KGFM 1: Macintyre on the impact of incompleteness on maths
I’m going to be reviewing the recently published collection Kurt Gödel and the Foundations of Mathematics edited by Baaz, Papadimitriou, Putnam, Scott and Harper, for Philosophia Mathematica. This looks to a really pretty mixed bag, as is usual with volumes generated by block-buster conferences: but there are some promising names among the contributors, and a quick initial browse suggests that some of the papers should be very worth reading. So, as I go through the twenty one papers over the coming few weeks, I will intermittently blog about them here.
First up is Angus Macintyre, writing on ‘The impact of Gödel’s Incompleteness Theorems on Mathematics’. His title is pretty much the same as that of a short and very readable piece by Feferman in the Notices of the AMS and his conclusion is also much the same: the impact is small. To be sure, “Some of the techniques that originated in Gödel’s early work (and in the work of his contemporaries) remain central in logic and occasionally in work connecting logic and the rest of mathematics.” But “[a]s far as incompleteness is concerned, its remote presence has little effect on current mathematics.” For example, “The long-known connections between Diophantine equations, or combinatorics, and consistency statements in set theory seem to have little to do with major structural issues in arithmetic” (p. 14). And similarly elsewhere in maths.
There’s a lot of reference to mathematical results, and nearly all of the detailed discussion is well beyond my comfort zone (or that of most readers of this blog, I’d guess: try, e.g., “Étale cohomology of schemes can be used to prove the basic facts of the coefficients of zeta functions of abelian varieties over finite fields”). So I can’t very usefully comment here.
Probably the most exciting and novel thing in this piece is the substantial appendix which aims to give an outline justification for Macintyre’s view that we have “good reasons for believing that the current proof(s) of FLT [Fermat’s Last Theorem] can be modified, without abandoning the grand lines of such proofs, to proofs in PA.” But again, I’m frankly outside my competence here, and I can only refer enthusiasts (or skeptics) about this project to the paper for the details, which look rather impressive to me.
A decidedly tough read for the opening piece!