Since about 2000, T. Gowers became a prominent advocate of two ideas. First, he works on changing the mathematical public opinion about relative merits of various mathematical results and branches of mathematics in favor of his own area of expertise. Second, he advocates the elimination of mathematics as a significant human activity, and a gradual replacement of mathematicians by computers and moderately skilled professionals assisting these computers. The second goal is more remote in time; he estimates that it is at least decades or even a century away. The first goal is already partially accomplished. I believe that his work toward these two goals perfectly fits the definitions 3a, 5a, and 5b from Merriam-Webster.
I would like to point out that public opinion about various branches of mathematics changes continuously and in a manner internal to the mathematics itself. An area of mathematics may be (or may seem to be) completely exhausted; whatever is important in it, is relegated to textbooks, and a research in it wouldn’t be very valued. Somebody may prove a startling result by an unexpected new method; until the power of this method is exhausted, using it will be a very fashionable and valuable direction of research. This is just two examples.
In contrast with this, T. Gowers relies on ideological arguments, and, as one may guess, on his personal influence (note that most of the mathematical politics is done behind the closed doors and leaves no records whatsoever). In 2000, T. Gowers published two essays: “Two cultures in mathematics” in a highly popular collection of articles “Mathematics: Frontiers and Perspectives” (AMS, 2000), and “Rough structures and classification” in a special issue “GAFA Vision” of purely research journal “Geometric and functional analysis”.
The first essay, brilliantly written, put forward a startling thesis of the existence of two different cultures in mathematics, which I will call the mainstream and the Hungarian cultures for short. Most mathematicians are of the opinion that (pure) mathematics is a highly unified subject without any significant division in “cultures”. The mainstream culture is nothing else as the most successful part of mathematics in the century immediately preceding the publication of the “Two cultures” essay. It encompasses almost all interesting mathematics of the modern times. The Hungarian culture is a very specific and fairly elementary (this does not mean easy) sort of mathematics, having its roots in the work of Paul Erdös.
The innocently titled “GAFA Visions” essay has as it central and most accessible part a section called “Will Mathematics Exist in 2099?” It outlines a scenario eventually leading to a replacement mathematicians by computers. The section ends by the following prediction, already quoted in this blog.
All arguments used to support the feasibility of this scenario are borrowed from the Hungarian culture. On the one hand, this is quite natural, because this is the area of expertise of Gowers. But then the conclusion should be “The work in the Hungarian culture would be simply to learn how to use Hungarian-theorems proving machines effectively”. This would eliminate the Hungarian culture, if it indeed exists, from mathematics, but will not eliminate pure mathematics."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 would hardly be pure mathematics as we know it today.”
This second project does not seem to be very realistic unless the mathematical community will radically change its preferences from favoring the mainstream mathematics to favoring the Hungarian one. And indeed, it seems that Gowers working simultaneously on both projects. He advocates Hungarian mathematics in his numerous lectures all over the world. He suddenly appears as the main lecturer on such occasions as the announcement of the Clay Institute million dollars prizes. It was a shock when he gave the main lecture about Milnor’s work at the occasion of the award of Abel prize to Milnor. Normally, such lectures are given by an expert in an area close to the area of the person honored. Gowers is in no way an expert in any of the numerous areas Milnor worked in. Moreover, he hardly had any understanding of the most famous results of Milnor; in fact, he consulted online (in a slightly veiled form at Mathoverflow.org) about some key aspects of this result. This public appearance is highly valuable for elevating the status of the Hungarian mathematics: a prominent representative of the last presents to the public some of the highest achievements of the mainstream mathematics.
The next year Gowers played the same role at the Abel prize award ceremony again. This time he spoke about his area of expertise: the award was given to a representative of Hungarian mathematics, namely, to E. Szemerédi. Be a presenter of a laureate work two year in a row is also highly unusual (I am not aware about any other similar case in mathematics) and is hardly possible without behind the closed doors politics. The very fact of awarding Abel prize to E. Szemerédi could be only the result of complicated political maneuvers. E. Szemerédi is a good and interesting mathematician, but not an extraordinary one. There are literally hundreds of better mathematicians. The award of the Abel prize to him is not an indicator of how good mathematician he is; it informs the mathematical community that the system of values of the mathematical establishment has changed.
How it could happen without politics that Gowers was speaking about the work of Milnor at the last year Abel prize ceremony? Gowers speaking about the work of Szemerédi is quite natural, but Gowers speaking about the work of Milnor (and preparing this presentation with the help of Mathoverflow) is quite bizarre. It is obvious that Gowers is the most qualified person in the world to speak about the works of Szemeredi, but there are thousands of mathematicians more qualified to speak about Milnor’s work.
Next post: The Politics of Timothy Gowers. 3.