KGFM 2, 3: Kreisel and Grattan-Guinness

The second paper in the collection is a seven-page ramble by Georg Kreisel, followed by twenty pages of mostly opaque endnotes. This reads in many places like a cruel parody of the later Kreisel’s oracular/allusive style. I lost patience very quickly, and got almost nothing from this. What were the editors doing, printing this paper as it is? (certainly no kindness to the author).

Something that struck me though, from the footnotes. Kreisel “saw a good deal of Bernays, who liked to remember Hilbert  …. According to Bernays … Hilbert was asked (before his stroke) if his claims for the ideal of consistency should be taken literally. In his (then) usual style, he laughed and quipped that the claims served only to attract the attention of mathematicians to the potential of proof theory” (pp. 42–43). And Kreisel goes on to say something about Hilbert wanting use consistency proofs to bypass “then popular (dramatized) foundational problems and get on with the job of doing mathematics”. Which chimes with Curtis Franks’s ‘naturalistic’ reading of Hilbert, which I discussed here.

The book’s next contribution couldn’t be more of a contrast, at least in terms of crisp clarity. Ivor Grattan-Guinness is his usual lucid and historically learned self when writing quite briefly about ‘The reception of Gödel’s 1931 incompletability theorems by mathematicians, and some logicians, to the early 1960s’. But in a different way I also got rather little out this paper. There are some interesting little anecdotes (e.g. Saunders Mac Lane studied under Bernays in Hilbert’s Göttingen in 1931 to 1933 — but writes that that he was not made aware of Gödel’s result). But the general theme that logicians got to know about incompleteness early (with some surprising little delays), and the word spread among the wider mathematical community much more slowly could hardly be said to be excitingly unexpected. Grattan-Guinness has J. R. Newman as a hero populariser (and indeed, I think I first heard of Gödel from his wonderful four-volume collection The World of Mathematics) — and Bourbaki is something of a anti-hero for not taking logic seriously. But, as they say, what’s new?

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!

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