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.