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.

Next post: Reply to a comment

ReplyDeleteWhen 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 question was reopened, thank G-d. Moreover, there is now a reply (by Vesselin Dimitrov) with a counterexample to one of Mochizuki's claims.

It seems not that bad after all...

wwwwww

Hi, wwwwww!

DeleteYes, I noticed that it was reopened.

It may be the case that my own intervention into this case played some role. The issue was controversial from the very beginning, both sides had fairly valid inside of the Mathoverflow arguments (in the bigger world, I believe, the arguments for closing are not valid at all). A small action, in principle, could change the equilibrium. I posted there a "question" (my intention was to use the title "This is not a question", but then I changed my mind) about the closing of this particular question, and about the policy for closings in general. I used a different nickname and an OpenID which tells nothing about me, except for few people.

The reaction was furious. The question was immediately closed by some first year graduate student (but he has a lot of reputation points on Mathoverflow; for me it would be a full time job to achieve a comparable level), then "removed" (technically, this means that the question is given the tag "removed", and then it does not shows up in any known to me way except knowing the direct link, but still exists in the system). I was amazed that even in this invisible state the post continued to slowly earn reputation points. Then it was deleted from the system completely.

And only after this the question about ABC was reopened.

Strangely, I do know personally 3 mathematicians involved in the story of the ABC question (voting to close, to reopen, etc.). One of them I know only online, but fairly well. I hope I am right about the latter. He voted for the most recent reopening.

I find your discussion on mathematical cultures somewhat interesting. Thurston and Shinichi Mochizuki cited here were trained in America and while Thurston is clearly a giant of 20th century mathematics, a lot of people have been interested in Mochizuki paper on ABC. This at least shows that the image of American Mathematics being oriented towards grants only while French Mathematics is about big theories not accurate. Also I do think that there are fewer people like Grothendieck who went to Hanoi during the Vietnam war to protest the war. I do not recall top Mathematician of any nationality these days taking a public stance on the war in Iraq for instance.

ReplyDeleteDear Jean Marcel,

DeleteThank you very much for your comment and for apparently going through the trouble of creating a Google account to be able to post it.

What I really like to do is to initiate, or provoke a discussion. It seems that my second attempt to do this at Gowers's blog was much more successful than the first. Now it looks like I have a chance to succeed here.

You pose really interesting questions.

They are important even for current graduate students. Should they be able to read at least mathematical papers in French? Usually they are not able; I cannot suggest to a student any topic for studying or research which require reading even few pages in French. Neither can suggest to most of my colleagues something in French. So, this is not simply another exercise in “France vs. US” game.

I think that there is some misunderstanding here. People sometimes reproach me for writing too heavily. I would love to write shorter, but then I am very often completely misunderstood.

Thurston and grants are disjoint issues. Grants were of no concern for Thurston for decades. At the same time he always had an NSF grant, as far as I know.

More important is the fact that the overarching importance of grants is a new phenomenon. Before 2000 it was widely understood that no more than 1/3 of mathematicians deserving government support got it. This was said by an official at the NSF, the only government agency in the US supporting pure mathematics. Nowadays mathematicians without NSF grants already have problems with promotions, a routine procedure in the past. For young people looking for job (or older ones interested in moving to a new place), grants and the

numberof publications is critically important, at least if he or she did not prove something like ABC-conjecture.Thurston got a highly unusual education, a one may see just from the Wiki article about him (few years ago). In the formal sense was trained in America, but his education is hardly an American education, especially the undergraduate education. I don’t know anything about Shinichi Mochizuki. But he went back to Japan. I am sure that he has much better working conditions in Japan than almost all US mathematicians (only permanent professors of IAS have comparable conditions; but this happens mostly after they are past their prime).

Of course, French mathematics is not only about building big theories. I don’t think that I need to prove my competence by listing other types of work of French mathematicians.

By mentioning Vietnam War and Iraq war you are entering the realm of politics. I am not inclined to do this in this blog. Let me say just that, in my opinion, it is better when

topmathematicians take apublicstance on political issues. The mysterious gift of ability to discover and prove new interesting theorems does not imply any other. There is no reason to believe that mathematicians have better insight into political or even moral issues. And usually they indeed do have the same level of understanding as an average citizen. But any fame results in a higher level of trust in general public.Sorry, in the last pargraph it shoul be:

Delete..., it is better when

topmathematiciansdo nottake apublicstance on political issues.Dear Owl,

ReplyDeleteI am not knowledgeable about Thruston ability to get grants, but I heard that S.T. Yau described some self-similar solutions of Ricci flow as solitons just because he was applying for a grant. If a recipient of the Fields Medal has to do this, just to get a grant, it shows that there are some issues in getting NSF grants. Now going back to Japan, France and the US. I could be wrong but the teaching load at a research university in the US, can be significantly higher than in France. For instance a top french mathematician once told me he can have all his classes taught in one day of the week. In America, the teaching load can be 9 hours a week and even at top school as much as 6 hours, a week. So yes, there is a lot of pressure to get grants in the US to obtain a reduced teaching load.

As for the difference between the french school and the american school, I think the main difference is that the american system gives room to second chances. In France there is an age limit to get into the best schools such as Ecole Normale or Polytechnique. So in essence, young people are pressure to pass competitive exams, leaving not much room for originality. I am not sure someone as Thurston could have succeeded in that system. I am not sure either Yitang Zhang whose papers on prime surprised the number theory community could have had a chance in France.

I also have a question. My background is in analysis and thus far away from algebraic geometry and number theory. When I look at the past 15 years, I see lots of french trained mathematician being recognized in algebraic geometry,Voisin,Lafforgues,Bau Chau but fewer in number theory, where Mochizuki,Yitang Zhang,Cildrim-Goldson were trained in North America. Would you conclude that the North American mathematical school is more pragmatic?

Thurston and Yau are not really good examples when discussing grants. The real issue is who gets the grants among young mathematicians, and the influence of grants on job offers to them, promotions to the tenured positions, etc. Unfortunately, the situation dramatically changed in the last 20 years or so, and now grants is a must, and very the NSF grant will not impress the university administrators anymore: they are too small. Now the university administrators see scientists simply as a source of income: more than 1/3 of any grant goes not to the researcher, but to the administration.

DeleteAlmost no grants reduce the teaching load in the US universities – in mathematics. On the other hand, it is still not very high. The standard teaching load in a so-called "research" university in the US is 3 semester courses per academic year, i.e. 3 hours of teaching in one semester, and 6 hours of teaching in the others. Of course, it takes much more times than the time in the class.

DeleteA big advantage of the French system is the CNRS positions which involve no teaching at all.

People like Thurston usually can pass any exam. In fact, this is true on much lover levels too. And the US exams are NOT competitive.

I do not buy the argument about the "second chances" anymore. These second chances lead to people spending decades, not years in graduate schools – I know more than one such student personally.

Academic career is only an option, and not a very attractive anymore. For most people who failed to succeed on this way on the first attempt choosing some other career would be much more sensible.

Shinichi Mochizuki works both in algebraic geometry and algebraic number theory. In fact, they are just two sides of the same thing, this idea is one of the main driving forces of both fields since Andre Weil.

DeleteYitang Zhang had a lot of difficulties within the US system. And I don't see anything pragmatic about his work. The gaps between the primes problem is not useful, even within mathematics itself. The same about Yildirim-Goldson.

In fact, there is no such thing as the American mathematical school. There are many mathematicians from all over the world, who were lured into the US by high salaries, or were looking for a refuge or plain political asylum.

Dear Owl,

ReplyDeleteYou are missing the point. Bill Gates when he was receiving his honorary degree from Harvard said that when he decided to drop out, he told his parents that if the business venture failed, he would come back to get his degree. The system gives second chances in America, and open more routes to originality. If Gates had been a student in Louis-Le-Grand in France, he would have had very limited second chances as the most prestigious schools have an age limit. Calderon, was an engineer first and came to math later in life and made important contributions. Grothendieck was reportedly having difficulties following the high powered seminars when he first came to Paris, and went to Nancy to do a PhD with Schwartz. I have some doubts he would have passed he competitive entrance exams at Ecole Normale. Starting slow at 18 is not an indication of future failure as a mathematician.

These examples cannot work. Bill Gates did not used any second chances.

Delete"Grothendieck was reportedly having difficulties following the high powered seminars when he first came to Paris." Well, this is what Wikipedia says, but it says about a young person who just arrived to Paris and the highest level seminar devoted the current research, Seminaire H. Cartan. And the French system did not failed Grothendieck, as everybody knows.

There are mathematicians whom the US failed. Of course, most of the failures remain unknown.

Grisha Perelman would fail in the US system for sure, and would, most like, succeed in the French system. In France he would get a CNRS position and would do the same as he did USSR-Russia. In the US he would be thrown out of the system very soon. No chances to get tenure. He understood well that the US system is not for him and declined offers from the Princeton U and the IAS.

In fact, the USSR is a very good example: very rigid in terms of age limits, time in the graduate school (3 years and you are out), etc., and very successful in mathematics.

Owl,

ReplyDeleteCould you indicate a school where the standard of 3 courses per year is typical. I am rather aware of typical 2 courses per semester being typical teaching load. A grant can reduce the teaching load, because the recipient of the grant can hire a post-doc to take some of teaching duties of the professor.

I would have to do some research to find a US university which qualifies as a "research" one, whatever it means, and which has standard teaching load 2 course per semester for faculty engaged in research. A standard research grant in (pure) mathematics from NSF has no provisions for hiring post-docs, and I am not aware of any NSF grants in mathematics allowing to hire anybody to *teach* instead of you. Such things exist in other sciences, and are a norm in biological sciences, but not in mathematics.

DeleteMay be I should explain what I mean by pragmatic. It is not necesserily applications in mathematics, but problems whose formulation is clear to the entire community, and motivate the creation of new mathematics. I will start by Yitang Zhang. The problem can be explained to a secondary school students, and yet to prove his theorem he used some tools from algebraic geometry. This is different from the Langlands program or the Hodge conjecture that people outside the area do not understand. Same for Mochizuki. From what I read on the web, when his advsior solve a problem related to Fermat last Theorem, another problem that could explained to a high school student, Grothendieck wrote him a letter about some of his thoughts. That problem was solved by Mochizuki. Here is another instance of what I meant by pragmatic mathematics. Clearly stated problem that can be understood by the entire community that lead to new mathematics. The same pragmatic thinking can be seen in his work on the abc conjecture. Here is a conjecture that people outside algebraic geometry can understand, which motivated the creation of new mathematics.

ReplyDeleteAs for Perelman, I have often wondered why he did not stay his the US after his postdoc. He has said that when he was invited at Princeton, nobody in the room understood the Soul Conjecture he had solved. In the US system when someone is believed to have high potential, he can be offered tenure, and even a reduced institution who believes in his potential. Did no-one in America understood his work on the Soul Conjecture at the time, you would have to ask the experts in differential geometry. Note that Perelman also rejected a prize from the European Mathematical Community for the same reason. In that regard the CNRS and the research institute in Russia provide room for such individual. I agree.

As for the refugee issue mathematics can not be separated from the whole society. If a society is more open to refugees, mathematics and other endeavours of life benefit. Grothendieck was a refugee in France, so the US is not the only country who has a history of accepting refugees, even as if the tendency seems changing these days.

To come back to Bill Gates, he knew that should the business venture fail, he would have a second chance at Harvard. So the fear of dropping out was not as big as it would have been in a country like France. In this sense the second chance opportunities provide by the system allows him to take more risks. The same can be said to the CNRS system.

Well, here is the definition of the word "pragmatic" in Merriam-Webster

Delete– relating to matters of fact or practical affairs often to the exclusion of intellectual or artistic matters : practical as opposed to idealistic (pragmatic men of power have had no time or inclination to deal with … social morality — K. B. Clark)

I believe that neither you had in mind neither the archaic one, nor the meaning "related to the philosophical pragmatism", but the above one.

I agree, most of the Americans hardly value anything intellectual or artistic. Since the universities are now in hands of career administrators, not of the faculty (apparently, the transitions was completed around 2007-2008), the support for anything of intellectual or artistic value is evaporating everywhere, including mathematics.

There is no need to tell me distorted web tales about various mathematical celebrities. The work of Gerd Faltings is related to the Fermat Last Theorem in the most superficial manner. The anabelian algebraic geometry of Grothendieck is not a problem, but a program, an outline of some ideas. Neither these ideas, nor Mochizuki contribution to this field can be understood by the "entire community". The ABC-conjecture is a French idea, and its significance cannot be understood by "entire community" either.

I knew Grisha Perelman personally. He did what was right thing for him to do. He wanted to work without distractions: no teaching, no paperwork and worries about getting a permanent position. But there are about 6 such positions in the US, and none was vacant. A little bit later, I believe, Voevodsky accepted an offer of a long term position at IAS (the same sort as was offered to Perelman), and in few years he was a permanent professor at IAS. But they have very very different personalities.

DeleteOf course, experts in differential geometry understood and valued his work. But the reception at particular talk could be quite different from Grisha's expectations. The talks are a sort of theater. There are rules. The purpose of talks in the US is not to communicate mathematics, but to register mutual respect: of the speaker by the audience and of the audience by the speaker.

This does not fit Grisha's standards of integrity.

Grothendieck survived despite the system not because of it. Virtually all the french fields medalist but Grothendieck are former students of Ecole Normale. The school was designed to train the elite and has rigid age limits. There is no room for trying new things at a young age. Either young people work night and day to enter Ecole Normale or thats it. Not good for originality. Bill Gates took advantage of the second chances in the system and took time off, form his last year in high school, before going to Harvard, to work for TRW.

ReplyDeleteUp to recent times, almost all French mathematicians were former students of ENS. Yes, ENS was designed to prepare the elite. It is only natural that most of the elite was indeed coming from this place. But the system was flexible enough to allow exceptions. And to say that the French system is "not good for originality" after so many original mathematicians and not only emerged from it is just... – I don't know what to say, I better put a period here.

DeleteI don't know where you got the idea that "young people work night and day to enter Ecole Normale". Surely, some do. As some worked night and day to pass the university entrance exams in the USSR, or as some kids in US work night and day to prepare for US equivalent of these exams. I don't know even a single case when these exams turned to be a real obstruction (I put aside the deliberate failing Jewish applicants to the best USSR universities). I don't know the content of ENS exams, but the entrance examinations to the USSR universities did not tested any knowledge or skills useful for a future mathematician. Except, perhaps, one: not to be afraid of a problem which appears to be difficult. By the way, typical students in the US freeze when they see a problem which they perceive as difficult.

Bill Gates did not took the advantage of second chances. Rather, the system forced him to waste some time at Harvard. And, frankly, I am not an admirer of Bill Gates. He was a great businessman, that's true to the extent this makes sense. But his product, Windows OS, is neither original nor good. It is a sense optimal: it is the worst OS which could be, in principle, widely accepted. That's quite an achievement, I admit.

The abc conjecture is indeed a french idea, but in my first remark, I talked about the last 15 years.

ReplyDeleteAs for Ecole Normale, they take if I am not mistaken, 30 students in mathematics a year. That is very small and they have an age limit. So the students have to study very hard for these exams. The intensity of the competition and the age limit, make it difficult for students to take time off, as they would not have second chances to get in, later in life.

My point was not to discuss whether Gates product was good or not but whether the system allowed him to do things he wanted, and if he failed provided him second chances in life.

I see. Your opinion about the French system is based on some aspects of its design, my is based on its success.

DeleteAnd your knowledge of the French system seems to be outdate. I failed to find immediately exactly how many student are admitted to ENS in mathematics (it seems that his number is not specified), but there are no age limits anymore (since 2005). It is not necessary to Ph.D. there for a successful career in mathematics. For example, Laurent Lafforgue and Ngô Bảo Châu did their Ph.D. work at the Université Paris-Sud.

If we ignore the quality of Bill Gates product, then I see no reason to talk about him. You mentioned only one thing afforded to him by the US system: the option to *tell* his parents that he will go back to Harvard if his business fails. Somebody who is not able to do what she/he wants over parents objection is not likely to succeed.

Going into such an obscure field as the pure mathematics, about which only few know that it exists (nowadays even deans of natural sciences don't) is not likely to inspire parents's enthusiasm. That's a common problem. And if parents's objections do prevent somebody from going into the pure mathematics, that's OK with me.

My knowledge of Ecole Normale was outdated. I was not aware of the end of age limits. That is a good development, it will give the freedom of students to take time off if they want, and do what they like and if it does not work out they could still come back. That is why I mentioned Gates not because of his promises to his parents, but because Harvard would take him back later on, if he wanted to come back. This is something that the old french system did not allow because of age limits.

ReplyDeleteLaurent Lafforgue is a former student of Ecole Normale and in the past students did not normally did their PhD there. It was a place to train young people, but they would do their PhD elsewhere. As for Bao Chau he came to France on a scholarship after competing in mathematics Olympiads. I was looking at the problem from the perspective of nurturing young domestic talents, who are not necesserily sure around age 18 of what they want to do in their life.

The system worked in mathematics, but I am looking at the education system as a whole.

Well, this blog is about the mathematics only.

DeleteAs of the education system as a whole, I saw the US education system from different sides. Before this, I would not believe to anybody who would attempt to explain me how bad it is. It was something beyond my imagination. Well, if immigrants from Nigeria sent their kids back to avoid the US schools, something must be wrong with it.

The "second chances" afforded by the US system are the second chances to waste years of your life. It would be better not take such "second chances".

If the US did not have second chances, the US may not have had companies like Microsoft,Facebook,Dell who employ hundred of thousand of people. Where do you think the monies to pay CNRS researchers come from?

ReplyDeleteI do not know how common is this phenomenum of immigrant sending back to Nigeria. One of the problems immigrants face in America, is that their foreign credentials understood by companies. Typically to get a better job they have to go back to school, which we know is very expensive in America. Now some of them prefer to send kids back home to grandparents so they can focus on school and later only bring them back to the US. This happen to native of the US too, Bill Clinton was sent to live with grandparents, for some time, while his mother studied nursing.

You don't have even a single example of somebody who did something noticeable – I am not even asking for "good" – by using that "second chance". I am feed up by your circular reasoning. You are just trolling. All this stuff is completely out of topic. If you want to talk about Microsoft, Facebook, etc. – please, find some other place. Get a blog, get a life, whatever.

DeleteThe above comment was the last comment from you which passed the moderation.