On the greats and mathematical style

Lex Fridman and I discuss who is the greatest mathematician in history, and what are the different mathematical styles of undertaking mathematical investigation.

I sat down a little while ago for a sweeping conversation with Lex Fridman on infinity, paradoxes, philosophy, mathematics, and more.

The conversation turned at one point to the question of who has been the greatest mathematician of all time. I demurred a bit at the question—explaining that I don’t organize my thinking about mathematicians in such a ranked list, and find insight wherever it might arise, which isn’t always only from the greats—but I did eventually give an answer, which you can find out below. The question was an opportunity to talk about differing mathematical styles, including my own mathematical style, which has served me very well in my mathematical investigations.

Please enjoy this excerpt from our extended conversation. The transcript is below.

Lex Fridman(03:12:59) Sorry to ask the ridiculous question, but who is the greatest mathematician of all time? Who are the possible candidates? Euler, Gauss, Newton, Ramanujan, Hilbert. We mentioned Gödel, Turing, if you throw him into the bucket.

Joel David Hamkins(03:13:14) So this is, I think, an incredibly difficult question to answer. Personally, I don’t really think this way about ranking mathematicians by greatness. Um…

Lex Fridman(03:13:28) So you don’t have, like… You know, some people have a Taylor Swift poster in their dorm room. You don’t have it.

Joel David Hamkins(03:13:33) I mean, if you forced me to pick someone, it would probably be Archimedes because…

Lex Fridman(03:13:37) Archimedes

Joel David Hamkins(03:13:37) …he had such incredible achievements in such an early era, which totally transcended the work of the other people in his era. But I also have the view that I want to learn mathematics and gain mathematical insight from whoever can provide it and wherever I can find it. And this isn’t always just coming from the greats. Sometimes the greats are doing things that are just first and not… You know, somebody else could have easily been first. So there’s a kind of luck aspect to it when you go back and look at the achievements. And because of this progress issue in mathematics that we talked about earlier, namely we really do understand things much better now than they used to.

Joel David Hamkins(03:14:22) And when you look back at the achievements that had been made, then maybe you can imagine thinking, “Well, somebody else could’ve had that insight also.” And maybe they would have… It’s already a known phenomenon that disparate mathematicians end up proving essentially similar results at approximately the same time. But, okay, the person who did it first is getting the credit and so on.

Lex Fridman(03:14:48) What do you make of that? Because I see that sometimes when mathematicians… This also applies in physics and science, where completely separately, discoveries are made…

Joel David Hamkins(03:14:58) Right. Yeah.

Lex Fridman(03:14:58) …maybe at a very similar time. What does that mean?

Joel David Hamkins(03:15:01) It’s relatively common. I mean, I think it’s like certain ideas are in the air and being thought about but not fully articulated, and so this is the nature of growth in knowledge.

Lex Fridman(03:15:13) Do you understand where ideas come from?

Joel David Hamkins(03:15:16) Not really.

Lex Fridman(03:15:17) I mean, what’s your own process when you’re thinking through a problem?

Joel David Hamkins(03:15:22) Yeah, that’s another difficult question. I suppose it has to do with… My mathematical style, my style as a mathematician, is that I don’t really like difficult mathematics. What I love is simple, clear, easy-to-understand arguments that prove a surprising result. That’s my favorite situation. And actually, the question of whether it’s a new result or not is somehow less important to me. And so that has to do with this question of the greats and so on, whoever does it first. Because I think, for example, if you prove a new result with a bad argument or a complicated argument, that’s great because you proved something new. But I still want to see the beautiful, simple, because that’s what I can understand.

Joel David Hamkins(03:16:16) Also, I’m kind of naturally skeptical about any complicated argument because it might be wrong. And… …If I can’t really understand it fully, like every single step all at once in my head, then I’m just worried maybe it’s wrong. And so these different styles, sometimes mathematicians get involved with these enormous research projects that involve huge numbers of working parts and… …Different technology coming together. I mean, mathematical technology, not physical technology.

Lex Fridman(03:16:48) And sometimes it actually involves now more and more something like the Lean programming language where some parts are automated, so you have this gigantic…

Joel David Hamkins(03:16:54) Yeah, yeah, I see. Well, that’s another issue because maybe those things are less subject to skepticism when it’s validated…

Lex Fridman(03:17:02) Sure

Joel David Hamkins(03:17:02) …by Lean. But I’m thinking about the case where the arguments are just extremely complicated, and so I sort of worry whether it’s right or not, whereas you know, I like the simple thing. So I tend to have often worked on things that are a little bit off the beaten path from what other people are working on from that point of view.

Lex Fridman(03:17:23) Your curiosity draws you towards simplicity.

Joel David Hamkins(03:17:25) Yeah. I want to work on the things that I can understand and that are simple. Luckily, I’ve found that I’ve been able to make contributions that other people seem to like, in this way, in this style. So I’ve been fortunate from that point of view. My process always, though, and I’ve recommended this always to my students, is just a kind of playful curiosity. So whenever I have…

Joel David Hamkins(03:17:55) Whenever there’s an idea or a topic then I just play around with it and change little things or understand a basic case and then make it more complicated or press things a little bit on this side or apply the idea to my favorite example that’s relevant, and see what happens, or you just play around with ideas, and this often leads to insights that then lead to more methods or more, then pretty soon you’re making progress on the problem. So this is basically my method, is I just fool around with the ideas until I can see a path through towards something interesting… …And then prove that, and that’s worked extremely well for me. So I’m pretty pleased with that method.

Lex Fridman(03:18:47) You do like thought experiments where you anthropomorphize like you mentioned?

Joel David Hamkins(03:18:51) Yeah, yeah. So this is a basic tool. I mean, I use this all the time. You imagine a set-theoretic model, a model of ZFC, as like a place where you’re living, and you might travel to distant lands by forcing. This is a kind of metaphor for what’s going on. Of course, the actual arguments aren’t anything like that because there’s not land and you’re not traveling and you’re not…

Lex Fridman(03:19:13) But you allow your mind to visualize that kind of thing-

Joel David Hamkins(03:19:15) Yeah

Lex Fridman(03:19:15) … in the natural real world.

Joel David Hamkins(03:19:16) And it helps you to understand. Particularly when there are parts of the argument that are in tension with one another, then you can imagine that people are fighting or something. And those kinds of metaphors, or you imagine it in terms of a game theoretic, you know, two players trying to win. So that’s kind of tension. And those kinds of metaphorical ways of understanding a mathematical problem often are extremely helpful in realizing, aha, the enemy is going to pick this thing to be like that because, you know, it makes it more continuous or whatever, and then we should do this other thing in order to… So it makes you realize mathematical strategies for finding the answer and proving the theorem that you want to prove because of the ideas that come out of that anthropomorphization.

Lex Fridman(03:20:01) What do you think of somebody like Andrew Wiles, who spent seven years grinding at one of the hardest problems in the history of mathematics? And maybe contrasting that a little bit with somebody who’s also brilliant, Terence Tao, who basically says if he hits a wall, he just switches to a different problem and he comes back and so on. So it’s less of a focused grind for many years without any guarantee that you’ll get there, which is what Andrew Wiles went through.

Joel David Hamkins(03:20:30) Right.

Lex Fridman(03:20:30) Maybe Grigori Perelman did the same.

Joel David Hamkins(03:20:32) I mean, Wiles proved an amazing theorem, Fermat’s Last Theorem result is incredible. This is a totally different style than my own practice, though, of working in isolation. For me, mathematics is often a kind of social activity. I have… I counted, I mean, it’s pushing towards a hundred collaborators, co-authors on various papers and so on. And, you know, if anybody has an idea they want to talk about with me, if I’m interested in it, then I’m going to want to collaborate with them and we might solve the problem and have a joint paper or whatever. You want to have a joint paper? Let me-

Lex Fridman(03:21:06) Yeah, exactly. Let’s go.

Joel David Hamkins(03:21:08) So my approach to making mathematical progress tends to involve working with other people quite a lot rather than just working on my…

Joel David Hamkins(03:21:17) …own, and I enjoy that aspect very much. So I, personally, I couldn’t ever do what Wiles did. Maybe I’m missing out. Maybe if I locked myself, you know, in the bedroom and just worked on whatever, then I would solve it. But I tend to think that no, actually, being on MathOverflow so much and I’ve gotten so many ideas, so many papers have grown out of the MathOverflow conversations and back and forth. Someone posts a question and I post an answer on part of it, and then someone else has an idea and it turns into a full solution, and then we have a three-way paper coming out of that. That’s happened many times. And so for me, I enjoy this kind of social aspect to it. And it’s not just the social part.

Joel David Hamkins(03:22:01) Rather, that’s the nature of mathematical investigation as I see it, is putting forth mathematical ideas to other people and they respond to it in a way that helps me learn, helps them learn, and I think that’s a very productive way of undertaking mathematics.

Lex Fridman(03:22:20) I think it’s when you work solo on mathematics, from my outsider perspective, it seems terrifyingly lonely. And because you’re, especially if you do stick to a single problem, especially if that problem has broken many brilliant mathematicians in the past, that you’re really putting all your chips in. And just the torment… …The rollercoaster of day to day. Because I imagine you have these moments of hopeful break, mini breakthroughs, and then you have to deal with the occasional realization that, no, it was not a breakthrough, and that disappointment.

Lex Fridman(03:23:00) And then you have to go, like, a weekly, maybe daily disappointment where you hit a wall, and you have no other person to brainstorm with. You have no other avenue to pursue. And it’s, I don’t know, the mental fortitude it takes to go through that. But everybody’s different. Some people are recluse and just really find solace in that lone grind. I have to ask about Grisha Grigori Perelman. What do you think of him famously declining the Fields Medal and the Millennial Prize? So he stated, “I’m not interested in money or fame. The prize is completely irrelevant to me. If the proof is correct, then no other recognition is needed.” What do you think of him turning down the prize?

Joel David Hamkins(03:23:52) I guess what I think is that mathematics is full of a lot of different kinds of people. And my attitude is that, hey, it doesn’t matter. Maybe they have a good math idea, and so I want to talk to them and interact with them. And so I think the Perelman case is maybe an instance where, you know, he’s such a brilliant mind and he solved this extremely famous and difficult problem, and that is a huge achievement. But he also had these views about, you know, prizes and somehow, I don’t really fully understand why he would turn it down.

Lex Fridman(03:24:33) I do think I have a similar thing, just observing Olympic athletes that are, in many cases, don’t get paid very much, and they nevertheless dedicate their entire lives for the pursuit… … Of the gold medal. I think his case is a reminder that some of the greatest mathematicians, some of the greatest scientists and human beings do the thing they do, take on these problems for the love of it, not for the prizes or the money or any of that. Now, as you’re saying, if the money comes, you could use it for stuff. If the prizes come, and the fame, and so on, that might be useful. But the reason fundamentally the greats do it is because of the art itself.

Joel David Hamkins(03:25:13) Sure, I totally agree with that. I mean, I share the view. That’s, you know, that’s why I’m a mathematician is because I find the questions so compelling and I’ve spent my whole life thinking about these problems. But, you know, but like if I won an award…

Lex Fridman(03:25:32) Yeah, it’s great. It’s great. I mean, I’m pretty sure you don’t contribute to MathOverflow for the wealth and the power. That you gain. I mean, it’s, yeah, genuine curiosity.

Joel David Hamkins(03:25:46) Well, you asked who the greatest mathematician is, and of course if we want to be truly objective about it, we would need a kind of an objective criteria…

Lex Fridman(03:25:55) Criteria, yeah.

Joel David Hamkins(03:25:55) …about how to evaluate the relative, you know, strength and the reputation of various mathematicians. And so, of course, we should use MathOverflow score… …Because…

Lex Fridman(03:26:06) That you’re definitively… I mean, nobody’s objectively the greatest mathematician of all time.

Joel David Hamkins(03:26:10) Yes, that’s true. I’ve also argued that tenure and promotion decisions should be based…

Lex Fridman(03:26:15) Based on MathOverflow.

Joel David Hamkins(03:26:16) …Yeah. So my daughter introduced me to her boyfriend. …And told me that she had a boyfriend. And I, um…

Lex Fridman(03:26:25) Asked him what his MathOverflow…

Joel David Hamkins(03:26:26) I wanted to know, first of all, what is his chess rating, and secondly, what is his MathOverflow score?

See the full transcript and watch the full video episode for more. I shall periodically be posting more excerpts like this one here on Infinitely More—find them in the lex-fridman tag.