Previous post: And who actually got Fields medals?
Alexandre Grothendieck, the greatest mathematician for the twenties century, passed away on November 13, 2014 at the Saint-Girons hospital (Ariège) near the village Lasserre.
Alexandre Grothendieck spent about the last 24 years of his life in this village in Pyrenees range of mountains in a self-imposed retirement avoiding all contacts with the outside world and the mathematical community.
He had good reasons for this, but till now the mathematical community does not want to listen, or, rather, to read his extensive partially autobiographical, partially philosophical texts.
Alexandre Grothendieck, with help of his pupils, collaborators, and admires, completely transformed mathematics. His best known contribution is the proof of most of the Andre Weil conjectures (with the last step done by his pupil Pierre Deligne). Much more important is his transformation of the algebraic geometry from relatively obscure branch of mathematics to its central part. Even more important is his most intangible contribution, the concept known as th "rising sea", the idea that every mathematical problem should be immersed in a sufficiently abstract theory, which will made the solution trivial. This theory should be, in a sense, trivial too - it should not involve any tricks or convoluted arguments. This was a drastic departure from the mathematical analysis, the central branch of mathematics at the time, which was dominated by proofs demonstrating not so much the vision, but the "executive power" of the authors (the concept introduced by G. Hardy, who valued the executive power most). These ideas are still far from being internalized or even understood by the mathematical community.
Despite his tremendous influence, surpassing by a large margin the influence of any mathematician after David Hilbert, Alexandre Grothendieck was at least about 100 years ahead of his time.
His integrity and his concern about the perils people put each other into are hardly matched by any other contemporary scientist. He did not succeed much in this respect, apparently because his concerns only appeared to be left wing politics, but in fact were not of political nature.
With Alexandre Grothendieck passing away we lost the last living giant in mathematics.
Here is a link to a memorial article Alexandre Grothendieck, le plus grand mathématicien du XXe siècle, est mort in Le Monde, France (in French).
Next post: Where one can find an autobiography of Alexander Grothendieck? Part 1.
Showing posts with label ethical issues. Show all posts
Showing posts with label ethical issues. Show all posts
Friday, November 14, 2014
Monday, July 29, 2013
Guessing who will get Fields medals - Some history and 2014
Previous post: 2014 Fields medalists?
This was a relatively easy task during about three decades. But it is nearly impossible now, at least if you do not belong to the “inner circle” of the current President of the International Mathematical Union. But they change at each Congress, and one can hardly hope to belong to the inner circle of all of them.
I would like to try to explain my approach to judging a particular selection of Fields medalists and to fairly efficiently guessing the winners in the past. This cannot be done without going a little bit into the history of Fields medals as it appears to a mathematician and not to a historian working with archives. I have no idea how to get to the relevant archives and even if they exist. I suspect that there is no written record of the deliberations of any Fields medal committee.
The first two Fields medals were awarded in 1936 to Lars Ahlfors and Jesse Douglas. It was the first award, and it wasn’t a big deal. It looks like that the man behind this choice was Constantin Carathéodory. I think that this was a very good choice. In my personal opinion, Lars Ahlfors is the best analyst of the previous century, and he did his most important work after the award, which is important in view of the terms of the Fields’ will. Actually, his best work was done after WWII. If not the war, it would be done earlier, but still after the award. J. Douglas solved the main problem about minimal surfaces (in the usual 3-dimensional space) at the time. He did with the bare hands things that we do now using powerful frameworks developed later. I believe that he became seriously ill soon afterward, but today I failed to find online any confirmation of this. Now I remember that I was just told about his illness. Apparently, he did not produce any significant results later. Would he continue to work on minimal surfaces, he could be forced to develop at least some of later tools.
The next two Fields medals were awarded in 1950 and since 1950 from 2 to 4 medals were awarded every 4 years. Initially the International Mathematical Union (abbreviated as IMU) was able to fund only 2 medals (despite the fact that the monetary part is negligible), but already for several decades it has enough funds for 4 medals (the direct monetary value remains to be negligible). I was told that awarding only 2 medals in 2002 turned out to be possible only after a long battle between the Committee (or rather its Chair, S.P. Novikov) and the officials of the IMU. So, I am not alone in thinking that sometimes there are no good enough candidates for 4 medals.
I apply to the current candidates the standard of golden years of both mathematics and the Fields medals. For mathematics, they are approximately 1940-1980, with some predecessors earlier and some spill-overs later. For medals, they are 1936-1986 with some spill-overs later. The whole history of the Fields medals can traced in the Proceedings of Congresses. They are interesting in many other respects too. For example, they contain a lot of very good expository papers (and many more of bad ones). It is worthwhile at least to browse them. Now they are freely available online: ICM Proceedings 1893-2010.
The presentation of work of 1954 medalists J.-P. Serre and K. Kodaira by H. Weyl is a pleasure to read. H. Weyl unequivocally tells that their mathematics is new and went into a new territory and is based on methods unknown to most of mathematicians at the time (in fact, this is still true). He even included an introduction to these methods in the published version.
The 1990 award at the Kyoto Congress was a turning point. Ludwig D. Faddeev was the Chairman of the Fields Medal Committee and the President of the IMU for the preceding 4 years. 3 out of 4 medals went to scientists significant part of whose works was directly related to his or his students’ works. The influence went in both directions: for one winner the influence went mostly from L.D. Faddeev and his pupils, for two other winners their work turned out to be very suitable for a synthesis with some ideas of L.D. Faddeev and his pupils. All these works are related to the theoretical physics. Actually, after reading the recollections of L.D. Faddeev and prefaces to his books, it is completely clear that he is a theoretical physicists at heart, despite he has some interesting mathematical results and he is formally (judging by the positions he held, for example) considered to be a mathematician.
The 1990 was the only year when one of the medals went to a physicist. Naturally, he never proved a theorem. But his papers from 1980-1994 contain a lot of mathematical content, mostly conjectures motivated by quantum field theory reasoning. There is no doubt that his ideas are highly original from the point of view of a mathematician (and much less so from the point of view of someone using Feynman’s integrals daily), that they provided mathematicians with a lot of problems to think about, and indeed resulted in quite interesting developments in mathematics. But many mathematicians, including myself, believe that the Fields medals should be awarded to outstanding mathematicians, and a mathematician should prove his or her claims. I don’t know any award in mathematics which could be awarded for conjectures only.
In 1994 one of the medals went to the son of the President of the IMU at the time. Many people think that this is far beyond any ethical norms. The President could resign from his position the moment the name of his son surfaced. Moreover, he should decline the offer of this position in 1990. It is impossible to believe that that guy did not suspect that his son will be a viable candidate in 2-3 years (if his son indeed deserved the medal). The President of IMU is the person who is able, if he or she wants, to essentially determine the winners, because the selection of the members of the Fields medal Committee is essentially in his or her hands (unless there is a insurrection in the community – but this never happened).
As a result, the system was completely destroyed in just two cycles without any changes in bylaws or procedures (since the procedures are kept in a secret, I cannot be sure about the latter). Still, some really good mathematicians got a medal. Moreover, in 2002 it looked like the system recovered. Unfortunately already in 2006 things were the same as in the 1990ies. One of the awards was outrageous on ethical grounds (completely different from 1994); the long negotiations with Grisha Perelman remind plays by Eugène Ionesco.
In the current situation I would be able to predict the winners if I would knew the composition of the committee. Since this is impossible, I will pretend that the committee is as impartial as it was in 1950-1986. This is almost (but not completely) equivalent to telling my preferences.
I would be especially happy if an impartial committee will award only 2 medals and Manjul Bhargava and Jacob Lurie will be the winners. I hope that their advisors are not on the committee. Their works look very attractive to me. I suspect that Jacob Lurie is the only mathematician working now and comparable with the giants of the golden age. But I do not have enough time to study his papers, or, rather, his books. They are just too long for everybody except people working in the same field. Usually they are hundreds pages long; his only published book (which covers only preliminaries) is almost 1000 pages long. Papers by Manjul Bhargava seem to be more accessible (definitely, they are much shorter). But I am not an expert in his field and I would need to study a lot before jumping into his papers. I do not have enough motivation for this now. An impartial committee would be reinforce my high opinion about their work and provide an additional stimulus to study them deeper. The problem is that I have no reason to expect the committee to be impartial.
Arthur Avila is very strong, or so tell me my expert friends. His field is too narrow for my taste. The main problem is that his case is bound to be political. It depends on the balance of power between, approximately, Cambridge, MA – Berkley and Rio de Janeiro – Paris. Here I had intentionally distorted the geolocation data.
The high ratings in that poll of Manjul Bhargava and Artur Avila are the examples of the “name recognition” I mentioned. I think that an article about Manjul Bhargava appeared even in the New York Times. Being a strong mathematician from a so-called developing country (it seems that the term “non-declining” would be better for English-speaking countries), Artur Avila is known much better than American or British mathematicians of the same level.
Most of mathematicians included in the poll wouldn’t be ever considered by anybody as candidates during the golden age. There would be several dozens of the same level in the same broadly defined area of mathematical. Sections of the Congress can serve as the first approximation to a good notion of an area of mathematics. And a Fields medalist was supposed to be really outstanding. Restricting myself by the poll list I prefer one of the following variants: either Bhargava, or Lurie, or both or no medals for the lack of suitable candidates.
Next post: Did J. Lurie solved any big problem?
This was a relatively easy task during about three decades. But it is nearly impossible now, at least if you do not belong to the “inner circle” of the current President of the International Mathematical Union. But they change at each Congress, and one can hardly hope to belong to the inner circle of all of them.
I would like to try to explain my approach to judging a particular selection of Fields medalists and to fairly efficiently guessing the winners in the past. This cannot be done without going a little bit into the history of Fields medals as it appears to a mathematician and not to a historian working with archives. I have no idea how to get to the relevant archives and even if they exist. I suspect that there is no written record of the deliberations of any Fields medal committee.
The first two Fields medals were awarded in 1936 to Lars Ahlfors and Jesse Douglas. It was the first award, and it wasn’t a big deal. It looks like that the man behind this choice was Constantin Carathéodory. I think that this was a very good choice. In my personal opinion, Lars Ahlfors is the best analyst of the previous century, and he did his most important work after the award, which is important in view of the terms of the Fields’ will. Actually, his best work was done after WWII. If not the war, it would be done earlier, but still after the award. J. Douglas solved the main problem about minimal surfaces (in the usual 3-dimensional space) at the time. He did with the bare hands things that we do now using powerful frameworks developed later. I believe that he became seriously ill soon afterward, but today I failed to find online any confirmation of this. Now I remember that I was just told about his illness. Apparently, he did not produce any significant results later. Would he continue to work on minimal surfaces, he could be forced to develop at least some of later tools.
The next two Fields medals were awarded in 1950 and since 1950 from 2 to 4 medals were awarded every 4 years. Initially the International Mathematical Union (abbreviated as IMU) was able to fund only 2 medals (despite the fact that the monetary part is negligible), but already for several decades it has enough funds for 4 medals (the direct monetary value remains to be negligible). I was told that awarding only 2 medals in 2002 turned out to be possible only after a long battle between the Committee (or rather its Chair, S.P. Novikov) and the officials of the IMU. So, I am not alone in thinking that sometimes there are no good enough candidates for 4 medals.
I apply to the current candidates the standard of golden years of both mathematics and the Fields medals. For mathematics, they are approximately 1940-1980, with some predecessors earlier and some spill-overs later. For medals, they are 1936-1986 with some spill-overs later. The whole history of the Fields medals can traced in the Proceedings of Congresses. They are interesting in many other respects too. For example, they contain a lot of very good expository papers (and many more of bad ones). It is worthwhile at least to browse them. Now they are freely available online: ICM Proceedings 1893-2010.
The presentation of work of 1954 medalists J.-P. Serre and K. Kodaira by H. Weyl is a pleasure to read. H. Weyl unequivocally tells that their mathematics is new and went into a new territory and is based on methods unknown to most of mathematicians at the time (in fact, this is still true). He even included an introduction to these methods in the published version.
The 1990 award at the Kyoto Congress was a turning point. Ludwig D. Faddeev was the Chairman of the Fields Medal Committee and the President of the IMU for the preceding 4 years. 3 out of 4 medals went to scientists significant part of whose works was directly related to his or his students’ works. The influence went in both directions: for one winner the influence went mostly from L.D. Faddeev and his pupils, for two other winners their work turned out to be very suitable for a synthesis with some ideas of L.D. Faddeev and his pupils. All these works are related to the theoretical physics. Actually, after reading the recollections of L.D. Faddeev and prefaces to his books, it is completely clear that he is a theoretical physicists at heart, despite he has some interesting mathematical results and he is formally (judging by the positions he held, for example) considered to be a mathematician.
The 1990 was the only year when one of the medals went to a physicist. Naturally, he never proved a theorem. But his papers from 1980-1994 contain a lot of mathematical content, mostly conjectures motivated by quantum field theory reasoning. There is no doubt that his ideas are highly original from the point of view of a mathematician (and much less so from the point of view of someone using Feynman’s integrals daily), that they provided mathematicians with a lot of problems to think about, and indeed resulted in quite interesting developments in mathematics. But many mathematicians, including myself, believe that the Fields medals should be awarded to outstanding mathematicians, and a mathematician should prove his or her claims. I don’t know any award in mathematics which could be awarded for conjectures only.
In 1994 one of the medals went to the son of the President of the IMU at the time. Many people think that this is far beyond any ethical norms. The President could resign from his position the moment the name of his son surfaced. Moreover, he should decline the offer of this position in 1990. It is impossible to believe that that guy did not suspect that his son will be a viable candidate in 2-3 years (if his son indeed deserved the medal). The President of IMU is the person who is able, if he or she wants, to essentially determine the winners, because the selection of the members of the Fields medal Committee is essentially in his or her hands (unless there is a insurrection in the community – but this never happened).
As a result, the system was completely destroyed in just two cycles without any changes in bylaws or procedures (since the procedures are kept in a secret, I cannot be sure about the latter). Still, some really good mathematicians got a medal. Moreover, in 2002 it looked like the system recovered. Unfortunately already in 2006 things were the same as in the 1990ies. One of the awards was outrageous on ethical grounds (completely different from 1994); the long negotiations with Grisha Perelman remind plays by Eugène Ionesco.
In the current situation I would be able to predict the winners if I would knew the composition of the committee. Since this is impossible, I will pretend that the committee is as impartial as it was in 1950-1986. This is almost (but not completely) equivalent to telling my preferences.
I would be especially happy if an impartial committee will award only 2 medals and Manjul Bhargava and Jacob Lurie will be the winners. I hope that their advisors are not on the committee. Their works look very attractive to me. I suspect that Jacob Lurie is the only mathematician working now and comparable with the giants of the golden age. But I do not have enough time to study his papers, or, rather, his books. They are just too long for everybody except people working in the same field. Usually they are hundreds pages long; his only published book (which covers only preliminaries) is almost 1000 pages long. Papers by Manjul Bhargava seem to be more accessible (definitely, they are much shorter). But I am not an expert in his field and I would need to study a lot before jumping into his papers. I do not have enough motivation for this now. An impartial committee would be reinforce my high opinion about their work and provide an additional stimulus to study them deeper. The problem is that I have no reason to expect the committee to be impartial.
Arthur Avila is very strong, or so tell me my expert friends. His field is too narrow for my taste. The main problem is that his case is bound to be political. It depends on the balance of power between, approximately, Cambridge, MA – Berkley and Rio de Janeiro – Paris. Here I had intentionally distorted the geolocation data.
The high ratings in that poll of Manjul Bhargava and Artur Avila are the examples of the “name recognition” I mentioned. I think that an article about Manjul Bhargava appeared even in the New York Times. Being a strong mathematician from a so-called developing country (it seems that the term “non-declining” would be better for English-speaking countries), Artur Avila is known much better than American or British mathematicians of the same level.
Most of mathematicians included in the poll wouldn’t be ever considered by anybody as candidates during the golden age. There would be several dozens of the same level in the same broadly defined area of mathematical. Sections of the Congress can serve as the first approximation to a good notion of an area of mathematics. And a Fields medalist was supposed to be really outstanding. Restricting myself by the poll list I prefer one of the following variants: either Bhargava, or Lurie, or both or no medals for the lack of suitable candidates.
Next post: Did J. Lurie solved any big problem?
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.
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Wednesday, August 15, 2012
The twist ending. 2. A Cambridge don
Previous post: The twist ending. 1.
Mel Nathanson made a right on the target comment "Mel Nathanson Says, July 8, 2012, 9:14 a.m." in Gowers's blog about ethical issues stemming from the fact that Timothy Gowers is a professor at Cambridge University, of which the publishing house Cambridge University Press, the publisher of his new journals, is a for-profit branch. The university as a whole is non-profit, i.e. cannot distribute profits to people not employed by it.
Next post: The twist ending. 3. R. Kirby.
Behind the jump break I posted the complete text of Mel Nathanson's comment as an insurance against the disappearance of the original. Nothing on the web is really permanent, and I hope that Professor Mel Nathanson will not object to this and will not consider this to be a copyright infringement (I am relying on the "fair use" doctrine, but will remove the text at his request immediately).
Mel Nathanson made a right on the target comment "Mel Nathanson Says, July 8, 2012, 9:14 a.m." in Gowers's blog about ethical issues stemming from the fact that Timothy Gowers is a professor at Cambridge University, of which the publishing house Cambridge University Press, the publisher of his new journals, is a for-profit branch. The university as a whole is non-profit, i.e. cannot distribute profits to people not employed by it.
Next post: The twist ending. 3. R. Kirby.
Behind the jump break I posted the complete text of Mel Nathanson's comment as an insurance against the disappearance of the original. Nothing on the web is really permanent, and I hope that Professor Mel Nathanson will not object to this and will not consider this to be a copyright infringement (I am relying on the "fair use" doctrine, but will remove the text at his request immediately).
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