The value of insights and the identity of the author

Previous post: Hard, soft, and Bott periodicity - Reply to T. Gowers.

This is partially a reply to a comment by Emmanuel Kowalski.

There is a phenomenon which I can hardly explain. For example, E. Kowalski said in the linked comment that he cannot comment on my statements (it seems that he is not addressing me at all, he is just commenting) without making assumptions about me, i.e. without using ad hominem arguments. Why he cannot write about my ideas without knowing my personal details?

It seems that E. Kowalski suspects that my opinions are somehow deducible from my personal life circumstances, my biography, etc.

In fact, it is possible that I have more experience due to my biography than most of other mathematicians. This is even partially the case, but only partially, and this does not affect my opinions about mathematical theories. These aspects of my life experience are quite obvious already in the discussion in the Gowers's blog.

But my opponents do not seem to adhere to this theory, which is obviously favoring me. Rather, it seems that they believe I am not knowledgeable enough or plain stupid. Would this be the case, my conclusions would be, most likely, wrong and, moreover, it would be quite easy to refute them without making any assumptions about me.

In fact, one of the main reasons for my semi-anonymity is that I would like to see my arguments and opinions evaluated on their intrinsic merits, without knowing if am I married or not, how good or bad is my employer - name anything you would like to know.

This phenomenon is not limited to my opponents. Somebody, apparently sympathetic to me, wrote: I’d be very interested in any small mathematical insight you might be willing to share, if you’re whom I conjecture you are". So, even my mathematical insights are interesting or not depending on who I am. For me, the interest of a mathematical (or “meta-mathematical”, like this discussion) insight does not depend on whom it belongs.

Of course, sometimes the authorship matters. But assumptions about the author still do not. Let us imagine that it is 1976 today (many other years will work also). Then any person interested in algebra, algebraic topology, or Grothendieck algebraic geometry knows that all papers by D. Quillen to date are very interesting and often contain incredibly deep insights. It is only natural to be interested in any new paper by Quillen. I don’t know anybody working now and comparable in this respect to 1976 Quillen; this is the reason for an exercise in time travel.

At the same time, if I see an interesting result, theory, insight, it does not matter for me if it is published in Annals or in Amer. Math. Monthly, who is the author, and what problems in life she or he has, if any.

In both situations the insights of a person lead to her or his reputation. The reputation itself does not make all insights of this person interesting. Only in rare cases the reputation may suggest that it is worthwhile to pay attention to works of a person.

Unfortunately, this seems to be not true nowadays at least in the West. The relatively recent cult of Fields medals makes the work and the area of any new winner instantly interesting. In the past the presenters of the awarded medals used to stress that there is at least 30-40 young mathematicians with comparable achievements. Not anymore. In the US, one will be monetarily rewarded for a trivial paper in Annals, but never for an expository paper (and no books, please, I was told many years ago), no matter how deep its insights. Papers in a European journal are treated by default as second rate papers. An insight of a person working in Ivy League is more valuable that a much deeper insight of a person working in Alabama. And so on.

Finally, I would like to make an offer to Emmanuel Kowalski (only to him).


Dear Emmanuel Kowalski,

You may ask me in comments here anything you would like to know. I do not promise to answer all the questions. I will evaluate to what extent my answers will help to sort out my real life identity, and will not answer to the questions which are really helpful in this respect. In particular, I will not tell what my area of research is. I will not answer to the questions which I will deem to be too personal. But if finding out my identity is not your goal, here is your chance to replace your assumptions by the actual knowledge.

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

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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