Philip Kitcher thinks that mathematics is surprisingly like empirical science. Few mathematicians would agree; philosophers too, from Socrates on, have held the opposite opinion. In mathematics, they have said, we are able to solve problems and construct proofs by pure thought, without any need to check out how the land lies. Yet we can use geometry for surveying; hence, somewhat mysteriously, the products of reasoning apply to the world. Philosophers have also said that anything that you prove in pure mathematics must be true. To put it metaphorically, not even God could create a world in which a theorem that we have demonstrated is false. Men such as Plato, Aquinas, Leibniz, Kant, Russell, and Wittgenstein have said such things. Almost the only famous dissenters from this established tradition were (until recently) Descartes and John Stuart Mill. Kitcher is on their side.
These recondite issues matter because theories about mathematics have had a big place in Western philosophy. All kinds of outlandish doctrines have tried to explain the nature of mathematical knowledge. Socrates set the ball rolling by using a proof in geometry to argue for the transmigration of souls. As reported by Plato in Meno, the boy who invents a proof of a theorem did not experiment on the physical world, but used only his mind in response to Socratic questions. Hence he must have had inborn knowledge of the proof and he must have got this knowledge in a previous incarnation.
Mathematics has never since been a subject for such philosophical levity. Russell taught that no philosophy is worth much unless it could first answer Kant’s question, “How is pure mathematics possible?” The liberal Mill suspected that the official view of mathematics was a Tory plot. As he wrote in his Autobiography, the notion that we can have knowledge
independently of observation and experience, is, I am persuaded, in these times, the great intellectual support of false doctrines and bad institutions. By the aid of this theory, every inveterate belief and every intense feeling, of which the origin is not remembered, is enabled to dispense with the obligation of justifying itself by reason, and is erected into its own all-sufficient voucher and justification. There never was such an instrument devised for consecrating all deep-seated prejudices. And the chief strength of this false philosophy in morals, politics and religion, lies in the appeal which it is accustomed to make to the evidence of mathematics….
Mill urged that the truths of mathematics are very general empirical facts about the world. We think that they are necessarily true, less because they are persuasive than because they are so pervasive.
Kitcher lacks Mill’s political passion but he knows more mathematics. Like Mill, he tries to deflate rationalist pretensions. How is pure mathematics possible? asks Kant. In just the same way as any other form of knowledge, Kitcher replies. Yet even he can suffer from the ancient philosophical fetishism that makes mathematics extraordinary. On the first page of his introduction he says …