Richard Marshall interviews Timothy Williamson in 3:AM Magazine:
3:AM: Alex Rosenberg thinks that understanding the nature of maths will rest on the success that science has had so far in making sense of everything else. He bets that an epistemology explaining how we know a priori synthetic truths will come from science not analytic philosophy. And he also points out that even if you’re right that naturalists have no handle on mathematical truths, neither do non-naturalists. How do you respond?
TW: Alex bravely defends what I think of as the worst beliefs associated with the word ‘naturalism’, the scientism and the reductionism. He claims that only the methodology of natural science produces genuine knowledge. I pointed to mathematics as a glaring counterexample. He admits that it’s a problem for his view, but has faith that somehow or other, he has no idea how, the methodology of natural science will solve the problem and show that mathematics isn’t really a counterexample after all. There is no symmetry between his view and mine with respect to mathematics. On the face of it, mathematics is a massive counterexample to his view. On the face of it, mathematics is perfectly consistent with my view. His view requires mathematics to be either reduced to natural science or ditched. He can’t ditch it, because natural science itself uses mathematics all the time, and he has no idea how to reduce it to natural science. My view involves no such dilemma. Mathematics is fine as it is, without being reduced to anything else. He can’t find the sort of handle on mathematics his view says he needs. My view says no such handle is needed.
Furthermore, it’s not even approximately true that natural science has had success in making sense of everything but mathematics, if that involves showing that other forms of genuine human knowledge resulted from the methodology of the natural sciences. We have massive knowledge of history that doesn’t come from that methodology. Of course, some naturalists in a watered-down sense would stretch their understanding of the methodology to cover past and present historical scholarship. But Alex Rosenberg doesn’t do that, he just dismisses history as a source of genuine knowledge. If he wants to enact a reductio ad absurdum of his own extremist brand of naturalism, that’s fine by me.