James Franklin in Aeon:
To the question: ‘Is mathematics about something?’ there are two answers: ‘Yes’ and ‘No’. Both are profoundly unsatisfying.
The ‘No’ answer, whose champions are known as nominalists, says that mathematics is just a language. On this view, it is just a way of talking about other things, or a collection of logical trivialities (as Singer claims), or a formal manipulation of symbols according to rules. However you cut it, it is not really about anything. Those whose encounter with mathematics at school was less than happy (‘Minus times minus equals plus/The reason for this we need not discuss’) might feel some sympathy with the nominalist picture. Then again, it is also a view that appeals to physicists and engineers who regard serious propositions about reality as their business. They look on tables of Laplace transforms and other such mathematical paraphernalia as, in the words of the German philosopher Carl Hempel, ‘theoretical juice extractors’: useful for getting extra sense out of meaty physical propositions, but not contentful in themselves.
Nominalism might have a certain down-to-earth appeal, but further reflection suggests that it can’t be right. Although manipulation of symbols is useful as a technique, we also have a strong sense that mathematics makes objective discoveries about a terrain that is in some sense ‘out there’. Take the subtleties of the distribution of primes. Some numbers are prime, some not. A dozen eggs can be arranged in cartons of 6 × 2 or 3 × 4, but eggs are not sold in lots of 11 or 13 because there is no neat way of organising 11 or 13 of them into an eggbox: 11 and 13, unlike 12, are prime, and primes cannot be formed by multiplying two smaller numbers. The idea is very easy to grasp. But this doesn’t mean there’s nothing to discover about it.