March 05, 2004

Fictionalism about Math

[UPDATE: Vera Tobin discusses this entry here.]

In the rationalism reading group we've been reading Jerrold Katz's Realistic Rationalism. In a brief account of fictionalism about mathematics, Katz (IIRC) contrasts fiction with mathematics in that fiction can sustain inconsistency, and concludes that the best explanation is that fiction is fictional and mathematics is real. I kind of like fictionalism about mathematics, so I'm going to try to object. [But I'm not sure these views hold up at all--that's part of the reason I didn't specialize in philosophy of math. So take 'em with a grain of salt, please.]

So: Katz's example of inconsistency in fiction is that Dr. Watson's war wound is said to be in two different places. I think the reason that the fiction can tolerate this inconsistency is that the inconsistency doesn't matter. The reader is never impelled to bring together the two statements about the location of the wound.

But some fictional inconsistencies do matter. In one mystery--I guess I can't say which--one of the details in the detective's summing up is that one character was born on Feb. 29. If you go back looking to see how he knew that, you see that the character does say that his birthday was Tuesday, which happens to be Feb. 29. Unfortunately, he also says that he turned thirty-four that day. Here the inconsistency matters--you have to retrospectively correct the story so that the character's age is divisible by four, or it won't make any sense.

I think that similar things can happen in mathematics. Most inconsistencies will get you in trouble. But, as Mark Wilson likes to point out, at various points in mathematical history people have used mathematical apparatus that on its face seems inconsistent. The trick is to avoid smashing the inconsistencies together--Wilson describes the pracititioners of the operational calculus using ad hoc contextual restrictions to avoid introducing error. And we use naive set theory all the time--we just need to make sure we don't run through the paradoxes.

Fictionalism appeals to me because it seems to account for mathematical evolution. In what sense do we have the same operation in a2 + b2 = c2 and in ei.pi = -1 ? I've never liked the idea that the operation is sitting there in Platonic heaven, waiting to be discovered. I don't see how proofs can provide knowledge then (Katz's book is meant to answer this question, but I'm not convinced), and I like Wittgenstein's statement that even God can only discover mathematical truth by doing mathematics.

So my version of the fictionalist answer is: People discovered that the best way to go on was to extend the definition of exponentiation so that eix = cos x + i sin x. In this group-authored fiction, no other extension worked, and false starts had to be excised from the canon as ruthlessly as Exploits of Moominpapa.

(There's an amazing passage in Cardano where, as far as I can tell, he takes the square root of a negative number even though negative numbers hadn't been invented yet. It contains what appears to be a pun based on the fact that the Latin phrase for "subtracting the cross-products" can also mean "dismissing the excruciating headaches involved." No, really.)

Of course fiction can tolerate inconsistency much better than mathematics can. Fiction is designed to--well, I don't know what it's designed to do, but one of the ways it can achieve this is to entrap the reader into contradictions. Mathematics must go ever onward, and occasionally be used to build bridges that won't collapse. So it's important that there be consensus in mathematics, and that doesn't usually go along with inconsistency. But I think you might be able to make the case that that's a matter of the purpose of mathematical inquiry rather than a question of its having more objective reality than fiction.

Posted by Matt Weiner at March 5, 2004 01:09 PM

I can't get my TrackBack pinging to work, so I just wanted to let you know that I've linked to this entry here.

Posted by: Vera Tobin at March 14, 2004 05:10 PM

You mean here. Cool (I'm putting this up in haste because the browser seems likely to crash, so I haven't read it yet.)

Posted by: Matt Weiner at March 15, 2004 08:49 AM