Tuesday, November 09, 2004

Review of Penelope Maddy's Realism in Mathematics

Maddy's Realism in Mathematics is a spirited defense of the position she calls “compromise Platonism” and “set theoretic realism” in the philosophy of mathematics. The position is that Mathematical objects, sets in particular, exist, just like ordinary medium size objects exist.

The book is really good, and I won’t really summarize it here. But basically what happens is that she defend this position on epistemological grounds. Maddy claims that we have the same epistemic access to mathematical objects as we do for any other. (This is a way of handling one of the famous Benacerraf problems.) That is we form beliefs about sets, from a psychological perspective, the same way we do about ordinary objects. We also perceive sets in the same way we perceive ordinary objects. There is also a section on defending the axioms of set theory. We need some intuitive grounds for choosing the axioms that set theory uses, as they are the other part of the problem. The sets are one thing, and the axioms for set theory another.

I have some problems with the book which I will outline below, however, despite the length of the critiques, the book was a good, and important read. It is definitely going to stay an important book in the philosophy of mathematics for a while.

Much of what follows are just random points that I wanted to make about the book.

My first problem with the book is that I really do not see the similarity between forming beliefs about objects and sets. They just seem way too different. The similarities are apparent, and I can’t help thinking that a question is being begged somewhere. We form beliefs about set, but that is if the sets exist. If not the beliefs we form are merely a convenient fiction, or a convenient way of talking.

What if all the sets disappeared and only their elements were left? What would the world look like? Maddy often jumps from elements to elements of a set. I find that jump a bit unintuitive.

Maddy often talks as if number supervenes on objects. (see for example on p 158.) As long as there are objects, there is number. I find that kind of talk odd.

On p 89 the following claim is made: “Knowledge of numbers is knowledge of sets because numbers are properties of sets.” Is that a valid inference? I do not see how that follows.

Axioms are not really all that intuitive. Their usefulness in math is not, contrary to Maddy’s claim, akin to indispensability arguments in physics. We would not loose arithmetic if the axiom of choice were forbidden. Mathematics would just be somewhat more impoverished. But we could still do much of it. There is nothing wrong with that. No one ever complained that there were too few fundamental forces or things that follow from them. The more axioms we have, the more robust our mathematics is. But adding an axiom because it is somewhat useful is not the same as adding something that fundamental that it is indispensable to the discipline.

Along with that, Maddy makes use of a notion of “intrinsic support” which seemed unclear. I thought the analogy with science was misleading too.

The sixth problem I had was that when talking about “extrinsic support” for axioms the phrase “verifiable consequences” is used in a very strange way. Maddy is using it to mean that an axiom can be said to have verifiable consequences if we show that the axiom that the theorem relies on can be proved without it. But as far the theorems that do not need the axiom, it shows us nothing, ie, at best the axiom is unnecessary, and at worst it is false. And what about the theorems that can’t be proved without the axiom? Are those the unverifiable ones? The existence of those theorems tells us that there are cases for which the axiom is “unverifiable.” And the verifiable ones tell us that the axiom is superfluous. Thus it seems that the condition is, at the very least, odd. So, in looking for “extrinsic conditions” for supporting a given axiom, being verifiable, is unhelpful. (And keep in mind that the way it appears, “verifiable” is only used as a partial analogy with science.)

“Natural” is also not used in the same way when assessing extrinsic support. Some mathematical consequences are considered “natural”. It is hard to say what that means though. The given example is the “zigzag pattern of separation properties in the projective hierarchy generated by projective determinacy (and hence SC) is considered more natural than the Pi-side pattern of V=L.” Without explaining what this means a pattern like: /\/\/\/\/\/\/\/\/\. . . . seems more natural than /\__________. . . . I personally do not see this. Certainly there is a sense in which the former has a certain appeal, but I am not sure if it is a mathematically significant appeal. Here is why. There is a very simple analogue to our “unnatural” pattern. That is the distribution of primes among the even versus odd numbers. That pattern is /\___________, where the first (and only) “peak” is the number 2, the only even number. The rest lie with the odds. So I am not sure what to make of the “natural” criterion and why it is valuable as a guide to valuable intuitions.

Next, a problem I had with chapter 2 is that just because for some reason we form set in our head, and we can generate a coherent epistemology, that does not show we have a coherent metaphysics. Isn’t the whole epistemological discussion contingent on the metaphysics being correct? If not it is like having an epistemology about unicorns. I can tell you how we see them, but only if they are really there. I can even tell you how to see them if they are not, and that is a problem.

Finally, and this is no fault of Maddy’s, and to her credit she deals with a nascent theory, the penultimate section deals with issues of structuralism. The section seems like it would benefit from more work on structuralism. What may stem from Shapiro, and the promising work emerging from Koslow’s logic were not available when she wrote the book, though it might be worth rewriting the section given new developments.

Sorry these were really schetchy notes on the book. But it was definitely worth the effort reading it.

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