I’m writing an interview with Kit Fine for the next issue of The Philosophers’ Magazine. Fine works at the seriously abstract end of logic and metaphysics, on such things as vagueness, parts and wholes, modal logic, and the objects of mathematics. Before you pass out, let me reassure you: it’s possible to get a grip on it, even if you couldn’t find your biconditionals in the dark.
What kind of things are numbers? Do they really exist independently of us or do we invent or construct them? There’s something to be said for both possibilities. On the one hand, numbers seem to have a “reality” that merely imagined objects lack. If you close your eyes, you can sort of “see” numbers, intuit them– they’re presented to us in a certain way, and there’s nothing we can do about it, just as objects are presented to us in vision. On the other hand, mathematicians have an extraordinary freedom to invent or extend the number series, and just “make up” such things as negative numbers or irrational numbers as and when they need them.
You’d think those two possibilities would be it – either we apprehend numbers or invent them – but Fine argues for something else entirely. Here’s part of what he said, “Mathematical objects are not exactly of our own making, but we actually have to do something to get them there. There’s something out there which we prod, but there’s the prodding that’s also required.” That’s marginally spooky, but maybe also right. What do you think — intuited, invented, or prodded?