2014 Fields medalists?

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

I was asked by Tamas Gabal about possible 2014 Fields medalists listed in an online poll. I am neither ready to systematically write down my thoughts about the prizes in general and Fields medals in particular, nor to predict who will get 2014 medals. I am sure that the world would be better without any prizes, especially without Fields medals. Also, in my opinion, no more than two persons deserve 2014 Fields medals. Instead of trying to argue these points, I will quote my reply to Tamas Gabal (slightly edited).

Would I know who the members of the Fields medal committee are, I would be able to predict medalists with 99% confidence. But the composition of the committee is a secret. In the past, the situation was rather different. The composition of the committee wasn't important. When I was just a second year graduate student, I compiled a list of 10 candidates, among whom I considered 5 to have significantly higher chances (I never wrote down this partition, and the original list is lost for all practical purposes). All 4 winners were on the list. I was especially proud of predicting one of them; he was a fairly nontraditional at the time (or so I thought). I cannot do anything like this now without knowing the composition of the committee. Recent choices appear to be more or less random, with some obvious exceptions (like Grisha Perelman).

Somewhat later I wrote:

In the meantime I looked at the current results of that poll. Look like the preferences of the public are determined by the same mechanism as the preferences for movie actors and actresses: the name recognition.

Tamas Gabal replied:

Sowa, when you were a graduate student and made that list of possible winners, did you not rely on name recognition at least partially? Were you familiar with their work? That would be pretty impressive for a graduate student, since T. Gowers basically admitted that he was not really familiar with the work of Fields medalists in 2010, while he was a member of the committee. I wonder if anyone can honestly compare the depth of the work of all these candidates? The committee will seek an opinion of senior people in each area (again, based on name recognition, positions, etc.) and will be influenced by whoever makes the best case... It's not an easy job for sure.

Here is my reply.

Good question. In order to put a name on a list, one has to know this name, i.e. recognize it. But I knew much more than 10 names. Actually, this is one of the topics I wanted to write about sometime in details. The whole atmosphere at that time was completely different from what I see around now. May be the place also played some role, but I doubt that its role was decisive. Most of the people around me liked to talk about mathematics, and not only about what they were doing. When some guy in Japan claimed that he proved the Riemann hypothesis, I knew about this the same week. Note that the internet was still in the future, as were e-mails. I had a feeling that I know about everything important going on in mathematics. I always had a little bit more curiosity than others, so I knew also about fields fairly remote from own work.

I do not remember all 10 names on my list (I remember about 7), but 4 winners were included. It was quite easy to guess 3 of them. Everybody would agree that they were the main contenders. I am really proud about guessing the 4th one. Nobody around was talking about him or even mentioned him, and his field is quite far from my own interests. To what extent I understood their work? I studied some work of one winner, knew the statements and had some idea about their proof for another one (later the work of both of them influenced a lot my own work, but mostly indirectly), and very well knew what are the achievements of the third one, why they are important, etc. I knew more or less just the statements of two main results of the 4th one, the one who was difficult to guess – for me. I was able to explain why this or that guy got the medal even to a theoretical physicist (actually did on one occasion). But I wasn’t able to teach a topic course about works of any of the 4.

At the time I never heard any complaints that a medal went to a wrong person. The same about all older awards. There was always a consensus in the mathematical community than all the people who got the medal deserved it. May be somebody else also deserved it too, but there are only 3 or 4 of them each time.

Mathematics is a human activity. This is one of the facts that T. Gowers prefers to ignore. Nobody verifies proofs line by line. Initially, you trust your guts feelings. If you need to use a theorem, you will be forced to study the proof and understand its main ideas. The same is true about the deepness of a result. You do not need to know all the proofs in order to write down a list like my list of 10 most likely winners (next time my list consisted of no more than 5 or 6, all winner were included). It seems that I knew the work of all guessed winners better than Gowers knew the work of 2010 medalists. But even if not, there is a huge difference between a graduate student trying to guess the current year winners, and a Fellow of the London Royal Society, a Fields medalist himself, who is deciding who will get 2010 medals. He should know more.

The job is surely not an easy one now, when it is all about politics. Otherwise it would be very pleasant.

Next post: Guessing who will get Fields medals - Some history and 2014.

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.

What is mathematics?

Previous post: D. Zeilberger's Opinions 1 and 62.


This is a reply to a comment by vznvzn to a post in this blog.

I am not in the business of predicting the future. I have no idea what people will take seriously in 2050. I do not expect that Gowers’s fantasies, or yours, which are nearly the same, will turn into reality. I wouldn’t be surprised that the humanity will return by this time to the Middle Ages, or even to the pre-historical level. But I wouldn't bet on this even a dollar. At the same time mathematics can disappear very quickly. Mathematics is an extremely unusual and fragile human activity, which appeared by an accident, then disappeared for almost a thousand years, then slowly returned. The flourishing of mathematics in 20^th century is very exceptional. In fact, we already have much more mathematicians (this is true independently of which meaning of the word “mathematician” one uses) than society, or, if you prefer, humanity needs.

The meaning of words “mathematics”, “mathematician” becomes important the moment the “computer-assisted proofs” are mentioned. Even Gowers agrees that if his projects succeeds, there will be no (pure) mathematicians in the current (or 300, or 2000 years old) sense. The issue can be the topic of a long discussion, of a serious monograph which will be happily published by Princeton University Press, but I am not sure that you are interested in going into it deeply. Let me say only point out that mathematics has any value only as human activity. It is partially a science, but to a big extent it is an art. All proofs belong to the art part. They are not needed at all for applications of mathematics. If a proof cannot be understood by humans (like the purported proofs in your examples), they have no value. Or, rather, their value is negative: a lot of human time and computer resources were wasted.

Now, a few words about your examples. The Kepler conjecture is not an interesting mathematical problem. It is not related to anything else, and its solution is isolated also. Physicists had some limited interest in it, but for them it obvious for a long time (probably, from the very beginning) that the conjectured result is true.

4 colors problem is not interesting either. Think for a moment, who may care  if every map can be colored by only 4 colors? In the real word we have much more colors at our disposal, and in mathematics we have a beautiful, elementary, but conceptual proof of a theorem to the effect that 5 colors are sufficient. This proof deserves to be studied by every student of mathematics, but nobody rushed to study the Appel-Haken “proof” of 4-colors “theorem”. When a graduate student was assigned the task to study it (and, if I remember correctly, to reproduce the computer part for the first time), he very soon found several gaps. Then Haken published an amusing article, which can be summarized as follows. The “proof” has such a nature that it may have only few gaps and to find even one is extremely difficult. Therefore, if somebody did found a gap, it does not matter. This is so ridiculous that I am sure that my summary is not complete. Today it is much easier than at that time to reproduce the computer part, and the human part was also simplified (it consists in verifying by hand some properties of a bunch of graphs, more than 1,000 or even 1,500 in the Appel-Haken “proof”, less than 600 now.)

Wiles deserved a Fields medal not because he proved LFT; he deserved it already in 1990, before he completed his proof. In any case, the main and really important result of his work is not the proof of the LFT (this is for popular books), but the proof of the so-called modularity conjecture for nearly all cases (his students later completed the proof of the modularity conjecture for the exceptional cases). Due to some previous work by other mathematicians, all very abstract and conceptual, this immediately implies the LFT. Before this work (mid-1980ies), there was no reason even to expect that the LFT is true. Wiles himself learned about the LFT during his school years (every future mathematician does) and dreamed about proving it (only few have such dreams). But he did not move a finger before it was reduced to the modularity conjecture. Gauss, who was considered as King of Mathematics already during his lifetime, was primarily a number theorist. When asked, he dismissed the LFT as a completely uninteresting problem: “every idiot can invent zillions of such problems, simply stated, but requiring hundreds years of work of wise men to be solved”. Some banker already modified the LFT into a more general statement not following from the Wiles work and even announced a monetary prize for the proof of his conjecture. I am not sure, but, probably, he wanted a solution within a specified time frame; perhaps, there is no money for this anymore.

Let me tell you about another, mostly forgotten by now , example. It is relevant here because, like the 3x+1 problem (the Collatz conjecture), it deals with iterations of a simple rule and by another reason, which I will mention later. In other words, both at least formally belong to the field of dynamical systems, being questions about iterations.

My example is the Feigenbaum conjecture about iterations of maps of an interval to itself. Mitchell Feigenbaum is theoretical physicist, who was lead to his conjecture by physical intuition and extensive computer experiments. (By the way, I have no objections when computers are used to collect evidence.) The moment it was published, it was considered to be a very deep insight even as a conjecture and as a very important and challenging conjecture. The Feigenbaum conjecture was proved with some substantial help of computers only few years later by an (outstanding independently of this) mathematical physicists O. Lanford. For his computer part to be applicable, he imposed additional restrictions on the maps considered. Still, the generality is dear to mathematicians, but not to physicists, and the problem was considered to be solved. In a sense, it was solved indeed. Then another mathematician, D. Sullivan, who recently moved from topology to dynamical systems, took the challenge and proved the Feigenbaum conjecture without any assistance from the computers. This is quite remarkable by itself, mathematicians usually abandon problem or even the whole its area after a computer-assisted argument. Even more remarkable is the fact that his proof is not only human-readable, but provides a better result. He lifted the artificial Lanford’s restrictions.

The next point (the one I promised above) is concerned with standards Lanford applied to the computer-assisted proofs. He said and wrote that a computer-assisted proof is a proof in any sense only if the author not only understands its human-readable part, but also understands every line of the computer code (and can explain why this code does that is claimed it does). Moreover, the author should understand all details of the operations system used, up to the algorithm used to divide the time between different programs. For Lanford, a computer-assisted proof should be understandable to humans in every detail, except it may take too much time to reproduce the computations themselves.

Obviously, Lanford understood quite well that mathematics is a human activity.

Compare this with what is going on now. People freely use Windows OS (it seems that even at Microsoft nobody understands how and what it does), and the proprietary software like Mathematica™, for which the code is a trade secret and the reverse engineering is illegal. From my point of view, this fact alone puts everything done using this software outside not only of mathematics, but of any science.

Next post: To appear.

Combinatorics is not a new way of looking at mathematics

Previous post: The value of insights and the identity of the author.


This is a partial reply to a comment by vznvzn in Gowers's blog.


Combinatorics is most resolutely not "a new way of looking at mathematics". It is very old, definitely known for hundreds years. Perhaps, it was known in the ancient Babylon already.

And Erdős is not a "contrarian". His work belongs to the most widely practiced tradition in analysis. As a crude approximation, one can say that this tradition originates in the calculus of Leibniz, which is quite different from the calculus of Newton. Even most of mathematicians are not aware of the difference between the Leibniz calculus and the Newton calculus. This is not surprising at all, since only the Leibniz calculus is taught nowadays.

It is the Grothendieck's way of looking at mathematics, the one which I advocate, which is new. This new, conceptual, way of doing mathematics immediately met strong resistance.

And in some cases its opponents won. For example, the early work of Grothedieck in functional analysis had no influence till analysts managed to translate part of his ideas into their standard language. It seems that only quite recently some of analysts realized that a lot was lost in this translation, and done a better translation, closer to the spirit of the original work of Grothendieck.

Another example is provided by the invasion of this new style and even some technical concepts developed in this style into the analysis of several complex variables. This was intolerable for the classical complex analysts, and they started to stress problems about which it was more or less clear that they can be approached by familiar methods. They succeeded, and already in the 1970ies a prominent representative of the classical school, W. Rudin, was able proudly say that Grothendieck's methods (he was more specific) disappeared into background. He did not publish his opinion at the time, but attempted to insult a prominent representative of the new style, A. Borel by such statements. A quarter of century (or more) later he told this story in an autobiographical book. (W. Rudin is a good mathematician and the author of several exceptionally good books, but A. Borel was a brilliant mathematician.)

Now we are observing a much broader attempt, apparently led by T. Gowers, to eliminate the conceptual way of doing mathematics completely. At the very least T. Gowers is the face of this movement for the mathematical public.  After this T. Gowers envisions an elimination of the mathematics itself by relegating it to computers. It looks like the second step is the one most dear to his heart (see the discussion in his blog about a year ago). It seems that combinatorics is much more amenable to the computerization (although I don't believe that even this is possible) than the conceptual mathematics.

Actually, it is not hard to believe that computers can efficiently produce proofs of a wide class of theorem (the proofs will be unreadable to humans, but still some will consider them as proofs). But for the conceptual mathematics it is the definition, and not the proofs, which is important. The conceptual mathematics is looking for new definitions interesting to humans. The proof and theorems serve as a stimulus for work and as a necessary testing ground for new definitions. If a new definition does not help to prove new theorems or to simplify the proofs of old ones, it is not interesting for humans.

There is only one way to get rid of the conceptual mathematics, namely, the Wigner shift of the second kind. The new generation should be told that combinatorics is new, that it is the field to work in, and very soon we will see the young people only the ones doing combinatorics. Since mathematics is to a huge extent "a young people’s game", such a shift can be accomplished very quickly.

P.S. It is worth to note that there are two branches of combinatorics, and one of them is already belongs to the conceptual mathematics. Some people (like D. Zeilberger) are intentionally ignoring this to promote the non-conceptual kind.



Next post: D. Zeilberger's Opinions 1 and 62.

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).


Next post: Reply to JSE.

The twist ending. 4

Previous post: The twist ending. 3. R. Kirby.


Finally, a few thoughts about what I see as the main problem with Gowers's new journals projects: the intended competition with “Annals of Mathematics” (Princeton UP), “Inventiones Mathematicae” (Springer) and “J. of the AMS” (AMS). These three journals are widely recognized as the main and the most prestigious journals in mathematics. As I mentioned already, only one of them, “Inventiones”, is expensive.

In fact, its real price is unknown, and in a sense does not exist. Nobody subscribes to this journal alone; it is way too expensive for individual researchers, and libraries nowadays subscribe to huge packages of Springer journals and electronic books in all sciences and mathematics. As is well known the price of such a package is substantially lower (may be by an order of magnitude) than the sum of list prices of subscribed journals. Of course, these package deals are one of the main problems with big publishers: most of journals in these packages are of very limited interest or just a plain junk. My point here is that this practice makes the list price of a journal irrelevant. But I do consider “Inventiones” as a very expensive journal.

The Gowers-Tao-Cambridge UP project is planned as a competitor not only to “Inventiones”, but to all top mathematics journals, both the general ones and specialized. If the project succeeds, the main and the most influential journal will be not “Annals”, but the new one. This would be very much like a corporate hostile takeover. The power will be shifted from the mathematicians at the Princeton University and the Institute for Advanced Studies (both of which hire just the best mathematicians in the world available, without any regard to country of origin, citizenship, and all other irrelevant for mathematics qualities) to a much more narrow circle of T. Gowers’s friends and admirers.

The choice of the managing editor is, probably, the best for achieving such a goal. R. Kirby is the only mathematician who attempted something similar and succeeded. This story is told in the previous post. The choice of R. Kirby as the managing editors raises strong suspicion that the Gowers’s goal is the same as Kirby’s one. Only Kirby’s ambitions at the time were much more moderate: to control the main journal in one branch of mathematics. Gowers aims higher: to control the main journal in whole (or may be only pure?) mathematics.

I do realize that Kirby will deny my explanations of his motives, and so will Gowers. Both will claim that their goal was and is to ease access to the mathematical literature. Neither me, nor anybody else has a way to know what was and is going on in their minds. This can be judged only by their actions and the results of their actions. The result of Kirby’s project is that he controls the main journal in his area, and nothing is cheaper than it was. I expect that the result of Gowers's initiative will be the same.

So, this is the sad twist in the story: the only thing done by T. Gowers in the last 10-15 years (after his work on Banach spaces) which I wholeheartedly approved only two months ago, now seems (to me) to be a supporting campaign for his attempt to get even more power and influence in mathematics. The attention he got by inspiring the boycott of Elsevier and the accompanying attention to the problems of scientific publishing allowed T. Gowers to present his new journals as a solution of these problems.

And one should never forget that one of his goals is the elimination of mathematics as we know it, and turning mathematicians into service personnel for computers.


Next post: William P. Thurston, 1946-2012.

Conclusion of the series about Timothy Gowers: To be written.

The twist ending. 2. A Cambridge don

Previous post: The twist ending. 1.

Mel Nathanson made a right on the target comment "Mel Nathanson Says, July 8, 2012, 9:14 a.m." in Gowers's blog about ethical issues stemming from the fact that Timothy Gowers is a professor at Cambridge University, of which the publishing house Cambridge University Press, the publisher of his new journals, is a for-profit branch. The university as a whole is non-profit, i.e. cannot distribute profits to people not employed by it.

Next post: The twist ending. 3. R. Kirby.


Behind the jump break I posted the complete text of Mel Nathanson's comment as an insurance against the disappearance of the original. Nothing on the web is really permanent, and I hope that Professor Mel Nathanson will not object to this and will not consider this to be a copyright infringement (I am relying on the "fair use" doctrine, but will remove the text at his request immediately).

The twist ending. 1

Previous post: T. Gowers about replacing mathematicians by computers. 2.



I thought that I more or less exhausted the topic of T. Gowers's mathematics and politics. I turned out to be wrong. The only aspect of Gowers's (quasi-)political activity which I supported was the initiated by him and supported by him boycott of Elsevier, the most predatory scientific publisher; namely the "Cost of Knowledge boycott". I had some reservations about the tactics (why Elsevier only, for example?), but felt that they are concerned with secondary issues and that the motives of Gowers are pure.

Well, in early July T. Tao published in his blog post "Forum of Mathematics, Pi and Forum of Mathematics, Sigma", which shed a lot of light on this political campaign. Further details were provided by T. Gowers himself in "A new open-access venture from Cambridge University Press".

It turned out that Gowers is also behind a project to establish a new electronic mathematical journal, or rather a system of new electronic journals, which will directly compete with the best existing journals, for example, with "Annals of Mathematics", which is usually regarded as simply the best one. In the words of T. Gowers:

"Thus, Pi papers will be at the level of leading general mathematics journals and will be an open-access alternative to them. Discussion is still going on about what precisely this means, but it looks as though the aim will probably be for Pi to be a serious competitor for Annals, Inventiones, the Journal of the AMS and the like."

Out of mentioned three journals, only the "Inventiones Mathematicae" (published by the second biggest scientific publisher after Reed-Elsevier, namely, Springer) is expensive. "Annals of Mathematics" is very cheap by any standards, and at the same time the most prestigious. One may suspect that it is subsidized by Princeton University, but I don't know. Why does it need any competition?


There is a buzz-word here: open access. Even the "Gold Open Access", which sounds great (this is what the buzz-words are for). Indeed, these journals are planned to be open for the readers, everybody will be able to download papers. But somebody is needed to pay at the very least for running a website, databases, for the servers. The "Gold" means that the authors pay. It is suggested that publishing an article in these "open" journals will cost the author $750.00 in current dollars, and the amount will be adjusted for inflation later. In order to attract authors, during the first three years this charge will be waived. Note that any new journal initially publishes mostly articles by the personal friends of the members of the editorial board; they will get a free ride. Gowers considers these three years free ride being really good news; I disagree and consider it to be a cheap trick to help launching his new journal(s).

I believe that it completely wrong to charge authors for publication. In the real world it is the authors who are paid if they done something good, be it a novel, a movie, or a painting. And what they will be paying for in this internet age? Not for the distribution of their papers, as before. Posting a paper at the ArXiv does this more efficiently than any journal. They will be paying for the prestige of the journal, i.e. for a line in CV which may increase their chances to get a good job, a salary raise, etc. This will introduce a new type of corruption into the mathematical community.

The idea of "gold open access" is very popular in the bio-medical sciences. If you work in a bio-med area, you need a big grant paying for your lab, equipment, lab technicians, etc. Adding to these huge costs only $750.00 per article is hardly noticeable (in fact, standard price for gold open access there is between two and three thousands depending on publisher). But mathematics is different. It is a cheap science. A lot of good mathematicians do not have any grants (about two thirds by an NSF estimate). In the current financial and political climate one cannot expect that their employers (the universities, except, perhaps, for a dozen of truly exceptional researches) will pay for publications. And $750.00 is not a negligible amount for a university professor, not to say about a graduate student.

I must mention that the idea of charging the author for the publication was realized in the past by some journals in the form of "page charges". The amount was proportional to the number of pages, since the typesetting costs were proportional; nowadays typesetting is done by the authors (which is, in fact, a hidden cost of publishing a paper), and only final touches are done by the publisher. Such journals existed about 30-something years ago. The author was never responsible for the payment, and if there was nobody to pay (no grant, the university has no such line in the budget) the paper was published anyhow. Still, the idea was abandoned in favor of the traditional publishing model: the one who wants to read a journal, pays for it. Exactly like in a grocery store: if you want an apple, then you pay for it, and not the farmer growing apple trees.


I believe that this idea of charging the authors for publications is much more morally reprehensible than anything done by Elsevier and is a sufficient ground for boycotting this Tao-Gowers initiative.

But this is not all...


Next post: The twist ending 2. A Cambridge don.

T. Gowers about replacing mathematicians by computers. 2

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


As we do know too well by now, not all scientific or technological progress is unqualifiedly beneficial for the humanity. As one of the results of scientific research the humanity now has the ability to exterminate not only all humans, but also all the life on Earth. Dealing with this problem determined to a big extent the direction of development of western countries since shortly after WWII. There are not so dramatic examples also; a scientific research about humans may damage only minor part of the population, or even just the subjects of this research (during the last decades, such a research is carefully monitored in order to avoid any harm to the subjects).

Gowers’s project is an experiment on humans. I believe that replacing mathematicians by computers will do a lot of harm at least to the people who could find their joy and the meaning of life in doing mathematics. But the results, if the project succeeds, are not predictable. If we agree, together with André Weil, that mathematics is an indispensable part of our culture, then it hardly possible to predict what will happen without it.


There is also question if Gowers’s goal is achievable at all. He limited it in at least two significant respects. First, he would be satisfied even if computer will not surpass humans (as opposed to the designers of “Deep Blue”, who wanted and managed to surpass the best chess players). Second, he always speaks about proving theorems, and never about discovering analogies, introducing new definitions, etc. These aspects are the most important part of mathematics, not the theorems (compare the already quoted maxim by Manin). But only theorems matter in the Hungarian-style mathematics. Perhaps, this is the reason why Gowers never mentions these aspects of mathematics. It is hard to tell if this limited goal can be achieved. Given a statement, a computer definitely able sometimes to find a proof of it (or disprove it) by a sufficiently exhaustive search. If it is not able to give an answer, the problem remains open, exactly as in human mathematics. What kind of statements a computer will be able to deal with, is another question.

Some of the best problems are not a true-false type of questions. For example, the problem of defining a “good” cohomology theory for algebraic varieties over finite fields (to a big extent solved by Grothendieck), or the problem of defining higher algebraic K-functors (solved by Quillen). It is impossible for me to imagine a computer capable to invent new definitions or suggest problems based on vague analogies like these two problems, responsible for perhaps a half of really good mathematics after 1950.


It seems that I could feel safe: even in the gloomy Gowers’s future, there will be place for human mathematicians. In fact, the future theorems, stated as conjectures, always served as one of the main, or simply the main stimulus to invention of new definitions. In addition, the success of Gowers’s project will mean the end of mathematics as a profession. There will be no new mathematicians, of Serre’s level, or any other, simply because there will be no way to earn a living by doing human mathematics.

Next post: The twist ending. 1

T. Gowers about replacing mathematicians by computers. 1

Previous post: The Politics of Timothy Gowers. 3.


Starting with his “GAFA Visions” essay, T. Gowers promotes the idea that it is possible and desirable to design computers capable of proving theorems at a very high level, although he will be satisfied if such computers still will be not able to perform at the level of the very best mathematician, for example, at the level of Serre or Milnor. I attempted to discuss this topic with him in the comments to his post about this year Abel prize.

I had no plans for such a discussion, and the topic wasn’t selected by me. I made a spontaneous comment in another blog, which was a reaction to a reaction to a post about E. Szemerédi being awarded this year Abel prize. But I stated my position with many details in Gowers’s blog. T. Gowers replied to only three of my comments, and only partially. It seems that for many people it is hard to believe that a mathematician of the stature of T. Gowers may be interested in eliminating mathematics as a human activity, and this is why my comments in that blog made their way to Gowers’s one (one can find links in the latter).

For Gowers, the goal of designing computers capable of replacing mathematicians is fascinating by itself. Adding some details to his motivation, he claims that such computers cannot be designed without deep understanding of how humans prove theorems. He will not consider his goal achieved if the theorem-proving computer will operate in the manner of “Deep Blue” chess-playing computer, namely, by a huge and a massively parallel (like “Deep Blue”) search. Without any explanation, even after directly asked about this, he claims that in fact a computer operating in the manner of “Deep Blue” cannot be successful in proving theorems. In his opinion, such a computer should closely imitate humans (whence we will learn something about humans by designing such a computer), and that it is much simpler to imitate humans doing mathematics than other tasks.

In addition, Gowers holds the opinion that elimination of mathematics would be not a big loss, comparing it to losing many old professions to the technology.


Gowers’s position contradicts to the all the experience of the humanity. None of successful technologies imitates the way the humans act. No means of transportation imitates walking or running, for example. On the other end and closer to mathematics, no computer playing chess imitates human chess players.

Note that parallel processing (on which “Deep Blue” had heavily relied) is exactly that Gowers attempts to do with mathematics in his Polymath project. It seems that this project approaches the problem from the other end: it is an attempt to make humans to act like computers. This will definitely simplify the goal of imitating them by computers. Will they be humans after this?


Gowers’s position is a position of a scientist interested in learning how something functions and not caring about the cost; in his case not caring about the very survival of mathematics. In my opinion, this means that he is not a mathematician anymore. Of course, he proves theorems, relies on his mathematical experience in his destructive project, but these facts are uninteresting trivialities. I expect from mathematician affection toward mathematics and a desire of its continuing flourishing. (How many nominal mathematicians such a requirement will disqualify?)


Next post: T. Gowers about replacing mathematicians by computers. 2.

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.

The times of André Weil and the times of Timothy Gowers. 3

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


Now we can hardly say that mathematics is a useless science in the sense of G.H. Hardy. It contributes to the exploitation in various ways. For example, the theory of stochastic differential equations, a highly sophisticated branch of mathematics, is essential for the financial manipulations leading to a redistribution of wealth from the middle class to the top 1% of the population. The encryption schemes, designed by mathematicians and implemented by software engineers, prevent access of the general public to all sorts of artistic and intellectual goods. This is a new phenomenon, a result of the development of the Internet.

There is no need to detail the enormous contribution of mathematics to the business of extermination; it is obvious now (this wasn’t known to the general public when A. Weil wrote his article).

Mathematicians are not as free now as they were at the times of André Weil. There are (almost?) no more non-mathematical jobs which will earn a decent livehood and will leave enough energy for mathematical research. This situation is aggravated by the fact that if someone is not employed by a sufficiently rich university, then he or she has no access to the current mathematical literature, which is mostly electronic now, and, if sold to individuals, then the prices are set to be prohibitive. The access to these electronic materials (which cost almost nothing to the publishers to produce) is protected by the above mentioned encryption tools. The industry of the scientific publishing does not have publishing as its main activity any more. Its main business now is the restricting access to scientific papers by a combination of encryption, software, and lobbying for favorable to this industry laws. The main goal pursued is the transfer of the taxpayers dollars to the pockets of its executives and shareholders (this topics deserves a separate detailed discussion).

There are no Nobel prizes in mathematics, but there are many others. The Norwegian Abel prize is specifically intended to be a “Nobel prize” in mathematics. Long before it was established (the first one was awarded in 2003), another prize, the Fields medal, achieved incredible prestige and influence in mathematics, despite the negligible monetary award associated with it. In contrast with the Nobel and Abel prizes, the Fields medal may be awarded only to “young” mathematicians. The meaning of the word “young” was initially not specified, but the mathematical establishment slowly arrived at a precise definition. Somebody is young for the purposes of awarding a Fields medal, if he did not achieved the age 41 in the year of the International Congress of Mathematicians, at which the medal is to be awarded. The Congresses are hold every 4 years (only World War II caused an interruption). So, the persons born in the year of a Congress have additional 4 year to work and to have their work recognized.

Even if this stupid rule would be discarded, the age limitation tends to reward fast people strong at applying existing methods to famous problems. The Fields medals (and many other prizes in mathematics) are usually awarded to the mathematician who made the last step in a solution of a problem, and only rarely to the one who discovered a new method or new line of thought. There are only little chances for “slow maturing work” to be rewarded by this most prestigious award (more prestigious by an order of magnitude than any other prize, except, may be, the Abel prize, which is up to now was awarded almost exclusively to the people of the retirement age).

It was possible to ignore all the prizes in 1948. The Fields medals were awarded only once, in 1936, to two mathematicians. Other prizes, where they existed, did not carry any serious prestige. But in 1950, 1954, and 1958 Fields medals went to exceptionally brilliant mathematicians, and since then this was a prize coveted by anybody who thought that there is a chance to get it.

Now there are many other prizes, each one striving to carry as much weight and influence as possible. An interesting example is the story of the Salem prize. The Salem prize was established by the widow of Raphaël Salem in order to encourage work in Salem's field of interest, primarily the theory of Fourier series. Note that Fourier series and their versions are used throughout almost whole mathematics; it is only natural to think that the prize was intended to mathematicians working on problems really close to Salem’s interests. The international committee (occasionally changing by an unknown to the public mechanism) gradually increased the scope of the prize. By 1991 no connection with Salem’s interests could be observed. Now it is the most prestigious prize for young analysts without any restrictions (and the analysis is understood in a very broad sense).

In fact, this change (as also a suspected preference for mathematicians belonging to one or two particular schools) was not welcomed by Salem’s family, and it withdraw the funding for the prize. The committee did not inform the mathematical public about these events and continued to award the prize with $0.00 attached. I am not aware of the current situation; may be the committee managed to raise some money. (Please, note that I cannot name my sources, as it is often the case in the news reporting, and hence cannot provide any proof. I can only vouch that my sources are reliable and well informed.)

The negligible monetary value of most mathematical prizes is not of any importance. The prestige is immediately transformed into the salary rises, offers from rich universities capable of doubling the salary, etc. The lifetime income could be increased by a much bigger amount than the monetary value of a Nobel Prize.

These are the signs of the lost innocence directly related to the article of André Weil. There are many other signs, and one can talk about them indefinitely. In any case, there are no more ivory towers for mathematicians; their jobs depend on many complicated and not always natural implicit agreements in the society, various laws and regulations detailing the laws, etc. From 1945 till about 1985 all these agreements and laws worked very favorably for mathematics. But, as it turned out, the same laws and understandings could be easily used to control mathematicians, sometimes directly, sometimes in hardly discernible ways, and the same arguments that were used to increase the number of jobs 60 or 50 years ago, could, in principle, be used to eliminate these jobs completely.


Next post: My affair with Szemerédi-Gowers mathematics.

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.