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This is something I've never been satisfied with. If two people are playing chess, and one person consistently beats the other person, then it makes sense to say that one is better than the other at chess. But in math, you have people who are working on different problems, so what metrics are used to rank mathematicians and say, "This mathematician is stronger than that one"?

I can think of a few:

  • Mathematician A published more papers.
  • Mathematician A published papers in more prestigious journals.
  • Mathematician A proved more results that are of interest to other mathematicians
  • Mathematician A solved a problem that other mathematicians have tried to solve.
  • Mathematician A is a better collaborator -- perhaps works more quickly or gets along well with other people.

For the last two, the issue seems to be that "quality" of mathematics depends on working on what others are interested in. The whole thing doesn't really make much sense to me. If you come up with a problem that interests you, solve it, and write it up, isn't that a success? The reason I ask is because mathematicians seem to care a lot about the rankings. I grew up with the idea that mathematicians at higher-ranked universities are "better" than mathematicians at lower-ranked universities. The only way I can think of that makes sense is if two mathematicians are working on the same problem and one mathematician comes up with the answer first.

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    Does it make sense to rank mathematicians at all? Just because someone decides to care about it doesn't mean it makes sense. Might be worth thinking as well about the models used by, say, US News to rank institutions; recently it's been revealed that the models are specifically designed to place certain institutions at the top, so they're entirely tautological.
    – Bryan Krause
    Commented May 4, 2023 at 17:23
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    @cgb5436 Hiring isn't the same as ranking. Tenure evaluates an individual, no need to rank. Grants are based on the content of the grant which includes the ability of the applicant to do what they say they'll do, but doesn't require you to rank people.
    – Bryan Krause
    Commented May 4, 2023 at 17:35
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    "The reason I ask is because mathematicians seem to care a lot about the rankings." Really? That hasn't been my experience at all. Certainly there may be some consensus about who the outstanding researchers are in a given subfield, but that doesn't mean that mathematicians generally go around ranking their colleagues on some imaginary leaderboard. I'm sorry, but this seems to be yet another variation of your persistently held idea (across many questions) that there is some sort of secret ranking/method/procedure that hiring committees use and being privy to it is key to success in academia. Commented May 4, 2023 at 17:49
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    @cgb5436 - the hiring example is not a strict ranking problem - there's no need for certainty in a strict linear order. Instead, it's a rough evaluation problem - you're just trying to give everyone a confidence interval on a line, a much easier problem because you have error tolerance built in. Commented May 4, 2023 at 19:30
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    @cgb5436 Questions usually work best when you ask about your actual problem, rather than your perceived solution. So, if your actual problem is "how can I know what a hiring department is looking for so that I can show them", asking instead "how do mathematicians rank other mathematicians" appears to be a case of the XY problem, and so far you've gotten a bunch of comments and at least one answer, but I don't think that answer will actually help you with your real question.
    – Bryan Krause
    Commented May 4, 2023 at 19:58

3 Answers 3

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Badly. Ranking people on producing useful things does not work with any kind of accuracy. It only works for games and sports because they are specifically designed to rank people. Even then, the rankings have poor predictive power in many cases.

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    :) Thinking of "moneyball" in baseball, for example? :) Commented May 4, 2023 at 17:50
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    Ranking by height can be done accurately. It just isn't useful by and large.
    – Jon Custer
    Commented May 4, 2023 at 18:44
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I have a feeling that the question was misunderstood. It is not uncommon for mathematicians to rank other mathematicians. In fact, almost anything in mathematics, whether it is universities, journals or people, are often being ranked for multiple purposes. For instance, most mathematicians will agree that the annals of mathematics, JAMS or inventiones are ranked higher than most other journals in pure mathematics, there are official rankings for universities (e.g. shanghai ranking), and, while ranking mathematicians is harder than ranking journals and universities, it is happening all the time (e.g. in hiring committees, prize committees, grants selections etc.).

It is not obvious how to rank one mathematician over another, and from my experience everyone has a different way of measuring the contribution of someone's work to mathematics. I think all of the metrics that you mentioned are evaluated and the preference of one over another depends completely on the individual (note that they may also depend on your field, for instance people who work in certain areas of combinatorics are often expected to publish more than people who work in, say, algebraic topology).

At the end of the day people look at your papers and where you published them, your grants and prizes, your university and often they also ask for opinions of other mathematicians in the field or the opinion of other mathematicians they trust. Therefore, anything you do to increase the quality of one of those aspect will be in your favor. Other than that you just need luck, you can never guarantee being the top choice of any committee.

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Let me begin by saying that in the phrase "rank other mathematicians" there are already two misconceptions. First, I don't think anyone tries to "rank", in terms of putting in an absolute order, except in the context of ranking candidates for a position, which is quite different and involves consideration of many factors, most of which are irrelevant to your question. But do we (informally and privately) evaluate? Yes, I imagine most people do.

Secondly, one can't really evaluate mathematicians, only their output. There are many reasons why someone who is (insofar as we can really make the comparison) a "better" mathematician than someone else might still achieve less, either in total or per unit time.


Having said all that, how do we evaluate other mathematicians' output? Any such evaluation is necessarily somewhat subjective - much more like comparing artists than comparing chess players.

As you suggest, it is about what they have published, and often where their papers appear, who else has worked on the same problems, etc. There are fairly good reasons for this.

It is much easier to make these evaluations for research that is close to your own area of expertise. Here you can read the papers and get a clear sense of how impressive they are, both in terms of significance of results and in terms of how difficult or brilliant the proofs are. Of course, there is still the difficulty with collaborative papers that you cannot tell who contributed what. But you can perhaps get a sense from the overall output, e.g. if it is almost all collaboration with one or two more senior people that it may not actually be as impressive as it appears.

Now if you are looking at someone further from your own area, you are probably reliant on quality of journals as a proxy for this. This is much better than trying to evaluate the papers yourself. The journal they end up in depends on the (subjective, somewhat unreliable, but informed) opinion of experts in the area, and this is normally a better guide than your own (subjective, somewhat unreliable and uninformed) opinion. There is the disadvantage, however, that it is much easier to get into top journals in some areas of math than in others.

Whether someone else has worked on the problem before, or is interested in the results, and who that someone else is, is very relevant. This is because the purpose of writing a paper is to advance mathematical knowledge, not to prove you can do mathematics. If other people are interested, then you are doing good mathematics, and if other (good) people have worked on the problem in the past without solving it, then you must be doing impressive mathematics. If you come up with your own problem and solve it - an "answer to a question no-one asked", to quote Ned Flanders - then that probably doesn't really advance mathematical knowledge, and gives no idea of how difficult it was. If you come up with enough problems, eventually one will turn out to be relatively easy, and if no-one else is interested there is no competition with anyone else trying to solve it first. It is the problems already known to be difficult and believed to be interesting (to paraphrase William Morris) that are worth solving.

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    " If you come up with your own problem and solve it - an "answer to a question no-one asked", to quote Ned Flanders - then that probably doesn't really advance mathematical knowledge" That is very much a matter of mathematical taste. Certainly coming up with a good question that no-one has thought of asking before can be a non-trivial contribution that advances mathematical knowledge. You're mainly taking the perspective of a problem-solver, which, while valid, is certainly not the only perspective one can take. (ctd) Commented May 6, 2023 at 13:02
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    You talk about "advancing mathematical knowledge", which I agree is the main goal here, but then you measure that by the difficulty of the problem being solved. These are two very different things. Certainly, if you solve a difficult problem that many people have tried to solve before, then you advance mathematical knowledge. But the converse is not true: you can equally well achieve this by doing something simple or by asking a new question, even if it turns out not to be difficult to answer. The idea that progress in mathematics is measured by the difficulty of proofs is very much false imo. Commented May 6, 2023 at 13:05

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