Previous post: About expository writing: a reply to posic
At 10:30 p.m. US Eastern Summer time, the winner of this (2014) year Fields medals will be announced in Seoul.
I would like to post my current guess, mostly to have a record of it with the date and time stamp from Google, at least for myself.
As I wrote about one year ago, I believe that I would be able to predict the actual winners if I would know the composition of the Fields medals committee. But I don't. I am not particularly interested in the names of the winners, so I did not attempted to find out the actual winners, who are known for at least three months already, and who are known to the press for at least two weeks already (if the practice of the last two congresses was continued). So, my guess is a guess and not based on any inside sources.
And the winner are (expected to be):
Artur Avila - my confidence is over 95-99%.
One of the winners will be a woman - my confidence is over 95%. This is a pure politics. This deserves a separate discussion. The main obstruction to the Fields medal for a woman is not the discrimination, but the absurd age restriction. Most likely, she is
Sophie Morel - my confidence is over 80%. There are political consideration against her. For example, she would be the 3rd medalist who was a student of Gérard Laumon.
Jacob Lurie - my confidence is about 60%. This is my favorite candidate. He will get it if Harvard has enough political clout now. So, it is a measure of the influence of the Harvard Department of Mathematics, and not of the level of J. Lurie as a mathematician.
Manjul Bhargava - my confidence is less than 50%. If Sophie Morel gets a medal, his chances are much lower than otherwise: two mathematicians from the same university (Princeton).
Following the tradition firmly established since 1990, one of the medals should go a "Russian" mathematician, no matter where she or he is working know and where she or he completed Ph.D. I don't see any suitable candidate. Some people were naming Alexei Borodin, but I was firmly told that he will not get one.
A couple of days ago a strange, apparently unmotivated idea come to my mind: one of the winners will be from Stanford. Some people were naming Maryam Mirzakhani, but, again, a couple of days ago was firmly told that she is not the winner. Her work is interesting and close to my own interests. In my personal opinion, she has some very good results, but nothing of the Fields medal level. I would estimate the number of mathematician of about her level or higher, working in closely related areas, as at least 2-3 dozens. Of course, I am not aware about her most recent unpublished (at least on the web) work.
Next post: And who actually got Fields medals?
Showing posts with label power. Show all posts
Showing posts with label power. Show all posts
Tuesday, August 12, 2014
Sunday, August 4, 2013
Did J. Lurie solved any big problem?
Previous post: Guessing who will get Fields medals - Some history and 2014.
Tamas Gabal asked the following question.
Short answer: I don't care. Here is a long answer.
Well, this is the reason why my opinion about Lurie is somewhat conditional. As I already said, if an impartial committee confirms the significance of Lurie’s work, it will remove my doubts and, very likely, will stimulate me to study his work in depth. It is much harder to predict what will be the influence of the actual committee. Perhaps, I will try to learn his work in any case. If he will not get the medal, then in the hope to make sure that the committee is wrong.
I planned to discuss many peculiarities of mathematical prizes in another post, but one of these peculiarities ought to be mentioned now. Most of mathematical prizes go to people who solved some “important problems”. In fact, most of them go to people who made the last step in solving a problem. There is recent and famous example at hand: the Clay $1,000,000.00 prize was awarded to Perelman alone. But the method was designed by R. Hamilton, who did a huge amount of work, but wasn’t able to made “the last step”. Perhaps, just because of age. As Perelman said to a Russian news agency, he declined the prize because in his opinion Hamilton’s work is no less important than his own, and Hamilton deserves the prize no less than him. It seems that this reason still not known widely enough. To the best of my knowledge, it was not included in any press-release of the Clay Institute. The Clay Institute scheduled the award ceremony like they knew nothing, and then held the ceremony as planned. Except Grisha Perelman wasn’t present, and he did not accept the prize in any sense.
So, the prizes go to mathematicians who did the last step in the solution of a recognized problem. The mathematicians building the theories on which these solutions are based almost never get Fields medals. Their chances are more significant when prize is a prize for the life-time contribution (as is the case with the Abel prize). There are few exceptions.
First of all, A. Grothendieck is an exception. He proved part of the Weil conjectures, but not the most important one (later proved by P. Deligne). One of the Weil conjectures (the basic one) was independently proved by B. Dwork, by a completely different and independent method, and published earlier (by the way, this is fairly accessible and extremely beautiful piece of work). The report of J. Dieudonne at the 1966 Congress outlines a huge theory, to a big extent still not written down then. It includes some theorems, like the Grothendieck-Riemann-Roch theorem, but: (i) GRR theorem does not solve any established problem, it is a radically new type of a statement; (ii) Grothendieck did not published his proof, being of the opinion that the proof is not good enough (an exposition was published by Borel and Serre); (iii) it is just a byproduct of his new way of thinking.
D. Quillen (Fields medal 1978) did solve some problems, but his main achievement is a solution of a very unusual problem: to give a good definition of so-called higher algebraic K-functors. It is a theory. Moreover, there are other solutions. Eventually, it turns out that they all provide equivalent definitions. But Quillen’s definitions (actually, he suggested two) are much better than others.
So, I do not care much if Lurie solved some “important problems” or not. Moreover, in the current situation I rather prefer that he did not solved any well-known problems, if he will get a Fields medal. The contrast with the Hungarian combinatorics, which is concentrated on statements and problems, will make the mathematics healthier.
Problems are very misleading. Often they achieve their status not because they are really important, but because a prize was associated with them (Fermat Last Theorem), or they were posed by a famous mathematicians. An example of the last situation is nothing else but the Poincaré Conjecture – in fact, Poincaré did not stated it as a conjecture, he just mentioned that “it would be interesting to know the answer to the following question”. It is not particularly important by itself. It claims that one difficult to verify property (being homeomorphic to a 3-sphere) is equivalent to another difficult to verify property (having trivial fundamental group). In practice, if you know that the fundamental group is trivial, you know also that your manifold is a 3-sphere.
Next post: New ideas.
Tamas Gabal asked the following question.
I heard a criticism of Lurie's work, that it does not contain startling new ideas, complete solutions of important problems, even new conjectures. That he is simply rewriting old ideas in a new language. I am very far from this area, and I find it a little disturbing that only the ultimate experts speak highly of his work. Even people in related areas can not usually give specific examples of his greatness. I understand that his objectives may be much more long-term, but I would still like to hear some response to these criticisms.
Short answer: I don't care. Here is a long answer.
Well, this is the reason why my opinion about Lurie is somewhat conditional. As I already said, if an impartial committee confirms the significance of Lurie’s work, it will remove my doubts and, very likely, will stimulate me to study his work in depth. It is much harder to predict what will be the influence of the actual committee. Perhaps, I will try to learn his work in any case. If he will not get the medal, then in the hope to make sure that the committee is wrong.
I planned to discuss many peculiarities of mathematical prizes in another post, but one of these peculiarities ought to be mentioned now. Most of mathematical prizes go to people who solved some “important problems”. In fact, most of them go to people who made the last step in solving a problem. There is recent and famous example at hand: the Clay $1,000,000.00 prize was awarded to Perelman alone. But the method was designed by R. Hamilton, who did a huge amount of work, but wasn’t able to made “the last step”. Perhaps, just because of age. As Perelman said to a Russian news agency, he declined the prize because in his opinion Hamilton’s work is no less important than his own, and Hamilton deserves the prize no less than him. It seems that this reason still not known widely enough. To the best of my knowledge, it was not included in any press-release of the Clay Institute. The Clay Institute scheduled the award ceremony like they knew nothing, and then held the ceremony as planned. Except Grisha Perelman wasn’t present, and he did not accept the prize in any sense.
So, the prizes go to mathematicians who did the last step in the solution of a recognized problem. The mathematicians building the theories on which these solutions are based almost never get Fields medals. Their chances are more significant when prize is a prize for the life-time contribution (as is the case with the Abel prize). There are few exceptions.
First of all, A. Grothendieck is an exception. He proved part of the Weil conjectures, but not the most important one (later proved by P. Deligne). One of the Weil conjectures (the basic one) was independently proved by B. Dwork, by a completely different and independent method, and published earlier (by the way, this is fairly accessible and extremely beautiful piece of work). The report of J. Dieudonne at the 1966 Congress outlines a huge theory, to a big extent still not written down then. It includes some theorems, like the Grothendieck-Riemann-Roch theorem, but: (i) GRR theorem does not solve any established problem, it is a radically new type of a statement; (ii) Grothendieck did not published his proof, being of the opinion that the proof is not good enough (an exposition was published by Borel and Serre); (iii) it is just a byproduct of his new way of thinking.
D. Quillen (Fields medal 1978) did solve some problems, but his main achievement is a solution of a very unusual problem: to give a good definition of so-called higher algebraic K-functors. It is a theory. Moreover, there are other solutions. Eventually, it turns out that they all provide equivalent definitions. But Quillen’s definitions (actually, he suggested two) are much better than others.
So, I do not care much if Lurie solved some “important problems” or not. Moreover, in the current situation I rather prefer that he did not solved any well-known problems, if he will get a Fields medal. The contrast with the Hungarian combinatorics, which is concentrated on statements and problems, will make the mathematics healthier.
Problems are very misleading. Often they achieve their status not because they are really important, but because a prize was associated with them (Fermat Last Theorem), or they were posed by a famous mathematicians. An example of the last situation is nothing else but the Poincaré Conjecture – in fact, Poincaré did not stated it as a conjecture, he just mentioned that “it would be interesting to know the answer to the following question”. It is not particularly important by itself. It claims that one difficult to verify property (being homeomorphic to a 3-sphere) is equivalent to another difficult to verify property (having trivial fundamental group). In practice, if you know that the fundamental group is trivial, you know also that your manifold is a 3-sphere.
Next post: New ideas.
Sunday, July 28, 2013
2014 Fields medalists?
Previous post: New comments to the post "What is mathematics?"
I was asked by Tamas Gabal about possible 2014 Fields medalists listed in an online poll. I am neither ready to systematically write down my thoughts about the prizes in general and Fields medals in particular, nor to predict who will get 2014 medals. I am sure that the world would be better without any prizes, especially without Fields medals. Also, in my opinion, no more than two persons deserve 2014 Fields medals. Instead of trying to argue these points, I will quote my reply to Tamas Gabal (slightly edited).
Somewhat later I wrote:
Tamas Gabal replied:
Here is my reply.
Good question. In order to put a name on a list, one has to know this name, i.e. recognize it. But I knew much more than 10 names. Actually, this is one of the topics I wanted to write about sometime in details. The whole atmosphere at that time was completely different from what I see around now. May be the place also played some role, but I doubt that its role was decisive. Most of the people around me liked to talk about mathematics, and not only about what they were doing. When some guy in Japan claimed that he proved the Riemann hypothesis, I knew about this the same week. Note that the internet was still in the future, as were e-mails. I had a feeling that I know about everything important going on in mathematics. I always had a little bit more curiosity than others, so I knew also about fields fairly remote from own work.
I do not remember all 10 names on my list (I remember about 7), but 4 winners were included. It was quite easy to guess 3 of them. Everybody would agree that they were the main contenders. I am really proud about guessing the 4th one. Nobody around was talking about him or even mentioned him, and his field is quite far from my own interests. To what extent I understood their work? I studied some work of one winner, knew the statements and had some idea about their proof for another one (later the work of both of them influenced a lot my own work, but mostly indirectly), and very well knew what are the achievements of the third one, why they are important, etc. I knew more or less just the statements of two main results of the 4th one, the one who was difficult to guess – for me. I was able to explain why this or that guy got the medal even to a theoretical physicist (actually did on one occasion). But I wasn’t able to teach a topic course about works of any of the 4.
At the time I never heard any complaints that a medal went to a wrong person. The same about all older awards. There was always a consensus in the mathematical community than all the people who got the medal deserved it. May be somebody else also deserved it too, but there are only 3 or 4 of them each time.
Mathematics is a human activity. This is one of the facts that T. Gowers prefers to ignore. Nobody verifies proofs line by line. Initially, you trust your guts feelings. If you need to use a theorem, you will be forced to study the proof and understand its main ideas. The same is true about the deepness of a result. You do not need to know all the proofs in order to write down a list like my list of 10 most likely winners (next time my list consisted of no more than 5 or 6, all winner were included). It seems that I knew the work of all guessed winners better than Gowers knew the work of 2010 medalists. But even if not, there is a huge difference between a graduate student trying to guess the current year winners, and a Fellow of the London Royal Society, a Fields medalist himself, who is deciding who will get 2010 medals. He should know more.
The job is surely not an easy one now, when it is all about politics. Otherwise it would be very pleasant.
Next post: Guessing who will get Fields medals - Some history and 2014.
I was asked by Tamas Gabal about possible 2014 Fields medalists listed in an online poll. I am neither ready to systematically write down my thoughts about the prizes in general and Fields medals in particular, nor to predict who will get 2014 medals. I am sure that the world would be better without any prizes, especially without Fields medals. Also, in my opinion, no more than two persons deserve 2014 Fields medals. Instead of trying to argue these points, I will quote my reply to Tamas Gabal (slightly edited).
Would I know who the members of the Fields medal committee are, I would be able to predict medalists with 99% confidence. But the composition of the committee is a secret. In the past, the situation was rather different. The composition of the committee wasn't important. When I was just a second year graduate student, I compiled a list of 10 candidates, among whom I considered 5 to have significantly higher chances (I never wrote down this partition, and the original list is lost for all practical purposes). All 4 winners were on the list. I was especially proud of predicting one of them; he was a fairly nontraditional at the time (or so I thought). I cannot do anything like this now without knowing the composition of the committee. Recent choices appear to be more or less random, with some obvious exceptions (like Grisha Perelman).
Somewhat later I wrote:
In the meantime I looked at the current results of that poll. Look like the preferences of the public are determined by the same mechanism as the preferences for movie actors and actresses: the name recognition.
Tamas Gabal replied:
Sowa, when you were a graduate student and made that list of possible winners, did you not rely on name recognition at least partially? Were you familiar with their work? That would be pretty impressive for a graduate student, since T. Gowers basically admitted that he was not really familiar with the work of Fields medalists in 2010, while he was a member of the committee. I wonder if anyone can honestly compare the depth of the work of all these candidates? The committee will seek an opinion of senior people in each area (again, based on name recognition, positions, etc.) and will be influenced by whoever makes the best case... It's not an easy job for sure.
Here is my reply.
Good question. In order to put a name on a list, one has to know this name, i.e. recognize it. But I knew much more than 10 names. Actually, this is one of the topics I wanted to write about sometime in details. The whole atmosphere at that time was completely different from what I see around now. May be the place also played some role, but I doubt that its role was decisive. Most of the people around me liked to talk about mathematics, and not only about what they were doing. When some guy in Japan claimed that he proved the Riemann hypothesis, I knew about this the same week. Note that the internet was still in the future, as were e-mails. I had a feeling that I know about everything important going on in mathematics. I always had a little bit more curiosity than others, so I knew also about fields fairly remote from own work.
I do not remember all 10 names on my list (I remember about 7), but 4 winners were included. It was quite easy to guess 3 of them. Everybody would agree that they were the main contenders. I am really proud about guessing the 4th one. Nobody around was talking about him or even mentioned him, and his field is quite far from my own interests. To what extent I understood their work? I studied some work of one winner, knew the statements and had some idea about their proof for another one (later the work of both of them influenced a lot my own work, but mostly indirectly), and very well knew what are the achievements of the third one, why they are important, etc. I knew more or less just the statements of two main results of the 4th one, the one who was difficult to guess – for me. I was able to explain why this or that guy got the medal even to a theoretical physicist (actually did on one occasion). But I wasn’t able to teach a topic course about works of any of the 4.
At the time I never heard any complaints that a medal went to a wrong person. The same about all older awards. There was always a consensus in the mathematical community than all the people who got the medal deserved it. May be somebody else also deserved it too, but there are only 3 or 4 of them each time.
Mathematics is a human activity. This is one of the facts that T. Gowers prefers to ignore. Nobody verifies proofs line by line. Initially, you trust your guts feelings. If you need to use a theorem, you will be forced to study the proof and understand its main ideas. The same is true about the deepness of a result. You do not need to know all the proofs in order to write down a list like my list of 10 most likely winners (next time my list consisted of no more than 5 or 6, all winner were included). It seems that I knew the work of all guessed winners better than Gowers knew the work of 2010 medalists. But even if not, there is a huge difference between a graduate student trying to guess the current year winners, and a Fellow of the London Royal Society, a Fields medalist himself, who is deciding who will get 2010 medals. He should know more.
The job is surely not an easy one now, when it is all about politics. Otherwise it would be very pleasant.
Next post: Guessing who will get Fields medals - Some history and 2014.
Wednesday, March 27, 2013
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.
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.
Tuesday, January 1, 2013
Reply to a comment
Previous post: Freedom of speech in mathematics
This is a reply to a recent comment by an Anonymous.
Dear Anonymous,
Thank you very much for reading these notes and for writing a serious and stimulating comment. I was thinking about it for a few days, and I am still not sure that my reply will be convincing. But let me try.
The initial goal of this blog was to provide a context for my arguments in Gowers's blog and in another blog. The latter was quoted by another anonymous in Gowers’s blog, and I had no other choice as to try to explain them (the original comment was addressed to people who know me and my views fairly well). So, the concentration on Gowers's views and activities was only natural; this was the intention. Now this topic is more or less exhausted, and expanding the scope of the blog, or even changing it completely may be in order.
Gowers himself described his goals and said that achieving them will eliminate mathematics as we know it. There is nothing apocalyptic in my discussion of his ideas. I do not anticipate that he or his followers will achieve his goals. I do not think that humans are some sort of computers, and I do not think that computers can do real mathematics (definitely, they can do a lot of mathematical things better than humans, but only when a class of problems was completely understood by humans). But he may achieve his goal in an Orwellian way by changing the meaning of the word “mathematics”. He already shifted the preferences of a big part of mathematical community. It took about ten years. If he will be able to do a comparable feat in the next ten years, and then, may be with his followers, once more, “mathematics” will mean “Hungarian-style combinatorics”. And I do believe that the Hungarian-style combinatorics is a field where computers eventually will be superior to humans because a nearly exhaustive search for a proof will be more efficient than human insights (like it happened with chess, which is actually a branch of combinatorics).
Independently of this scenario, I believe that having a person intended to eliminate mathematics (even if his goal is not achievable) in position of such influence as Gowers is extremely unhealthy and dangerous.
I agree that an “open and critical analysis of very influential individuals or groups in the mathematics research community” is highly desirable. But I am not quite comfortable with the way you wrote about this: “what the discussion should be about”. First of all, I am not comfortable with writing this either, but this is my blog and I write about whatever I like and whenever I like. Perhaps, you meant to apply “should” not to me, but to the mathematical community itself. I will assume that this is what you meant.
I believe that such a discussion is hardly possible. As a rule, mathematicians strongly dislike to openly discuss any contentious issues. One may see this everywhere: from insignificant issues on a department of mathematics to major decision made by the AMS or Fields Medals Committee, for example. I would suggest these posts for publication in the Notices of the AMS (under my real name) if I would believe that there is some chance for them to be published.
To Editors of the Notices of the AMS: this is a challenge. Prove that I am wrong!
Anyhow, I am willing to participate in such an open discussion. There is no venue for this now. Still, I would be glad to expand this blog into such a venue. For example, it may be easily transformed into a collective blog, and, for example, you will be able to post here. You will need only a Google account for this, and I will need to know the name of the account and some way of verification that it belongs to the Anonymous who wrote the comment. I reserve the right to be the owner of the blog and its moderator (right now it is not moderated, comments appear without my approval). And, may be, I will eventually transfer it to somebody else.
Finally, I disagree that mathematics was “long dominated by geometry, topology, arithmetic geometry, etc.” By some reason the word “geometry” is very popular for a few decades, and topology is usually considered to be a branch of geometry. So, the word “geometry” was appended to almost any good mathematical theory. Say, Non-Commutative Geometry is actually a branch of functional analysis. Arithmetic Geometry is a branch of algebraic number theory. Topology is not a branch of geometry in the classical sense of the word “geometry”. Algebraic Topology is a branch of algebra. Well, I realize that this is a sort of scandalous statement. It took me many years to come to this conclusion. There was a subfield of topology called Geometric Topology (it is hardly alive by now); this would be a nonsense would topology be indeed a branch of geometry.
So, in my opinion mathematics was long dominated by good mathematics, but this is not the case anymore.
Next post: Happy New Year!
This is a reply to a recent comment by an Anonymous.
Dear Anonymous,
Thank you very much for reading these notes and for writing a serious and stimulating comment. I was thinking about it for a few days, and I am still not sure that my reply will be convincing. But let me try.
The initial goal of this blog was to provide a context for my arguments in Gowers's blog and in another blog. The latter was quoted by another anonymous in Gowers’s blog, and I had no other choice as to try to explain them (the original comment was addressed to people who know me and my views fairly well). So, the concentration on Gowers's views and activities was only natural; this was the intention. Now this topic is more or less exhausted, and expanding the scope of the blog, or even changing it completely may be in order.
Gowers himself described his goals and said that achieving them will eliminate mathematics as we know it. There is nothing apocalyptic in my discussion of his ideas. I do not anticipate that he or his followers will achieve his goals. I do not think that humans are some sort of computers, and I do not think that computers can do real mathematics (definitely, they can do a lot of mathematical things better than humans, but only when a class of problems was completely understood by humans). But he may achieve his goal in an Orwellian way by changing the meaning of the word “mathematics”. He already shifted the preferences of a big part of mathematical community. It took about ten years. If he will be able to do a comparable feat in the next ten years, and then, may be with his followers, once more, “mathematics” will mean “Hungarian-style combinatorics”. And I do believe that the Hungarian-style combinatorics is a field where computers eventually will be superior to humans because a nearly exhaustive search for a proof will be more efficient than human insights (like it happened with chess, which is actually a branch of combinatorics).
Independently of this scenario, I believe that having a person intended to eliminate mathematics (even if his goal is not achievable) in position of such influence as Gowers is extremely unhealthy and dangerous.
I agree that an “open and critical analysis of very influential individuals or groups in the mathematics research community” is highly desirable. But I am not quite comfortable with the way you wrote about this: “what the discussion should be about”. First of all, I am not comfortable with writing this either, but this is my blog and I write about whatever I like and whenever I like. Perhaps, you meant to apply “should” not to me, but to the mathematical community itself. I will assume that this is what you meant.
I believe that such a discussion is hardly possible. As a rule, mathematicians strongly dislike to openly discuss any contentious issues. One may see this everywhere: from insignificant issues on a department of mathematics to major decision made by the AMS or Fields Medals Committee, for example. I would suggest these posts for publication in the Notices of the AMS (under my real name) if I would believe that there is some chance for them to be published.
To Editors of the Notices of the AMS: this is a challenge. Prove that I am wrong!
Anyhow, I am willing to participate in such an open discussion. There is no venue for this now. Still, I would be glad to expand this blog into such a venue. For example, it may be easily transformed into a collective blog, and, for example, you will be able to post here. You will need only a Google account for this, and I will need to know the name of the account and some way of verification that it belongs to the Anonymous who wrote the comment. I reserve the right to be the owner of the blog and its moderator (right now it is not moderated, comments appear without my approval). And, may be, I will eventually transfer it to somebody else.
Finally, I disagree that mathematics was “long dominated by geometry, topology, arithmetic geometry, etc.” By some reason the word “geometry” is very popular for a few decades, and topology is usually considered to be a branch of geometry. So, the word “geometry” was appended to almost any good mathematical theory. Say, Non-Commutative Geometry is actually a branch of functional analysis. Arithmetic Geometry is a branch of algebraic number theory. Topology is not a branch of geometry in the classical sense of the word “geometry”. Algebraic Topology is a branch of algebra. Well, I realize that this is a sort of scandalous statement. It took me many years to come to this conclusion. There was a subfield of topology called Geometric Topology (it is hardly alive by now); this would be a nonsense would topology be indeed a branch of geometry.
So, in my opinion mathematics was long dominated by good mathematics, but this is not the case anymore.
Next post: Happy New Year!
Friday, August 17, 2012
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.
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.
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