious. How about adding yet another arrow?

3^^^3 = 3^^(3^^3) = 3^^7,625,597, 484, 987

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.

But ok, let’s quit trying to imagine and just add one more arrow.
Let’s look at 3^^^^3

3^^^^3 = 3^^^3^^^3 = 3^^3^^......3^^3

where the ^^ operation is done 3^^^3 times.

Take another quick read through the part where I describe 3^^^3. Now take a very deep breath.

Imagine you are doing the evaluation of the right hand side in the expression above. Remember it’s done from right to left.

So, at step 1, you get 3^^3 which is kid-stuff.

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....

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.

Graham’s Number

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.”

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!).

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.

So, what is this number? Let’s define a sequence as follows.

G1 = 3^^^^3, our humongous old friend.

Now let G2 = 3^^...^^3

where - and read this bit very carefully – the number of arrows is G1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (What I really feel like doing is adding at least a googol exclamation marks here, or maybe a googolplex.)

G3 = 3^^...^^3 where the number of arrows is G2.

Still hanging on? Ok then, so we define G3, G4, G5 etc in the same style.
Graham’s number is G64.


Infinity

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.”

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.

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.

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.)

However, compared to Infinity, there is no difference between Graham’s number and zero.

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.

That’s what Infinity is. Treat it with respect.