Guessing who will get Fields medals - Some history and 2014

Previous post: 2014 Fields medalists?

This was a relatively easy task during about three decades. But it is nearly impossible now, at least if you do not belong to the “inner circle” of the current President of the International Mathematical Union. But they change at each Congress, and one can hardly hope to belong to the inner circle of all of them.

I would like to try to explain my approach to judging a particular selection of Fields medalists and to fairly efficiently guessing the winners in the past. This cannot be done without going a little bit into the history of Fields medals as it appears to a mathematician and not to a historian working with archives. I have no idea how to get to the relevant archives and even if they exist. I suspect that there is no written record of the deliberations of any Fields medal committee.

The first two Fields medals were awarded in 1936 to Lars Ahlfors and Jesse Douglas. It was the first award, and it wasn’t a big deal. It looks like that the man behind this choice was Constantin Carathéodory. I think that this was a very good choice. In my personal opinion, Lars Ahlfors is the best analyst of the previous century, and he did his most important work after the award, which is important in view of the terms of the Fields’ will. Actually, his best work was done after WWII. If not the war, it would be done earlier, but still after the award. J. Douglas solved the main problem about minimal surfaces (in the usual 3-dimensional space) at the time. He did with the bare hands things that we do now using powerful frameworks developed later. I believe that he became seriously ill soon afterward, but today I failed to find online any confirmation of this. Now I remember that I was just told about his illness. Apparently, he did not produce any significant results later. Would he continue to work on minimal surfaces, he could be forced to develop at least some of later tools.

The next two Fields medals were awarded in 1950 and since 1950 from 2 to 4 medals were awarded every 4 years. Initially the International Mathematical Union (abbreviated as IMU) was able to fund only 2 medals (despite the fact that the monetary part is negligible), but already for several decades it has enough funds for 4 medals (the direct monetary value remains to be negligible). I was told that awarding only 2 medals in 2002 turned out to be possible only after a long battle between the Committee (or rather its Chair, S.P. Novikov) and the officials of the IMU. So, I am not alone in thinking that sometimes there are no good enough candidates for 4 medals.

I apply to the current candidates the standard of golden years of both mathematics and the Fields medals. For mathematics, they are approximately 1940-1980, with some predecessors earlier and some spill-overs later. For medals, they are 1936-1986 with some spill-overs later. The whole history of the Fields medals can traced in the Proceedings of Congresses. They are interesting in many other respects too. For example, they contain a lot of very good expository papers (and many more of bad ones). It is worthwhile at least to browse them. Now they are freely available online: ICM Proceedings 1893-2010.

The presentation of work of 1954 medalists J.-P. Serre and K. Kodaira by H. Weyl is a pleasure to read. H. Weyl unequivocally tells that their mathematics is new and went into a new territory and is based on methods unknown to most of mathematicians at the time (in fact, this is still true). He even included an introduction to these methods in the published version.

The 1990 award at the Kyoto Congress was a turning point. Ludwig D. Faddeev was the Chairman of the Fields Medal Committee and the President of the IMU for the preceding 4 years. 3 out of 4 medals went to scientists significant part of whose works was directly related to his or his students’ works. The influence went in both directions: for one winner the influence went mostly from L.D. Faddeev and his pupils, for two other winners their work turned out to be very suitable for a synthesis with some ideas of L.D. Faddeev and his pupils. All these works are related to the theoretical physics. Actually, after reading the recollections of L.D. Faddeev and prefaces to his books, it is completely clear that he is a theoretical physicists at heart, despite he has some interesting mathematical results and he is formally (judging by the positions he held, for example) considered to be a mathematician.

The 1990 was the only year when one of the medals went to a physicist. Naturally, he never proved a theorem. But his papers from 1980-1994 contain a lot of mathematical content, mostly conjectures motivated by quantum field theory reasoning. There is no doubt that his ideas are highly original from the point of view of a mathematician (and much less so from the point of view of someone using Feynman’s integrals daily), that they provided mathematicians with a lot of problems to think about, and indeed resulted in quite interesting developments in mathematics. But many mathematicians, including myself, believe that the Fields medals should be awarded to outstanding mathematicians, and a mathematician should prove his or her claims. I don’t know any award in mathematics which could be awarded for conjectures only.

In 1994 one of the medals went to the son of the President of the IMU at the time. Many people think that this is far beyond any ethical norms. The President could resign from his position the moment the name of his son surfaced. Moreover, he should decline the offer of this position in 1990. It is impossible to believe that that guy did not suspect that his son will be a viable candidate in 2-3 years (if his son indeed deserved the medal). The President of IMU is the person who is able, if he or she wants, to essentially determine the winners, because the selection of the members of the Fields medal Committee is essentially in his or her hands (unless there is a insurrection in the community – but this never happened).

As a result, the system was completely destroyed in just two cycles without any changes in bylaws or procedures (since the procedures are kept in a secret, I cannot be sure about the latter). Still, some really good mathematicians got a medal. Moreover, in 2002 it looked like the system recovered. Unfortunately already in 2006 things were the same as in the 1990ies. One of the awards was outrageous on ethical grounds (completely different from 1994); the long negotiations with Grisha Perelman remind plays by Eugène Ionesco.

In the current situation I would be able to predict the winners if I would knew the composition of the committee. Since this is impossible, I will pretend that the committee is as impartial as it was in 1950-1986. This is almost (but not completely) equivalent to telling my preferences.

I would be especially happy if an impartial committee will award only 2 medals and Manjul Bhargava and Jacob Lurie will be the winners. I hope that their advisors are not on the committee. Their works look very attractive to me. I suspect that Jacob Lurie is the only mathematician working now and comparable with the giants of the golden age. But I do not have enough time to study his papers, or, rather, his books. They are just too long for everybody except people working in the same field. Usually they are hundreds pages long; his only published book (which covers only preliminaries) is almost 1000 pages long. Papers by Manjul Bhargava seem to be more accessible (definitely, they are much shorter). But I am not an expert in his field and I would need to study a lot before jumping into his papers. I do not have enough motivation for this now. An impartial committee would be reinforce my high opinion about their work and provide an additional stimulus to study them deeper. The problem is that I have no reason to expect the committee to be impartial.

Arthur Avila is very strong, or so tell me my expert friends. His field is too narrow for my taste. The main problem is that his case is bound to be political. It depends on the balance of power between, approximately, Cambridge, MA – Berkley and Rio de Janeiro – Paris. Here I had intentionally distorted the geolocation data.

The high ratings in that poll of Manjul Bhargava and Artur Avila are the examples of the “name recognition” I mentioned. I think that an article about Manjul Bhargava appeared even in the New York Times. Being a strong mathematician from a so-called developing country (it seems that the term “non-declining” would be better for English-speaking countries), Artur Avila is known much better than American or British mathematicians of the same level.

Most of mathematicians included in the poll wouldn’t be ever considered by anybody as candidates during the golden age. There would be several dozens of the same level in the same broadly defined area of mathematical. Sections of the Congress can serve as the first approximation to a good notion of an area of mathematics. And a Fields medalist was supposed to be really outstanding. Restricting myself by the poll list I prefer one of the following variants: either Bhargava, or Lurie, or both or no medals for the lack of suitable candidates.



Next post: Did J. Lurie solved any big problem?

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 conceptual mathematics vs. the classical (combinatorial) one.

Previous post: Simons's video protection, youtube.com, etc.

This post is an attempt to answer some questions of ACM in a form not requiring knowledge of Grothendieck ideas or anything simlilar.

But it is self-contained and touches upon important and hardly wide known issues.

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It is not easy to explain how conceptual theorems and proofs, especially the ones of the level close to the one of Grothendieck work, could be at the same time more easy and more difficult at the same time. In fact, they are easy in one sense and difficult in another. The conceptual mathematics depends on – what one expect here? – on new concepts, or, what is the same, on the new definitions in order to solve new problems. The hard part is to discover appropriate definitions. After this proofs are very natural and straightforward up to being completely trivial in many situations. They are easy. Classically, the convoluted proofs with artificial tricks were valued most of all. Classically, it is desirable to have a most elementary proof possible, no matter how complicated it is.

A lot of efforts were devoted to attempts to prove the theorem about the distribution of primes elementary. In this case the requirement was not to use the theory of complex functions. Finally, such proof was found, and it turned out to be useless. Neither the first elementary proof, nor subsequent ones had clarified anything, and none helped to prove a much more precise form of this theorem, known as Riemann hypothesis (this is still an open problem which many consider as the most important problem in mathematics).

Let me try to do this using a simple example, which, perhaps, I had already mentioned (I am sure that I spoke about it quite recently, but it may be not online). This example is not a “model” or a toy, it is real.

Probably, you know about the so-called Fundamental Theorem of Calculus, usually wrongly attributed to Newton and Leibniz (it was known earlier, and, for example, was presented in the lectures and a textbook of Newton's teacher, John Barrow). It relates the derivatives with integrals. Nothing useful can be done without it. Now, one can integrate not only functions of real numbers, but also functions of two variables (having two real numbers as the input), three, and so on. One can also differentiate functions of several variables (basically, by considering them only along straight lines and using the usual derivatives). A function of, say, 5 variables has 5 derivatives, called its partial derivatives.

Now, the natural question to ask is if there is an analogue of the Fundamental Theorem of Calculus for functions of several variables. In 19^th century such analogues were needed for applications. Then 3 theorems of this sort were proved, namely, the theorems of Gauss-Ostrogradsky (they discovered it independently of each other, and I am not sure if there was a third such mathematician or not), Green, and Stokes (some people, as far as I remember, attribute it to J.C. Maxwell, but it is called the Stokes theorem anyhow). The Gauss-Ostrogradsky theorem deals with integration over 3-dimensional domains in space, the Green theorem with 2 dimensional planar domains, and the Stokes theorem deals with integration over curved surfaces in the usual 3-dimensional space. I hope that I did not mix them up; the reason why this could happen is at the heart of the matter. Of course, I can check this in moment; but then an important point would be less transparent.

Here are 3 theorems, clearly dealing with similar phenomena, but looking very differently and having different and not quite obvious proofs. But there are useful functions of more than 3 variables. What about them? There is a gap in my knowledge of the history of mathematics: I don’t know any named theorem dealing with more variables, except the final one. Apparently, nobody wrote even a moderately detailed history of the intermediate period between the 3 theorems above and the final version.

The final version is called the Stokes theorem again, despite Stokes has nothing do with it (except that he proved that special case). It applies to functions of any number of variables and even to functions defined on so-called smooth manifolds, the higher-dimensional generalization of surfaces. On manifolds, variables can be introduced only locally, near any point; and manifolds themselves are not assumed to be contained in some nice ambient space like the Euclidean space. So, the final version is much more general. And the final version has exactly the same form in all dimension, but the above mentioned 3 theorems are its immediate corollaries. This is why it is so easy to forget which names are associated to which particular case.

And – surprise! – the proof of general Stokes theorem is trivial. There is a nice short (but very dense) book “Calculus on manifolds” by M. Spivak devoted to this theorem.  I recommend reading its preface to anybody interested in one way or another in mathematics. For mathematicians to know its content is a must. In the preface M. Spivak explains what happened. All the proofs are now trivial because all the difficulties were transferred into definitions. In fact, this Stokes theorem deals with integration not of functions, but of the so-called differential form, sometimes called also exterior forms. And this is a difficult notion. It requires very deep insights to discover it, and it still difficult to learn it. In the simplest situation, where nothing depends on any variables, it was discovered by H. Grassmann in the middle of 19^th century. The discoveries of this German school teacher are so important that the American Mathematical Society published an English translation of one of his books few years ago. It is still quite a mystery how he arrived at his definitions. With the benefits of hindsight, one may say that he was working on geometric problems, but was guided by the abstract algebra (which did not exist till 1930). Later on his ideas were generalized in order to allow everything to depend on some variables (probably, E. Cartan was the main contributor here). In 1930ies the general Stokes theorem was well known to experts. Nowadays, it is possible to teach it to bright undergraduates in any decent US university, but there are not enough of such bright undergraduates. It should be in some of the required course for graduate students, but one can get a Ph.D. without being ever exposed to it.

To sum up, the modern Stokes theorem requires learning a new and not very well motivated (apparently, even the Grassmann did not really understood why he introduced his exterior forms) notion of differential forms and their basic properties. Then you have a theorem from which all 19^th century results follow immediately, and which is infinitely more general than all of them together. At the same time it has the same form for any number of variables and has a trivial proof (and the proofs of the needed theorems about differential forms are also trivial). There are no tricks in the proofs; they are very natural and straightforward. All difficulties were moved into definitions.

Now, what is hard and what is difficult? New definitions of such importance are infinitely rarer than new theorems. Most mathematicians of even the highest caliber did not discover any such definition. Only a minority of Abel prize winner discovered anything comparable, and it is still too early to judge if their definitions are really important. So, discovering new concepts is hard and rare. Then there is a common prejudice against anything new (I am amazed that it took more than 15 years to convince public to buy HD TV sets, despite they are better in the most obvious sense), and there is a real difficulties in learning these new notions. For example, there is a notion of a derived category (it comes from the Grothendieck school), which most of mathematicians consider as difficult and hardly relevant. All proofs in this theory are utterly trivial.

Final note: the new conceptual proofs are often longer than the classical proofs even of the same results. This is because in the classical mathematics various tricks leading to shortcut through an argument are highly valued, and anything artificial is not valued at all in the conceptual mathematics.

Next post: The Hungarian Combinatorics from an Advanced Standpoint.

D. Zeilberger's Opinions 1 and 62

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

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

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

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

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

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

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

Opinion 62. First, the factual mistakes.

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

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

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

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

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

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

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

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

Next post: What is mathematics?

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.