From a recent post at Siris:
"Logic makes us reject certain arguments, but it cannot make us believe any argument." Lebesgue
- the editors of Lakatos, Proofs & Refs (p. 53n4) claim that modern logic shows this is false if taken literally; we can determine, precisely, that some arguments are valid, & therefore logic can make us believe the argument even if not the conclusion.
- But what we can characterize precisely is validity for a domain; and thus we are back at Lebesgue, for one can say that we still have the question of whether the domain is rightly chosen. The editors have slipped, either they have forgotten Lakatos for the moment or think logic works differently from mathematics.
-I see by their further note on Lakato's historical note (56n) that this is their considered opinion. Disappointingly unimaginative and uncritical; what is worse, they think they can have this for free: infallible arguments without infallible principles. This is simply absurd; it is pulling certainty out of a hat.
Incidentally, if anyone was wondering about the short piece on Lakatos in the recent Quine collection, here's a summary: "This is a book about Euler's formula. It is a lot of fun and I enjoyed it." About the only point of substance was: Quine liked that mathematics looked like it was being revised like happens in the other sciences. Apart from that, it's pretty much "This is fun, you should read the book if you like things that are fun." Which is a reasonable way to do a book review.
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3 comments:
Yeah, the Quine review of Lakatos was really disappointing. It seemed like a great platform for Quine to talk about the development and revision of concepts. Since the subject was mathematical concepts, it seems like Quine would want to say something since on his view we are more reluctant to revise those concepts.
Are you reading other bits of that book?
I read other bits of the book (like the Davidson conversation and "Let Me Accentuate the Positive") at the Seminary Co-Op; I've not yet purchased the book. I wouldn't be surprised if I looked at more of it when I go back next.
Quine did at least note that when Euler's formula hit difficulties, it was saved by restricting the domain to which it was applicable -- we didn't toss out the formula, but merely shifted when we invoked it. He seemed fine with that being all there was to say about our resistance to revise mathematical truths. It does seem like there has to have been more he could've said.
I know in one or two of the short pieces I read he repeated the claim that we don't revise mathematical truths because the revisions would "undesirably reverberate throughout science". It seemed like he just liked letting that claim stand alone. (It's explicitly all C.I. Lewis says in "A Pragmatic Conception of the A Priori" -- we don't revise math because that would be too much trouble, so we don't want to do it -- and Lewis was Quine's teacher. But this seems to make Quine's views of maths less interesting than they seem like they should be.)
...this seems to make Quine's views of maths less interesting than they seem like they should be.
That's called nominalism, puto. Quine has no "a priori truths," Kantian, Fregean, or otherwise (tho' many varsity boys misread him as Fregean).
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