This post was started as a reply to a comment by vznvzn. It had quickly overgrown the comment format, but still is mostly a reply to vznvzn's remarks.

Gowers did not identify any “new mathematical strand/style”, and did not even attempt this. The opposition “conceptual” mathematics vs. “Hungarian” combinatorics was well known for quite a long time. It started to be associated with Hungary only after P. Erdös started to promote an extreme version of this style; but it was known for centuries. When I was in high school, it was known to any student attending a school with teaching of mathematics and physics on a fairly advanced level and having some interest in mathematics. Of course, this is not about UK (Gowers is a British mathematician). I don’t know enough about the schools there.

There is nothing new in looking at the big picture and doing what you called “mathematical anthropology” either. It is just an accident that you encountered such things in Gowers’s two essays first. I doubt that you are familiar with his writing style in mathematics, and even in more technical parts of his essay “Rough Structure and Classification” (by the way, it is available not only as a .ps file; I have a .pdf file in my computer and a hard copy). Gowers’s writing style and his mathematics are very left-brained. I saw no evidence that he even understands how right-brained mathematicians are working. Apparently he does not like the results of their thinking (but carefully tries to hide this in his popular writings). This may be the main reason why he believes that computers can do mathematics. It seems to me that his post-1998 kind of mathematics (I am not familiar enough with his work on Banach spaces, for which he was awarded Fields medal) indeed can be automated. If CS people do need this, then, please, go ahead. This will eliminate this kind of activities from mathematics without endangering the existence of mathematics or influencing its core.

But when Gowers writes some plain English prose, he is excellent. Note that the verbal communication is associated with the left half of the brain.

The left-right brain theory is not such a clear-cut dichotomy as it initially was. But I like it not so much as a scientific theory, but as a useful metaphor. Apparently, you are right and these days most of mathematicians are left-brained. But this is an artifact of the current system of education in Western countries and not an inherent property of mathematics. Almost all mathematics taught in schools and in undergraduate classes of universities is left-brained. This bias reaches its top during the first two years of undergraduate education, when students are required to take the calculus courses (and very often there are no other options). Only the left-brained aspect of calculus is taught in the US universities. Students are trained to perform some standard algorithms (a task which can be done now, probably, even by a smart phone). The calculus taught is the left-brained Leibniz’s calculus, while the right-brained Newton’s calculus is ignored. So, right-brained people are very likely not to choose mathematics as a career: their experience tells them that this is a very alien to them activity.

In fact, a mathematician usually needs both halves of the brain. Some people flourish using only the left half – if their abilities are very high. Others flourish using only right half. But the right half flourishing is only for geniuses, more or less. With all abilities concentrated in the right half, a mathematician is usually unable to write papers in a readable manner. If the results are extremely interesting, other will voluntarily take the job of reconstructing proofs and writing them down. (It would be much better if such work was rewarded in some tangible sense.) Otherwise, there will be no publications, and hence no jobs. The person is out of profession. On a middle level one can survive mostly on the left half by writing a huge amount of insignificant papers (the barrier to “huge” is much lower in mathematics than in other sciences). Similar effects were observed in special experiments involving middle school students. Right-brained perform better in mathematics in general, but if one considers only mathematically gifted students, both halves are equally developed.

What you consider as Gowers’s “project/program of analysis of different schools of thought” is not due to Gowers. This is done by mathematicians all the time, and some of them wrote very insightful papers and even books about this. His two essays are actually a very interesting material for thinking about “different schools”; they provide an invaluable insight into thinking of a partisan of only one very narrow school.

You are wrong in believing that history of mathematics has very long cycles. Definitely, not cycles, but let us keep this word. Mathematics of 1960 was radically different from mathematics of 1950. I personally observed two hardly predictable changes.

There is no “paradigm shift identified” by Gowers. Apparently, Kuhn's concept of paradigm shift does not apply to mathematics at all. The basic assumptions of mathematics had never changed, only refined.

There is another notion of a “shift”, namely, Wigner’s shift of the second kind. It happens when scientists lose interest in some class of problems and move to a different area. This is exactly what Gowers tries to accomplish: to shift the focus of mathematical research from conceptual (right-brained) one to the one that needs only pure “executive power” (left-brained, the term belongs to G. Hardy) at the lowest level of abstraction. If he succeeds, the transfer of mathematics from humans to computers will be, probably, possible. But it will be another “mathematics”. Our current mathematics is a human activity, involving tastes, emotions, a sense of beauty, etc. If it is not done by humans and especially if the proofs are not readable by humans (as is the case with all computer-assisted proofs of something non-trivial to date), it is not mathematics. The value for the humanity of theorems about arithmetic progressions is zero if they are proved by computers. It is near zero anyhow.

Here all three main directions of Gowers’s activities merge: the promotion of combinatorics; the attempt to eliminate human mathematics; his drive for influence and power.

Thanks for appreciating my comments as “visionary”, no matter of that kind. But they are not. What I was doing in my comments to two Gowers’s posts and in this blog is just pointing out some facts, which are, unfortunately, unknown to Gowers’s admirers, especially to the young ones or experts in other fields. Hardly anything mentioned is new; recent events are all documented on the web. I intentionally refrain from using ideas which may be interpreted as my own – they would be dismissed on this ground alone.

I agree that the discussion in Gowers’s blog eventually turned out to be interesting. But only after the people who demanded me to identify myself and asked why I allow myself to criticize Gowers have left. Then several real mathematicians showed up, and the discussion immediately started to make sense. I hope that the discussion in Gowers’s blog was useful at least for some people. The same about this blog. Right now it shows up as 7

^{th}entry in Google search on “t gowers mathematics” (the 2^{nd}entry is Wiki; other five at the top are his own blogs, pages, etc.) It will go down, of course: I have no intention to devote all my life to an analysis of his mathematics and his personality. And, hopefully, he will eventually cease to attract such an interest as now.In any case, at least one person definitely benefitted from all this – myself. These discussions helped me to clarify my own views and ideas.

Next post: What is combinatorics and what this blog is about according to Igor Pak.