Mathematics and reputation

This great lecture by Terrence Tao on Mathematics in the age of AI has a lot of value. I agree with probably everything said there (how could I not 🙂) but I’d also add that a big part of mathematical practice is about rituals (giving invited lectures, organising conferences, giving medals/awards, …) which very often are about reputation. For better or worse, reputation is key in mathematics. If someone reputable says that somebody else did a good job with some paper then that makes it stand out and, consequently, the person who wrote the paper is now considered more reputable. Similarly, if somebody reputable says that some problem is difficult then that problem becomes more important (in the eyes of other mathematicians) and so on.

Reputation is hard fought, fragile (e.g. some people would not make public half-made results, being afraid that found errors might impact their reputation) and mathematicians fight viciously to protect it, with endless debates about authorship of mathematical theorems. Notably, some aspects of reputation are sometimes disputed. For example, there is an uneven representation of maths fields among Fields medallists. When was the last time a price was awarded for logic, btw? Cohen, you say? Why Shelah never got one? And what about all the brilliant people working in type theory, whose work essentially enabled Lean to exist? This has a lot to do with the fact that reputation is build on reputation itself. There might be a professor in some small university in Italy who is doing extremely difficult mathematics, making ingenuous constructions but she might never get recognised if her work never gets praised by some highly reputable mathematician. This makes sense, people can only understand how hard a problem is if they (or many others at least) tried to solve it and failed. Therefore, the professor in a small university in Italy cannot get recognised if she does not impress other mathematicians with something she could do and others couldn’t or if she comes up with tools that helps others solve their problems.

With all its flaws, I think reputation might be key ingredient in “the age of AI”. Mathematicians have something to learn from the other areas build on reputation. For example, AI has essentially “solved” Chess (and Go, poker, Arimaa, …) and still people still talk about the best human players, they compete in championships, the same as before and, similarly to mathematics, reputation plays a massive role in Chess. Notably, the situation in Chess is way simpler because there is a clear measure for reputation: the ELO system, tournament victories, recognitions as grandmasters and so on. However, I can imagine some pre-selection process to exist where in prestigious journals only accept papers for review from reputable enough authors. In fact, such process probably exists informally, but it might have to become formal. I am not sure, thought, how to go about it. Making reputation formal (e.g. by formalising what a “Math grandmaster” is), the same way as in Chess is probably the wrong approach and it might backfire quickly. On the other hand, reputation is an important aspect of mathematical practice and it will show up, whether hidden or explicitly mentioned, in the considerations of how to deal with AI.