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.

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