Inspired by this post I would like to ask for an explanation of a remark often attributed to David Kazhdan concerning the fruitfulness of applications of mathematical logic to domains in mathematics not directly related to logic as traditionally conceived, and comparing those to applications of physics in mathematics in the sense of providing new intuitions that are not ordinarily accessible to practicing mathematicians via their traditional training. So there are really two separate questions here: (1) How valid is the claim of such effectiveness? and (2) How valid is the comparison of physics to logic in this sense?
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4$\begingroup$ Will you align the title of the post and the questions in the body? As it stands now, the body asks two questions, while the title seems to answer one of them affirmatively and then ask a third question. $\endgroup$– user44143Commented Jul 10, 2017 at 20:56
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3$\begingroup$ @MikhailKatz It was understood long before 2006 (the earliest reference in your notes) that quantifier elimination for algebraically closed fields is basically equivalent to Chevalley's theorem in algebraic geometry. (see A. Joyal, Les Théorèmes de Chevalley-Tarski et Remarques sur l’Algèbre Constructive , Cah. Top. Géom. Diff. Cat. XVI (1975) pp.256-258, specifically its title, for evidence of this). $\endgroup$– Will SawinCommented Jul 12, 2017 at 7:25
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5$\begingroup$ @MikhailKatz I never said as such, and my paraphrase may have been unfair anyways - it's been a few years. My only claim is that 1) the effectiveness of logic in mathematics is reasonable and 2) the theory that logic exists only to bring rigor to mathematics is unreasonable. $\endgroup$– Will SawinCommented Jul 12, 2017 at 13:22
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4$\begingroup$ @GeraldEdgar This seems to be a take on the following quote by Richard Hamming: "Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane. " $\endgroup$– Michael GreineckerCommented Jul 16, 2017 at 13:26
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3$\begingroup$ @GeraldEdgar A similar sentiment relating calculation and bridges is much earlier than 1980. The book "Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939" contains an exchange between Turing and Wittgenstein. Turing says that if you build a bridge that depends on a wrong calculation, it will fall down. Wittgenstein replies that the wrong calculation doesn't cause the bridge to fall down, rather the bridge falling down serves as a definition of what it means for the calculation to be wrong. $\endgroup$– R HahnCommented Jul 19, 2017 at 4:04
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3 Answers
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I will not dwell on "unreasonableness" because in my opinion, the word is being used in an emotional way to express appreciation and wonder, not to assert a factual claim that the precise amount of effectiveness quantitatively exceeds some rigorously defined threshold of reasonableness. And de gustibus non est disputandum.
However, I think that the part about "providing new intuitions that are not ordinarily accessible to practicing mathematicians via their traditional training" is fairly easy to explain. Many applications of logic to other areas of mathematics center around some kind of transfer principle. One way to think about transfer principles is as follows: We are studying some area of mathematics, and we are able to formalize not just the mathematical objects themselves, but everything we can say about the objects. That is, we are able to rigorously define a formal language that is able to capture (virtually) everything we want to say about the objects. Then by analyzing the formal language, we are able to draw conclusions about some other domain that is not quite the same as our original domain, but to which the formal language applies equally well.
This kind of argument does indeed involve a type of abstraction that is different from "usual" mathematical argumentation, because instead of examining the mathematical objects themselves, we examine the language that we are using to talk about the objects. Examining mathematical language is a natural thing to do when considering "meta-mathematical" questions such as consistency; after all, how else can you analyze the limits of mathematical reasoning other than by formalizing mathematical language? But the part that surprises some people is that the move from studying mathematical objects to studying the language used to talk about the objects can yield concrete results about the mathematical objects themselves, and not just abstract meta-mathematical results. Without detracting from the awe and joy that we feel when we contemplate mathematical beauty, I would submit that this should not really be any more surprising than the general fact that mathematical abstraction—at least, the right kind of abstraction—can yield concrete consequences.
As for the analogy to physics, I personally don't think it goes beyond the truism that a different perspective can yield new insights. For the parallel to be more than that, I think we would have to argue that the use of physical intuition amounts to an unusual process of abstraction, and this does not seem plausible to me.
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$\begingroup$ Out of curiosity, would you place Hrushovski's proof of function-field Mordell-Lang in this category? Personally, I find these transfer principle results (I'm thinking of e.g., Ax-Kochen here) to be "reasonable", in the sense that while the central idea is certainly surprising, once one thinks about it one sees why the idea should work. On the other hand, though I can't claim to fully understand the proof, I really find it hard to explain why Hrushovski's line of argument should work. $\endgroup$– dhyCommented Jul 19, 2017 at 10:39
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$\begingroup$ @dhy : I don't understand Hrushovski's work very well but I would not describe it as being centered around a transfer principle. I'm not even sure that I would say that Hrushovski applies logic to other branches of mathematics; to a "traditional" mathematician, model theory and logic might seem like the same thing, but I tend to use "logic" to refer to arguments with a more syntactic flavor. Having said that, I do think that one could argue that model-theoretic arguments derive much of their power from generalizing to an abstract setting that most mathematicians are not accustomed to. $\endgroup$ Commented Jul 19, 2017 at 18:08
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The most striking thing I have seen for many decades, and perhaps throughout the last 200+ years of "modern" mathematics, is the use of model theory by F. Loeser and others, and then Ngo, to prove a certain form of Langlands' notorious "Fundamental Lemma" by model-theoretic means, quasi-magically transferring a function-field version of the result (proven highly-non-trivially, but, still, more physically-conceptually, by algebraic-geometric methods) to the number field context. Amazing!
But/and I do not know of any other recent (last 30 years?) results, though I would not claim scholarship here. (The Ax-Kochen things are a bit older, and perhaps do not have the same impact...?)
(I am acquainted with David Kazhdan a little, but have not heard direct comments from him in such direction. In fact, given my acquaintance with his general mathematical operational style, I would tend to think that any comments from him in such direction might indeed refer to the relatively recent application of model theory to prove a form of The Fundamental Lemma.)
EDIT: so, yes, as suggested by @Matt F., this is indeed an example of some magical/unreasonable power of logic/model-theory in (the rest of) mathematics.
For that matter, the Robinson's non-standard analysis, especially as nicely packaged by E. Nelson, is pretty magical and explanatory, in a way that seemed impossible by "direct mathematics".
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4$\begingroup$ To be fair, Waldspurger (maybe with collaborators, I am not sure) had first proved (without model theory) that the function field version of the fundamental lemma implies the number field version. At least at first, I think this version of the result was "better" in that it didn't involve things working for "sufficiently large" primes, which is what the model theory version gave. See arxiv.org/pdf/0712.0708.pdf for the paper of Cluckers, Hales and Loeser in this direction. $\endgroup$ Commented Jul 18, 2017 at 11:53
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$\begingroup$ This is a great example on the topic. Will you also give an answer to one of the questions in the post or the title? For instance, it would be consistent with your comments to say "I think this magic shows the unreasonable effectiveness of logic in mathematics to be valid in this case." $\endgroup$– user44143Commented Jul 18, 2017 at 14:51
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$\begingroup$ @DenisChaperondeLauzières, thanks for that comment. I was not aware of, or had forgotten, Waldspurger's work on this. $\endgroup$ Commented Jul 18, 2017 at 16:08
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3$\begingroup$ I'm not sure if it's in quite the same direction, but Ehud Hrushovski's use of model theory to say things about approximate groups was very significant, and led ultimately to Breuillard--Green--Tao's general classification of those objects. (If I understand correctly, their result is a strengthening of Hrushovski's original theorem in which the relevant constants are made effective.) This use of logic to say something really ground-breaking about the structure of groups seems to me to have a similar flavour to your example in number theory. $\endgroup$ Commented Jul 18, 2017 at 21:09
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2$\begingroup$ @NickGill That seems worthy of an answer on its own. $\endgroup$– j.c.Commented Jul 19, 2017 at 3:38
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Since mathematics is powerful on the mathematical level, it is only natural that it is potentially even more so on meta-mathematical level. Thus embedding metamathematical constructions back into mathematics, as mathematics, should indeed be extra impressive. Thus all this is about mathematicians getting into the habit of applying meta to mathematics, the difficulty is partly psychological.
A similar situation is still about the theory of categories. It should be more common to apply definitions for functions to functors, etc.