Monday, January 26, 2009

Parsons's Mathematical Thought: Sec 55, Set theory

The final(!) section of Parsons's book is one of the briefest, and its official topic is about the biggest -- the question of the justification of set-theoretic axioms. But, reasonably enough, Parsons just offers here some remarks on how the case of justifying set theory fits with his remarks in the preceding sections.

First, on "rational intuition" again. We can work ourselves into sufficient familiarity with ZFC for its axioms to come to seem intrinsically plausible -- but such rational intuitions (given the questions than have been raised, by mathematicians and philosophers) "fall short of intrinsic evidence". Which isn't very helpful.

And what about Parsons's modified holism? In the case of set theory, is there "a dialectical relation of axioms and their consequences such as our general discussion of Reason would suggest"? We might suppose not, given that (equivalents of) the standard axioms were already "essentially in place in Skolem's address of 1922". Nonetheless, Parsons suggests, we do find such a dialectical relation, historically in the reception of the axiom of choice, and perhaps now in continuing debates about large cardinal axioms, etc., where "the role of intrinsic plausibility" is much diminished, and having the right (or at least desirable) consequences are an essential part of their justification. But, Parsons concludes -- the final sentence of his book -- "apart from the purely mathematical difficulties, many problems of methodology and interpretation remain in this area". Which is, to say the least, a rather disappointing note of anti-climax!

Afterword Later this week, I'll post (a link to) a single document wrapping up all the blog-posts here into a just slightly more polished whole, and then I must cut down 30K words to a short critical notice for Analysis Reviews. I feel I've learnt a lot from working through (occasionally, battling with) Parsons -- but in the end I suppose my verdict has to be a bit lukewarm. I'm unconvinced about his key claims on structuralism, on intuition, on the impredicativity of the notion of number, in each case in part because, after 340 pages, I'm still not really clear enough what the claims amount to.

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