The times of André Weil and the times of Timothy Gowers. 3

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Now we can hardly say that mathematics is a useless science in the sense of G.H. Hardy. It contributes to the exploitation in various ways. For example, the theory of stochastic differential equations, a highly sophisticated branch of mathematics, is essential for the financial manipulations leading to a redistribution of wealth from the middle class to the top 1% of the population. The encryption schemes, designed by mathematicians and implemented by software engineers, prevent access of the general public to all sorts of artistic and intellectual goods. This is a new phenomenon, a result of the development of the Internet.

There is no need to detail the enormous contribution of mathematics to the business of extermination; it is obvious now (this wasn’t known to the general public when A. Weil wrote his article).

Mathematicians are not as free now as they were at the times of André Weil. There are (almost?) no more non-mathematical jobs which will earn a decent livehood and will leave enough energy for mathematical research. This situation is aggravated by the fact that if someone is not employed by a sufficiently rich university, then he or she has no access to the current mathematical literature, which is mostly electronic now, and, if sold to individuals, then the prices are set to be prohibitive. The access to these electronic materials (which cost almost nothing to the publishers to produce) is protected by the above mentioned encryption tools. The industry of the scientific publishing does not have publishing as its main activity any more. Its main business now is the restricting access to scientific papers by a combination of encryption, software, and lobbying for favorable to this industry laws. The main goal pursued is the transfer of the taxpayers dollars to the pockets of its executives and shareholders (this topics deserves a separate detailed discussion).

There are no Nobel prizes in mathematics, but there are many others. The Norwegian Abel prize is specifically intended to be a “Nobel prize” in mathematics. Long before it was established (the first one was awarded in 2003), another prize, the Fields medal, achieved incredible prestige and influence in mathematics, despite the negligible monetary award associated with it. In contrast with the Nobel and Abel prizes, the Fields medal may be awarded only to “young” mathematicians. The meaning of the word “young” was initially not specified, but the mathematical establishment slowly arrived at a precise definition. Somebody is young for the purposes of awarding a Fields medal, if he did not achieved the age 41 in the year of the International Congress of Mathematicians, at which the medal is to be awarded. The Congresses are hold every 4 years (only World War II caused an interruption). So, the persons born in the year of a Congress have additional 4 year to work and to have their work recognized.

Even if this stupid rule would be discarded, the age limitation tends to reward fast people strong at applying existing methods to famous problems. The Fields medals (and many other prizes in mathematics) are usually awarded to the mathematician who made the last step in a solution of a problem, and only rarely to the one who discovered a new method or new line of thought. There are only little chances for “slow maturing work” to be rewarded by this most prestigious award (more prestigious by an order of magnitude than any other prize, except, may be, the Abel prize, which is up to now was awarded almost exclusively to the people of the retirement age).

It was possible to ignore all the prizes in 1948. The Fields medals were awarded only once, in 1936, to two mathematicians. Other prizes, where they existed, did not carry any serious prestige. But in 1950, 1954, and 1958 Fields medals went to exceptionally brilliant mathematicians, and since then this was a prize coveted by anybody who thought that there is a chance to get it.

Now there are many other prizes, each one striving to carry as much weight and influence as possible. An interesting example is the story of the Salem prize. The Salem prize was established by the widow of Raphaël Salem in order to encourage work in Salem's field of interest, primarily the theory of Fourier series. Note that Fourier series and their versions are used throughout almost whole mathematics; it is only natural to think that the prize was intended to mathematicians working on problems really close to Salem’s interests. The international committee (occasionally changing by an unknown to the public mechanism) gradually increased the scope of the prize. By 1991 no connection with Salem’s interests could be observed. Now it is the most prestigious prize for young analysts without any restrictions (and the analysis is understood in a very broad sense).

In fact, this change (as also a suspected preference for mathematicians belonging to one or two particular schools) was not welcomed by Salem’s family, and it withdraw the funding for the prize. The committee did not inform the mathematical public about these events and continued to award the prize with $0.00 attached. I am not aware of the current situation; may be the committee managed to raise some money. (Please, note that I cannot name my sources, as it is often the case in the news reporting, and hence cannot provide any proof. I can only vouch that my sources are reliable and well informed.)

The negligible monetary value of most mathematical prizes is not of any importance. The prestige is immediately transformed into the salary rises, offers from rich universities capable of doubling the salary, etc. The lifetime income could be increased by a much bigger amount than the monetary value of a Nobel Prize.

These are the signs of the lost innocence directly related to the article of André Weil. There are many other signs, and one can talk about them indefinitely. In any case, there are no more ivory towers for mathematicians; their jobs depend on many complicated and not always natural implicit agreements in the society, various laws and regulations detailing the laws, etc. From 1945 till about 1985 all these agreements and laws worked very favorably for mathematics. But, as it turned out, the same laws and understandings could be easily used to control mathematicians, sometimes directly, sometimes in hardly discernible ways, and the same arguments that were used to increase the number of jobs 60 or 50 years ago, could, in principle, be used to eliminate these jobs completely.


Next post: My affair with Szemerédi-Gowers mathematics.

The times of André Weil and the times of Timothy Gowers. 2

Previous post: The times of André Weil and the times of Timothy Gowers. 1.


Different people hold different views about the future of humankind, even about the next few decades. No matter what position is taken, it is not difficult to understand the concerns about the future of the human race in 1948. They are still legitimate today.

It seems to me that today we have much more evidence that we may be witnessing an eclipse of our civilization than we had in 1948. While the memories of two World Wars apparently faded, these wars are still parts of the modern history. The following decades brought to the light many other hardly encouraging phenomena. Perhaps, the highest point of our civilization occured on July 20, 1969, the day of the Apollo 11 Moon landing. While the Apollo 11 mission was almost purely symbolic, it is quite disheartening to know that nobody can reproduce this achievement today or in a near future. In fact, the US are now not able to put humans even on a low orbit and have to rely on Russian rockets. This does not mean that Russia went far ahead of the US; it means only that Russians preserved the old technologies better than Americans. Apparently, most of western countries do not believe in the technological progress anymore, and are much more willing to speak about restraining it, in contrast with the hopes of previous generations. Approximately during the same period most of arts went into a decline. This should be obvious to anybody who visited a large museum having expositions of both classical and modern arts. In particular, if one goes from expositions devoted to the classical arts to the ones representing more and more modern arts, the less people one will see, until reaching totally empty halls. It is the same in the New York Museum of Modern Art and the Centre Pompidou in Paris.

Mathematics is largely an art. It is a science in the sense that mathematicians are seeking truths about some things existing independently of them (almost all mathematician feel that they do not invent anything, they do discover; philosophers often disagree). It is an art in the sense that mathematician are guided mainly by esthetic criteria in choosing what is worthwhile to do. Mathematical results have to be beautiful. As G.H. Hardy said, there is no permanent place in the world for ugly mathematics. In view of this, the lesson of the art history are quite relevant for mathematicians.


How Timothy Gowers sees the future of mathematics? He outlined his vision in an innocently entitled paper “Rough structure and classification” in a special issue “Visions in Mathematics” of “Geometric and Functional Analysis”, one of the best mathematical journals (see Geom. Funct. Anal. 2000, Special Volume, Part I, 79–117). Section 2 of this paper is entitled “Will mathematics exists in 2099?” and outlines a scenario of gradual transfer of the work of mathematicians to computers. He ends this section by the following passage.

“In the end, the work of the mathematician would be simply to learn how to use theorem-proving machines effectively and to find interesting applications for them. This would be a valuable skill, but it would hardly be pure mathematics as we know it today.”

Surely, this will be not mathematics. This prognosis of T. Gowers is even gloomier than the one which was unthinkable to A. Weil. The destiny of mathematics, as seen by Gowers, is not to be just a technique in the service of other techniques; its fate is non-existence. The service to other techniques will be provided by computers, watched over by moderately skilled professionals.

We see that nowadays even mathematicians of his very high stature do not consider mathematics as necessary, and ready to sacrifice it for rather unclear goals (more about his motivation will be in the following posts). Definitely, an elimination of mathematics as a human activity will not improve the conditions of human life. It will not lead to new applications of mathematics, because for applications mathematics is not needed at all. Mathematics is distinguished from all activities relaying on it by the requirement to provide proofs of the claimed results. But proofs are not needed for any applications; heuristic arguments supported by an experiment are convincing enough. André Weil and, in fact, most of mathematicians till recently considered mathematics as an irreplaceable part of our culture. If mathematics is eliminated, then a completely different sort of human society will emerge. It is far from being clear even that the civilization will survive. But even if it will, are we going to like it?

This is the main difference between the times of André Weil and the times of Timothy Gowers. In 1948 at least mathematicians cared about the future of mathematics, in 2012 one of the most influential mathematicians declares that he does not care much about the very existence of mathematics. Timothy Gowers is not the only mathematician with such views; but nobody of his stature in the mathematical community expressed them so frankly and clearly. He is a very good writer.


Next post: The times of André Weil and the times of Timothy Gowers. 3.

The times of André Weil and the times of Timothy Gowers. 1

Previous post: A reply to some remarks by André Joyal.


This is the first in a series of posts prompted by the award of 2012 Abel Prize to E. Szemerédi. He is, perhaps, the most prominent representative of what is often called the Hungarian style combinatorics or the Hungarian style mathematics and what until quite recently never commanded a high respect among mainstream mathematicians. At the end of the previous millennium, Timothy Gowers, a highly respected member of the mathematical community, and one of the top members of the mathematical establishment, started to advance simultaneously two ideas. The first idea is that mathematics is divided into two cultures: the mainstream conceptual mathematics and the second culture, which is, apparently, more or less the same as the Hungarian style combinatorics; while these two styles of doing mathematics are different, there is a lot parallels between them, and they should be treated as equals. This is in a sharp contrast with the mainstream point of view, according to which the conceptual mathematics is incomparably deeper, and Hungarian combinatorics consists mostly of elementary manipulations with elementary objects. Here “elementary” means “of low level of abstraction”, and not “easy to find or follow. The second idea of Gowers is to emulate the work of a mathematician by a computer and, as a result, replace mathematicians by computers and essentially eliminate mathematics. In fact, these two ideas cannot be completely separated.

In order to put these issues in a perspective, I will start with several quotes from André Weil, one of the very best mathematicians of the last century. Perhaps, he is one of the two best, the other one being Alexander Grothendieck. In 1948 André Weil published in French a remarkable paper entitled “L’Avenier des mathémathiques”. Very soon it was translated in the American Mathematical Monthly as “The Future of Mathematics” (see V. 57, No. 5 (1950), 295-306). I slightly edited this translation using the original French text at the places where the translation appeared to be not quite clear (I don’t know if it was clear in 1950).

A. Weil starts with few remarks about the future of our civilization in general, and then turns to the mathematics and its future.

“Our faith in progress, our belief in the future of our civilization are no longer as strong; they have been too rudely shaken by brutal shocks. To us, it hardly seems legitimate to “extrapolate” from the past and present to the future, a Poincaré did not hesitate to do. If the mathematician is asked to express himself as to the future of his science, he has a right to raise the preliminary question: what king of future is mankind preparing for itself? Are our modes of thought, fruits of the sustained efforts of the last four or five millennia, anything more than a vanishing flash? If, unwilling to stumble into metaphysics, one should prefer to remain on the hardly more solid ground of history, the same question reappear, although in different guise, are we witnessing the beginning of a new eclipse of civilization. Rather than to abandon ourselves to the selfish joys of creative work, is it not our duty to put the essential elements of our culture in order, for the mere purpose of preserving it, so that at the dawn of a new Renaissance, our descendants may one day find them intact?”

“Mathematics, as we know it, appears to us as one of the necessary forms of our thought. True, the archaeologist and the historian have shown us civilizations from which mathematics was absent. Without Greeks, it is doubtful whether mathematics would ever have become more than a technique, at the service of other techniques; and it is possible that, under our very eyes, a type of human society is being evolved in which mathematics will be nothing but that. But for us, whose shoulders sag under the weight of the heritage of Greek thought and walk in path traced out by the heroes of the Renaissance, a civilization without mathematics is unthinkable. Like the parallel postulate, the postulate that mathematics will survive has been stripped of its “obviousness”; but, while the former is no longer necessary, we couldn't do without the latter.”

““Mathematics”, said G.H. Hardy in a famous inaugural lecture “is a useless science. By this I mean that it can contribute directly neither to the exploitation of our fellowmen, nor to their extermination.

It is certain that few men of our times are as completely free as the mathematician in the exercise of their intellectual activity. ... Pencil and paper is all the mathematician needs; he can even sometimes get along without these. Neither are there Nobel prizes to tempt him away from slowly maturing work, towards brilliant but ephemeral result.”

One of the salient points made by A. Weil in this essay (and other places) is the fragility of mathematics, its very existence being a result of historical accident, namely of the interest of some ancient Greeks in a particular kind of questions and, more importantly, in a particular kind of arguments. Already in 1950 we could not take for granted the continuing existence of mathematics; it seems that the future of mathematics is much less certain in 2012 than it was in 1950.


Next post: The times of André Weil and the times of Timothy Gowers. 2.

A reply to some remarks by André Joyal

Previous post: The first post.


I am a fairly surprised by your criticism of Bourbaki work (I would be not surprised by a critique of Bourbaki from many other mathematicians). I hardly can say anything new in the defense of Bourbaki. The best defense was, probably, provided by J. Dieudonne. And one should not forget that, as some other former member of Bourbaki pointed out, that Bourbaki books succeeded in changing the style in which books in mathematics are written. There is no more such an urgent need in rigorous and systematic expositions as was in France before WWII, because many books by other authors are both rigorous and systematic.

Of course, applications are absent from Bourbaki books, as they are absent from almost all books in pure mathematics. I doubt that I ever read or used any book in pure mathematics discussing applications. And since I think that pure mathematics is not needed for any applications, I believe that it is better to separate them.

The issue of the presumed absence of motivation in Bourbaki books is more subtle. There are several sorts of motivations. A reference to applications is a common but very poor motivation to study proofs, because only statements and formulas are needed for applications. It seems that the most widespread sort of motivation is a pseudo-historical one: one invents some way how mathematicians could in principle arrive at a result or definition; in the best cases it resembles the actual history. Very often such a motivation uses tools developed only later and, moreover, under the influence of the original discovery. I consider such motivations as very misleading. The only true motivation in mathematics is the real history of the problem or of the theory at hand. I learned this point of view from my Ph.D. thesis adviser in the context of writing introductions to research papers, and later I saw that it applies to expository writings, textbooks, everywhere. Using it in the teaching or learning is possible, but by purely practical reasons only occasionally. This approach requires such amount of time that it will defer own research by decades. Another effective form of motivation is apparently already forgotten in the western countries. It is simply the trust to a more experienced person than you. If you are told that you will need to know the derived categories or it just worth to know about them, this should be a sufficient motivation and you will study the theory of derived categories. Finally, the organization of the material could be self-motivating. I think that Bourbaki books are superb in this respect. They are, in fact, much more readable than most of mathematicians think. Many other books are also self-motivating without any special efforts.


Next post: The times of André Weil and the times of Timothy Gowers. 1.

The first post

This is mainly a test of the feel and look of this particular Google template and my customizations of it. A part of the test is the following picture of a little piece of my life.

 

This close-up photograph was done by me in my office by an outdated cell phone.

So, I am the © copyright holder of this silly picture!

Most of this stuff should either go trash, or be destroyed because it may contain some private information (not about me), or returned to library (now I am very reluctant to return books to the library; they have very high chances to end up in a trash truck there), etc. And some things are valuable. The main problem is to sort out what belong to what category.

Probably, this is sufficient for a test. Good luck!

Next post: A reply to some remarks by André Joyal.