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.




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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.


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