One of the major highlights of MWPM was getting to hear eminent set theorist Hugh Woodin. He gave a great talk about his program of investigating large cardinal axioms, looking for a characterization of the sets that's as informative as our understanding of the numbers etc. I didn't quite buy his case for the truth of large cardinal axioms, for reasons which I present here with some hesitation (I mean, who is more likely to be wrong about explanation in set theory - me or Woodin?)

One of Woodin's main arguments seems to be that if you don't believe in large cardinals, you can't explain the fact that various large cardinal axioms turn out to be consistent. I'm not sure whether the explanation required here is mathematical (how come there's this pattern whereby all these different con sentences happen to be true?), or epistemic (how come thinking about large cardinals/the possibility of non-trivially mapping the universe into itself leaving a certain initial segment fixed reliably leads us to consistent theories, if it's not the case that in so thinking, we are seeing how the universe of sets actually is). But:

-If the explanation desired is mathematical, then it seems like there might be a purely number theoretic explanation for each of the con(ZF+{some large cardinal axiom}) statements. Why wouldn't this be explanation enough? (Indeed, I thought [?] each large cardinal axioms implied the existence of the smaller large cardinals, so giving a number theoretic explanation for some strongest axiom might simultaniously explain the others?)

-If the explanation desired is epistemic, you might think that people are reasoning about what's metaphysically/mathematically POSSIBLE - e.g. that a structure satisfying ZF and containing a large cardinal is metaphysically possible. We clearly do have mathematical/metaphysical intuitions about when a given body of claims are incoherent/couldn't possibly all be true. And, claims that are logically inconsistent are paradigmatic cases of claims that couldn't all be true. What's possible has to be *at least* logically consistent.

Thus, one might explain the fact we've got whole strings of large cardinal axioms A that are consistent by saying not that mathematicians saw that the sets really were A, but that they saw that objects *could possibly be* as required by A.

I think you are confusing explanation with justification.

ReplyDeleteWhat woodin is asking for isn't a justification of each of these truths. He grants, in fact may even be asserting, that we might justify each of these statements considering only arithmetic. I mean any Con statement is simply universal quantified arithmetic claim so at the very least we could get arbitrarily good inductive eviidence for the truth of any of these claims.

Rather he wants an explanation in the sense that a physicist might want an explanation for why the gravitational constant has such and such a value. He doesn't need evidence that it really has such a value, rather he wants to find a simple and compelling theory which entails that G would have this value. Or in other words a simple rule that captures a broad class of phenomena.

Similarly what woodin is asking for is a tidy way to organize all the results from set theory. His argument then is something like this:

In physics we think it's justified to infer the existence of a class of objects because they allow us to organize many phenomena under a simple theory. In particular we feel it's justified to infer the actual existence of the objects rather than merely saying 'the world acts as if such and such exists', e.g., if the Michelson-Morley experiment had gone the other way we would think there really was an ether out there not merely that light acted as if there was. Therefore, since set theory is necessary to organize these mathematical results we can infer the existence of sets.

There are several problems with this argument:

1) I'm sucpiscious of the move in science already.

2) The force of the argument in science often seems to rely on our intuition that things could have turned out differently so there has to be a reason they turned out as they did. In mathematics there is no sense in which things could have turned out differently to get the argument off the ground.

3) We don't feel it's justified to infer that our simply unifiying theory straightforwardly explain the phenomena it unifies. For instance if string theory does turn out to predict all our observations in physics we wouldn't regard it as less likely to be true because it took us so long to prove the observational consequences. As long as the unifying theory actually entails all the phenomena we seek to unify it's fine.

This is a devastating problem for Woodin's theory because I can unify all the number theoretic consequences of large cardinals by saying, "These are true statements about the numbers."

4) It's not at all clear what the phenomena that requires explanation even is.

Surely the principle is not that every collection of arithmetical truths is the result of some elegant mathematical theory about other objects. In fact there are continuum many sets of arithmetic truths so the HUGE majority of them must simply be 'true' without any elegant unifying theory.

So the phenomena can't simply be "here is a class of true sentences." Is it the fact that it's a class of true sentences that we came up with? That's a matter of psychology and history. Are you really going to claim that vaguries of human nature authorize this kind of ontological inference?

Thanks a lot for a bunch of good tips. I look forward to reading more on the topic in the future. Keep up the good work! This blog is going to be great resource. Love reading it.

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