Written: Aug. 12, 2018
Less than two weeks ago, the "World Cup" of Mathematics, took place in Rio, where four new inductees got Fields medals. Less than four weeks ago, the real World Cup, that of Soccer, took place, where France was declared the best team in the world. While billions of people could enjoy the Soccer World Cup, regardless of their soccer-playing ability, very few could appreciate and enjoy the "game" of the Math champions. At least, I could not. Below are links to talks of the four new Fields medalists that are easily found on YouTube.
While I could see that the brilliant speakers made a great effort to be lucid, (and except for Figalli, to my pleasant surprise, all were blackboard talks), I could not really see "what it is all good for". Of course the official citations, and the ICM press releases made something up, but the works of the four (admittedly brilliant) new Fields medalists are really championship play in four completely different games, with their own rules and cultures, and each of them can be really appreciated and judged by ≤ 100 people in the whole world (and that is probably not a sharp inequality).
I am willing to bet that each of these Fields medalists' understanding of the other ones' work is not far greater than mine. In our sins, the God of mathematics made mathematics into a Babel of an almost disjoint union of little mathematical tribes, where each tribe can't really understand, let alone appreciate, any of the languages of the other tribes.
I also went to arxiv.org and browsed a random sample of the new winners' papers. I did not understand any details, and while the content of their articles were very different, they all shared two things in common.
Lemma, Theorem, Proposition, Corollary, Remark, ...
we all grew up with, and forces the careful reader to linearly, i.e. line-by-line and word-by-word, go through the very difficult argument. A much more user-friendly format, and also more efficient for checking proof-correctness, is to use the structured format adopted in this paper. This paper is also accompanied by a software program that enables the reader to check every node empirically. This way, readers do not have to read every word, but can pick a random set of "nodes" (statements), and then randomly decide whether to follow the logical proof, or test it empirically on the computer.
These latter works combine human ingenuity (until further notice we still need it!) assisted by computer work, at least at the "end-game". Hence they are much deeper than the work of our new Fields medalists, and for that matter of all purely human-generated math (including Wiles'). More importantly, unlike the latter results, whose statements one can explain to five-year-olds, and get them to try them out themselves, the mainstream-highbrow human math output of the Fields medalists is so esoteric. Also, in my humble opinion, while purely human-generated, Peter Keevash should have gotten a Fields medal for this exciting work, whose statement (the definition of a [block] design) can be understood by everyone, proving Steiner's conjecture from 1853.
And this brings us to the most important point. Because of the high prestige of the Fields medal and of Fields medalists, we got stuck in a non-optimal vicious cycle, where brilliant young mathematicians are lured into working on these esoteric, and often contrived, in-bred problems worked by past Fields medalists, rather than working on truly fundamental problems that everyone can appreciate. For example, while I admire Robert Langlands, I am so tired of hearing about his program! It garnered too many Fields medals (at the expense of even more interesting subjects). Ditto for Optimal Control, algebraic geometry, topology, and the more esoteric parts of number theory. Also note that the members of the Fields medal committee are former Fields medalists (or people with the same mind-set), and naturally, they prefer work similar to theirs.
Today's mathematics is the way it is due to a historical accident: it started out without computers. Since humans can only do so much, when they could not do something, they made up things that they can do, and this lead to today's mainstream pure math culture, with its high priests who still get hung up on these esoteric problems. Let me make a suggestion: A Fields medal can only be awarded for work that can be explained to an advanced undergraduate, and if possible, a five-year-old!
Analogous things apply to the more minor ICM honors of Plenary and Invited Speakers. For every one who got this honor, there exist at least two people who deserve it even more, and once again the selection committees perpetuate their own culture and pick people with similar results and agendas as their own.
Let's break-up this vicious cycle, and make math fun again (sorry of the stupid allusion.)
P.S. See also my opinion written eight years ago
Doron Zeilberger's Opinion's Table of Content