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I just checked the first version of Green-Tao paper in arXiv (the file is in my computer). The Introduction presents paper as a paper proving a long-standing conjecture about prime numbers. The Erdős' conjecture on arithmetic progressions is not even mentioned. Of course, most of the non-experts read only the introduction.

Your impression could (and, actually, should) be different from that of a layman mathematician: you are an expert in the field. And my claim wasn’t that nobody realized that the Green-Tao paper is not a work about primes at all. I claim only that this is far from being obvious, and a lot of mathematicians thought that it is a work about primes. Primes have a special (and well-deserved) status in mathematics, everything new about primes seems to be much more valuable than some result about a class of subsets of the set of natural numbers.

Then you make a quite interesting claim, and even using ALL CAPS. I must admit that I also do not care now about existence of arbitrarily long arithmetic progression of primes after the spell of Szemerédi's theorem and Gowers’s work about it disappeared. We differ in that you are interested in arithmetic progressions in other sets, despite being not interested in the set of primes. I am not. I am not interested in arithmetic progressions in other groups even more definitely. Pretending for a moment that I still believe in the theory of “Two cultures”, I see such questions as an easy way to turn some conceptual notions (the notions of primes and groups in this case) into a playground for the “second culture” mathematicians, and an opportunity for them to mingle with the ones working in the “first culture”. Another standard way to do this is to ask about “best estimates” or simply the existence of any estimate for an existence theorem. (It is hardly known that most “pure existence” proofs can be transformed into proofs with estimates according to a hardly known result of logician G. Kreisel – known so little that I am going to have some quite difficult time looking for a reference.)

Let us to step back to the source of all these questions about arithmetic progressions, to the theorem of van der Warden. I never thought that its statement is important or interesting. But I found the proof being interesting (in an agreement with the maxim that proofs are more important than theorems). It was the most complicated and powerful use of (iterated) mathematical induction that I saw at time I learned it. I still think that this aspect of the proof is interesting. Of course, the real questions are concerned not with the usual induction but with the transfinite induction. To the best of my knowledge, Martin’s proof of the determinacy of Borel games still holds the place of a purely mathematical theorem (in contrast with advanced set theory) requiring most complicated form of the transfinite induction. Apparently, it is also the only mathematical result which needs the axiom of replacement for its proof (namely, F of ZF, Fraenkel’s axiom of the Zermelo-Fraenkel set theory). This is hardly a mainstream topic nowadays (for either of “cultures”), but for me it is really deep and interesting.


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