What is combinatorics and what this blog is about according to Igor Pak

Previous post: About Timothy Gowers.

I came across the post “What is Combinatorics?” by Igor Pak. His intention seems to be refuting what is, in his opinion, a basic fault of my notes, namely, the lack of understanding of what is combinatorics.

“While myself uninterested in engaging in conversation, I figured that there got to be some old “war-time” replies which I can show to the Owl blogger.  As I see it, only the lack of knowledge can explain these nearsighted generalizations the blogger is showing.  And in the age of Google Scholar, there really is no excuse for not knowing the history of the subject, and its traditional sensitivities.”

Unfortunately, he did not show me anything. I come across his post while searching other things by Google. May be he is afraid that giving me a link in a comment will engage him in conversation? I would be glad to discuss these issues with him, but if he is not inclined, how can I insist? My intention was to write a comment in his blog, but for this one needs to be registered at WordPress.com. Google is more generous, as is T. Gowers, who allows non-WordPress comments in his blog.

Indeed, I don't know much about “traditional sensitivities” of combinatorics. A Google search resulted in links to his post and to numerous papers about “noise sensitivity”.

Beyond this, he is fighting windmills. I agree with most of what he wrote. Gian-Carlo Rota is my hero also. But I devoted a lot of time and space to explaining what I mean by "combinatorial" mathematics, and even stated that I use this term only because it is used by Gowers (and all my writings on this topics have a root in his ones), and I wasn't able to find quickly a good replacement (any suggestions?). See, for example, the beginning of the post “The conceptual mathematics vs. the classical (combinatorial) one” , as also other posts and my comments in Gowers's blog. In particular, I said that there is no real division between Gowers's “second culture” and “first culture”, and therefore there is no real division between combbinatorics and non-combinatorics.

So, for this blog the working definition of combinatorics is “branches of mathematics described in two essays by T. Gowers as belonging to the second culture and opposed in spirit to the Grothendieck's mathematics”.

I don't like much boxing of all theorems or papers into various classes, be they invented by AMS, NSF, or other “authorities”. I cannot say what is my branch of mathematics. Administrators usually assign to me the field my Ph.D. thesis belongs to, but I did not worked in it since then. I believe that the usual division of mathematics into Analysis, Algebra, Combinatorics, Geometry, etc. is hopelessly outdated.


Next post: New comments to the post "What is mathematics?"

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.

The Hungarian Combinatorics from an Advanced Standpoint

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

Again,  this post is a long reply to questions posed by ACM. It is a complement to the previous post "Conceptual mathematics vs. the classical (combinatorial) one". The title is intentionally similar to the titles of three well known books by F. Klein.

First, the terminology in “Conceptual mathematics vs. the classical (combinatorial) one” is my and was invented at the spot, and the word "classical" is a very bad choice. I should find something better. The word "conceptual" is good enough, but not as catchy as I may like. I meant something real, but as close as possible to the Gowers's idea of "two cultures". I do not believe in his theory anymore; but by simply using his terms I will promote it.

Another choice, regularly used in discussions in Gowers's blog is "combinatorial". It looks like it immediately leads to confusion, as one may see from your question (but not only). First of all (I already mentioned it in Gowers's blog or here), there two rather different types of combinatorics. At one pole there is the algebraic combinatorics and most of the enumerative combinatorics. R. Stanley and the late J.-C. Rota are among the best (or the best) in this field. One can give even a more extreme example, mentioned by M. Emerton: symmetric group and its representations. Partitions of natural numbers are at the core of this theory, and in this sense it is combinatorics. One the other hand, it was always considered as a part of the theory of representations, a highly conceptual branch of mathematics.

So, there is already a lot of conceptual and quite interesting combinatorics. And the same time, there is Hungarian combinatorics, best represented by the Hungarian school. It is usually associated with P. Erdös and since the last year Abel prize is also firmly associated with E. Szemerédi. Currently T. Gowers is its primary spokesperson, with T. Tao serving as supposedly independent and objective supporter. Of course, all this goes back for centuries.

Today the most obvious difference between these two kinds of combinatorics is the fact that the algebraic combinatorics is mostly about exact values and identities, and Hungarian combinatorics is mostly about estimates and asymptotics. If no reasonable estimate is in sight, the existence is good enough. This is the case with the original version of Szemerédi's theorem. T. Gowers added to it some estimates, which are huge but a least could be written down by elementary means. He also proved that any estimate should be huge (in a precise sense). I think that the short paper proving the latter (probably, it was Gowers's first publication in the field) is the most important result around Szemerédi’s theorem. It is strange that it got almost no publicity, especially if compared with his other papers and Green-Tao's ones. It could be the case that this opinion results from the influence of a classmate, who used to stress that lower estimates are much more deep and important than the upper ones (for positive numbers, of course), especially in combinatorial problems.

Indeed, I do consider Hungarian combinatorics as the opposite of all new conceptual ideas discovered during the last 100 years. This, obviously, does not mean that the results of Hungarian combinatorics cannot be approached conceptually. We have an example at hand: Furstenberg’s proof of Szemerédi theorem. It seems that it was obtained within a year of the publication of Szemerédi’s theorem (did not checked right now). Of course, I cannot exclude the possibility that Furstenberg worked on this problem (or his framework for his proof without having this particular application as the main goal) for years within his usual conceptual framework, and missed by only few months. I wonder how mathematics would look now if Furstenberg would be the first to solve the problem.

One cannot approach the area (not the results alone) of Hungarian combinatorics from any conceptual point of view, since the Hungarian combinatorics is not conceptual almost by the definition (definitely by its description by Gowers in his “Two cultures”). I adhere to the motto “Proofs are more important than theorems, definitions are more important than proofs”. In fact, I was adhering to it long before I learned about this phrase; this was my taste already in the middle school (I should confess that I realized this only recently). Of course, I should apply it uniformly. In particular, the Hungarian style of proofs (very convoluted combinations of well known pieces, as a first approximation) is more essential than the results proved, and the insistence on being elementary but difficult should be taken very seriously – it excludes any deep definitions.

I am not aware of any case when “heuristic” of Hungarian combinatorics lead anybody to conceptual results. The theorems can (again, Furstenberg), but they are not heuristics.

I am not in the business of predicting the future, but I see only two ways for Hungarian combinatorics, assuming that the conceptual mathematics is not abandoned. Note that still not even ideas of Grothendieck are completely explored, and, according to his coauthor J. Dieudonne, there are enough ideas in Grothendieck’s work to occupy mathematicians for centuries to come – the conceptual mathematics has no internal reasons to die in any foreseeable future. Either the Hungarian combinatorics will mature by itself and will develop new concepts which eventually will turn it into a part of conceptual mathematics. There are at least germs of such development. For example, matroids (discovered by H. Whitney, one of the greatest topologists of the 20^th century) are only at the next level of abstraction after the graphs, but matroids is an immensely useful notion (unfortunately, it is hardly taught anywhere, which severely impedes its uses). Or it will remain a collection of elementary tricks, and will resemble more and more the collection of mathematical Olympiads problems. Then it will die out and forgotten.

I doubt that any area of mathematics, which failed to conceptualize in a reasonable time, survived as an active area of research. Note that the meaning of the word “reasonable” changes with time itself; at the very least because of the huge variations of the number of working mathematicians during the history. Any suggestions of counterexamples?

Next post: About Timothy Gowers.

Reply to Timothy Gowers

Previous post: Happy New Year!


Here is a reply to a comment by T. Gowers about my post My affair with Szemerédi-Gowers mathematics.

I agree that we have no way to know what will happen with combinatorics or any other branch of mathematics. From my point of view, your “intermediate possibility” (namely, developing some artificial way of conceptualization) does not qualify as a way to make it “conceptual” (actually, a proper conceptualization cannot be artificial essentially by the definition) and is not an attractive perspective at all. By the way, the use of algebraic geometry as a reference point in this discussion is purely accidental. A lot of other branches of mathematics are conceptual, and in every branch there are more conceptual and less conceptual subbranches. As is well known, even Deligne’s completion of proof of Weil’s conjectures was not conceptual enough for Grothendick.

Let me clarify how I understand the term “conceptual”. A theory is conceptual if most of the difficulties were moved from proofs to definitions (i.e. to concepts), or they are there from the very beginning (which may happen only inside of an already conceptual theory). The definitions may be difficult to digest at the first encounter, but the proofs are straightforward. A very good and elementary example is provided by the modern form of the Stokes theorem. In 19th century we had the fundamental theorem of calculus and 3 theorems, respectively due to Gauss-Ostrogradsky, Green, and Stokes, dealing with more complicated integrals. Now we have only one theorem, usually called Stokes theorem, valid for all dimensions. After all definitions are put in place, its proof is trivial. M. Spivak nicely explains this in the preface to his classics, “Calculus on manifolds”. (I would like to note in parentheses that if the algebraic concepts are chosen more carefully than in his book, then the whole theory would be noticeably simpler and the definitions would be easier to digest. Unfortunately, such approaches did not found their way into the textbooks yet.) So, in this case the conceptualization leads to trivial proofs and much more general results. Moreover, its opens the way to further developments: the de Rham cohomology turns into the most natural next thing to study.

I think that for every branch of mathematics and every theory such a conceptualization eventually turns into a necessity: without it the subject grows into a huge body of interrelated and cross-referenced results and eventually falls apart into many to a big extent isolated problems. I even suspect that your desire to have a sort of at least semi-intelligent version of MathSciNet (if I remember correctly, you wrote about this in your GAFA 2000 paper) was largely motivated by the difficulty to work in such a field.

This naturally leads us to one more scenario (the 3rd one, if we lump together your “intermediate” scenario with the failure to develop a conceptual framework) for a not conceptualized theory: it will die slowly. This happens from time to time: a lot of branches of analysis which flourished at the beginning of 20th century are forgotten by now. There is even a recent example involving a quintessentially conceptual part of mathematics and the first Abel prize winner, J.-P. Serre. As H. Weyl stressed in his address to 1954 Congress, the Fields medal was awarded to Serre for his spectacular work (his thesis) on spectral sequences and their applications to the homotopy groups, especially to the homotopy groups of spheres (the problem of computing these groups was at the center of attention of leading topologists for about 15 years without any serious successes). Serre did not push his method to its limits; he already started to move to first complex manifolds, then algebraic geometry, and eventually to the algebraic number theory. Others did, and this quickly resulted in a highly chaotic collection of computations with the Leray-Serre spectral sequences plus some elementary consideration. Assuming the main properties of these spectral sequences (which can be used without any real understanding of spectral sequences), the theory lacked any conceptual framework. Serre lost interest even in the results, not just in proofs. This theory is long dead. The surviving part is based on further conceptual developments: the Adams spectral sequence, then the Adams-Novikov spectral sequence. This line of development is alive and well till now.

Another example of a dead theory is the Euclid geometry.

In view of all this, it seems that there are only the following options for a mathematical theory or a branch of mathematics: to continuously develop proper conceptualizations or to die and have its results relegated to the books for gifted students (undergraduate students in the modern US, high school students in some other places and times).


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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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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.

The times of André Weil and the times of Timothy Gowers. 2

Previous post: The times of André Weil and the times of Timothy Gowers. 1.


Different people hold different views about the future of humankind, even about the next few decades. No matter what position is taken, it is not difficult to understand the concerns about the future of the human race in 1948. They are still legitimate today.

It seems to me that today we have much more evidence that we may be witnessing an eclipse of our civilization than we had in 1948. While the memories of two World Wars apparently faded, these wars are still parts of the modern history. The following decades brought to the light many other hardly encouraging phenomena. Perhaps, the highest point of our civilization occured on July 20, 1969, the day of the Apollo 11 Moon landing. While the Apollo 11 mission was almost purely symbolic, it is quite disheartening to know that nobody can reproduce this achievement today or in a near future. In fact, the US are now not able to put humans even on a low orbit and have to rely on Russian rockets. This does not mean that Russia went far ahead of the US; it means only that Russians preserved the old technologies better than Americans. Apparently, most of western countries do not believe in the technological progress anymore, and are much more willing to speak about restraining it, in contrast with the hopes of previous generations. Approximately during the same period most of arts went into a decline. This should be obvious to anybody who visited a large museum having expositions of both classical and modern arts. In particular, if one goes from expositions devoted to the classical arts to the ones representing more and more modern arts, the less people one will see, until reaching totally empty halls. It is the same in the New York Museum of Modern Art and the Centre Pompidou in Paris.

Mathematics is largely an art. It is a science in the sense that mathematicians are seeking truths about some things existing independently of them (almost all mathematician feel that they do not invent anything, they do discover; philosophers often disagree). It is an art in the sense that mathematician are guided mainly by esthetic criteria in choosing what is worthwhile to do. Mathematical results have to be beautiful. As G.H. Hardy said, there is no permanent place in the world for ugly mathematics. In view of this, the lesson of the art history are quite relevant for mathematicians.


How Timothy Gowers sees the future of mathematics? He outlined his vision in an innocently entitled paper “Rough structure and classification” in a special issue “Visions in Mathematics” of “Geometric and Functional Analysis”, one of the best mathematical journals (see Geom. Funct. Anal. 2000, Special Volume, Part I, 79–117). Section 2 of this paper is entitled “Will mathematics exists in 2099?” and outlines a scenario of gradual transfer of the work of mathematicians to computers. He ends this section by the following passage.

“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 it would hardly be pure mathematics as we know it today.”

Surely, this will be not mathematics. This prognosis of T. Gowers is even gloomier than the one which was unthinkable to A. Weil. The destiny of mathematics, as seen by Gowers, is not to be just a technique in the service of other techniques; its fate is non-existence. The service to other techniques will be provided by computers, watched over by moderately skilled professionals.

We see that nowadays even mathematicians of his very high stature do not consider mathematics as necessary, and ready to sacrifice it for rather unclear goals (more about his motivation will be in the following posts). Definitely, an elimination of mathematics as a human activity will not improve the conditions of human life. It will not lead to new applications of mathematics, because for applications mathematics is not needed at all. Mathematics is distinguished from all activities relaying on it by the requirement to provide proofs of the claimed results. But proofs are not needed for any applications; heuristic arguments supported by an experiment are convincing enough. André Weil and, in fact, most of mathematicians till recently considered mathematics as an irreplaceable part of our culture. If mathematics is eliminated, then a completely different sort of human society will emerge. It is far from being clear even that the civilization will survive. But even if it will, are we going to like it?

This is the main difference between the times of André Weil and the times of Timothy Gowers. In 1948 at least mathematicians cared about the future of mathematics, in 2012 one of the most influential mathematicians declares that he does not care much about the very existence of mathematics. Timothy Gowers is not the only mathematician with such views; but nobody of his stature in the mathematical community expressed them so frankly and clearly. He is a very good writer.


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

The times of André Weil and the times of Timothy Gowers. 1

Previous post: A reply to some remarks by André Joyal.


This is the first in a series of posts prompted by the award of 2012 Abel Prize to E. Szemerédi. He is, perhaps, the most prominent representative of what is often called the Hungarian style combinatorics or the Hungarian style mathematics and what until quite recently never commanded a high respect among mainstream mathematicians. At the end of the previous millennium, Timothy Gowers, a highly respected member of the mathematical community, and one of the top members of the mathematical establishment, started to advance simultaneously two ideas. The first idea is that mathematics is divided into two cultures: the mainstream conceptual mathematics and the second culture, which is, apparently, more or less the same as the Hungarian style combinatorics; while these two styles of doing mathematics are different, there is a lot parallels between them, and they should be treated as equals. This is in a sharp contrast with the mainstream point of view, according to which the conceptual mathematics is incomparably deeper, and Hungarian combinatorics consists mostly of elementary manipulations with elementary objects. Here “elementary” means “of low level of abstraction”, and not “easy to find or follow. The second idea of Gowers is to emulate the work of a mathematician by a computer and, as a result, replace mathematicians by computers and essentially eliminate mathematics. In fact, these two ideas cannot be completely separated.

In order to put these issues in a perspective, I will start with several quotes from André Weil, one of the very best mathematicians of the last century. Perhaps, he is one of the two best, the other one being Alexander Grothendieck. In 1948 André Weil published in French a remarkable paper entitled “L’Avenier des mathémathiques”. Very soon it was translated in the American Mathematical Monthly as “The Future of Mathematics” (see V. 57, No. 5 (1950), 295-306). I slightly edited this translation using the original French text at the places where the translation appeared to be not quite clear (I don’t know if it was clear in 1950).

A. Weil starts with few remarks about the future of our civilization in general, and then turns to the mathematics and its future.

“Our faith in progress, our belief in the future of our civilization are no longer as strong; they have been too rudely shaken by brutal shocks. To us, it hardly seems legitimate to “extrapolate” from the past and present to the future, a Poincaré did not hesitate to do. If the mathematician is asked to express himself as to the future of his science, he has a right to raise the preliminary question: what king of future is mankind preparing for itself? Are our modes of thought, fruits of the sustained efforts of the last four or five millennia, anything more than a vanishing flash? If, unwilling to stumble into metaphysics, one should prefer to remain on the hardly more solid ground of history, the same question reappear, although in different guise, are we witnessing the beginning of a new eclipse of civilization. Rather than to abandon ourselves to the selfish joys of creative work, is it not our duty to put the essential elements of our culture in order, for the mere purpose of preserving it, so that at the dawn of a new Renaissance, our descendants may one day find them intact?”

“Mathematics, as we know it, appears to us as one of the necessary forms of our thought. True, the archaeologist and the historian have shown us civilizations from which mathematics was absent. Without Greeks, it is doubtful whether mathematics would ever have become more than a technique, at the service of other techniques; and it is possible that, under our very eyes, a type of human society is being evolved in which mathematics will be nothing but that. But for us, whose shoulders sag under the weight of the heritage of Greek thought and walk in path traced out by the heroes of the Renaissance, a civilization without mathematics is unthinkable. Like the parallel postulate, the postulate that mathematics will survive has been stripped of its “obviousness”; but, while the former is no longer necessary, we couldn't do without the latter.”

““Mathematics”, said G.H. Hardy in a famous inaugural lecture “is a useless science. By this I mean that it can contribute directly neither to the exploitation of our fellowmen, nor to their extermination.

It is certain that few men of our times are as completely free as the mathematician in the exercise of their intellectual activity. ... Pencil and paper is all the mathematician needs; he can even sometimes get along without these. Neither are there Nobel prizes to tempt him away from slowly maturing work, towards brilliant but ephemeral result.”

One of the salient points made by A. Weil in this essay (and other places) is the fragility of mathematics, its very existence being a result of historical accident, namely of the interest of some ancient Greeks in a particular kind of questions and, more importantly, in a particular kind of arguments. Already in 1950 we could not take for granted the continuing existence of mathematics; it seems that the future of mathematics is much less certain in 2012 than it was in 1950.


Next post: The times of André Weil and the times of Timothy Gowers. 2.