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Joel David Hamkins

lines on the record: 286
first: 2025-12-31
most recent: 2025-12-31
sittings or episodes: 1
sources: podcast

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“The nature of the world and the way things are. It's about objective reality in a sense, whereas proof is about our understanding of the world and about how we come to know the things that we know about the world. And so to focus on proof is to focus on the interaction that we have with the objective reality. Okay, I'm talking about the reality of mathematics, not the physical world, because as I said, I live in the platonic realm and I interact with mathematical reality. And so proof is about the interaction and how we come to know the facts that are true in this mathematical reality. Whereas truth is about what's really the case, sort of apart from our knowledge of it. And this is, I think, such a core way of... That I have of understanding the world and the nature of logic and reasonings”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Okay, but unfortunately I don't really have to choose between them. So you ask about the most beautiful idea in philosophy. And I would have to say that I think it's the distinction between truth and proof, the one that we discussed already. It's so profound and gets at the heart of so many philosophical issues. I mean, of course, this is a distinction that's maybe born in mathematics or mathematical logic, but that's already philosophical to a degree. And it's fundamentally a philosophical distinction. The truth is about the”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Care to decide whether I'm a mathematician or philosopher and my work is engaging with mathematics and with philosophical issues in mathematics and with plain philosophy and this ample region between these two subjects so it's not necessary to choose. I remember when I first went to Oxford and I told my daughter that I was going to become professor of philosophy in Oxford and she looked at me plaintively and said but papa you're not a philosopher Because in her mind, you know, her father was the mathematician and her mother was the philosopher because my wife, Barbara, is a philosopher. Now, also at Notre Dame, we're together there.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So I have a foot in both Fields Philosophy and Mathematics. Some contexts, I seem to be required to choose whether I'm a mathematician or a philosopher. I mean, my training is in mathematics, my PhD, all my degrees are mathematics, but somehow I turned myself into a philosopher over the years because my mathematical work was engaging with these philosophical issues. And so when I went in New York, I had appointments first in mathematics only, but then eventually I was also joining the philosophy faculty at the Graduate Center. And when I went to Oxford for the first time, my main appointment was in philosophy, and that's also true now at Notre Dame, although I'm also concurrent professor in mathematics. I have math PhD students still and philosophy PhD students. And so I don't really...”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Continuing with the construction of the V hierarchy and girdle. Zermelo's proof of the well order principle using the axiom of choice is a transfinite recursive construction. And so the idea of just counting past infinity is so simple and elegant and has led to so much fascinating mathematics.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“First number that comes after all, those numbers will be omega squared. And this one is the first compound limit ordinal because it's a limit ordinal is one of these numbers, an ordinal, that doesn't have an immediate predecessor like omega and omega times 2, omega times 3, those are all limit ordinals. But omega squared is the limit ordinal, but it's also a limit of limit ordinals because the omega times three, omega times four, and so on, those are all limit ordinals that limit up to omega squared. And then, of course, you form omega squared plus one, and then omega squared plus two, and so on. And it never stops. And it's just absolutely beautiful and amazing. And furthermore, forms the foundation for these transfinite recursive constructions that came later, I mean, starting with the canorbendixon theorem that I mentioned.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“The most beautiful idea in mathematics is the transfinite ordinals. These were the number system invented by Georg Kentor about counting beyond infinity, just the idea of counting beyond infinity. I mean, you count through the ordinary numbers, the natural numbers 0, 1, 2, 3, and so on, and then you're not done because after that comes omega and then omega plus 1 and omega plus 2 and so on. And you can always add one. And so of course, after you count through all those numbers of the form omega plus n, then you get to omega plus omega, the first number after all those. And then comes omega plus omega plus one and so on. You can always add one. And so you can just keep counting through the ordinals. It never ends. Eventually you get to omega times 3, omega times four, and so on. And then the limit of those numbers.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Right, it's probably true. I also find it likely that a lot of the, as far as mathematical training data is concerned, I just have to assume that math overflow answers part of the trading data. It's so.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“To come up with a mathematical argument, I think it's a dangerous source of error if you're not especially attuned to this very issue that the AI is going to produce something that's not grounded in mathematical understanding, but rather something that is trying to look like something that is grounded in mathematical understanding. And those are not the same thing at all. And furthermore, I really wonder if one can make a kind of system for producing genuine mathematical insight that isn't based in what I would view as mathematical understanding as opposed to the text generation systems, the methods that are used, yeah, they don't seem close enough grounded in understanding of the underlying mathematical concepts, but rather grounded in the way words appear on a page in arguments about those concepts which are not the same.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Yes, and so I think that the chat programs and so on are producing these arguments that look really, they look like that's what they're striving to do. It's what they're designed to do. They're not designed to make a logically correct argument. They're designed to make something that looks like a logically correct argument. And it's easy to get fooled if you're not skeptical. And so that's why I worry a bit when people rely on AI for mathematical arguments. I mean, using tying them to lean in the formal proof verification systems and so on, this is a totally different way of operating. But for the sort of ordinary person sitting down and using chat.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Except at the time, you know, I didn't know anything, I was an undergrad and LaTeX was sort of unheard of. And so I was producing these beautifully typeset, you know, problem set solutions and so on. And I would print it up and submit it and so on. And the grades would come back terrible grades. And I realized what was happening is that The copy was so beautiful mathematically typeset in this way. It looked like the kind of mathematics you find in a book, you know, because basically that's the only time you saw that kind of mathematical typesetting was in a In a professional published book. And those mathematics was almost always correct in a book, right? And so I had somehow lost my Critical because it was so beautiful, and I'm used to only seeing that kind of typesetting when an argument was totally right. I wasn't critical enough.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Rather than the AI. And so I tend to be kind of skeptical, but also. Skeptical for another reason, and that is. Because of the nature of the large language model approach to AI doing mathematics, I recognized that the AI is trying to give me an argument that sounds like a proof rather than an argument that is a proof. The motivation is misplaced. And so I worry that this is a very dangerous source of error because it often happens in mathematics that, I mean, if I think back to when I was an undergrad, you know, here at Caltech and I was a math major eventually. And at that time, LaTeX was a pretty new thing. And I was learning LaTeX. And so I was typing up my homeworks in LaTeX. And they looked beautiful. Actually, they looked like garbage. From my current standards, I'm sure it was terrible.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Often very surprised to hear that based on my own experience, which is quite the opposite. And so maybe my process isn't any good, although, you know, I use it for other things, like, you know, for programming things or for image generation and so on. It's amazingly powerful and helpful. But for mathematical arguments, I haven't found it helpful. And maybe I'm not interacting with it in the right way. Yet, or it could be. And so maybe I just need to improve my skill. But also, maybe I wonder like these examples that are provided by other people maybe involved quite a huge amount of interaction. And so I wonder if maybe the mathematical ideas are really coming from the person, you know, these great mathematicians who are doing it.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“If I were having such an experience with a person, I would simply refuse to talk to that person again. But okay, one has to overlook these kind of flaws. And so I tend to be a kind of skeptic about The current value of the current AI systems as far as mathematical reasoning is concerned, it seems not reliable. Okay, but I know for a fact that many that there are several prominent mathematicians who I have enormous respect for who are saying that they are using it in a way that's helpful.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“I guess I want to draw a distinction between what we have currently and what might come in future years. I've played around with it and I've tried experimenting, you know, but I haven't found it helpful at all. Basically zero, it's not. It's not helpful to me. And, you know, I've used various systems and so on, the paid models and so on. And my typical experience is interacting with AI on a mathematical question is that it gives me garbage answers that are not mathematically correct. And so I find that not helpful and also frustrating. Like if I was interacting with a person The frustrating thing is when you have to argue about whether or not that argument that they gave you is right and you point out exactly the error and the AI saying, oh, it's totally fine”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“And so at the time, this was the best known result, the sort of state of the art. But since that time, it's been improved now dramatically. And in fact, we know now that every countable ordinal arises as the game value of a position in infinite chess. So it's fantastic result.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Work because of this and that, and so on. And so, this kind of back and forth was extremely helpful to me. And eventually we converged on arguments that were correct. And so it was quite interesting. Also, maybe another thing to say is that the follow-up paper to this one was a three-way paper with also Corey and myself and my PhD student Norman Promoter in which we improved the bounds. So we were aiming to produce more and more transpositions with higher and higher ordinal values. So the initial position was value omega, and then we made omega squared and omega cubed in the first paper. And then in this three-way collaboration, we made omega to the fourth.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“But the details of the argument have to do with kind of chess reasoning, you know, like, and my chess reading wasn't quite up to it because I would create the positions, almost all the positions are ones that I made, but this is like after many generations of being corrected by Corey because Corey would come and say, hey, this pond is hanging and it breaks your argument or, you know, this bishop can leak out of the cage or whatever. And so... And so the process was I knew kind of in terms of these ordinals what we needed to create with the position. And I would struggle to do it and create something that sort of had the features that I wanted and then I would show it to Corey and he would say, look, it doesn't.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So I met him because he was a grad student at CUNY, where I was at the time in New York. And also, he was my son's chess coach when my son was playing chess competitively in elementary school. Then Corey was the coach. And so we knew him that way. And that was right around the time when I was getting interested in Infinite Chess. And I knew I needed a chess knowledgeable partner. And so Corey was invaluable for the paper because The proofs in this chess infinite chess are extremely finicky because you create these positions.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“No, this is an act of mathematical creativity, really, to come up with I had a co-author, my co-author Corey Evans, he's a national US national master chess player, a very strong chess player. He's also a philosophy professor of law”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“And then after that, it's going to be at most that many numbers afterwards to count down, right? So the nature of counting down from omega is that you take this giant step on the first count, and then after that, you subtract one each time. You can't subtract one from omega because that's not inordinal. So if you come down from omega, you have to go to some finite number. And then if you just subtract one each time, then that's how many more moves you get. So that's the sense in which black can make it take as long as he wants because he can pick his initial number to be whatever he wants.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“And there's no promotion because there's no edge, right? Exactly. And this position that we were just talking about is a position with game value omega, which means that because it has an ordinal value, white is going to win, but black can play as though counting down from omega. What is the nature of counting down from omega? If you're black and you need to count down from omega, then you have to say a finite number.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“The one thing is that, okay, so the white pawns always move upwards and the black ponds always move downwards, but when they're capturing the ponds, capture on the diagonal. So I think the peace movement is pretty clear. There's a couple of differences that you have to pay attention to from ordinary chess. For example, there's this threefold repetition rule in ordinary chess, but we just get rid of this for infinite chess because, of course, threefold repetition is just a proxy for infinite play. The real rule is infinite play as a draw, not threefold repetition is a draw. That's just a kind of convenient approximation to what I view as the actual rule, which is that infinite play is a draw. So the only way to win is to make checkmate on the board at a finite stage of play. And if you play infinitely, you haven't done that, and so it's a draw.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Right, so the rules of Infinite Chess, there's just the ordinary pieces, and they move on this infinite board, which is just a chessboard but extended in all directions, infinitely, with no edge. So there's no boundary. But the pieces move just like you'd expect. So the knights move just the same, and the rooks move on the ranks and files, and the bishops move on the same color diagonals. And just like you would expect, except they can move as far as they want, you know, if there's no intervening piece in the way.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“But it's doomed. Black can say, Well, I know you're going to win, but this time you're going to take a thousand moves at least, or maybe in a different way of playing black and say, well, I know you're going to win, but this time you're going to have to take a million moves. For any number, black can say that. So it's these really interesting positions. It's the position in my first infinite chess paper. So it's black to play in this position. And if black doesn't move that rook there Then White is going to checkmate pretty quickly.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“No end. So it's not made an end for any end, but it's a white win infinitely many. The way to think about it is white is going to win, but black controls how long it takes.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“White has a winning strategy that will win in finitely many moves. In other words Let me say it again. There are positions in infinite chess that white can definitely win in finitely many moves. White is going to make checkmate. But there's no particular end for which white can guarantee to win in end moves.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“The interesting situation is that you present a position where there's a lot of pieces already on the board in a complicated way, and you say, what would it be like to start from this position or from that one? And we want to produce positions that have interesting features, meaning mathematically interesting features. And so I can tell you, for example, Probably a lot of people are familiar with the, say, the made-in two genre of chess problem. You know, you have a chess problem and it's white to maiden too, which means That white is going to make two moves, but the second move is going to be a checkmate. Or maybe made in three or made in five or whatever, we can have mate in N positions for any n. I mean, in infinite chess You can create a position which is not made in N for any n, but”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, absolutely. Infinite chess, fantastic. Chess ordinarily is played on this tiny, tiny board this eight by eight board, right? So when you play chess, normally it's on the eight by eight board. But we want to play infinite chess. So on the integer board, it's infinite in all four directions, but it still has the chessboard pattern. And maybe there's pieces on this board, maybe infinitely many pieces we allow. But one difference from finite ordinary chess in infinite chess, we don't Play from a standard starting position. Rather, you”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“I mean, Of all time I've also argued that tenure and promotion decisions should be based on So, my daughter introduced me to her boyfriend and told me that she had a boyfriend. And I”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“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 objective criteria about how to evaluate the relative strength in the reputation of various mathematicians. And so, of course, we should use math overflow score because...”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Sure, I totally agree with that. I mean, I share the view. That's why I'm a mathematician is because I find the question so compelling and I've spent my whole life thinking about these problems. But if I won an award”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Guess what I think is that mathematics is full of a lot of different kinds of people. In 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, you know, is maybe an instance where he's Such a brilliant mind, and he saw 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.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“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.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So, my approach to making mathematical progress tends to involve working with other people quite a lot rather than just working on my Own and I enjoy that aspect very much. So personally, I couldn't ever do what Wiles did. Maybe I'm missing out. Maybe if I locked myself in the bedroom and just worked on whatever, then I would help it. But I tend to think that, no, actually, like being math overflow so much and I've gotten so many ideas, so many papers have grown out of the math overflow 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 it's, I enjoy this kind of social aspect to it and it's not just the social part rather that's the nature of mathematical investigation as I see it.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“I mean, Wiles proved an amazing theorem, the Fermosles theorem result is incredible. This is a totally different cell than my own practice, though, of working in isolation. I mean, for me, mathematics is often a kind of social activity. I counted, I mean, it's pushing towards 100 collaborators, co-authors on various papers, and so on. And 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, whatever. You want to have a joint paper? Yeah, exactly. Let's go.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“And it helps you to understand, particularly when there's parts of the argument that are in tension with one another, then you can imagine that people are fighting or something and those kind of metaphors, you know, 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 kind 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.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, yeah. So, this is a basic tool. I mean, I use this all the time. You know, 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. And this is a kind of metaphor for what's going on. Of course, you know, the actual arguments aren't anything like that because there's not land and you're not traveling and you're not.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“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, you know, that's relevant or 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, you know, then pretty soon you're making progress on the problem. And 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 proof that. And that's worked extremely well for me. So I'm pretty pleased with that method.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“I want to work on the things that I can understand and that are simple and luckily I've found that I've been able to make contributions that other people seem to like in this way, in this style. And so I've been kind of fortunate from that point of view. I mean, my process always, though, and I've recommended this always to my students, Is just a kind of playful curiosity. So, whenever I've an idea or a topic,”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Are automated so you see. Well, that's another issue because maybe those things are less subject to skepticism when it's validated by Lean. But I'm thinking about the case where the arguments are just extremely complicated.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“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 there's different styles. Sometimes mathematicians get involved with this enormous research project that involve huge numbers of working parts and different technology coming together. I mean, mathematical technology, not physical technology.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“That's another difficult question. I suppose it has to do with, I mean, 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, so 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 complicated argument. That's great because you prove something new, but I still want to see the beautiful, simple, because that's what I can understand. I mean, I'm kind of naturally skeptical about any complicated argument because it might be wrong.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“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.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“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.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“I mean, if you forced me to pick someone, it would probably be Archimedes. Archimedes, he is such incredible achievements in such an early 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. And sometimes the greats are doing things that are just first and not, you know, somebody else could have easily been first. And so there's a kind of luck aspect to it. When you go back and look at, you know, 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. And when you look back at the achievement that had been made, then maybe you can imagine, you know, thinking, well, you know, somebody else could have had that insight also. And maybe they would have, it's already a known phenomenon.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So, this is, I think, an incredibly difficult question to answer. I mean, personally, I don't really think this way about sort of ranking the mathematicians by greatness.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“The set solvers work amazingly well in lots and lots of cases, even though we can prove that we don't expect if p is not equal to NP, then there won't be a polynomial time sat solver, but actually the SAT solver approximations are really quite amazing.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“You have to temper those remarks by the realization that P and P equal NP or P naught equal NP are not about these practical things at all because of the asymptotic nature of the question itself. That's on the one hand. But on the second hand, we already have the algorithm, so we could use it already, except it's a terrible algorithm because it involves all this incredible amount of code and so on.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Because it's an asymptotic property. It's really in the limit as the size of the inputs goes to infinity. That's the only time that polynomial or NP becomes relevant. And so maybe it's important to keep that in mind. Sometimes you find kind of overblown remarks about made about if P equals NP, then this will be incredibly important for human civilization because it means that we'll have feasible algorithms for solving these incredibly important problems in NP that it would cause immense wealth for human societies and so on because we would be able to solve these otherwise intractable problems and that would be the basis of new technology and industry and so forth. I mean people make these kind of remarks but”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source