tag:blogger.com,1999:blog-8062754651764668865.post5502535653479805737..comments2023-12-09T04:56:42.838-04:00Comments on Stop Timothy Gowers! !!!: Did J. Lurie solved any big problem?Owlhttp://www.blogger.com/profile/13083493391293465339noreply@blogger.comBlogger16125tag:blogger.com,1999:blog-8062754651764668865.post-79317503666200999552017-05-15T01:30:55.678-03:002017-05-15T01:30:55.678-03:00Unknown:
About right, except it is not clear what...Unknown:<br /><br />About right, except it is not clear what do you mean by a "computation".<br /><br />A computation in a proper sense is not mathematics. At the same time mathematicians often use it in a very wide sense, like in "computing the homotopy groups of spheres" – such things may involve very deep proofs and may require new definitions. In this sense "to Owlhttps://www.blogger.com/profile/13083493391293465339noreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-16285560248703187622017-04-28T17:37:57.108-03:002017-04-28T17:37:57.108-03:00There is a hierarchy of things a mathematician can...There is a hierarchy of things a mathematician can do<br /><br />(1) A computation<br />(2) A proof<br />(3) A useful definition <br /><br />It is (3) which is the most difficult and the people who don't get the point just don't have the experience. Anonymoushttps://www.blogger.com/profile/14544168852872718580noreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-7274703767493228832013-08-20T20:45:38.317-03:002013-08-20T20:45:38.317-03:00Dear Tamas Gabal,
I tried to answer your question...Dear Tamas Gabal,<br /><br />I tried to answer your question in the next post <a href="http://owl-sowa.blogspot.com/2013/08/previous-post-did-j.html" rel="nofollow">New ideas</a>.Owlhttps://www.blogger.com/profile/13083493391293465339noreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-43432928573420633312013-08-18T22:46:04.507-03:002013-08-18T22:46:04.507-03:00Dear Sowa, in your own experience, how often genui...Dear Sowa, in your own experience, how often genuinely new ideas appear in an active field of mathematics and how long are the periods in between when people digest and build theories around those ideas? What are the dynamics of progress in mathematics, and how various areas are different in this regard?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-14645701568976684472013-08-14T07:58:16.810-03:002013-08-14T07:58:16.810-03:00Dear Pete,
It is never an issue if some abstract ...Dear Pete,<br /><br />It is never an issue if some abstract theory can be replaced by "elementary" arguments. First, you always can unravel the abstract definitions and get a proof on a lower level. Then you remove everything not related to your theorem, and you may get a shorter proof in the end. There are also some genuinely elementary proofs, like the proof of the Prime Number Owlhttps://www.blogger.com/profile/13083493391293465339noreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-31042936583863515152013-08-13T07:44:17.839-03:002013-08-13T07:44:17.839-03:00The very basic applications of the probabilistic m...The very basic applications of the probabilistic method (such as the famous lower bound on diagonal Ramsey numbers) fit the pure-counting pattern you mention, and of course you can equally well write these arguments as counting arguments without bringing probability in.<br /><br />But almost everything proved via the probabilistic method does not reduce to counting structures, but rather weightedPetehttps://www.blogger.com/profile/02123359695496629332noreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-27535252346427022402013-08-08T04:20:09.394-03:002013-08-08T04:20:09.394-03:00Well, I believe that Alon-Spencer has that status ...Well, I believe that Alon-Spencer has that status of the best book in the field, or at least had it several years ago. This is one of the books I attempted to read. The reviews at Amazon describe the book correctly. The number of stars does not matter. They like such book, I don't. Everything is in an agreement: the status of the authors and of their book, Amazon reviews, my own experience Owlhttps://www.blogger.com/profile/13083493391293465339noreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-22358157885097237222013-08-07T21:26:13.803-03:002013-08-07T21:26:13.803-03:00Dear Sowa,
I was talking about the results in th...Dear Sowa, <br /><br />I was talking about the results in the Geometry chapter of "The Probabilistic Method" by Alon and Spencer. Other applications of nontrivial concentration inequalities are included in some other chapters as well. Unfortunately, this does not mean that I recommend this book to those who just want to expand their mathematical knowledge. In my opinion, the style of Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-83983814476545235512013-08-07T16:52:04.763-03:002013-08-07T16:52:04.763-03:00Dear Tamas Gabal,
There is no real need to “defen...Dear Tamas Gabal,<br /><br />There is no real need to “defend” the probabilistic method. My claim is not that it is some junk, but that it is what the term “probabilistic method” means. Namely, it is a <b>method</b>, and not a new concept. It is a technique useful for proving some theorems. I tried to study it at least two times from books considered as the best (the first one is, probably, not Owlhttps://www.blogger.com/profile/13083493391293465339noreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-65764197336601939762013-08-07T11:39:25.301-03:002013-08-07T11:39:25.301-03:00Dear Sowa,
Let me say a few words in defense of t...Dear Sowa,<br /><br />Let me say a few words in defense of the probabilistic method. I met applications of the probabilistic method when it is not applied to prove that an object satisfying certain properties exists but to estimate a cardinality of a collection of objects that satisfy certain properties. In such cases, we do not consider the objects being counted as random, then "count"Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-18147193030938438792013-08-06T00:42:30.461-03:002013-08-06T00:42:30.461-03:00Dear shopkins,
First of all, I do not believe any...Dear shopkins,<br /><br />First of all, I do not believe anymore that mathematics can be divided into "two cultures" of T. Gowers. I wrote about this already. A branch of mathematics can be more or less conceptual, like it can be more geometrical (i.e. based on the spatial intuition), or less geometrical. "Hungarian combinatorics" is a widely used phrase, describing an Owlhttps://www.blogger.com/profile/13083493391293465339noreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-85318853734634346942013-08-05T22:26:24.720-03:002013-08-05T22:26:24.720-03:00Sowa, do you believe that what makes a part of mat...Sowa, do you believe that what makes a part of math "conceptual" are only and ultimately definitions? Because I believe that techniques can themselves be conceptual. For example (and in contrast to your depiction of Hungarian combinatorics as antithetical to conceptual math), I see the probabilistic method as a conceptual advance. Its main content- that in order to show a configuration shopkinshttps://www.blogger.com/profile/02320295517689731413noreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-73959986059143880252013-08-05T21:54:49.406-03:002013-08-05T21:54:49.406-03:00Well, may be it is provocative. I just stated my p...Well, may be it is provocative. I just stated my preferences, and, moreover, preferences not intrinsic to the mathematics. But for whom this could be challenge? Lurie did what he did. Even if he proves Riemann Hypothesis next week, there will be no time to verify his solution before 2014 Congress. It is definitely not a challenge for Lurie. <br /><br />Probably, it is a challenge for the Owlhttps://www.blogger.com/profile/13083493391293465339noreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-25490656490383692952013-08-05T20:21:05.394-03:002013-08-05T20:21:05.394-03:00Dear Sowa, in my opinion, the most provocative sen...Dear Sowa, in my opinion, the most provocative sentence in your post was: "Moreover, in the current situation I rather prefer that he did not solve any well-known problems, if he will get a Fields medal." It almost sounds like a challenge! Also, I meant that the hand of the Committee are tied in the same sense as you described - it is a part of modern mathematical culture that "Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-35301162805154960392013-08-05T17:47:56.810-03:002013-08-05T17:47:56.810-03:00Dear Tamas Gabal,
I intentionally write in a pro...Dear Tamas Gabal,<br /><br />I intentionally write in a provocative manner quite often. This is the result of observing the online reaction to unconventional ideas of many people, including my. I decided to keep the current title of this blog much longer than I initially planned by similar reasons.<br /><br />The word "provocative" doesn't seem to be a purely negative Owlhttps://www.blogger.com/profile/13083493391293465339noreply@blogger.comtag:blogger.com,1999:blog-8062754651764668865.post-21054962157731469272013-08-05T10:05:02.964-03:002013-08-05T10:05:02.964-03:00Sowa, despite the provocative nature of your words...Sowa, despite the provocative nature of your words, I agree with you. On the other hand, perhaps, the hands of the Fields Committee are tied and they have to select the big-problem-solvers. But this is exactly the argument against the Fields medal.Anonymousnoreply@blogger.com