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.