Cambridge Foundations, 1

I am taking some time off from revising the Gödel book, going to (most of) a local conference, Foundations of Mathematics: What are they and what are they for? Yes, it really is all fun, fun, fun at Logic Matters.

The conference kicked off with Tim Gowers (that’s Sir Tim to you) talking about discovering proofs by breaking down proof-tasks into simpler and simpler ones. He talked through a couple of cases: for example, how would you tackle the question “Find the 2012th digit after the decimal point in the expansion of $latex (\sqrt 2 + \sqrt 3)^{4024}$”? As an exercise by a wonderful teacher leading an audience to see that there is a “natural” way of getting to an answer, just brilliant. As (part of) a gesture towards evidence for a general thesis Gowers likes, namely that the role of “flashes of intuition” in mathematical discovery is much exaggerated, interesting and suggestive. As a hopeful illustration of the kind of heuristics we could give a computer to discover the proof, not so convincing. And as far as the official theme of the conference goes, I suppose somewhat off-topic.

Brendan Larvor’s talk was about the so-called “Philosophy of Mathematical Practice” and the supposed rise and fall of a certain conception of the architectural organisation of mathematics as having/needing “foundations”. But this was at a level of arm-waving generality that I find entirely uncongenial.

Philip Welch talked about a General Reflection Principle in set theory — fine, as you would expect, on the technicalities, but this got bound up with some entirely unclear remarks about why it might help to move from talking about sets qua objects to talking about the concept of sets (which turned out to be talking about a structure). I was also unclear why he thought that talk about “parts” of the set-theoretic universe rather than of sub-classes of the universe was a gain.

The best talk of the day was by Patricia Blanchette, essentially on the Frege-Hilbert dispute about theories, independence proofs, etc. This was extremely clear and convincing, but she has written more than once before about this (see her terrific entry in the Stanford Encyclopedia), and I’m not really sure what was new.

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