About Timothy Gowers

Previous post: The conceptual mathematics vs. the classical (combinatorial) one.

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.

D. Zeilberger's Opinions 1 and 62

Previous post: Combinatorics is not a new way of looking at mathematics.

While this is a reply to a comment by  Shubhendu Trivedi in Gowers's blog, I hope that following is interetisting independently of the discussion there.

Opinion 1. Zeilberger admits there that he has no idea about the methods used even in his examples (the 4th paragraph).

He is correct that Jones polynomial is to a big extent a combinatorial gadget. Probably, he is not aware that this gadget applies to topology only if you have a purely topological theorem at your disposal (proved by Reidemeister in 1930s, it remains to be a non-trivial theorem). He may be not aware also of the fact that Jones polynomial did not led to solution of any problem of interest to topologists at the time. The proof of the so-called Tait conjecture was highly publicized, and many people believe that this was an important conjecture. Fortunately, there is a document proving that this is not the case. Namely, R. Kirby with the help of many other topologists compiled around 1980 a list of problems in topology. About 15 years later he published an updated and expanded version. Both editions consist of several parts, one of which is devoted to problems in knot theory. Tait conjecture is about knots and it is not in the 1980 list (by time Kirby started to prepare the new expanded list, it was already proved). Nobody was interested in it, and its solution has no applications.

Eventually, the theory of Jones polynomial and its generalizations turned into an independent self-contained field, desperately searching for connections with other branches of mathematics or at least with topology itself.

But D. Zeilberger should be aware that the Tutte polynomial belongs to the conceptual mathematics. It is one of the precursors of one of the main ideas of Grothendieck, namely, of K-theory. There is no reasons to think that Grothendieck was aware of Tutte's work, but Tutte polynomial is still an essentially a K-theoretic construction.

The Seiberg-Witten ideas have nothing to do with combinatorics. The Seiberg-Witten invariants are based on topology and some advanced parts of the theory of nonlinear PDE. In the last decade some attempts to get rid of PDE in this theory were partially successful. They involve some rather combinatorics-like looking pictures. I wonder if Zeilberger wrote anything about this. But the situation is essentially the same as with the Tutte polynomial. These quite remarkable attempts are inspired, not always directly, by such abstract ideas as 2-categories, for example. Note that the category theory is the most abstract part of mathematics, except, may be, modern set theory (which is a field in which only very few mathematicians are working).

Opinion 62. First, the factual mistakes.

Grothendieck did not dislike other sciences. In particular, at the age of approximately 42-46 he developed a serious interest in biology. Ironically, in the same paragraph Zeilberger commends I.M. Gelfand for his interest in biology.

Major applications of the algebraic geometry were not initiated by the “Russian” school, but the soviet mathematicians indeed embraced this field very enthusiastically. And initial applications did not involve any Grothendieck-style algebraic geometry.

More important is the fact that Zeilberger’s opinions are self-contradicting. He dislikes the abstract (in fact, the conceptual) mathematics, and at the same time praises the “Russian” school for applications of exactly the same abstract conceptual methods.

Zeilberger writes: “Grothendieck was a loner, and hardly collaborated”. Does he really knows at least a little about Grothendieck and his work? Grothendieck’s rebuilding of algebraic geometry in an abstract conceptual framework was a highly collaborative enterprise. He has almost no papers in algebraic geometry published by him alone. The foundational text EGA, Elements of Algebraic Geometry, has Grothendieck and Dieudonne as authors (in this order, violating the tradition to list the authors of mathematical papers in alphabetic order) and was written by Dieudonne alone. More advanced things were published as SGA, Seminar on Algebraic Geometry, and most of this series of Springer Lecture Notes in Mathematics Volumes is authored by Grothendieck and various collaborators. Some present his ideas, but don’t have him as an author. One of them is written by P. Deligne and authored by P. Deligne alone.

Zeilberger has no idea about what kind of youth was given to Grothendieck and presents some (insulting, I would say) conjectures about it. Grothendieck was always concerned with injustice done to other people, in particular within mathematics. His elevated sense of (in)justice eventually led him to (fairly misguided, I believe, but sincere and well-intentioned) political activity. He was initially encouraged by colleagues, who abandoned him when this enterprise started to require more than a lip service.

The phrase “...was already kicked out of high-school (for political reasons), so could focus all his rebellious energy on innovative math” is obviously absurd to everyone even superficially familiar with the history of the USSR. If someone was persecuted on political grounds, then (he could by summarily executed, but at least) any mathematical or other scientific activity would be impossible for him for life. There would be no ways to be a professor of Moscow State University, or taking part in the soviet atomic-nuclear project.

Surely, Gelfand said something like Zeilberger writes about the future of combinatorics. I never was at the Gelfand seminar, neither in Moscow, nor in Rutgers. But there are his publications, from which one can get the idea what kind of combinatorics Gelfand was interested in. Would Zeilberger attempted to read any of these papers, he would hardly see there even a trace of what is so dear to him. All works of Gelfand are highly conceptual.

Finally, it is worth to mention that Gelfand always wanted to be the one who determines the fashion, not the one who follows it. Of course, I see nothing wrong with it. In the late 60ies he regretted that he missed the emergence of a new field: algebraic and differential topology. He attempted to rectify this by two series of papers (with coauthors, by this time he did not published anything under only his name), one about cohomology of infinitely dimensional Lie algebras, another about a (conjectural) combinatorial definition of  Pontrjagin classes (a basic notion in topology). It is very instructive to see what was a “combinatorial definition” for I.M. Gelfand.

Next post: What is mathematics?

Reply to a comment

Previous post: Freedom of speech in mathematics

This is a reply to a recent comment by an Anonymous.

Dear Anonymous,

Thank you very much for reading these notes and for writing a serious and stimulating comment. I was thinking about it for a few days, and I am still not sure that my reply will be convincing. But let me try.

The initial goal of this blog was to provide a context for my arguments in Gowers's blog and in another blog. The latter was quoted by another anonymous in Gowers’s blog, and I had no other choice as to try to explain them (the original comment was addressed to people who know me and my views fairly well). So, the concentration on Gowers's views and activities was only natural; this was the intention. Now this topic is more or less exhausted, and expanding the scope of the blog, or even changing it completely may be in order.

Gowers himself described his goals and said that achieving them will eliminate mathematics as we know it. There is nothing apocalyptic in my discussion of his ideas. I do not anticipate that he or his followers will achieve his goals. I do not think that humans are some sort of computers, and I do not think that computers can do real mathematics (definitely, they can do a lot of mathematical things better than humans, but only when a class of problems was completely understood by humans). But he may achieve his goal in an Orwellian way by changing the meaning of the word “mathematics”. He already shifted the preferences of a big part of mathematical community. It took about ten years. If he will be able to do a comparable feat in the next ten years, and then, may be with his followers, once more, “mathematics” will mean “Hungarian-style combinatorics”. And I do believe that the Hungarian-style combinatorics is a field where computers eventually will be superior to humans because a nearly exhaustive search for a proof will be more efficient than human insights (like it happened with chess, which is actually a branch of combinatorics).

Independently of this scenario, I believe that having a person intended to eliminate mathematics (even if his goal is not achievable) in position of such influence as Gowers is extremely unhealthy and dangerous.

I agree that an “open and critical analysis of very influential individuals or groups in the mathematics research community” is highly desirable. But I am not quite comfortable with the way you wrote about this: “what the discussion should be about”. First of all, I am not comfortable with writing this either, but this is my blog and I write about whatever I like and whenever I like. Perhaps, you meant to apply “should” not to me, but to the mathematical community itself. I will assume that this is what you meant.

I believe that such a discussion is hardly possible. As a rule, mathematicians strongly dislike to openly discuss any contentious issues. One may see this everywhere: from insignificant issues on a department of mathematics to major decision made by the AMS or Fields Medals Committee, for example. I would suggest these posts for publication in the Notices of the AMS (under my real name) if I would believe that there is some chance for them to be published.

To Editors of the Notices of the AMS: this is a challenge. Prove that I am wrong!

Anyhow, I am willing to participate in such an open discussion. There is no venue for this now. Still, I would be glad to expand this blog into such a venue. For example, it may be easily transformed into a collective blog, and, for example, you will be able to post here. You will need only a Google account for this, and I will need to know the name of the account and some way of verification that it belongs to the Anonymous who wrote the comment. I reserve the right to be the owner of the blog and its moderator (right now it is not moderated, comments appear without my approval). And, may be, I will eventually transfer it to somebody else.

Finally, I disagree that mathematics was “long dominated by geometry, topology, arithmetic geometry, etc.” By some reason the word “geometry” is very popular for a few decades, and topology is usually considered to be a branch of geometry. So, the word “geometry” was appended to almost any good mathematical theory. Say, Non-Commutative Geometry is actually a branch of functional analysis. Arithmetic Geometry is a branch of algebraic number theory. Topology is not a branch of geometry in the classical sense of the word “geometry”. Algebraic Topology is a branch of algebra. Well, I realize that this is a sort of scandalous statement. It took me many years to come to this conclusion. There was a subfield of topology called Geometric Topology (it is hardly alive by now); this would be a nonsense would topology be indeed a branch of geometry.

So, in my opinion mathematics was long dominated by good mathematics, but this is not the case anymore.


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