Previous post: The value of insights and the identity of the author.
This is a partial reply to a comment by vznvzn in Gowers's blog.
Combinatorics is most resolutely not "a new way of
looking at mathematics". It is very old, definitely known for hundreds
years. Perhaps, it was known in the ancient Babylon already.
And Erdős is not a "contrarian". His work belongs to the
most widely practiced tradition in analysis. As a crude approximation, one can
say that this tradition originates in the calculus of Leibniz, which is quite
different from the calculus of Newton. Even most of mathematicians are not
aware of the difference between the Leibniz calculus and the Newton calculus.
This is not surprising at all, since only the Leibniz calculus is taught
nowadays.
It is the Grothendieck's way of looking at mathematics, the one which I
advocate, which is new. This new, conceptual, way of doing mathematics immediately met strong resistance.
And in some cases its opponents won. For example, the early work of
Grothedieck in functional analysis had no influence till analysts managed to
translate part of his ideas into their standard language. It seems that only quite
recently some of analysts realized that a lot was lost in this translation, and
done a better translation, closer to the spirit of the original work of
Grothendieck.
Another example is provided by the invasion of this new style and even some
technical concepts developed in this style into the analysis of several complex
variables. This was intolerable for the classical complex analysts, and they
started to stress problems about which it was more or less clear that they can
be approached by familiar methods. They succeeded, and already in the
1970ies a prominent representative of the classical school, W. Rudin, was able
proudly say that Grothendieck's methods (he was more specific) disappeared into
background. He did not publish his opinion at the time, but attempted to insult
a prominent representative of the new style, A. Borel by such statements. A quarter
of century (or more) later he told this story in an autobiographical book. (W.
Rudin is a good mathematician and the author of several exceptionally good
books, but A. Borel was a brilliant mathematician.)
Now we are observing a much broader attempt, apparently led by T. Gowers, to
eliminate the conceptual way of doing mathematics completely. At the very
least T. Gowers is the face of this movement for the mathematical
public. After this T. Gowers envisions an elimination of the mathematics
itself by relegating it to computers. It looks like the second step is the one
most dear to his heart (see the discussion in his blog about a year ago). It
seems that combinatorics is much more amenable to the computerization
(although I don't believe that even this is possible) than the conceptual
mathematics.
Actually, it is not hard to believe that computers can efficiently
produce proofs of a wide class of theorem (the proofs will be unreadable to
humans, but still some will consider them as proofs). But for the conceptual
mathematics it is the definition, and not the proofs, which is important. The
conceptual mathematics is looking for new definitions interesting to humans.
The proof and theorems serve as a stimulus for work and as a necessary testing
ground for new definitions. If a new definition does not help to prove new
theorems or to simplify the proofs of old ones, it is not interesting for
humans.
There is only one way to get rid of the conceptual mathematics, namely, the
Wigner shift of the second kind. The new generation should be told that combinatorics
is new, that it is the field to work in, and very soon we will see the
young people only the ones doing combinatorics. Since mathematics is to a huge
extent "a young people’s game", such a shift can be
accomplished very quickly.
P.S. It is worth to note that there are two branches of combinatorics, and one of them is already belongs to the conceptual mathematics. Some people (like D. Zeilberger) are intentionally ignoring this to promote the non-conceptual kind.
Next post: D. Zeilberger's Opinions 1 and 62.
sowa, you crazy troll you. couldnt believe you actually replied to my gowers offhand remark, taking it seriously.
ReplyDeleteastonishing! this rant actually makes quite an alarming bit of sense in places! its actually exactly along the lines Ive been thinking of how there could be an interplay between machine proofs and human conceptualization. except I dont imagine it as competitive, but in fact as complementary and a synergistic/symbiotic relationship. "cross pollination"! the essence of math.
as for machine proofs, what a coincidence, think I am very close to a machine-driven proof of the collatz conjecture, but havent found anyone interested in being the 1st to review it. from the way you write, you are surely an old geezer, verging on a curmudgeon, but I say to you, do you have the cojones to try something new? if so reply on my blog :p
Dear vznvzn,
DeleteI am glad to see you here.
But I have no idea why you are constantly trying to offend me? Try to count how many words having no other purpose but to offend you managed to insert in our 8 (or, rather, 7) lines. If you really believe that I am a troll, your visit here makes no sense. Even following the link to this post makes no sense, not to say about commenting.
Please, notice that there is no reason to think I took your remarks seriously. As I said in Gowers's blog, I was interested in commenting only one aspect of your rant. You are not the only one who expressed the opinion that combinatorics is a new way of thinking. Later on A. Granville posted a comment containing a more elaborate version of the same idea. The question if Gowers-like mathematics is something new or just a continuation of centuries, if not thousands, years tradition, is important. This post was written, frankly, not for you, but for very young people, who often uncritically trust any such declaration, But at least one phrase in your comment here indicates that, maybe, you are sufficiently open-minded to discuss such questions. May be you even can do it seriously, without futile attempt to offend me.
I would be very happy to try something new. But if I decide to quit mathematics (not just move to another branch of it), I will move to something much more distant than machine-driven proofs. Most likely, it will be not mathematics and even not science or engineering. And I see no reasons to be interested in Collatz conjecture. It looks like a nice problem for a mathematical Olympiad, but, apparently, it is too difficult for this. Well, it is a nice toy for high school kids, but what else? One can invent zillions of such problems, and if you attach some monetary value for one of them, and do a decent PR work, it will be popular at the periphery of mathematics, like the Collatz conjecture is.
May be you are just celebrating April Fouls’ Day?
no, comment defn is not an april fools joke! but the same assertion would be questionable as to your entire blog site here!
ReplyDeleteas for computer-assisted proofs, someday only fools will not take it seriously! actually that day is already here as critical and deep proofs already exist that cant be done without computers. (kepler conjecture, four color theorem, etc).
if you will humor me for just a moment, think of the most trivial problem you can think of, such that solving it, led to some of the deepest new theory possible. you know, exactly what you just said of collatz could have been said about FLT at any point in its 3.5 century unresolved lifetime and yet you cry elsewhere that wiles was not awarded a fields medal. right?
so, in this vein, am convinced that a particular technique Ive discovered to solve the collatz problem will lead to grander vistas. put a comment about it on gowers blog on the szemeredi award post you pointed to. [skimmed the dialog on that]. may write a new blog on all this at some pt.
I meant not the comment in Gowers's blog, but your comment here, which was posted on April 1, according to Google. My blog was created not at the April 1. Whatever. I doubt that it looks like a joke to anyone.
DeleteThe rest is a separate post What is mathematics?
ReplyDeleteIn combinatorics (like other areas of mathematics) there are both deep conceptual aspects and hard technical issues. Combinatorialists mainly study concepts and problems that arise in combinatorics itself, but there are many connections with essentially all other areas of mathematics. Methods of probabilistic and extremal combinatorics not only revolutionized our own field but they are also fundamental in computer science. One of the wonderful aspects of combinatorial reasoning is that many deep phenomena from analysis, geometry, algebra, probability and other areas have remarkably simple combinatorial manifestations, and are often pre-discovered in combinatorics. Euler's famous formula (V-E+F=2) is an early example, and Tutte's polynomials (already mentioned here) which gave early manifestation of crucial ideas from the later K-theory, and the much later spectacular Jones polynomial, as well as Tutte's theory for enumerating planar graphs are more recent examples. (Both these works of Tutte were motivated by the four-color conjecture.) Like other areas, combinatorics contains major developed theories on the one hand, and sporadic gems on the other. An example of a major theory within combinatorics that can be compared in term of breath, depth and difficulty to central developments in topology or in algebra, is the theory of graph minors including the Roberston-Seymour well-quasi-ordering theorem and subsequent developments. Gil Kalai
Dear Gil Kalai,
DeleteApparently, you comment is not related to this particular post, in which I argue that combinatorics is not a new way of looking at mathematics, and the way of Grothendieck is new.
It looks like you a simply defending combinatorics. Well, there is no need to defend it from me. I like it, although I am deeply convinced (and was convinced even during “affair with Szemerédi-Gowers mathematics") that there is nothing in combinatorics comparable with most of achievements of conceptual branches of mathematics. I will comment on your specific examples.
I discuss all these issues from the position of a pure mathematician. I don’t care about the computer science. Moreover, I suspect the computer science is neither science, nor mathematics, and is hardly relevant for designing and building computers. The physics of condensed matter is the science most useful for the progress of computers.
Euler’s formula is not combinatorics at all. Some its proofs have some combinatorial flavor, but, in fact, most of proofs are such, as I wrote already (in “My affair...”). Apparently, B. Grünbaum’s proof of Euler’s formula for convex polyhedra is pure geometry, without any combinatorics or topology hidden. And without serious restrictions on the considered polyhedra we have no other choice but to turn to algebraic topology (cf. the famous book of I. Lakatos).
Tutte’s polynomial is a very nice topic. But I wouldn’t say that it “gave early manifestation of crucial ideas from the later K-theory”. Tutte’s polynomial is a predecessor of only one of the ideas of K-theory, and not the only predecessor. The work of Witt uses the same idea and precedes Tutte by at least 10 years. Actually, the standard construction of integer numbers for the natural ones is based on the same idea. The rest of K-theory is due to Grothendieck. And without these further ideas of Grothendieck K-theory is just a useless toy.
I disagree with your opinion about the Jones polynomial. By an unfortunate twist of the fate it is now “remarkable”. If it would be discovered without going through von Neumann algebras hardly anybody would know about it. L. Kaufmann was quite close to doing this (so close that it seems that he indeed discovered it, but did not hurried enough with the publication). Anyhow, Jones polynomial is a toy elevated to the status of a deep theory, and struggling already almost 30 years to find connections with another branches of mathematics or at least topology appropriate for such a presumably deep theory. I like this toy too. Nice topic for any kind of introductory lectures.
The theory of graph minors looks rather mysterious. I tried to find any explanation of why it is deep or important. I looked for answers in textbooks and papers. The best was a lecture by nobody else but László Lovász. At least he clearly explained what is the Robertson-Seymour well-quasi-ordering theorem. Surely, its proof is difficult. Actually, too difficult to be of the depth comparable with works of Serre, Grothendieck, etc.