Freedom of speech in mathematics

Previous post: Who writes about big questions?

Behind the popular site Mathoverflow there is a less known site meta.mathoverflow.net, having a definitely postmodernist spirit: this is a place where people discuss not the mathematical questions, but what mathematical questions are allowed to be discussed on the front site (other issues about the front site too, of course).

Oops! I said "discussions"! No, discussions are not allowed on Mathoverflow at all. They pretend that the software is not suitable for discussions; in fact it is as suitable as any blog. So, at Metamathoverflow some people (I have no idea who qualifies for participation in Metamathoverflow) discuss what questions may be asked and answered at Mathoverflow. For example, it is not allowed to ask if some (at least some recent) paper is believed to be correct by the experts in the field.

Here is the link to a quite remarkable discussion "Is this question acceptable?: Mochizuki proof of ABC". The AMS reported that Shinichi Mochizuki claimed that he has proved the famous ABC Conjecture; as a place to find some additional information, they referred to the question "What is the underlying vision that Mochizuki pursued when trying to prove the abc conjecture". The part in italics can be deduced from the URL; I just rounded it off in the shortest possible way.

When you follow the AMS link, you will get to a slightly different question "Philosophy behind Mochizuki’s work on the ABC conjecture [closed]". "[Closed]" means that it is impossible to post any answer. The body of the question is:

“Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?”

This question was classified as "subjective and argumentative" and closed by this reason. After reading the postmodernist metadiscussion I realized that the original question was somewhat different, and, moreover, had a different author. Still, it is closed.

Some answers were posted there before the question was closed; they are interesting and informative. Why these people do not allow more answers?

Well, one of the answers sheds some light on how the modern mathematical society functions. Despite Shinichi Mochizuki is highly regarded for his earlier achievements, and despite it was known for quite a while that he is working on the ABC conjecture (unlike A. Wiles or G. Perelman, he wasn't hiding this) almost nobody was reading his papers. So, almost all experts cannot say anything about his solution because they cannot start reading with his last paper.

Looks like nowadays mathematicians are not interested in mathematics for its own sake, they care only about publications and grants. And the specific questions which one may encounter trying to finish the proof of the last lemma in a paper are the most welcome at Mathoverflow.


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Who writes about big questions?

Previous post: Slavic soul? The interest in big questions about mathematics.


Reply to a comment to the pervious post.

Well, your question makes it very tempting to write a long essay about the current state of mathematics. But this is not really needed.

I mentioned the most famous people; I hope that all names are immediately recognizable. Our contemporaries cannot be so famous just because they are our contemporaries. Let me try, but first I would like to say a few words about Thurston.

Yes, his essay is not quite about this topic. But there are hints, and his reply at Mathoverflow, which is reproduced in this blog is worth of 100s pages of other writers. A. Weil did not write a book on philosophy of mathematics, he just mentioned such issues here and there, and we know a lot about his views from his Bourbaki collaborators. The article I quoted in my first posts was published in French in an obscure (at least for non-French mathematicians) place. The “Monthly” translation expanded the audience, but the translation appears to be a not very good one. My point is that he wasn’t concerned much about dissemination these ideas. It looks like Thurston was more concerned about his ideas.

The most obvious example is, of course, T. Gowers. He wrote two essays, the one about “two cultures”, the other about replacing mathematicians by computers (this is, in fact, a section in his GAFA Visions paper), and he writes about such things in his blog. I don’t like his ideas, but if somebody outlines a project of replacing mathematicians by computers and offers a justification for such a project, he is definitely writing about the larger place of mathematics in the world in the most dramatic way: do we need mathematics or not? His answer is “no, we don’t”. His writings are definitely related to his own work: all examples are taken from his corner of mathematics.

Even the n-Category Café itself is an example, and one of the persons running it, David Corfield, wrote a book about philosophy of mathematics (I only browsed through it but plan to read it; it seems to be quite interesting). There is Colin McLarty, who writes about the implications of Grothendieck’s way of thinking. Of course, there is an autobiographical text (or, rather, several texts) of Grothendieck himself, which nobody dares to publish for 30 years already. On a much less abstract level, there is Neal Koblitz, who wrote about the role of mathematics in the society and criticized (largely from a political perspective) the way the “help” is given to the developing countries (and wrote an autobiographical book).

On the other hand, one can easily speak about mathematicians with Soviet-Russian upbringing, but how many texts written by them and worth reading can you suggest? Manin is excluded. Borovik’s book did not impress me enough to read it. Is my impression wrong? Honestly, I don’t know, but it is the business of the author to attract readers.

Finally, and perhaps most importantly, nowadays mathematics has a quite respectable place in the society. This allows us to earn a living by doing mostly the things we like to do no matter what. It would be too dangerous to try to insert a controversy about the larger role of mathematics in the society. Almost nobody dares to say anything nontrivial about this. Paul Halmos once said that NSF grants or any other government financing is not needed, because it does not matter when a theorem is proved, tomorrow or in 300 years. I don’t know any other comparable statement.

Personally, I believe that the government financing had already damaged mathematics too much and should be eliminated or, at least, radically reformed. Looking at the whole society, I believe that the teaching of mathematics in high schools and to almost all college students (in the US) is a serious damage to the society. But, this gives us our jobs! And these are just the most obvious issues.




Next post: To appear

The interest in big questions about mathematics in different communities

Previous post: William Thurston about the humanity and mathematics


Reply to the (first) comment to the pervious post.

Even if there is such a phenomenon as a tendency of some ethnic groups to speculate about the larger place of mathematics, it is unlikely to be either Russian or Slavic phenomenon. The word "Russian" usually used in the USA to designate anybody or anything coming from Russia or the (former) USSR. Most of "Russians" in the USA and other Western countries are, in fact, of Jewish extraction (usually not practicing any religion, including Judaism), and therefore are neither "Russians" in the USSR sense (this one is purely ethnical), nor Slavic. May be “European” would be more correct, but this would eliminate the very appealing reference to “the mysterious Russian (or Slavic) soul”.

Some of the most important writings about mathematics and its role for the humanity due to H. Poincaré (French), F. Klein (German), N. Bourbaki (French, or French-Jewish if we turn to the ethnicity), A. Weil (French, ethnically Jewish, and deeply influenced by Bhagavad Gita and related philosophy), just to give the most prominent examples. I quoted A. Weil a lot in the first posts in this blog. I believe that these examples alone are sufficient to dispel any myths about “the mysterious Russian/Slavic soul” at least in this question.

It seems to me that the opposition of the American and the European cultures is more relevant. Americans are much more focused on “practical” things of immediate importance. I mean very immediate: say, having a grant is more important than proving good theorems. This is not specific for mathematics and shows up everywhere, from arts to Hollywood to highways repairs. Naturally, “Russian” mathematicians transplanted to the US soil stand out. So would be French mathematicians, but there is virtually none of them in the US.

The late William Thurston was an example of an American mathematician paying attention to the larger issues. But he was too exceptional (even his education was rather unusual; one can read in Wiki about the undergraduate school he attended) to serve as an example.

Much more typical is a comment I once come across in T. Tao’s blog. This was an advice to young mathematicians: do not try to understand big general theories; use them as black boxes to solve specific narrow problems (and then soon you will have publications, grants, etc. – Owl). This was a big shock for me despite I knew personally people working in this manner. This approach, in particular, makes American mathematical literature less reliable than, say, the French one. The Soviet/Russian mathematical literature is also not very reliable sometimes, but by different reasons: some people write for their close friends only (but expect and usually get a universal recognition).

Perhaps, it is worthwhile to find this comment and write more extensively about it.

Another manifestation of the American attitude is the fact that general (especially partially philosophical) questions are regularly closed at Mathoverflow.

I agree that the n-categories are one of the most interesting things happening in mathematics now, perhaps the most interesting. But with the current pace of the development, they are still decades away from recognition by the whole mathematical community (if it will survive).


P.S. The title and the tags are slightly modified on March 13, 2013 in order to avoid at least some spam.


Next post: Who writes about big questions?

William Thurston about the humanity and mathematics

Previous post: William P. Thurston, 1946-2012

To my regret, I found the included below short essay by Thurston at Mathoverflow only today. It is quite remarkable, presenting some not quite conventional ideas in a brilliant and succinct form. I do not think that many mathematicians will (be able to) argue that these ideas are wrong; they usually do not think about such matters at all. A young mathematician is usually so concentrated on technical work, proving theorems, publishing them, etc. that there is no time to think about if all this is valuable for the humanity, and if it is valuable, then why. Later on she or he is either still preoccupied with writing papers, despite the creative power had significantly diminished (there is at least a couple of ways to deal with this problem), or turns to a semi-administrative or purely administrative career and seeks the political influence and power.

I started to think about these questions long ago (my interest in semi-philosophical questions goes back to high school and owes a lot to a remarkable high school teacher) and eventually have come to essentially the same conclusions as Thurston. But I was unable to put them in writing with such clarity as Thurston did.

Bill Thurston's answer to the question "What can one contribute to mathematics?" October 30, 2010 at 2:55.

It's not mathematics that you need to contribute to. It's deeper than that: how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics? Such a question is not possible to answer in a purely intellectual way, because the effects of our actions go far beyond our understanding. We are deeply social and deeply instinctual animals, so much that our well-being depends on many things we do that are hard to explain in an intellectual way. That is why you do well to follow your heart and your passion. Bare reason is likely to lead you astray. None of us are smart and wise enough to figure it out intellectually.

The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.

The world does not suffer from an oversupply of clarity and understanding (to put it mildly). How and whether specific mathematics might lead to improving the world (whatever that means) is usually impossible to tease out, but mathematics collectively is extremely important.

I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind. There are many ideas in mathematics that may be hard to get, but are easy once you get them. Because of this, mathematical understanding does not expand in a monotone direction. Our understanding frequently deteriorates as well. There are several obvious mechanisms of decay. The experts in a subject retire and die, or simply move on to other subjects and forget. Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual -> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replaces the conceptual forms of understanding. And mathematical conventions and taken-for-granted knowledge change, so older texts may become hard to understand.

In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining --- they depend very heavily on the community of mathematicians.

(Note that this short essay is quite relevant to the discussion of T. Gowers in this blog.)

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William P. Thurston, 1946-2012

Previous post: The twist ending. 4


William P. Thurston passed away yesterday (August 21) at 8:00 p.m. at a hospital in Rochester, NY, surrounded by the members of his family.

From the announcement of the Cornell Department of Mathematics (there is no permalink for this):

"All those who knew Bill, especially his many students and collaborators, know that nothing can replace his insight and personality. We are all terribly saddened by this loss."

American Mathematical Society posted a short obituary: William P. Thurston, 1946-2012.


William Thurston was the greatest geometer of the last century. The word "geometry" is very fashionable since about fifty years ago, and this phrase now calls for a clarification. William P. Thurston was able to see unexpected, remarkable, beautiful pictures hidden from all other mathematicians. After he showed them to other mathematicians, they saw and appreciated them also. His thinking was predominantly visual. Perhaps, in this respect he was the greatest geometer of all times.

It is extremely difficult to convey any visual concept by the means of a conventional mathematical text, even with a lot of illustrations. Some visual concepts are too complicated or too many dimensional (here the usual 3 dimensions are often already too many) to be adequately explained by a 2-dimensional picture. Apparently, mathematics lacks a proper language to efficiently describe visions of Thurston's level of complexity and originality. Perhaps, this is the main reason why Thurston did published only sketches or partial expositions of his results (his own published explanation is different, but compatible with this one). Some of his ideas were successfully translated into the conventional language and written down by other mathematicians. But some others are not, and the results themselves are reproved using different means. Of course, Thurston's visions are more important than his theorems, and I am afraid that some were lost completely already in the last century. I hope that his students and collaborators will write down and publish everything they learned from Thurston.


William P. Thurston was looking for alternative ways to convey visual concepts; his prize-winning book ''Three-Dimensional Geometry and Topology'' is an attempt to deal with this problem, but covers mostly pre-Thurston ideas.

Creating a proper language to describe visual mathematical concepts is, may be, the most important problem for the future generations. The layman language has the same deficiency; every description of a tree or a landscape relies on the previous experience of the readers with trees and landscapes in the real world. This recourse is not available to mathematicians creating new visual concepts, and by this reason the problem is so difficult that it is extremely rarely even acknowledged as a problem.


The passing away of William Thurston created a hole in the mathematical community which cannot be filled. Everybody who was happy enough to talk with him at least ten minutes knows how remarkable person he was. We lost not only an exceptional mathematician; we lost also an exceptional person. This deepens the feelings of loss, emptiness, sadness, and sorrow.


Next post: William Thurston about the humanity and mathematics.

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. 3. R. Kirby

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


R. Kirby (UC at Berkeley) is the Managing Editor of Gowers's journals. This justifies the following digression into Kirby's past achievements in scientific publishing.

In the 90ies he declared a war on the main journal in his field, namely “Topology”. Originally published by “Pergamon Press”, it was sold in early 90ies to Elsevier by late Robert Maxwell when his financial empire started to face serious problems. Of course, this wasn’t a good development, but it remained an excellent journal due, most likely, to its excellent and small editorial board. It was moderately expensive. I still fail to see any reason to single it out (as I don’t see any convincing justification for singling out the Elsevier in the recent boycott).

R. Kirby launched a new journal “Geometry&Topology” specifically intended to compete with “Topology” (and to eventually bring it down). It was published both online and in paper version. Online version was free; the paper version was very cheap initially. In contrast with “Topology”, the editorial board of “Geometry&Topology” was big and growing with time. The journal was also growing, and with the number of pages the price of the paper version was growing (the libraries were encouraged to subscribe to it; technically, for libraries the electronic access never was free). “Geometry&Topology” succeeded in diverting a lot of papers from “Topology”, and the editorial board of “Topology” was constantly pressured to attempt to lower the price (even when the individual subscription price to the paper version of “Topology&Geometry” surpassed that of “Topology”). Elsevier argued that the list price of a subscription is not relevant anymore (by the reasons I explained above using the example of “Inventiones”). The purpose of a relatively high list price, I believe, was to encourage participation in “package deals”. Eventually, the editorial board and Elsevier made a quite reasonable deal substantially lowering the price, but it was too late (and the list price already was not relevant).

On August 10, 2006 the whole editorial board of “Topology” resigned. Elsevier continued to publish the already accepted papers and managed to fill by them the 2007 volume. The subscription to 2007 volume was free for subscribers to the 2006 one. But the journal was, of course, dead.

Within a month (if I remember correctly, already in August 2006) “Geometry&Topology” closed free access to its electronic version or at least announced the imminent closing. Since then the access to the electronic version is by subscription only. Well, this is how much one can trust promises to be freely accessible in perpetuity. At the same time, “Geometry&Topology” doubled the subscription price, and invented some convoluted reason for quadrupling the subscription price for year 2007 (for any form of subscription, electronic or paper, individual or library). Being a member of our Library Committee, I attempted to understand their reasoning, but failed.

Nobody saved any money as a result of success of Kirby’s project. A slightly modified editorial board of “Topology” launched a replacement, “Journal of Topology” (apparently, Elsevier own the rights to the trademark “Topology”). “Geometry&Topology” is not a part of any package deal. I don’t know if the Oxford UP, the publisher of “Journal of Topology”, offers package deals, but our library had to subscribe to it as a standalone journal. So, the cost of subscription to specialized journals in the field of topology for our library substantially increased. If anybody was subscribing to “Geometry&Topology” or “Topology” as an individual, she or he, most likely, lost these subscriptions because of much higher prices.

During this struggle with “Topology” mathematicians gradually started to consider “Geometry&Topology” as the journal of choice for paper in topology and related fields.

So, the main result of Kirby’s ten-year effort is the fact that he now controls the main journal in his field (topology, of course). It seems that he had no chances to get into the editorial board of “Topology”. The co-author of his most famous papers, L. Siebenmann, was a member of the editorial board for decades, first as a regular member, then as a honorary one.


Next post: The twist ending. 4.

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.

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

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


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

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

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

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

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

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

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

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


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

A reply to some remarks by André Joyal

Previous post: The first post.


I am a fairly surprised by your criticism of Bourbaki work (I would be not surprised by a critique of Bourbaki from many other mathematicians). I hardly can say anything new in the defense of Bourbaki. The best defense was, probably, provided by J. Dieudonne. And one should not forget that, as some other former member of Bourbaki pointed out, that Bourbaki books succeeded in changing the style in which books in mathematics are written. There is no more such an urgent need in rigorous and systematic expositions as was in France before WWII, because many books by other authors are both rigorous and systematic.

Of course, applications are absent from Bourbaki books, as they are absent from almost all books in pure mathematics. I doubt that I ever read or used any book in pure mathematics discussing applications. And since I think that pure mathematics is not needed for any applications, I believe that it is better to separate them.

The issue of the presumed absence of motivation in Bourbaki books is more subtle. There are several sorts of motivations. A reference to applications is a common but very poor motivation to study proofs, because only statements and formulas are needed for applications. It seems that the most widespread sort of motivation is a pseudo-historical one: one invents some way how mathematicians could in principle arrive at a result or definition; in the best cases it resembles the actual history. Very often such a motivation uses tools developed only later and, moreover, under the influence of the original discovery. I consider such motivations as very misleading. The only true motivation in mathematics is the real history of the problem or of the theory at hand. I learned this point of view from my Ph.D. thesis adviser in the context of writing introductions to research papers, and later I saw that it applies to expository writings, textbooks, everywhere. Using it in the teaching or learning is possible, but by purely practical reasons only occasionally. This approach requires such amount of time that it will defer own research by decades. Another effective form of motivation is apparently already forgotten in the western countries. It is simply the trust to a more experienced person than you. If you are told that you will need to know the derived categories or it just worth to know about them, this should be a sufficient motivation and you will study the theory of derived categories. Finally, the organization of the material could be self-motivating. I think that Bourbaki books are superb in this respect. They are, in fact, much more readable than most of mathematicians think. Many other books are also self-motivating without any special efforts.


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

The first post

This is mainly a test of the feel and look of this particular Google template and my customizations of it. A part of the test is the following picture of a little piece of my life.

 

This close-up photograph was done by me in my office by an outdated cell phone.

So, I am the © copyright holder of this silly picture!

Most of this stuff should either go trash, or be destroyed because it may contain some private information (not about me), or returned to library (now I am very reluctant to return books to the library; they have very high chances to end up in a trash truck there), etc. And some things are valuable. The main problem is to sort out what belong to what category.

Probably, this is sufficient for a test. Good luck!

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