Again, this post is a long reply to questions posed by ACM. It is a complement to the previous post "Conceptual mathematics vs. the classical (combinatorial) one". The title is intentionally similar to the titles of three well known books by F. Klein.

First, the terminology in “Conceptual mathematics vs. the classical (combinatorial) one” is my and was invented at the spot, and the word "classical" is a very bad choice. I should find something better. The word

*"conceptual"*is good enough, but not as catchy as I may like. I meant something real, but as close as possible to the Gowers's idea of*"two cultures"*. I do not believe in his theory anymore; but by simply using his terms I will promote it.Another choice, regularly used in discussions in Gowers's blog is

*"combinatorial"*. It looks like it immediately leads to confusion, as one may see from your question (but not only). First of all (I already mentioned it in Gowers's blog or here), there two rather different types of combinatorics. At one pole there is the algebraic combinatorics and most of the enumerative combinatorics. R. Stanley and the late J.-C. Rota are among the best (or the best) in this field. One can give even a more extreme example, mentioned by M. Emerton: symmetric group and its representations. Partitions of natural numbers are at the core of this theory, and in this sense it is combinatorics. One the other hand, it was always considered as a part of the theory of representations, a highly conceptual branch of mathematics.So, there is already a lot of conceptual and quite interesting combinatorics. And the same time, there is Hungarian combinatorics, best represented by the Hungarian school. It is usually associated with P. Erdös and since the last year Abel prize is also firmly associated with E. Szemerédi. Currently T. Gowers is its primary spokesperson, with T. Tao serving as supposedly independent and objective supporter. Of course, all this goes back for centuries.

Today the most obvious difference between these two kinds of combinatorics is the fact that the algebraic combinatorics is mostly about exact values and identities, and Hungarian combinatorics is mostly about estimates and asymptotics. If no reasonable estimate is in sight, the existence is good enough. This is the case with the original version of Szemerédi's theorem. T. Gowers added to it some estimates, which are huge but a least could be written down by elementary means. He also proved that any estimate should be huge (in a precise sense). I think that the short paper proving the latter (probably, it was Gowers's first publication in the field) is the most important result around Szemerédi’s theorem. It is strange that it got almost no publicity, especially if compared with his other papers and Green-Tao's ones. It could be the case that this opinion results from the influence of a classmate, who used to stress that lower estimates are much more deep and important than the upper ones (for positive numbers, of course), especially in combinatorial problems.

Indeed, I do consider Hungarian combinatorics as the opposite of all new conceptual ideas discovered during the last 100 years. This, obviously, does not mean that the

**results**of Hungarian combinatorics cannot be approached conceptually. We have an example at hand: Furstenberg’s proof of Szemerédi theorem. It seems that it was obtained within a year of the publication of Szemerédi’s theorem (did not checked right now). Of course, I cannot exclude the possibility that Furstenberg worked on this problem (or his framework for his proof without having this particular application as the main goal) for years within his usual conceptual framework, and missed by only few months. I wonder how mathematics would look now if Furstenberg would be the first to solve the problem.One cannot approach the area (not the results alone) of Hungarian combinatorics from any conceptual point of view, since the Hungarian combinatorics is not conceptual almost by the definition (definitely by its description by Gowers in his

*“Two cultures”*). I adhere to the motto*“Proofs are more important than theorems, definitions are more important than proofs”*. In fact, I was adhering to it long before I learned about this phrase; this was my taste already in the middle school (I should confess that I realized this only recently). Of course, I should apply it uniformly. In particular, the Hungarian style of proofs (very convoluted combinations of well known pieces, as a first approximation) is more essential than the results proved, and the insistence on being elementary but difficult should be taken very seriously – it excludes any deep definitions.I am not aware of any case when

*“heuristic”*of Hungarian combinatorics lead anybody to conceptual results. The theorems can (again, Furstenberg), but they are not heuristics.I am not in the business of predicting the future, but I see only two ways for Hungarian combinatorics, assuming that the conceptual mathematics is not abandoned. Note that still not even ideas of Grothendieck are completely explored, and, according to his coauthor J. Dieudonne, there are enough ideas in Grothendieck’s work to occupy mathematicians for centuries to come – the conceptual mathematics has no internal reasons to die in any foreseeable future. Either the Hungarian combinatorics will mature by itself and will develop new concepts which eventually will turn it into a part of conceptual mathematics. There are at least germs of such development. For example, matroids (discovered by H. Whitney, one of the greatest topologists of the 20

^{th}century) are only at the next level of abstraction after the graphs, but matroids is an immensely useful notion (unfortunately, it is hardly taught anywhere, which severely impedes its uses). Or it will remain a collection of elementary tricks, and will resemble more and more the collection of mathematical Olympiads problems. Then it will die out and forgotten.I doubt that any area of mathematics, which failed to conceptualize in a reasonable time, survived as an active area of research. Note that the meaning of the word

*“reasonable”*changes with time itself; at the very least because of the huge variations of the number of working mathematicians during the history. Any suggestions of counterexamples?Next post: About Timothy Gowers.

A minor remark: when talking about (conceptual) combinatorics, one should also mention André Joyal's work on species, which can be seen as a conceptual foundation for combinatorics.

ReplyDeleteRecently, Jacob Lurie has taught a course on combinatorics (lecture notes are available from his homepage), and unsurprisingly he started with Joyal's species.

Of course. I would mention it if the completness would be one of my goals. This theory only supports my position that conceptual combinatorics already exists - it is in no way Hungarian.

DeleteDear sowa,

DeleteThanks for the answer. I now understand better your point of view. If you let me I would like to include the theory of cluster algebras (and also cluster categories) as an example of conceptual combinatorics. Actually, one of the motivations of my question was that I believe in the existence of conceptual combinatorics and wanted to clarify what your idea is in order to fix the discussion and now I see we agree in this point. I would like to make the following metaphor: mathematics is an alive organism which controls its own evolution even in the case when mathematicians insists to go against of what the mathematics are saying to us. This is my way to say that I agree with you when you say that a mathematical area which is not conceptualized eventually will die. In other words, mathematical ideas are in a first place than politics. Certainly, I am quite platonic in my understanding of mathematics.

As I said above, the completeness wasn't my goal. I usually try to use the most well known and accesiible examples. Perhaps, the work of H. Nakamura also should be mentioned (it may be even the best example). I don't know enough about cluster algebras in order to freely use them in examples. I still do not quite understand what is raison d'être for cluster algebra (I know that they appeared as a tool to solve a problem in representation theory, but, to the best of my knowledge, the problem is still not solved).

DeleteI agree with the rest, but not without qualifications. While the mathematics may live in a platonic world, we are definitely living in this one, in which politics plays a huge role and detremines what people do. We should not forget that for about 1000 years there was no mathematics in Europe at all. And mathematics in the proper sense, the subject for which the proofs are the feature determinig all others did not existed in other cultures till late 1800ies.

Given that the change of the meaning of words is one of the main political tools now, I wouldn't be surprized if in 2020 or 2030 the word "mathematician" will mean "the system administrator" assisting computers in proving theorems in combinatorics. There will be no word for conceptual mathematics at all. And if there is no word, there is no corresponding notion.

Is "H.Nakamura" intended to be "H.Nakajima"? Or else what works of Nakamura are you referring to?

DeleteDear Volodya,

DeleteI was asked about this already. In comment below there is a reference to a (freely accessible) talk by Procesi. Other references are there. Probably, I should mention only M. Haiman and H. Nakajima, but does it really matter?

My only concern was that the only H.Nakamura I am aware of has no connections to combinatorics whatsoever, should it be Hungarian combinatorics or any other one. I.Nakamura, on the other hand, may be an example, alongside with Haiman and Nakajima ;)

DeleteDear Volodya,

DeleteYou are right. I made a mistake/misprint. I always had in mind that Hakamura, which is referenced to in Procesi's talk (reference [IN]). He is I. Nakamura, not H. Nakamura. I thought that the reference to Procesi would be sufficient. Comments cannot be edited, so I cannot correct this mistake where it is first occurred.

As a combinatorialist, I'd add a few comments (mainly on the bits that aren't accurate).

ReplyDeleteFirst, the original bounds for Szemeredi's Theorem weren't so totally ridiculous (they were I think primitive recursive). Gowers' bound is something one can write down easily. The truth is likely quite close to the known (geometric) lower bound constructions (Behrend, improved by Elkin). A proof of this would imply the Green-Tao theorem directly from the Prime Number Theorem (though not accurate estimates on the number of APs in the primes, which they obtained more recently).

The lower bound Gowers proved (which is important) is on Szemeredi's Regularity Lemma. This is a lemma in the proof of Szemeredi's Theorem, and (in my opinion) is more important. It is (per your definition) conceptual combinatorics: it's trivial to prove once you have the right definition, and that definition is far from obvious (it's not obvious that any statement like the Regularity Lemma should even be true, and such statements weren't in any case formulated before Szemeredi). But all of the standard applications of the Regularity Lemma are either known or believed to have much better bounds than Gowers' lower bound. So the main point of that lower bound is as a no-go result: if you want good dependencies, don't use that proof method.

Your idea of the 'ideals' of 'Hungarian combinatorics' is plain nonsense. I don't care about proofs being 'elementary', however you define it. I certainly prefer simple proofs with clear ideas to difficult or complicated proofs (a corollary to this is that Furstenberg's proof is by any reasonable measure nicer than Szemeredi's, except that it doesn't contain a certain useful Lemma). But, I do not always know how to prove something simply, I can only see how to get there by a complicated route. And while I have some idea about how one can put a fair bit of the combinatorics I care about into a more conceptual framework (Razborov's flag algebras; Lovasz, Szegedy et al's graph limits) for the time being I do not know how to do this in a way that preserves most of the features I (personally) care about. Quite a few things I want to prove don't (yet?) have exact equivalents in these frameworks, and the approximate equivalent is not (as) interesting reinterpreted as a graph theory question.

As to the whole computer proofs stuff, I think it's very interesting to try, but I don't really expect any serious progress any time soon (this side of 2050 at least, say). Maybe we can get the computers to do (more of) the routine stuff for us, and perhaps it is true that this will benefit combinatorics more than algebraic geometry. But any important result contains some serious idea (and, even in `Hungarian combinatorics' this idea is often a definition), not just some old tricks put together a bit differently: and I don't think anyone can really do a good job of explaining to another person how to come up with these sorts of ideas, let alone explaining it to a computer.

Peter

Dear Pete,

DeleteIt seems to me that your position is close enough to my one. I don’t see any justification for using words like “plain nonsense”. I do not see what is the point of calling my post "not accurate" when you would like to see more details (which are known to you anyhow).

I did not say that Szemerédi’s estimates are “ridiculous”, but, apparently, everybody agrees that they are useless, and even T. Gowers uses this as a motivation of his work. The primitive recursive bound is not the one which one gets from Szemerédi’s proof. Sh. Shelah gave such a bound for the van der Warden’s theorem in 1987. Since Szemerédi’s proof is based on the van der Warden’s theorem (for which only the Ackermann-type estimates were available before Shelah), Shelah’s theorem automatically improves Szemerédi’s one. The Gowers’s estimate is given by a simple formula and by this reason alone is a huge improvement of previous ones.

I wrote in this post that I consider the Gowers’s lower estimate to be the most important result around Szemerédi’s theorem. And at some other point of this discussion I wrote that, apparently, Szemeredi's Regularity Lemma is much more useful (and hence more important) than Szemeredi's theorem itself. It seems that you agree. The proof of Szemeredi's Regularity Lemma is fairly accessible, and I studied it. I have to admit that I did not notice in the exposition I studied any definition trivializing the proof. It is rather convoluted. I suspect that I even do not understand what you mean by a “definition”. Could you elaborate?

My picture of the combinatorics is based on what Gowers wrote about it, in addition to what I actually studied in details. You call this picture “plain nonsense”, and the only arguments you offer are based on your own tastes. You see, for mathematicians working in fields even remotely related to my own, any interest in combinatorics, especially of the Hungarian type, is highly unusual. I was already reproached for such interests. I the past I was reproached for my interest in the mathematical logic and the set theory. I wouldn’t even try to offer my tastes as an argument. And, please forgive me that I trust Gowers’s description more than yours.

Finally, I agree with you opinion about computers. If you would like to have an opponent, you should look somewhere else. Of course, T. Gowers is the most natural choice.

I didn't mean that I think your entire post was not accurate, rather that a few points weren't. In particular, when you spend a couple of lines discussing bounds on Szemeredi's Theorem and then say 'He [Gowers] also proved that any estimate should be huge' it really looks like you think Gowers proved something about lower bounds in Szemeredi's Theorem, which isn't true.

DeleteThe 'nonsense' reference was to your assertion that Hungarian combinatorialists 'insist on' elementary but difficult proofs. This just is not true. Maybe we often (not always) cannot find a better way to prove what we want to prove, but if we use some complicated method it means we didn't see any other way, not that we somehow reject it as 'not elementary'.

With Szemeredi's theorem, you're right about the order of events, apologies. For my taste, I am not particularly interested in explicit bounds of tower type or larger, at this point I don't see too much difference between explicit but very far from true bounds and existence results such as Furstenberg's (In particular I wouldn't find the recent improvement to the Removal Lemma bounds due to Fox interesting except in that they show that one really doesn't need the full strength of the Regularity Lemma to prove the Removal Lemma).

As to the Regularity Lemma, the definitions relevant are of epsilon-regular pairs and partitions, and of the energy (or mean square density) of a partition. Of course you cannot even state the Lemma without the first two, though you could state for example the triangle removal lemma (for every epsilon there is delta such that all graphs on n vertices which cannot be made triangle-free by removing epsilon n^2 edges, contain delta n^3 triangles) without it. I (and many others) would be very interested if you could find a proof of this triangle removal lemma which doesn't use anything like the Regularity Lemma.

I don't really see how one can call the proof of the Regularity Lemma convoluted. Start with an arbitrary balanced partition into 1/epsilon parts. If your partition is not epsilon-regular take the Venn diagram of witnesses of irregularity as a second partition. Observe that (and this is really a simple calculation of a few lines with the defect Cauchy-Schwarz inequality) the energy of the second partition is greater by at least epsilon^5 than that of the first. Iterate until you reach an epsilon-regular partition, which iteration terminates since the energy of any partition is trivially bounded above by one.

I know there are various refined forms of the Regularity Lemma that ask for a lot of extra properties of the partition (which don't follow directly from the above argument) but these are not properties one ever really needs. It's just more convenient to work with the extra properties, and so it's worth once doing some routine extra work (within the argument) to establish those properties. Maybe this sort of routine work is what Gowers would like to be done by a computer in the future.

Dear Pete,

DeleteWe can try to start to split some hairs and argue about that phrase, for example. I thought that the word "around" indicates that the Gowers's lower bound is not about Szemerédi’s theorem, but about something around. But is this really important? Experts like you know the precise statements, and who else cares? On the other hand, I will be only thankful for friendly corrections. Such comments are bound to have mistakes.

You probably know about short popular article of T. Tao in the Notices of the AMS full of wrong attribution of results. I. Laba even wrote two pages of corrections.

It seems that for you the Regularity Lemma is your bread and butter. Sure, you do not think that its proof is convoluted. OK, here is one of the reasons why I think so. It is a lemma, not a Big Theorem. It is included in at least one graduate level textbook. I had plans to include it in a graduate course about graphs. After I studied the proof in details, I realized that I will ruin the whole course in this way. One term will be not enough for it alone. At the same time it is quite possible to use the same time to prove some nontrivial theorems, not lemmas, in homological algebra or in several complex variables. And I did some nontrivial stuff from graph theory. Perhaps, it is trivial for you, but I went as far as students let me.

So, experts suggest the Regularity Lemma as one of many topics suitable for a graduate course. I compare it not with the last papers in Annals, but with other topics for graduate courses.

Alternatively, I may compare this Lemma with the strongest form of the Atiyah-Singer index theorem. The latter has a very natural proof, easy to follow - if your background allows you to understand its statement.

For me epsilon-regular pairs and partitions, and the energy do not qualify as definitions. They are just some tools internal to a particular proof (and, of course, later – to its generalizations). The definitions I am taking about are very rare. I discussed a couple of them in a previous post, but from a slightly different angle. I mean the definitions of exterior forms on a vector space, and of differential forms. The construction of Lebesque measure is a definition. The notions of a category, of a functor, and of a natural tranformation of functors are definitions. The Tutte polynomial, if it is defined not by an explicit construction, but by few axioms, is a definition in the graph theory. Probably, the most important definition related to the graph theory is that of a matroid (and they are even not graphs!)

Regarding the comparison between combinatorics and other fields, I never cared too much about it. No matter how the comparison goes, the conclusion is that we, combinatorialists, should do our best, and if so why should we care too much about it. It is nice that Owl/Sowa has opinions about combinatorics but I dont see any reason to give these opinions much weight. I like both extremal/probabilistic combinatorics and algebraic combinatorics and I don't think Sowa's distinctions between and evaluation of different areas of combinatorics have much merit. I also like matroid theory but disagree with it being the "most important definition related to graph theory." In any case, since you, Sowa, are excited with Grothendieck's mathematics, why not write about it rather than (or at least in addition to) writing about combinatorics.

DeleteDear Unknown,

DeleteI do not understand to whom you writing, to Pete or to me. In any case, it is quite strange that you "don't see any reason to give these (i.e., my - Owl) opinions much weight" and still write here. Or you worry so much that I may convert Pete in my faith? If Pete knows who you are, why not to save him by writing to him directly? If not, your opinions, unsupported by any arguments and signed by "Unknown", do not carry any weight.

I truly like one thing in online discussions. Namely, the suggestions what I should write about. Probably, it will be a big surprize for you. You know, I write about whatever I like to. In fact, in this blog I was writing a lot about Grothendieck mathematics, but you did not even noticed this.

If you would like me to write about a topic of your choice, you need to hire me. For topics at least remotely related to this blog I currently charge $100.00 per hour.

So, please, keep in mind that the no more comments suggesting me what I should do will pass the moderation.

Hi Sowa, the unknown comment was by me and it was written to you. (The site did not give me the opportunity to put in my wordpress signature, but I did not mean to be unknown.) I went back quickly over your blog and did not find your earlier writings on Grothendieck's mathematics; maybe you can kindly give some links. I don't see anything wrong with raising suggestions for what you might write about, but if it bothers you I will not make any more suggestions. --Gil

DeleteDear Gil Kalai,

DeleteI do not care much about real names (but it is always pleasant to talk online with a well known by publications mathematician). The nicknames like "anonymous", "unknown" are simply confusing. By the way, Google accepts Wordpress as an OpenID, and here the link directs to your blog.

Well, this blog essentially starts with 3 posts "The times of André Weil and the times of Timothy Gowers 1, 2, 3." André Weil, of course, represents Grothendieck-Bourbaki-French style mathematics. When I speak about Gowers vs. Gronthendieck mathematics, I cannot do this without taking about the latter. So, there are many remarks here and there.

This blog was started as a place to present some background to my 2012 comments in Gowers's blog (to the post about the Abel prize, naturally). I did not finished this job yet when Gowers announced his publishing initiative. This announcement turned this blog into a blog about Timothy Gowers. See 4 posts in the last August, "The twist ending 1, 2, 3, 4." Taking about Grothendieck and his mathematics now fits this blog only to the extent it would be relevant to the main topic. So, if I will be inclined to do something in this direction, I will need to find another outlet. Perhaps, just my professional page.

And here I encounter a problem much more serious than my opinions about this or that sort of mathematics. A week or two ago I got from my employer (everybody got) a document substantially restricting our freedom of speech on the internet. This document is certainly illegal, but I don't have resources to bring this issue before the Supreme Court (any decision on a lower level will be short-living). And anyhow, many things are ilegal, but are done and go unpunished. We all are losing our freedoms, including academic freedoms, and losing them very fast. In the coming society neither Grothendieck's, nor Hungarian mathematics will matter. But FRS, Fields medallist Gowers will. I know, I already lived in such a society.

@Sowa: is Nakamura above a slip of the keyboard for Nakajima?

ReplyDelete@Pete: what do you mean by your use of the word combinatorialist? As others have already pointed out, Gel'fand's interests certainly included (what is widely regarded as) combinatorics, as represented, for example, by his book (with Kapranov and Zelevinsky) on discriminants etc. Of course, his interests also included algebraic geometry. Voisin has a personal but beautiful essay on being an algebraic geometer that describes her willingness to change her viewpoint on what she is studying; for me, it is this willingness to change viewpoints that characterizes the best people in that subject, and maybe the subject itself.

I am not convinced by "plain nonsense". What is your argument?

@xerxes: Partially, yes. It should be M. Haiman in the first place, as also I. Nakamura and H. Nakajima> I had in mind the work reported by C.Procesi in his talk "On the n!-conjecture", Séminaire Bourbaki, 44 (2001-2002) and later developments.

DeleteI find your opinions interesting, and agree with most of them. But I was wondering what you have to say on this post of Gowers http://gowers.wordpress.com/2013/04/14/answers-results-of-polls-and-a-brief-description-of-the-program/

ReplyDeleteDear Ravi,

DeleteWow! It will take some time to read this post, and much more time to find time to read it.

I the meantime, I would like to say that this "experiment" (it was not up to even very relaxed standards of social sciences) is mostly an experiment in psychology. Moreover, it is concerned mostly with the same issues as the mentioned by Gowers famous Milgram’s experiment. Milgram investigated the obedience to authority figures. The Gowers’s posts are ideal for a similar purpose: to investigate how much people trust to a person in an authority position; this time to a recognized expert in something they may even have no idea about.

I looked at the proofs in the first post with the intention to take part in the poll. I immediately realized that I cannot do this. I don’t see any difference between these proofs, much less any difference in exposition. The reason is that the statements and these proofs are uttely trivial, and I can comprehend each of proofs only as a whole. They are part of my most basic baggage for many years. I do not think why 2+2=4. The only thing I can tell you that this is correct. The same with these proofs.

My working hypothesis is that the people who so any differences are either not mathematicians or are at a very early stage of their mathematical education (normally, this stage should occur no later than at the age 12-14, but this is quite distorted in the modern society). Or they trust Gowers and see things that are not there.

hi "sowa". not surprised you devalue the recent gowers work! not really sure if the gowers code for ATP is valuable yet, unfortunately gowers seems not to have figured out how it differs from the very large body of work in the area. however, think his interest and promotion of the area is "way cool". heres a post on the subject which cites gowers, haken, your standout reactionism, the combinatorics controversy etc... adventures and commotions in automated theorem proving

ReplyDeleteSorry for the delay with publishing of your comment.

DeleteThanks for advertisement of my blog!

Unfortunately, you continue to ignore some key facts about which I told you. May be you not reading even the titles of my posts? You give a reference to my post "Combinatorics is not a new way of looking at mathematics", but still call combinatorics

"a relatively recent mathematical invention". May be I just don't understand the style of computer scientists: immediately after these word you give a reference to a paper by P. Cameron"Combinatorics entering the third millennium". By the way, if one expands the concept of combinatorics to the extent P. Cameron does in this note, about half of my work would be in combinatorics. Neither the work R. Borcherds, nor even the work for which T. Gowers was awarded Fields medals belong to combinatorics. The same applies to the work of Tits, to the ubiquity of Coxeter-Dynkin diagrams (their graph theory characterization is known from the very beginning and is of no help for addressing Arnold’s problem), etc. This is just an illustration of how desperate is the situation of people doing combinatorics: they need to claim that results from rather different, even in spirit, fields are combinatorics.You continue to call me a reactionary, while I am supporting a revolutionary way to do mathematics, still accessible to only few, and Gowers wants to refocus the mathematical community back to using very old ways of thinking, and, moreover, to the problems which are not interesting even by "pre-revolution" standards. 4-colors theorem is a great example. Its supposed solution did not resulted in any new developments in mathematics in general or in combinatorics properly.

To be continued in the next comment.

As of Gowers’s recent experiment in psychology, I did not found anything deserving a commentary. Initially I was inclined to take part in his poll. It turned out that I cannot do this. His presumably different proofs looked identical to me. The facts proved are utterly trivial and were known to me back in high school or even earlier. They are completely internalized long ago. As a result, I see each of these proofs as a single block and I am able to comprehend them only as a whole. As a result, the only thing I can say about each of these proofs is that it is correct. I am unable to distinguish between them. Apparently, Gowers thinks differently, in a linear manner, and comprehends proofs line after line. This way of thinking may be closer to CS.

DeleteIf Gowers made a striking new advance in CS and I did not react, this does not mean that I was

“stunned into total silence”. I am interested in CS only slightly more than any other person using computers in daily life. Neither I am interested in Gowers to the extent of following everything he writes. Before making any pronouncements about “total silence”, you could check the facts. Between September 20, 2012 and March 24, 2013 there was only 1 post in this blog, namely, “Happy New Year!”.Also, I would like to point out that I did not started the blog “Stop Timothy Gowers! !!!” about one year ago. I started a blog “Notes of an owl”, and changed the title on the day of presentation by Gowers of the works of an Abel Prize winner 3rd time in a row. And for the second time he was speaking about an area of mathematics about which he knew nothing by his own admission. This part is “reactionary” in a very direct sense: it is a reaction, namely, a part of my reaction to highly unethical behavior of FRS T. Gowers.

“Gowers is mostly refusing to take the bait...”Gowers never replies if he is unable to win immediately. The most notable example is not related with combinatorics vs. Grothendieck controversy at all. It is his reply to a question how he can take part in the selection of Fields medalists when he has no idea about the work of any of them even after the medals were handed out. His reply:“No comments”.hi again sowa. I think TG has done some awesome early work in attempting to identify a new mathematical strand/style but which is very subtle to discriminate because it is so recent, just like CS. his writing is controversial even within the mathematical community, agreed. but hes doing something fundamentally different than mathematics, yet it is valuable. hes like a mathematical commentator, a mathematical anthropologist. he looks at the big picture. hes probably a highly right-brained thinker, it shows in his style of writing in many ways, whereas much mathematics is left-brain focused.

ReplyDeleteso, I have sympathy for his project/program of analysis of different schools of thought. it is a very difficult undertaking, a very difficult project, but mathematics overall is strengthened by this analysis. its similar to doing mathematical history, but the problem is that history goes in long cycles and he's attempting to seize on a shorter cycle (less than a few decades, and bursting out/forth as we are alive right now). the shape is not yet distinct or clear, its partial, tentative, even embryonic at times. I cant see it fully clearly myself, nor can gowers, but hes succeeded in mapping it out more than anyone else, and his immense creativity [in writing, not merely confined to proofs] continues to pull it out.

it is not finished yet. it will not finish in our lifetimes probably. it appears hes identified a paradigm shift in mathematics before anyone else is much aware of it or able to remark on it. that is visionary! as the saying goes "the pioneers are the ones with the arrows in their back".

yet sowa, I admit you are a visionary too, but more of a reactionary visionary. you are brilliant & well versed enough with detailed math history to argue with his grand theme using specifics, and deep history, when nobody else really has much extended reaction or comment [yes there is a lot of good debate in comments on his blog]....

Hi! My reply is posted as About Timothy Gowers.

DeleteMany people said me that combinatorics has applications in computer science. However, do you know any application of recent (not results before personal computer was invented) combinatorics results of mathematicians (not computer scientists though technically one can say computer scientists are mathematicians)?

ReplyDeleteOh, why this comment stays here for years without a reply? Sorry.

DeleteNo, I don't. But, for one things, often it takes centuries for a mathematical result to find applications. For another, my standards are, probably, too high.

What is your name, please? Your anonymity is cowardly, not helpful. Shame on you.

ReplyDeleteShelley Seymour:

DeleteThat's too bad. I was about to post my real name here, but after your comment I will be not able to do this for a while.

And why do you care? According to your profile, your interests are very far from mathematics.

Anyhow, a lot of people know who is the author of this blog, and for a sufficiently broadly educated mathematician it is not very hard to guess (with some help from Google search).

Shelley Seymour:

DeleteIt would be much better if your husband would show up here himself, and with arguments instead of accusations.

Here my name is "Owl", and I don't have to tell any other name to anybody.

Dear Owl,

ReplyDeleteAs somebody who studies combinatorics and has some interest in CS, it seems obvious to me that the 'Hungarian combinatorial' perspective is and will continue to be useful to mathematics, and more generally the sciences, and more generally still, humanity. If you do not see the value of this perspective then that is ok, but don't you realize that there are many people that do see value in it? If you do realize this, then are you trying to discredit their belief, or are you trying to better understand where they are coming from?

Igor Balla:

ReplyDeleteWhy? I do see a value in 'Hungarian combinatorics'. I studies some parts of it not for the sake of writing this blog 10 or 20 years later. I know where they come from.

But I do not want it take even whole combinatorics, not to say about whole mathematics. I mean, to take over by political and administrative tools.

There was a Section of Combinatorics at 2014 Congress – as usual. There was 10 talks. 9 of them were devoted to the 'Hungarian combinatorics'. Only one talk wasn't clearly belonging to this type of combinatorics, but it was close in spirit to it. Other types of combinatorics were not represented.

Congresses have huge influence on what is considered to be important, and what to be not. And Sir Gowers certainly has a huge influence on the selection of speakers.

So, "they" managed to get rid of conceptual mathematics in one of the sections of the congresses. There is conceptual combinatorics, but many its practitioners are working now at fairly (or even very) obscure places. Not at Oxbridge.