bigideas

A recent Big Ideas podcast asked whether mathematics is an art or a science. Here’s a write-up of some of my thoughts on the subject, including some things I’d have liked to say at the time but didn’t get around to.

Mathematical Aesthetics

Many, probably most, mathematicians talk about an aesthetic beauty in their subject that can leave the uninitiated staring blankly in incomprehension. As G H Hardy famously said,

The mathematician’s patterns, like the painter’s or poet’s, must be beautiful. The ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: There is no permanent place in the world for ugly mathematics.

People whose only experience of the subject is solving lists of algebra problems in high school could be forgiven for finding such a suggestion odd. But mathematics isn’t really about rote calculation; it’s about coming up with elegant solutions to puzzles nobody has solved before, and coming up with new puzzles as well.

The aesthetic values that mathematicians value seem to include:

• Elegance, simplicity, avoidance of waste
• Explanatory power — we prefer proofs that say what makes a theorem true
• Intuitive appeal
• Neatness — everything “fits together”

These certainly sound like aesthetic values; indeed they fall roughly into the “Classical” style that Nietzsche identified as Dionysian. They’re optional and somewhat subjective things that we choose to value in mathematics. We can do mathematics that has none of these qualities, and it’s still correct, but we prefer it to have at least some of them. This is the kind of thing that makes mathematics seem like an artistic practice.

But these values are also applied in the empirical sciences, increasingly as the science gets more theoretical, and the existence of aesthetic preferences doesn’t make something an art, or stop it being a science. And in fact there’s also an empircal side to mathematics that seems quite alien to the arts.

Empirical Methods in maths

Consider Appel & Haken’s famous proof of the 4-colour theorem. The proof involves a number of clever steps that reduce the infinity of possible maps to a finite subset. It then proceeds by checking that the theorem si true for each one.

This is a bit like establishing how many people are in your house by checking each room, except in this case there were 1476 rooms, and a computer was used to check them all. The procedure doesn’t tell you why those people are there, or what they’re doing, and there’s nothing elegant or pleasing or symmetrical about it.

This part of the proof feels empirical because it establishes truth by looking at evidence. Many other proofs are empirical in this sense, too. The Riemann Hypothesis, for instance, has been computer-checked for (at least) the first ten trillion cases. There’s no proof yet for the hypothesis, but a single counterexample would serve to diprove it. We might have no idea why this counterexample exists; it would just be something we’d found that we couldn’t explain. It wouldn’t be beautiful or explanatory, but it would still be maths.

Yet such “empirical” exercises are necessarily finite. True empiricism generalises from finite evidence to infinite conclusions; laws that always apply in all cases. There’s no place for this in mathematics. Even if Riemann is false, the first counterexample could be billions of orders of magnitude away, and the human race might never find it. Ten trillion cases verified doesn’t allow us to make a general pronouncement, even though far fewer observations would suffice in the empirical sciences.

All we can do is either prove it abstractly, disprove it with a single counterexample or make the resricted claim that the hypothesis if true for the finite number of cases we’ve tested. This is a long way from other kinds of artistic practice, but a long way from science, too.

Platonism

Platonists characteristically take the idea of maths as a science quite seriouslty. For them, the mathematical truths are all “out there”, and always have been. They’re discovered, not invented.

As a consequence, Platonists also tend to believe that we don’t have any choice about what mathematics is right. This is in contrast to, say, what the right way is to paint a picture, which is somethign culturally-determined that changes with time and place.

In maths, on the other hand, aesthetics comes second; no journal would publish an incorrect proof just because it was elegant, and an inelegant proof can still be accepted.

The Quine-Putnam indispensability argument is a modern defence of Platonism. The argument springs from matephysical Naturalism, and specifically the idea that metaphysics should get its objects from empirical science. In the language of analytical philosophy, we ought to be ontologically committed to the existence of all the things that make successful science work, and nothing more. That includes mathematical objects.

Say someone holds these three things to be true:

• 1. A solution to this equation exists
• 2. If this equation has a solution, it will be a number
• 4. Numbers don’t exist

That certainly seems to be an inconsistent set of beliefs. If the solution is a number, and the solution exists, then numbers must exist.

This argument seems to me to equivocate on “existence”. Of course scientists need mathematics to “exist” as a body of knowledge, but not necessarily as a bunch of independently-existing Platonic objects. How many classical physicists were ontologically committed to the existence of differential equations, over and above the existence of a theory of differential equations? Quine might dismiss my question a mere metaphysicsl, but it seems to me that metaphysics is exactly what this is about, and anyway that field is perhaps a bit less disreputable than it was in Quine’s day.

All successful scientific theories, however, need mathematics. There seems to be no substitute for it, and wrong maths (usually) gives wrong science. That’s a strong indication that maths is “out there” in the same way that physics is.

There may be no rational explanation for the effectiveness of mathematics in science. So can believing in it be called an act of faith? Theoretical physicist Eugene Wigner’s essay The Unreasonable Effectiveness of Mathematics in the Natural Sciences ends with this:

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

I’d like to come back to this in a future post, but let’s leave Platonism there fore now.

Nominalism

We’ve said that mathematics has an empirical feel about it — a sense of discovery rather than invention. But some mathematical problems don’t seem to have this flavour at all.

Think of the Continuum Hypothesis, a technical but important question in axiomatic set theory. In 1940, Kurt Godel showed that it couldn’t be disproved within standard set theory, and in 1963 Paul Cohen showed it couldn’t be proved either.

Both Godel and Cohen preferred to reject the hypothesis, largely on aesthetic grounds, making appeals to elegance and intuitive appeal. In 1986 Chris Freiling showed that the hypothesis could be disproved if you introduced a new axiom, Freiling’s Axiom of Symmetry.

Aside from being strong enough, taken together, to prove everything you want to prove, which axioms you choose is a matter of what seems

• intuitively satisfying
• simple
• “right” in a very ill-defined sense

and these are all aesthetic qualities. Assuming for a moment that all maths can be developed axiomatically, our choice of foundations is ultimately aesthetic. A key question remains, though: if our aesthetic preferences changed, would the resulting maths be equivalent?

So those are some of the observations that stuck with me about the topic of the podcast. As Phil pointed out, the question invites us to assume that maths must be either an art or a science, when it isn’t clearly either. But the presence of both scientific and artistic practices within the discipline is, for me, at least as interesting.