Science and Math

My 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 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.

The scientist does not forever create new theories 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.

For instance, the astronomer looks up on a clear night, and whoa! There’s a new star in the sky! (I must confess I am very partial to astronomy, far and away my favourite science. Aficionados of other sciences can undoubtedly find analogous examples.)

Similarly, the mathematician draws a triangle and all its medians, and hey, look ! They all pass through the same point!

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 comet or something nearby.

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 because his pencil is blunt.

But no, it’s authentic. The astronomer focuses a telescope, calls her colleagues, takes parallax measurements and everything confirms that what is being 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 naked eye, at that exact location in the sky.

Meanwhile the mathematician has sharpened his pencils, and redrawn the diagram. Nope, no mistake.

Now at this point, the astronomer will not try to pull a brilliant 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 existing theory.

For starters, it looks like this wasn’t a new star really, but an existing one which has suddenly brightened drastically. Can existing theories of stellar structure and evolution explain how this may happen? Time to solve some equations and run simulations...

The mathematician, meanwhile, doesn’t have the luxury of inventing new theories. Math starts from the bedrock, remember? Triangles are triangles, medians are medians. Their properties are clearly given. All one can do is try to see what consequences follow.

Maybe he can work this out just by calculating side lengths and angles. Maybe not...

Success!! The astronomer discovers that it is possible for massive stars to explode cataclysmically at the end of 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.

A new observation has been given a satisfying theoretical explanation. She decides to call this phenomenon a supernova.

Our mathematician is also very pleased. It wasn’t trivial. He had to make some constructions, which weren’t at all apparent at the 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.

At this point, our stories diverge. The mathematician’s colleagues have all verified his proof, and he can forever rest easy.

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.

Nobody knows what the Universe really is and the possibilities are endless.

All she can do is use a principle called Occam’s razor – go for the simplest explanation which fits the facts.

Now what if we don’t have the happy ending?

What if the new star defies explanation? What if the result can’t be proved?

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.

But if the evidence keeps piling up, it’s time to look for a new theory!!

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...

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 counterexample – a triangle whose medians don’t intersect. In this case, the search will be futile because the result is true.

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.

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.

Currently, there is a one million dollar prize offered for a correct proof or counterexample.

I might have given the impression that the astronomer’s task is simple. Just pull a theory out of thin air and voila!

In reality, it is extremely difficult to come up with a successful new theory – particularly if the current theory is well-established. Not only must a new theory explain the new observations, it must also be consistent with everything that is already known and explained by current theory.

Theoretical physicists know this very well.

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.

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.

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.

Meanwhile, in 2004, mathematics saw the solution of one of the most famous unsolved math problems of the past century – the Poincare conjecture.

Life goes on for science and math – similar but different.

What is Mathematics

Is it a science, an art, or something else altogether?
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.

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.”

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.
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 theory, of interest only to historians of science.

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.

Theories versus Theorems

So does that end the argument? Not quite.
While truth may be central to both science and math, a vital difference exists regarding how truths are established.

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.

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 (http://en.wikipedia.org/wiki/Riemann_hypothesis). 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.

On the flip side, none of the confirmations establishes general relativity beyond doubt.
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!

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.

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?
I will argue that the reason is the essence of the difference between science and mathematics.

Top-down versus Bottom-up

Nobody knows what the Universe really is.
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.

A scientific theory is a statement about fundamental properties of the universe, inferred from observed properties of objects within it.
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.
But ultimately, we want to “get to the bottom” - to work out the ultimate structure of the universe from our observed data.
Science works from the top down.

And this is why a scientific theory can never be proved beyond doubt.
It is ultimately a guess.
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.

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.

Mathematics, by contrast, works from the bottom up.
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.

A mathematical theorem is a statement about properties of mathematical objects, deduced from their fundamental properties.
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.

So where does this leave us?
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.