The Politics of Timothy Gowers. 3

Previous post: The Politics of Timothy Gowers. 2.


The preparations of Gowers to the elimination of mathematics are not limited to the elevation of the status of the most amenable to the computerization part of mathematics. T. Gowers uses other means also. His web page "Mathematical discussions” aims at developing some “more natural” ways to discover key mathematical ideas. By “more natural” Gowers apparently means “not requiring a sudden insight”. Some titles of his mini-articles are very telling. A good example is “How to solve basic analysis exercises without thinking.” To do mathematics without thinking is exactly what is needed for replacing mathematicians by computers. I consider this project as a failed one: no real way to discover key ideas without insight, not to say without thinking, is even hinted at in these notes. One of approaches used by Gowers is to reverse the history and shows how to use more recent ideas to discover the older ones, like in his note about the zeta-function. Euler’s and Riemann’s work on the zeta-function stimulated a lot of developments in analysis, and to use these developments to rediscover the main result of Euler looks like cheating. In other cases, like in his note about cubic equations, Gowers more or less rediscovers the original approaches. His approach to the cubic equations is very close to the one presented in every book about Galois theory paying some attention to the history.

My favorite part of this page is entitled "Topology”. It consists of only one phrase: "Watch this space”. This did not change at least since April 18, 2001 (according to the web archive). So, I am watching this space for more than a decade. Topology is the quintessential “first culture” mainstream mathematics, mathematics of Serre and Milnor. Completely missing, even without such a phrase, is algebraic geometry. It seems that the two most important developments in the twenties century mathematics are not amenable even to an attempt to eliminate or at least reduce the roles of insights and thinking.

The posts under the tag ‘Demystifying proofs’ in Gowers blog have the same goal and overlap with his “Mathematical discussions”.


Another project Gowers is actively promoting is called the “Polymath”. See posts in his blog under tags like “polymath”,  “polymath1”,  etc. Perhaps, the best place to start is the posts “Is massively collaborative mathematics possible?” and “Background to a polymath project”. The idea is, apparently, to prove theorems not by the usual process of an individual discovery or close interaction of few mathematicians, but by a massively parallel working of many mathematicians interacting on a special web site. This immediately brings to the memory famous computer “Deep Blue”, who won (at the second attempt) a chess match with Garry Kasparov, perhaps the best chess player of all times. “Deep Blue” relied on massively parallel computation, combined with the chess players’ insights (it turned out that without substantial help from human chess players the computer cannot beat Kasparov). Gowers attempt to arrange something similar but using only humans. This is, clearly, could be a good step toward replacing human mathematicians by computers, if successful.

To the best of my knowledge, the first attempt was somewhat successful, in the sense that it resulted in a published paper. But the result proved was not surprising at all, and the main contributions to the proof were made by very few mathematicians (perhaps, no more than three). The result was certainly accessible to a good mathematician working alone.

Much more can be found at “The polymath blog” and the Polymath1wiki (a Wiki-like site). Amazingly, 1/3 of the described there nine “Polymath projects” are devoted to solving specific problems from International Mathematical Olympiads. As is well known, a sufficiently bright and trained high school student can solve such problem in one-two hours.

So, it seems that the idea failed.

(It is worth to note that the domain name michaelnielsen.org, a subdomain of which is the Polymath1wiki, belongs to Michael Nielsen, who presents himself as “a writer, scientist, and programmer”. Perhaps, he is neither a writer, nor a scientist, nor a programmer, if these notions are understood in a sufficiently narrow sense. But we are not living at the times of André Weil, and nowadays he is definitely all of the above. The point is that even nowadays he is not classified as a mathematician.)


I think that all this gives a good idea of what I understand by the politics of Gowers.

He is also actively involved in a battle with big publishers over the prices of scientific journals. In this case his goals are quite close to my heart (in contrast with the already discussed activities), and I even signed an inspired by Gowers declaration of non-collaboration with the infamous publisher Reed-Elsevier. Still, I believe that his approach is misguided. Elsevier may be the most evil scientific publisher, but not the only one evil, and even the university presses and learned societies act in an evil manner more often than one may expect. From my point of view, the root of the problem is in the scientific community itself, and the solution can be found also only within this community. Everything depends on the transfer of the copyright from authors to publishers. Scientists need to refuse to transfer the copyright. But this is another topic.


Next post: T. Gowers about replacing mathematicians by computers. 1

The Politics of Timothy Gowers. 2

Previous post: The Politics of Timothy Gowers. 1.


Since about 2000, T. Gowers became a prominent advocate of two ideas. First, he works on changing the mathematical public opinion about relative merits of various mathematical results and branches of mathematics in favor of his own area of expertise. Second, he advocates the elimination of mathematics as a significant human activity, and a gradual replacement of mathematicians by computers and moderately skilled professionals assisting these computers. The second goal is more remote in time; he estimates that it is at least decades or even a century away. The first goal is already partially accomplished. I believe that his work toward these two goals perfectly fits the definitions 3a, 5a, and 5b from Merriam-Webster.

I would like to point out that public opinion about various branches of mathematics changes continuously and in a manner internal to the mathematics itself. An area of mathematics may be (or may seem to be) completely exhausted; whatever is important in it, is relegated to textbooks, and a research in it wouldn’t be very valued. Somebody may prove a startling result by an unexpected new method; until the power of this method is exhausted, using it will be a very fashionable and valuable direction of research. This is just two examples.

In contrast with this, T. Gowers relies on ideological arguments, and, as one may guess, on his personal influence (note that most of the mathematical politics is done behind the closed doors and leaves no records whatsoever). In 2000, T. Gowers published two essays: “Two cultures in mathematics” in a highly popular collection of articles “Mathematics: Frontiers and Perspectives” (AMS, 2000), and “Rough structures and classification” in a special issue “GAFA Vision” of purely research journal “Geometric and functional analysis”.

The first essay, brilliantly written, put forward a startling thesis of the existence of two different cultures in mathematics, which I will call the mainstream and the Hungarian cultures for short. Most mathematicians are of the opinion that (pure) mathematics is a highly unified subject without any significant division in “cultures”. The mainstream culture is nothing else as the most successful part of mathematics in the century immediately preceding the publication of the “Two cultures” essay. It encompasses almost all interesting mathematics of the modern times. The Hungarian culture is a very specific and fairly elementary (this does not mean easy) sort of mathematics, having its roots in the work of Paul Erdös.

The innocently titled “GAFA Visions” essay has as it central and most accessible part a section called “Will Mathematics Exist in 2099?” It outlines a scenario eventually leading to a replacement mathematicians by computers. The section ends by the following prediction, already quoted in this blog.

"In the end, the work of the mathematician would be simply to learn how to use theorem-proving machines effectively and to find interesting applications for them. This would be a valuable skill, but would hardly be pure mathematics as we know it today.”

All arguments used to support the feasibility of this scenario are borrowed from the Hungarian culture. On the one hand, this is quite natural, because this is the area of expertise of Gowers. But then the conclusion should be “The work in the Hungarian culture would be simply to learn how to use Hungarian-theorems proving machines effectively”. This would eliminate the Hungarian culture, if it indeed exists, from mathematics, but will not eliminate pure mathematics.

This second project does not seem to be very realistic unless the mathematical community will radically change its preferences from favoring the mainstream mathematics to favoring the Hungarian one. And indeed, it seems that Gowers working simultaneously on both projects. He advocates Hungarian mathematics in his numerous lectures all over the world. He suddenly appears as the main lecturer on such occasions as the announcement of the Clay Institute million dollars prizes. It was a shock when he gave the main lecture about Milnor’s work at the occasion of the award of Abel prize to Milnor. Normally, such lectures are given by an expert in an area close to the area of the person honored. Gowers is in no way an expert in any of the numerous areas Milnor worked in. Moreover, he hardly had any understanding of the most famous results of Milnor; in fact, he consulted online (in a slightly veiled form at Mathoverflow.org) about some key aspects of this result. This public appearance is highly valuable for elevating the status of the Hungarian mathematics: a prominent representative of the last presents to the public some of the highest achievements of the mainstream mathematics.

The next year Gowers played the same role at the Abel prize award ceremony again. This time he spoke about his area of expertise: the award was given to a representative of Hungarian mathematics, namely, to E. Szemerédi. Be a presenter of a laureate work two year in a row is also highly unusual (I am not aware about any other similar case in mathematics) and is hardly possible without behind the closed doors politics. The very fact of awarding Abel prize to E. Szemerédi could be only the result of complicated political maneuvers. E. Szemerédi is a good and interesting mathematician, but not an extraordinary one. There are literally hundreds of better mathematicians. The award of the Abel prize to him is not an indicator of how good mathematician he is; it informs the mathematical community that the system of values of the mathematical establishment has changed.

How it could happen without politics that Gowers was speaking about the work of Milnor at the last year Abel prize ceremony? Gowers speaking about the work of Szemerédi is quite natural, but Gowers speaking about the work of Milnor (and preparing this presentation with the help of Mathoverflow) is quite bizarre. It is obvious that Gowers is the most qualified person in the world to speak about the works of Szemeredi, but there are thousands of mathematicians more qualified to speak about Milnor’s work.


Next post: The Politics of Timothy Gowers. 3.

The Politics of Timothy Gowers. 1

Previous post: My affair with Szemerédi-Gowers mathematics.


I mentioned in a comment in a blog that a substantial part of activity of Timothy Gowers in recent ten or more years is politics. It seems that this claim needs to be clarified. I will start with the definitions of the word “politics” in Merriam-Webster online. There are several meanings, of which the following (3a, 5a, 5b) are the most relevant.

3
a : political affairs or business; especially : competition between competing interest groups or individuals for power and leadership (as in a government).

5
a : the total complex of relations between people living in society
b : relations or conduct in a particular area of experience especially as seen or dealt with from a political point of view .

It seems that the interpretation of T. Gowers himself is based only on the most objectionable meaning, namely:

3
c : political activities characterized by artful and often dishonest practices.

I am not in the position to judge how artful the politics of Gowers is; its results suggest that it is highly artful. But I have no reason to suspect any dishonest practices.

With only one exception, I was (and I am) observing Gowers activities only online (this includes preprints and publications, of course). I easily admit that in this way I may get a distorted picture. But this online-visible part does exist, and this part is mostly politics of mathematics, not mathematics itself.

I do classify as politics things like “The Princeton Companion to Mathematics”, which do not look as such at the first sight. This particular book gives a fairly distorted and at some places an incorrect picture of mathematics, and this is why I consider it as politics – it is an attempt to influence both the wide mathematical public and the mathematicians in power.

I was shocked by Gowers reply to the anonym2’s comment to his post “ICM2010 — Villani laudatio” in his blog. The Gowers blog at the time of 2010 Congress clearly showed that he has almost no idea about the work of mathematicians awarded Fields medals that year. But Gowers was a member of the committee selecting the medalists. “How it could be?” asked anonym2. The reply was very short: “No comment”. This lack of a response (or should I say “this very telling response”?) and the following it explanations of T. Tao clearly showed that the work of the Fields medals committee is now a pure politics, contrary to Tao’s assertion of the opposite. If the members of the committee do not understand the work of laureates, they were not able to base their choices on the substance of the works considered, and only the politics is left. In fact, nowadays it is rather easy to guess which member of the committee was a sponsor for which medalist. This was not the case in the past, and the predictions of the mathematical community were very close to the outcome. I myself, being only a second year graduate student, not even suspecting that there is any politics involved, was able to compile a list of 10 potential Fields medalist for that year, and all four actual medalists were on the list. The question of anonym2 "How could the mathematical community be so wrong in their predictions?" could not even arise at these times.


Next post: Part 2.

My affair with Szemerédi-Gowers mathematics

Previous post: The times of André Weil and the times of Timothy Gowers. 3.


I learned about Szemerédi’s theorem in 1978 from the Séminaire Bourbaki talk by Jean-Paul Touvenot “La démonstration de Furstenberg du théorème de Szemerédi sur les progressions arithmétiques”. As it is clear already from the title, the talk was devoted to the work of Furstenberg and not to the work of Szemerédi.

The theorem itself looked amusing, being a generalization of a very well known theorem of van der Warden. The latter one was, probably, known to every former student of a mathematical school in USSR and was usually considered as a nice toy and a good way to show smart and mathematically inclined kids how tricky the use of the mathematical induction could be. Nobody considered it as a really important theorem or as a result comparable with the main work of van der Warden.

But the fact that such a statement can be proved by an application of the theory of dynamical systems was really surprising. It looks like Bourbaki devoted a talk to this subject exactly for the sake of this unusual at the time application and not for the sake of the theorem itself. According to a maxim attributed to Yu.I. Manin, proofs are more important than theorems, and definitions are more important than proofs. I wholeheartedly agree. In any case, the work of Szemerédi was not reported at the Séminaire Bourbaki. I also was impressed by this application of dynamical systems and later read several initial chapters of Furstenberg’s book. But when I told about this to a young very promising expert in my area of mathematics, I got very cold reception: “This is not interesting at all”. Even references to Bourbaki and to the dynamical systems did not help. Now I think that we were both right. The theorem was not interesting because it was (and, apparently, still is) useless for anything but to proving its variations, and it is not sufficiently charming by itself (I think that the weaker van der Warden’s theorem is more charming). The theorem is interesting because it can be proved by tools completely alien to its natural context.

Then I more or less forgot about it, with a short interruption when Furstenberg’s book appeared.

Many years later I learned about T. Gowers from a famous and very remarkable mathematician, whom I will simply call M, short for Mathematician. In 1995 he told me about work of Gowers on Banach spaces, stressing that a great work may be completely unnoticed by the mathematical community. According to M, Gowers solved all open problems about Banach spaces. I had some mixed feelings about this claim and M’s opinion. May be Gowers indeed solved all problems of the Banach spaces theory (it seems that he did not), but who cares? For outsiders the theory of Banach spaces is a dead theory deserving a chapter in Bourbaki’s treatise because its basic theorems (about 80 years old) are exceptionally useful. On the other hand, Gowers was a Congress speaker in 1994, and this means that his work did not went unnoticed. In 1998 Gowers was awarded one of the four Fields medals for that year, quite unexpectedly to every mathematician with whom I discussed 1998 awards (M is not among them). It was also surprising that in his talk on the occasion of the award Gowers spoke not about his work on Banach spaces, but about a new approach to Szemerédi’s theorem. The approach was, in fact, not quite new: it extended the ideas of an early paper by K.-F. Roth on this topic (the paper is a few years earlier than his proof of what is known now as the Tue-Siegel-Roth theorem).

I trusted enough to M’s opinion to conclude that, probably, all work by Gowers deserves attention. So, I paid some attention to his work about Szemerédi’s theorem, but his paper looked technically forbidding (especially given that my main interests always were more or less at the opposite pole of pure mathematics). Then Gowers published a brilliantly written essay “Two cultures in mathematics”. He argued that the mainstream mathematics, best represented by the work of Serre, Atiyah, Grothendieck and their followers (and may be even Witten, despite he is not really a mathematician) is no more than a half of mathematics, “the first culture”, as he called it. Usually it is called “the conceptual mathematics”, since the new concepts are much more important to it than solutions of particular problems (as was already mentioned, the definitions are more important than proofs and theorems). Gowers argued that there is an equally important “second culture”. Apparently, it is best represented by the so-called “Hungarian combinatorics” and the work of Erdös and his numerous collaborators. In this mathematics of “the second culture”, the problems are stressed, the elementary (not involving abstract concepts, but may be very difficult) proofs are preferred, and no rigid structures (like the structure of a simple Lie algebra) are visible. Moreover, Gowers argued that both cultures are similar in several important aspects, despite this is very far from being transparent. A crucial part of his essay is devoted to outlining these similarities. All this was written in an excellent language at the level of best classical fiction literature, and appeared to be very convincing.

I decided to at least attempt to learn something from this “second culture”. Very soon I have had some good opportunities. T. Gowers was giving a series of lectures about his work on Szemerédi’s theorem in a not very far university. I decided to drive there (a roundtrip for each lecture) and attend the lectures. The lectures turned out to be exceptionally good. Then, after I applied some minor pressure to one of my colleagues, he agreed to give a series of lectures about some tools used by Gowers in his work. His presentation was also exceptionally good. I also tried to read relevant chapters in some books. All this turned out to be even more attractive than I expected. I decided to teach a graduate course in combinatorics, and attempted to include some Gowers-style stuff. The latter wasn’t really successful; the subject matter is much more technically difficult (and I do not mean the work of Szemerédi and Gowers) than would be appropriate. Anyhow, over the years I devoted significant time and efforts to familiarize myself with this “second culture” mathematics. This was interrupted both by mathematical reasons (it is nearly impossible to completely switch areas in the western mathematical community), and by some completely external circumstances.

When later I looked anew both at the “second culture” mathematics and at the theory of the “Two cultures in mathematics”, I could not help but to admit that they both lost their appeal. There is no second culture. The fact is that some branches of mathematics are not mature enough to replace assembling long proofs out of many similar pieces by a conceptual framework, making them less elementary, but more clear. The results of the second culture still looked isolated from the mainstream mathematics. I realized that the elementary combinatorial methods of proofs, characteristic for the purported second culture, occur everywhere (including my own work in “the first culture”). I would not say that they are always inevitable, but very often it is simpler to verify some fact by a combinatorial argument than to find a conceptual framework trivializing it.

Perhaps, my opinion about the “second culture” reached its peak on the day (April 8, 2004) of posting to the arXiv of the Green-Tao paper about arithmetic progression of primes. Prime numbers are the central notion of mathematics, and every new result about them is interesting. But gradually it became clear that the Green-Tao paper has nothing to do with primes. Green and Tao proved a generalization of Szemerédi’s theorem. By some completely independent results about primes due to Goldston and Yildirim, the set of primes satisfies the assumptions of the Green-Tao theorem. The juxtaposition of these two independent results leads to a nicely looking theorem. But anything new about primes is contained in the Goldston-Yildirim part, and not in Green-Tao part. This was a big disappointment.

So, the affair ended without any drama, in contrast with the novel “The End of the Affair” by Graham Greene.


Next post: The politics of Timothy Gowers. 1.