I’m in Vienna for the Gödel Centennial conference, Horizons of Truth. Day 1 featured talks by:
- Angus MacIntyre, How much has mathematics been affected by Gödel’s work? His answer: not much (yet). He surveyed the developments arising from Gödel’s work (recursion theory, definability theory, results on noncomputability of classical problems such as the undecidability of the word problem for groups and Hilberts 10th problem). Then he talked about the reactions of “real” mathematicians to these results and pointed out that: a. the dimensions of the number theoretic varieties we get from, e.g., the results on Hilbert’s 10th problem are higher than anything number theorists are usually interested in, and b. the unsolvability statements obtained from consistency and the like have no interesting arithmetical or geometric structure. He said that the Birch/Swinnerton-Dyer conjecture implies that logical independence results are irrelevant to number theory, and that this was shown by Manin — anyone know what he meant and where Manin proved this? There was a little exchange between him and Sol Feferman, with MacIntyre claiming that it’s “clear” that the Weil ocnjectures can be proved in PA, and Feferman saying that it’s “conjecture”. He went on to talk about group theory, and talked about how the group theorists (Higman, Gromov) transformed the original logical results into group theoretic results which exhibited useful algebraic structures. On set theory, he said that it’s a new branch of advanced mathematics, but its impact on most older areas of mathematics is negligible. He concluded by saying that for a statement to “merely to involve sin, or polynomials, or a Ramsey Principle is no guarantee of relevance.” It’s “too soon to say” what the impact of Gödel’s work on mathematics is.
- Georg Kreisel. Logical hygiene, foundations, and abstractions. Kreisel didn’t have slides, so I had a bit of trouble following him. Well, it might just have been that it was Kreisel talking. He talked about foundations, Hilbert’s program, consistency proofs as “purification rituals”, etc. He did say two things which stuck in my head, though: a. He said that Gödel dictated the material on the second incompleteness theorem in Hilbert-Bernays, vol. 2, to Bernays — so the derivability conditions are reall due to Gödel himself? and b. he said that Bernays told him, in the 70s, that he had asked Hilbert before his stroke in the 1920s whether he meant his [Hilbert’s] claims about consistency to be taken literally. Hilbert, according to Kreisel, laughed, and said “Of course not. They are just to attract the attention of mathematicians.”
- Ivor Grattan-Guinness. The Reception of Gödel’s Work by Logicians and Mathematicians. Ivor gave a tour of general works (textbooks, popular expositions) of important figures up to the late 1950s whith an eye to who talked about Gödel’s theorem. In the immediate vicinity of the results (the 1930s), he pointed to four striking examples of logicians who failed to mention Gödel’s work, even though it would have been important and relevant to what they were doing: Hans Hahn, in a popular lecture in Vienna in the early 1930s, Quine in his System of logicistic of 1934, MacLane in his 1934 dissertation (written in Göttingen under Weyl in 19934, when Gentzen and Bernays were still there — when asked, MacLane responded, “I did not mention the [incompleteness] theorems because noone told me about it” !), and Russell). He went on to talk about lots of other mathematicians and expositors — the conclusion was that it took a good while until Gödel’s theorems made it into mainstream expositions, that this was faster and more complete in the English-speaking world, and that the Bourbakis were especially guilty of not even mentioning Gödel’s accomplishments in their work. Note to self: must look up Dubislav’s 1931 review of Gödel’s paper in the Jahrbuch fuer Fortschritte der Mathematik).
- Juliette Kennedy, Three Moments in the Philosophical Life of Kurt Gödel. The three moments were the remarks at the beginning of his dissertation, the 1964 supplement to “What is Cantor’s continuum problem”, and conversations from 1975 with Sue Toledo. The last one was especially interesting: Gödel and Toledo talked, among lots other things, about Plato’s Eutyphro — Juliette said that he noticed something that experts on ancient philosophy have missed, but then didn’t get around to talking about that; I hope she’ll tell me today what that was!
- Sol Feferman. Lieber Herr Bernays! Lieber Herr Gödel! Sol talked about the correspondence between Gödel and Bernays, starting with the first letter from 1930 requesting an offprint of the incompleteness paper, and spanning their respective careers, with emphasis on the question of what the reach of finitism is. Read the correspondence and Sol’s introduction in the Collected Works, vol. 4. (Sol dedicated his lecture to the memory of Torkel Franzen.)
- Christos Papadimitriou. Computation and Intractability. Echoes of Kurt Gödel. Christos started with a bit of history on the influence of Gödel’s work on the development of computation and complexity theory, including the letter to von Neumann where he anticipated the P =? NP problem. The second part of the talk was about some of Christos’ recent work on complexity of games, specifically computation of Nash equilibria. Turns out that Nash’s proof reduced the problem of the existence to Brouwer’s fixed point theorem; Christos (and his students) give a reduction in the other direction — and that requires constructing certain games which operate on real numbers, a sort of arithmetization. He also sang us a few bars from Bobby Darin’s “Multiplication” and told us this funny story at the beginning of his talk: He started by saying “I don’t know how many of you in the audience have actually talked to Gödel — I have!” — and went on to tell the story of his converstaion with Gödel: Christos was a graduate student at Princeton in the 70’s. His office mate once left a note on his desk with his phone number and a second number, next to which was written “Gödel number”. Since the number wasn’t large enough to be the actual Gödel number of anything, Christos didn’t quite know what to make of it, so he picked up the phone dialled it. An elderly gentleman answered, “Hello?” Christos replied, “Sorry, wrong number.”
- B. Jack Copeland. From the Entscheidungsproblem to the Personal Computer. This was mainly a story about the development of electronic computers in Britain in the 40s and 50s — the story of Turing, and Bletchley Park, Colossus, ACE, and the engineers behind their development, especially Tommy Flowers.
I forgot to write down all the good comments made by Dana Scott in discussion, sorry.
I skipped the last talk, but went back for the Young Scholars Competition, where the 10 finalists for the 20,000 EUR prize each gave super-stressful 10-minute talks. The finalists are: Lorenzo Carlucci, Andrey Bovykin, Lutz Strassburger, Laurentiu Leustean, Mark van Atten, Hannes Leitgeb, Itay Neeman, Justin Moore, Eli Ben-Sasson, and Russell O’Connor. They were all excellent! I’m rooting for Mark and Hannes, but if I had to place a bet, I guess I would go with Itay. We’ll know tonight who the lucky winners are.