Thursday, March 11, 2010

"Coherence" and Mathematical Existence

When I say that "the more practically benign a system of proto-mathematics is, the more likely it is to count as expressing largely true claims about some domain of objects", I realize that this sounds horribly woolly. People naturally ask me: but how practically benign does a practice have to be to guarantee that it succeeds in talking about some object? (not to mention: how do you measure `largely true'?)

Why Be Woolly

Here's a classic example of a claim that isn't woolly:

H: Any logically consistent system of mathematical beliefs counts as expressing truths about some suitable domain of objects.

We can see H is false because it implies that if I believe ZFC+{X} and you believe ZFC+{~X} where X is some statement about number theory independent of ZFC, since we both have logically consistent systems of belief, we will both be right - just talking about different objects.

But what goes wrong?

Note the problem isn't that there aren't enough mathematical objects (if we just have sets every first order consistent theory has a model). Rather (I claim) it's because actual people will use words in the mathematical theory like 'finite' or 'smallest' or 'number' which have meaning that goes beyond their role in this first order logical stipulation.

When we both say that by the "numbers" we mean (among other things) the smallest collection containing 0 and closed under successor, smallest (intuitively) means the same thing for both of us, so it is NOT correct to then interpret each of us as talking about whatever larger non-standard model makes our claim true.

Hence our informal use of the words like "smallest" or "all possible collections" imposes constraints on interpreting us which go beyond the first order logical content of our mathematical statements.

If you buy this, here's why you should be wooly. In general there will be some vagueness with regard to how wrong you can be about Madagascar, Christmas or to use Quine's famous example, atoms, and still count as talking about these things. Once a theory is sufficiently wrong it can be a tossup whether to say that the objects in question are real, and the person is wrong about them, or that there are no such objects. But this is exactly what we face with regard to mathematical objects as well! We have an amorphous informal practice, and a norm that people count as referring to whatever the most natural object is that best satisfies their methods of reasoning about these putative objects, provided there is one that matches suitably well.

There's no bright line about how wrong your various formal and informal beliefs about some putative object can be while you still still count as referring - for the same reason in math as in physics or history.

Hence, I don't try to draw one, and that's why I'm woolly on this issue, and why you should be to!

All we can say generally is: Mathematicians can posit new objects, and the more logically consistent their reasoning about these objects is, and the less their intuitions about consequences of reasoning about these objects lead to false conclusions about other things, the more likely it is that they will count as expressing largely true claims about some suitable piece of the mathematical universe.

The other non-woolly alternative is to give a list: a mathematician counts as referring if they have x beliefs (which are true of the integers), y beliefs (true of the reals), z which are true of imaginary numbers, w for quaterinians, v for sets, k for arrows ... and thats all the mathematical objects that it is metaphysically possible to think about! But surely this is insane.

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