## Saturday, December 10, 2005

### Contingent Numbers

While in Dunedin I got a chance to discuss some philosophy of mathematics with Mark Colyvan and Charles Pigden. Charles argued that, at best, numbers are merely contingent objects, and ones with causal powers besides! I'll make a rough attempt at reconstructing the argument...

First suppose that the only decent argument for realism about numbers is Quine's indispensibility argument. Since our best scientific theories quantify over numbers, this presents us with an ontological commitment to such entities. But Hartry Field, in Science Without Numbers, has successfully reformulated Newtonian physics without quantifying over numbers. That is, numbers are dispensible in a Newtonian universe. So we have no reason to think that they really exist there. We can opt for some form of nominalism or fictionalism about numbers instead.

Of course, our universe is not a Newtonian universe, so where does that leave us? Well, there are two possibilities. Either numbers are dispensible in our universe too, or else they are not. If they are, then nominalism wins. So let's instead suppose that numbers really are indispensible to our science. This is still not particularly good news for the Platonist. After all, it is a merely contingent fact about our natural laws that numbers play an indispensible role. Further, it might be thought that numbers play some sort of weird causal role, since, after all, we learnt about them empirically, by learning that we are in an Einsteinian universe rather than a Newtonian one. If numbers are required to explain how the one universe works but not the other, then they must be doing something in the former. And something is a terribly odd thing for numbers to do.

Categories:

1. This comment follows from the post by a rather convoluted causal chain, but I hope it's not despensible. Metaphysics, as I understand it, has a lot to do with "what exists", but in my (very limited) experience Metaphysics attempts to say an awful lot about "existence" without ever saying what it means by the word "existence." So I have two questions: firstly, is there a widely accepted definition of "existence"; and, secondly, if there is not such a definition, what kind of answer can we expect when we ask the question "do numbers exist?"

2. Bogus. "First suppose pigs can fly. Then, make the obviously true claim that we learn about numbers empirically. Why, when I first was told that there were an infinity of primes, I wasn't sure, but eventually, as I found more and more of them, I ended up believing it. At the same time, I strongly confirmed Fermat's last theorem and also, yeah, Goldbach's Conjecture. Some guy wasted his time 'proving' this stuff, but look how look it took him. Oh, he hasn't even managed to proof the last one yet!"

3. Anonymous, it's one thing to say "bogus", and quite another to show what's wrong with the argument. Your silly rant doesn't actually offer any reasons to doubt Pigden's argument. Indeed, judging by your insertion of first-order mathematical claims, you don't seem to understand what this argument is even about. (Everyone can agree to the first-order mathematical claim that there are infinitely many primes, and that we prove this a priori. But the present discussion is about metaphysics, and the ontological status of numbers, i.e. whether they genuinely exist. The meta-level dispute is about what makes first-order mathematical claims true.)

The indispensibility argument is generally considered the best argument for realism, and unless you can point to any others, I really don't think you're in any position to dispute that premise, let alone compare it to something obviously false ("suppose pigs can fly").

Finally, quit cowering behind the cover of anonymity. If you want to comment on my blog again, you can click the 'other' identity option to enter your name.

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