Theories of Everything with Curt Jaimungal - Richard Borcherds (Fields Medalist) on the Monster Group, String Theory, Self Studying and Moonshine

Starting point is 00:00:00 Alright, hello to all listeners, Kurt here. That silence is missed sales. Now, why? It's because you haven't met Shopify, at least until now. Now that's success. As sweet as a solved equation. Join me in trading that silence for success with Shopify. It's like some unified field theory of business.

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Starting point is 00:00:57 All lowercase, that's shopify.com slash theories. Today's guest is Professor Richard Borchardt of UC Berkeley. Richard is one of the brightest people, if not the brightest person, that I've been lucky to speak to on this channel. He's a Fields Medalist, and a Fields Medal is essentially the Nobel Prize equivalent of mathematics. He was awarded this medal due to him proving a speculative assertion at the time, put forward by Conway, that there's a deep

Starting point is 00:01:26 connection between the monster group and the J-function, which is a simple, or one of the simplest modular forms. This is called the monstrous moonshine conjecture, and it was proven by Professor Borchardt in the early 1990s when he was in his early 30s. The monster group is the largest of all the sporadic simple groups, and its order is of the size of pretty much all the atoms that comprise the earth. So around 10 to the 53 if I'm not mistaken. Borchardt also runs a wonderful YouTube channel, which the link is in the description. It's essentially teaching graduate mathematics. Over the next few months, I'll be asking mathematicians and physicists about how they think and how they study because if you're watching this, you're likely a sagacious individual who knows the importance of self-study.

Starting point is 00:02:10 This is because it's often more important to know how to think about a problem, rather than exactly how to solve this particular problem, because the former generalizes to more problems and more solutions. If you'd like to see more conversations like this, then please consider supporting at patreon.com slash kurtjaimungle. Right now, there's the goal of hosting multiple intellectuals on simultaneously, almost like Theomaki, so that they can battle it out in real time. The first will be Bernardo Kastrup and Donald Hoffman and Jonathan Vervaeke simultaneously. If I'm lucky enough to reach around 50 patrons, I'll be able to host this conversation. Thank you so much to

Starting point is 00:02:47 all my current patrons. I hope you enjoy. Part of this was filmed as a live stream, so if it looks like I'm reading, I am. I'm most likely reading an audience question. Enjoy. First of all, why don't you tell the audience a little bit about yourself, your interest, what field of math you study? Math professor, working mainly on algebra and number theory. I did some stuff on monstrous moonshine about 20 or 30 years ago and have been struggling to do something similar ever since. Similar as in you can't announce it or you can? Well, I feel like one of these people, you know, said that the worst thing that can happen to a musician is they have a big hit early in their career because they spend the rest of their time trying to equal that. And I feel I'm in sort of that position that I did Monstrous Moonshine when I was young and I've never managed to do anything similar since.

Starting point is 00:03:39 How did you get interested in mathematics? I've always been as long as I can remember. I mean, I think someone was asking me this question, why did I decide to become a mathematician? And I never did decide. I just was a mathematician. Right. You equated it to being a dolphin or a dolphin deciding to be a dolphin. Something like that. Yeah. How did you become interested in the particular field that you're interested in? Oh, moonshine. that was sort of by accident um i i went to a talk by john conway as an undergraduate once you know he he he would sometimes give undergraduate talks and he mentioned this it was about finite simple groups which i had a sort of vague idea of what they were

Starting point is 00:04:20 and he said well there was this monster group with this dimension living in dimension one nine six eight eight three and he also mentioned this is the same as the coefficient of the elliptic modular function i was just you know just completely blown away by this it was just you know that these were two areas of mathematics that had absolutely nothing to do with each other and there was this bizarre coincidence going on. So that's how I got interested in it. Do you find that there's some connection between unifying different fields in math and unifying different fields in physics? Because as you know, unification happens frequently in physics, but it seems to also happen in math. There's a sort of vague analogy. I mean, one of the things I've discovered in math is you really

Starting point is 00:05:03 want to know little bits of lots of different areas, because every now and then you find some bizarre connection between two apparently different areas. And I would guess the same is sort of true in physics. For instance, you know, Maxwell unified light and electromagnetism. And there were two apparently completely different areas of physics. But Maxwell happened to know about both of them, so he was able to unify them. I guess gravitation and quantum field theory is another very topical example. General relativity looks quite different from quantum mechanics, and it's a big problem trying to connect them.

Starting point is 00:05:43 What do your average days look like? Do you wake up at a particular time? Do you meditate? I wake up at random times and frankly most of my days spent procrastinating about various things I ought to be doing. I mean, I'm supposed to be doing mathematics research but mathematics research consists mainly of trying to do things that don't work. And what do you procrastinate with? What are your vices? Oh, I can procrastinate by making YouTube videos on mathematics or something like that. Do you play any video games or do you go for walks? Do you have pets or kids? I go for walks. I used to have chickens, but they all died a few years ago.

Starting point is 00:06:29 How much time do you spend studying versus, well, let's say you're doing research. Some of the research is studying. Some of the research is just writing. Some of it is thinking. Some of it is collaborating. What does that look like for you? And how much time do you get to spend in each, let's say, on a weekly basis? Yeah.

Starting point is 00:06:46 I knew an accountant once who said they had to spend, for every 15 minutes of the day, they had to write down what they were doing. And I've never done anything like that. And I never dared to because if I did, I would probably be horrified at what I was wasting my time on all the time. I just don't know. I probably spend far less time doing useful stuff than I like to imagine. So you don't feel like you manage your time well? Dreadful at it. I'm poorly organized and have atrocious time management skills. Let's say that you had an ideal day in terms of process, so not in terms of outcome, because your ideal day may be solve the Riemann hypothesis in the morning, move on to so-and and solve it so forget about

Starting point is 00:07:25 that process that is spend four hours writing spend two hours answering email i don't know whose ideal day would be emails i can't what would your idea look like i can't spend four hours of a day working on anything i i just get too fidgety and impatient um I mean, ideal day would be if I was able to focus on something that well. Yes, I mean, sometimes in the past when mathematics, when research was going well, I could spend, you know, 12 or 16 hours a day working on a calculation. Straight or with breaks? Well, I'd have lunch occasionally and things like that.

Starting point is 00:08:04 But yeah, when I was younger, I really was able to spend many hours straight working on something. Now, is that a function of age, that decline, or is it because of interest? I don't know. Probably just getting less energetic as you get older or something. Are you more creative now than you were when you were younger? I don't seem to be. I mean, when I was in my 20s and 30s, I was getting all these ideas and they all worked. And these days I'm getting loads and loads of ideas and they all fail. Can you give an example?

Starting point is 00:08:42 Of what? An idea that worked, an idea that failed. Recently, let's say in the past two months. Past two months. Well, nothing very easy to explain. I mean, they're all rather technical ideas. I mean, I can tell you, I mean, I was trying to find an extension of the Smith-Minkowski-Ziegler formula to give an exact number for class numbers rather than a weighted sum, which probably means absolutely nothing to most people. You'll find that much like the audience of yours, the audience of this channel, they comprise extremely bright people, some postdocs, some even professors. So don't worry if you feel like you're going above someone's head.

Starting point is 00:09:24 It seems like one of the appeals of this podcast is the fact that we do go into some technical professors. So don't worry if you feel like you're going above someone's head. It seems like one of the appeals of this podcast is the fact that we do go into some technicalities, unlike other podcasts, which tend to stay at a fluffy or cosmetic level. Well, I can go into technicalities if you want, but I suspect your audience is going to plummet. Whatever. What do you feel like is your greatest strength and your greatest weakness? whatever. What do you feel like is your greatest strength and your greatest weakness? Greatest strength? I'm really obstinate that I can keep working on a problem for, I can keep coming back to it year after year. And sometimes, you know, when you come back to a problem for the 10th time, you finally make some progress on it. Weakness, there are too many to think of. As I said,

Starting point is 00:10:08 I've got lousy time management and work skills. I don't collaborate with people nearly as much as I ought to. Most of the top mathematicians have this fantastic network of talking to other good mathematicians. I do far too little of that. Why is that? Poor social skills, I guess. Social anxiety or just poor social skills or disagreeableness? Just not very good at interacting with people and I never quite figured out how to do it. I was watching one of your Q&As and someone said, what's or who is the nicest mathematician? And you said, they're all pretty much nice. So then if they're all pretty much nice, that to me implies that you get along with them unless you're not being nice in return.

Starting point is 00:10:52 So why is it that you don't get along with them? Well, I do get along with them just fine. It's just kind of stressful interacting with people. So I just tend to not do it unless I have to. What are the advantages of that? See, there's the sole mathematician or the sole physicist who in isolation comes up with a brilliant idea. I believe Max Planck was a proponent of that, but then there are obviously others. People have different styles. I mean, there are some people like Atir and

Starting point is 00:11:20 Sayer who are fantastic at collaboration, and there are other people. I mean, Perelman is an obvious example he just went off by himself didn't talk to anybody else worked for 10 or 20 years and solved this absolutely fantastic problem I mean that that's that's the sort of ideal of what that's the sort of one ideal of mathematics research. You just solve a problem by yourself by brute force. But it's actually, it's very rare that that happens. Nearly all mathematics is done very collaboratively these days. Are there any advantages to studying math later in life? So, for example, starting in your 20s or starting in your 30s or even 40s? We can take those as three different

Starting point is 00:12:05 questions. Yeah, I haven't heard of any mathematicians who started in their 30s or 40s. There have been a few really good mathematicians who started in their 20s. I mean, I think Ed Witten was doing history as an undergraduate or something and only started taking math seriously in his 20s. So, I mean, it might be possible. I just don't know of any cases. I mean, it may be something like learning languages. If you learn languages when you're young, you're much, much better at it than if you start when you're older. Did you enjoy your graduate studies more than your undergrad? They were both fun. I mean, I wouldn't say I enjoyed either more than the other. When I was watching one of your talks, you mentioned quite a few times, and probably

Starting point is 00:12:54 even in some of your non-Q&A videos, you mentioned that interest matters. So if you're going to pick a problem, pick one that you're terribly interested in, because over the long run, you'll be faced with with you'll be set with problems and so on and it had me wondering did you either voluntarily like force yourself or it was forced upon you have to study some piece of math that you absolutely hated and now you abhor studying whatever you're not interested in well i i've never actually bothered studying math I didn't like. I mean I don't remember anybody past high school telling me to study math that I didn't want to. I mean at university you can very much pick what you want to study. I mean high school yes I was forced to

Starting point is 00:13:42 study various bits of applied math that I can no longer stand. Often solving mathematical problems requires a certain state of mind. Do you find yourself going through periods where you feel like your mind is not creative enough, not focused enough, and how do you solve that? Yeah, of course. Well, you just have to go off and do something else. So there are several things you can do for this. I mean, John Conway told me that what he did is he always had a sort of background calculation. So he'd be calculating tables of knots or tables of lattices or something. And you can do that even if you're not feeling inspired. And I think Richard Feynman had this idea that you could always do teaching as a

Starting point is 00:14:25 sort of background thing and the point in both cases is so that you have something where you can see that you're making progress because I mean the trouble with research is most of the time you're working hard and making no progress at all and this is psychologically kind of a bit depressing so you need to have something else you can do where you can actually make visible progress on it. What's the longest lull or the longest involuntary cognitive hiatus you've experienced? Well, I'd say, hard to say. I mean, it might be now. I've been working on a problem for several years and I don't know what the progress is because you know 20 times I think I've solved it and got really excited and then I start to write a job and geez there's this

Starting point is 00:15:14 huge hole I somehow overlooked so I go back to being depressed for a bit and so it keeps on switching backwards and forwards between thinking I've solved and thinking I haven't. And I sometimes think that sooner or later I'm going to get so senile that I can't actually figure out what's wrong with my solution. And then I can publish it. And, you know, this is actually unfortunately what happens to some mathematicians when they get old. They sort of lose this capacity to see what's wrong with what they're doing. You mentioned depression. I don't know if you just meant that as a throwaway comment, but do you feel like you're characterized by depression or are you just using that in an informal sense? I don't feel I've ever had anything I'd call depression. I mean,

Starting point is 00:15:56 of course you go through phases when you're feeling a little bit less happy than usual. less happy than usual. Actually, I don't know. I mean, maybe I'm just not very good at spotting symptoms of depression. I remember seeing a poster once of saying, you know, look out for these typical symptoms of depression. I remember being really unimpressed by that because I was thinking that was sort of my normal state as a graduate student. I didn't count that as being depressed. So maybe I just don't recognize depression. Someone asked, what's your IQ and do you have any methods of increasing IQ? I've no idea what my IQ is. And as far as I know, the only method of increasing IQ is to go back in time because your IQ steadily decreases as you get older. Have you taken any nootropics or supplements to try and either stave off cognitive deterioration or to improve your cognitive state?

Starting point is 00:16:54 No, I don't. Besides caffeine. I don't take caffeine and I don't know what a nootropic is. And I stay well clear of anything that's remotely mind altering. You don't drink coffee? I don't drink coffee, no. Tea? You're British. No tea.

Starting point is 00:17:09 Well, I drink tea out of politeness if someone puts it in front of me. Do you find that depression is higher in mathematicians than in the general population? Way outside my expertise. I've no idea what the incidence of depression is in either the general population or mathematicians. I've never noticed what the incidence of depression is in either the general population or mathematicians. I've never noticed mathematicians being particularly depressed,

Starting point is 00:17:29 but I'm so clueless, I probably wouldn't notice even if they were. The reason is that you mentioned you read the signs of depression one time on a bus ad or somewhere, and you thought, oh, that's my entire department. oh, that's my entire department. Well, yeah, but I mean, it's not like I'm qualified to actually diagnose people with depression or not. It was just, you know,

Starting point is 00:17:52 you go through periods where you, you know, depression is, you know, they have things like you go through periods when you're getting nowhere, feel you're getting nowhere and so on. And that's just normal for research if you're researching anything serious. on and that that's just normal for research if you're

Starting point is 00:18:05 researching anything serious what's your creative process like is it different depending on the problem or do you have a general framework um a lot of randomness in it um you you just i mean i think what's the creative process like it It's sort of like, you know, imagine you're on one of these huge garbage dumps that you sometimes see in countries. There are these garbage dumps of people picking over them, looking for something interesting. And research is a bit like that. You're wandering around this area, looking out for something that somebody else has overlooked. looking out for something that somebody else has overlooked and nearly all the time you don't find anything because most areas in math research have been pretty well picked over by everybody so you're continually trying out new things and 99% of the time the new thing you've tried out either doesn't work or if you work on it and discover that someone's already

Starting point is 00:19:05 thought of it creative process is just trying to generate enormous numbers of ideas and throwing nearly all of them away actually I've heard the same something like this is actually supposedly true in business that I think um what was it the guy who does the Dilbert cartoons was saying somewhere, you know, he had tried 30 ideas for making money and 27 of them failed or something like that. Scott Adams? Yeah. So you just keep on generating new ideas and discard the ones that don't work. Do you find you think more geometrically or algebraically or you think in words or you think in concepts? I try to wonder about that. And it's amazingly difficult to actually figure out what's going on in your own brain.

Starting point is 00:19:52 It's something that's very difficult to describe. I seem to have sort of some sort of algorithms built into my brain and I sort of know how they're going to work. And I just don't know how to describe this. of know how they're going to work and I just don't know how to describe this. I mean there's a bit of geometric imagination going on in there and there's a bit of algebraic manipulation but I don't know how the algebraic manipulation works because I'm doing things with formulas that are far too complicated for me to actually visualize. I mean, I can't visualize the formula itself in my mind, but somehow I still know what it's doing. And I've no idea how this works.

Starting point is 00:20:33 Actually, the mathematician Jacques Adamar once wrote to a lot of mathematicians, asked them to describe how they thought. And the problem is it's so long ago since I looked at this that I can't remember what conclusions he came to. But there have been some attempts by mathematicians to research this question. I was reading Penrose's Emperor's New Mind.

Starting point is 00:20:55 And for anybody who is a mathematician or a physicist that wants to read that book, just read the last third. It was a slug to get through. It was quite an ordeal to get through the first bit because it's so much retreading of similar ground. And The Last Third is actually where it's interesting. But in The Last Third, he talks about when he was in high school, he's noticed that he thought differently than his peers. And then when he got to university,

Starting point is 00:21:20 he thought, okay, great, I'll be around other mathematicians and physicists, so we'll think alike. But then he found that there was even more diversity in terms of how people think. What I'm wondering is, have you encountered that? Yeah, I noticed this too, that has sort of gradually dawned on me that lots of people learn or think in completely different ways. I mean, one example is lectures. I mean, I've always found lectures. If I go to a lecture, I fall asleep after 10 minutes. I just can't stay awake and I do all my learning from books. And I sort of assumed everyone was like this. But no, an awful lot of people do most of their learning from lectures and just hardly open the textbooks. So there are very different styles. And again, some people obviously have very powerful geometric intuition and other people have

Starting point is 00:22:09 some sort of very good intuition about something abstract like set theory and a very little geometric intuition and so on. When you work, do you have to have uninterrupted bouts of time exclusive to yourself? So what I mean is there's this concept called deep work. And I'm wondering if you instituted that where you tell, I don't know if you're married. Are you married first of all? Yeah. Okay. So do you tell your wife, don't bother me for four hours because I'm working or do you just, you start working and then she interrupts you and that's okay because that's your process? Well, yeah. I mean, if I get interrupted, which happens quite regularly, I just pick it up later.

Starting point is 00:22:47 It just varies. I mean, I think you have to be a bit like Jane Austen. I just sort of remember she she wrote her novels in, you know, whenever she could snatch 15 minutes from her social life or something like that. So you you get so you can work in whatever spare time you have. Do you read any non-mathematical books? Yeah, I read plenty of novels and things and books about physics and general interest science books and so on. What would be an example of one that you particularly liked besides Lord of the Rings? Well, there's, I don't know, Game of Thrones and,

Starting point is 00:23:26 I don't know, Game of Thrones and, hang on, fairly eclectic reading. I mean, I like Agatha Christie novels or, my mind is blanking out, or Terry Pratchett or any light reading. I mean, I don't go around reading sort of the sort of deep literature that people in English literature classes write essays about. I've got a copy of Finnegan's Wake Upstairs, and you can open a random page and I can't understand a single sentence. Mandalore here wants to know if you've read or heard of Jordan Peterson and what you think of him or his work. Jordan Peterson. Yeah, I've got one of his books about, I had the word 12 in it, I seem to remember.

Starting point is 00:24:09 That's the most important part, just that number. Yeah, yeah. And I sort of read it. For most of it, I can't remember an awful lot about it anymore. Do you find that you read slow when it's a non-mathematical book or even when it's a mathematical book? I read math books slowly, other books fairly fast. I mean, math books very slowly. It takes a long time to absorb what's in it if it's anything

Starting point is 00:24:33 new. Okay, I'm going to take a question from the audience here. Oh, okay. Primetime Durkheim says, has he experienced any sort of moments of madness or psychological aberration being someone who has a lifelong habit of working in high math in isolation? Nothing that I've noticed, but I mean, if maybe I'm mad and so mad that I don't notice I'm mad, who knows? Maybe doing mathematics obsessively is itself a mad thing to do. Socolo isIsStan96 says, he said he avoids anything that is mind-altering. So do you also avoid certain foods or do you optimize your diet in general? Do you do intermittent fasting? I tried intermittent fasting for about three or four hours once. I don't think that counts. Yeah, well, yeah.

Starting point is 00:25:25 That's infinitesimal fasting. Well, yeah, I mean, I think I tried it for a day or two, and it didn't seem to make much difference, so I gave up. No, I don't have any particular food habits. It's mostly vegetarian, partly for health reasons, but other than that, nothing interesting. How is it that you go about learning a new math, either learning from a new field or learning math in general?

Starting point is 00:25:50 It's very slow. You have to, I mean, as I said, I learned by reading. So what you have to do is to, well, I learned by reading and by calculating examples. And it's a very long, slow process learning anything. I mean, mean generally you know try reading about it in a dozen different books working out lots of examples and you know the first half dozen times I try and learn it I just forget it again.

Starting point is 00:26:17 I mean I suspect there are a lot of other good mathematicians who are much more efficient at learning things I mean I get the impression there are some people who can learn things just by reading about it once, but I've never been able to do that. It's kind of like learning, you know, the way people learn languages. You learn a language by, you have to do a lot of repetition of basic stuff before it sinks in, at least for most people. Again, there are some people who seem to be able to learn 10 or 20 languages. I've no idea how they do it. In one of your Q&As, you mentioned when someone asked you about setbacks and difficulties, what do you do about them? You said, often you just ignore them because you'll be surprised how many problems disappear with time.

Starting point is 00:27:00 Yeah, this is my fundamental problem- technique, especially when someone else comes up with a problem for me to do. I sort of figure, well, if it was really important, they would have done it so I can just ignore it. Do you mind giving a specific example of a problem you ignored and then it was solved? Or that you realized you didn't even have to solve it to begin with? to begin with um actually i find that's that's sort of true of a surprising number of problems actually in math research if i if i don't try and understand a problem and just sort of don't try and work hard on it and just sort of put it aside that then you find a month or two later you suddenly realize it's kind of obvious what you ought to be doing. I mean, maybe it's just my subconscious is working on it or something.

Starting point is 00:27:50 I don't really know. I mean, since it's subconscious, I don't know what's going on in there. Akash says, what advice does he have for someone in their mid-20s to learn math and physics? I work in law and I'm interested in physics. I would like to get a better understanding of physics. Now I know Fermat and Cayley, I believe, were lawyers in their mid-20s when they started physics. Sorry, when they started math. I mean, yeah, sure. As you point out, there are a couple of rather prominent examples of lawyers who end up being really rather good at math. Advice, I I'd just say follow whatever you're interested in

Starting point is 00:28:27 I mean and what one thing you can do for example is you just go into a math library and just browse at random you know you could pick out books at random look at the first few pages see if it's interesting and if it's interesting you could go deeper into it. So I think this is an example of what I was saying of generating random ideas, how do you do it? Well one way is just to pick out books at random in a math library and see what they say and again you know 90 or 99 percent of the time there won't be anything interesting there. But every now and then you run into something really good. What is it about infinity categories that you find confounding or troublesome? The definition to start off with.

Starting point is 00:29:15 I mean, it's one of these definitions. I must have read it half a dozen times. And half an hour later, I just totally forgot what the definition is. It just doesn't seem to stick in my mind at all. I mean, most things, you know, I have to read them, you know, half a dozen times before they stick. Infinity categories, I don't know. Half a dozen times is nowhere near enough. Have you read any of Emily Real's work on infinity categories? No, I don't know any of that. Okay.

Starting point is 00:29:49 Does he somehow systematically order his knowledge, mathematical or other, in something like a tree of knowledge? Does he explicitly memorize theorems or proof to make use of them more easily? Absolutely not. As I said, I'm really disorganized. I've never bothered trying to learn the proof of a theorem or memorize the theorem. I just sort of reread them several times. So I probably eventually do sort of know the proofs, but I never made a conscious effort to memorize the proofs. What I do try and do is instead of memorizing a proof so I can write it out word for word which is useless I mean a photocopier can do that you somehow want to

Starting point is 00:30:33 internalize the ideas behind the essential ideas in a proof and it's sort of very hard to explain exactly what this is I mean you somehow want to know the theorem well enough that you can actually use the ideas for something else, which, of course, a photocopier can't do. And what's going on in my mind when this is going on, I have just no idea. Do you find that you're happy with your life? Yeah, sure. Maybe I'm just too dumb to notice I shouldn't be happy, but yeah, I'm generally happy. Are you increasingly so? That is, with time you find it more meaningful? About the same. Not something I pay a lot of attention to, but... What are the most meaningful aspects of your life? Doing math mainly. Relationships? Well, I've got two kids and I'm married, yeah.

Starting point is 00:31:32 Do you find the relationship with your kids to be more meaningful than the mathematics? It's different. I mean, I'm not sure you can really compare them. They're quite different sorts of relationships. Okay, how about this? All the major math theorems that you want solved are solved. And you have as much money as you like. What do you do with your time? Sleep, I don't know.

Starting point is 00:31:59 I've no idea. I mean, I spend all my life, basically my spare time has been spent thinking about mathematics. If there was no mathematics left to do, I really don't know. It's sort of like maybe just read math for fun or something. I mean, it's still quite, I mean, even if you're not actually doing research, it's still quite interesting just reading other people's theorems and their research. So it would be a bit like me asking a dolphin what if you weren't a dolphin anymore and it can't conceptualize it? Yeah, yeah, I mean it's just the question doesn't actually mean very much.

Starting point is 00:32:37 What was it like to win the Fields Medal? Rather an anti-climax. I mean it didn't really mean very much, I mean it was just a sort of rather embarrassing ceremony where I sort of froze as far as I remember. Proving the theorems was was fun you feel very excited when you've done that but but you know getting a medal or something for it is just just sort of nothing the the only advantage of it is is is you stop is you stop worrying about getting these these things when you realize how unimportant they are. When you talked about math, you mentioned that it's meaningful to you, maybe one of the most meaningful, if not the most meaningful

Starting point is 00:33:28 aspect of your life. So what is it particularly about math that you find meaningful? It's understanding, it's part of understanding how the universe works. You know, there are all these fundamental questions like, why does the the universe exist and what are the fundamental laws of physics and mathematics is part of that. So whenever you're doing mathematics, you're sort of, every piece of mathematics is helping you a little bit to understand how the universe works, maybe not very much, but works, maybe not very much, but the universe, as far as we can tell from physics, the universe is running mainly on mathematics, although we haven't yet quite figured out the exact

Starting point is 00:34:12 mathematics behind it. Does that mean that you have the sense that when you're performing math, you're engaged in an act of discovery rather than invention? Oh, very much so. Mathematics is, yeah yeah there's this question about do you invent or discover math and I'd say it's entirely discovered not invented well there are one or two areas I won't mention where I would say it's invented and totally useless but almost all mathematics it's you're discovering something that's already there. So a classical example of

Starting point is 00:34:45 this is the monster sporadic simple group. This has always really amazed me because you know the axioms for a finite simple group are so trivial you can write them out in one line and they're very trivial and natural and then the monster simple group is somehow hidden in this single line so we've got this monster simple group with what 10 to the 54 elements and this fantastically intricate structure somehow hiding in this trivial one line definition and a lot of the best mathematics is like that you you you know you you you've got some sort of box that looks just like a plain small box without anything figuring it out. You open it up and somehow there's this massive diamond inside it with no indication on the

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Starting point is 00:36:45 for free. Just make sure to add them to the cart. Plus 100 free blades when you head to h-e-n-s-o-n-s-h-a-v-i-n-g.com slash everything and use the code everything. What is it about the monster group or the sporadic monster group that you find fascinating? Because to an outsider, let's say someone who's not interested in math, to say that it's interesting is almost as arbitrary as saying, why are there six platonic solids or why does the number 1740 exist? Yeah, because we don't understand why the monster exists. I mean, you know, so there's this classification of the finite simple groups and the classification says, well, there are these infinite families and

Starting point is 00:37:31 okay, infinite families of things are kind of uninteresting. Well, okay, they're not uninteresting. Most mathematicians find them very interesting, but finding an infinite family of things isn't too surprising. But the classification says there are these infinite families like alternating groups, and then there are 26 others left over, which there's no obvious reason why they should exist. And the infinite families, you can sort of think of a reason why they exist. I mean, you know, they're general linear groups and they exist because they're symmetries of vector spaces. It's a very natural explanation. And then you have these sporadic symbol groups exist and they exist because nobody really has any idea why they exist.

Starting point is 00:38:15 There's no explanation for them beyond going through, you know, 10,000 pages of the classification. you know, 10,000 pages of the classification. So this may be one of these unanswerable questions like, you know, what I mentioned earlier is why does the universe exist? I mean, maybe it just doesn't have an answer. What would an answer look like? This is the problem. I just cannot imagine how you could even answer the question, why does the universe exist? I mean... Imagine an answer that would satisfy you as to why the monster group exists. What would that even look like? One thing might be that, you know, that there's some very natural class of geometric objects whose symmetries are the sporadic simple groups.

Starting point is 00:39:03 And people have been searching for this for decades and no one has ever found any plausible candidates. So I mean for instance the infinite families of groups there is a sort of natural answer that they all act that they all have these structures called BN pairs. And this sort of gives an explanation of them. There's a uniform construction of all the simple groups that aren't Spradic. You know, there's sort of one construction that works for all of them. You just twiddle a few parameters and you get out all of them. But for the Spradics, there's nothing like that.

Starting point is 00:39:46 What is it about quantum field theory that you find difficult? In our email exchange, you mentioned that it gives you a headache. Yeah, I think the problem is that mathematicians are trained that everything should be really precisely defined and you shouldn't go into the next step before you've completely understood

Starting point is 00:40:04 all the previous steps. And quantum field theory just isn't like that. Things just aren't precisely defined. And if you try and really find out what the definition is, you find there's nothing there and it just doesn't make sense. So do you mind giving an example? Yeah, sure. If you look at Feynman diagrams, for example, they give you a sort of asymptotic expansion of something like a scattering coefficient or whatever. So it's a well-defined asymptotic expansion. You can sort of make sense of that. And it's an asymptotic expansion of, and at that point you suddenly get stuck. expansion of and at that point you suddenly get stuck. No one has actually figured out what it's an asymptotic expansion of so in some sense it doesn't really make sense and what is even more

Starting point is 00:40:55 annoying is that I mean if it didn't make sense I could cope with that. The trouble is that it doesn't make sense and it gives answers that you can check in experiments and this is really annoying because you know theoretically it doesn't make sense but experimentally it does so there's this horrible conflict um um so and so i mean in some sense the fundamental problem of quantum field theory is to give a rigorous definition of it. There still isn't one. Well, people have given rigorous definitions. The trouble is the quantum field theories that people actually use in the standard model don't fit these rigorous definitions, or at least no one has managed to prove that they do. And so this has been an open problem for maybe 90 years or so.

Starting point is 00:41:46 Do you have any thoughts on the existence of free will? Do you think you have free will? And if not, then why not? If so, why so? And how are you defining it? I thought about this a bit. And what I eventually came to the conclusion was that I didn't even know how to define what free will was. So if you can't define something precisely, it makes it very hard to discuss. And in fact, this is a big problem with an awful lot of philosophical questions. If you look at them close enough, you discover that the terms in them are simply not well defined enough to talk about. So can you even define free will? Can you talk about it, not as a mathematician, but let's say as a regular person? About what? Free will? About free will. Now, what you're saying is that there's the slippery definition.

Starting point is 00:42:31 But so there is also a slippery definition as to what constitutes a chair or I or you. And presumably you've used those in sentences before with facility. So can you do the same with free will? I can sort of vaguely define a chair in some way that i can most of the time recognize a chair i've i've tried to define free will and i just cannot come up with a definition i've poked around a bit and as far as i can figure out nobody else has come up with a convincing definition either i mean you certainly have this feeling in inside yourself that you can choose what to do but you know maybe this feeling is an illusion and everything we do is determined by some laws of physics and we just have this illusion

Starting point is 00:43:15 that we're choosing what these laws of physics do but I don't know I mean I can't even define what is meant by having a feeling that we can choose what to do. So you're agnostic on the issue? I think it's even worse than being agnostic. I'd say I don't, I'm not saying I don't know what the answer to the question is. I'm saying I don't even know what the question means. And I'm not at all sure that it even makes sense. What about God? Do you believe in God? And do you have similar feelings there? Before we start discussing God, I would like you to define what you mean by the term God, because there are so many different definitions. So you

Starting point is 00:43:56 could have a God of your favorite religion saying it's this guy who maybe zaps people with thunderbolts or whatever. Or you could say, you know, God is just a name for the underlying essence of the universe, which is sort of maybe meaningless, but you could say in that sense, God exists because God is just a name for whatever the explanation of the universe is. So there are all these different possible definitions of God, and some of them, all these different possible definitions of God. And some of them, you could reasonably say they do exist. Some of them you could say there seems to be very little evidence for it. And others, it's just not clear what the question means. So before discussing this philosophical question, let's first have some precise definitions. You mentioned one, that God is the universe

Starting point is 00:44:42 in some way, shape, or form. Now, why is that meaningless? Well, because it's just words. It doesn't actually tell you anything more about the universe that you didn't already know. So there's a whole lot of questions like this. You could say, you know, the universe is really a simulation or the universe is a cellular automaton, or maybe there are lots of parallel universes, and none of these actually say anything whatsoever about the universe. I mean, they're all great plots for comic book stories or whatever, but what's the difference between the universe being a cellular automaton and the universe not being a cellular automaton?

Starting point is 00:45:21 Well, there just doesn't seem to be any difference. Or what's the universe? What's the difference between there being a sort of pantheistic God underlying everything or there not being a pantheistic God? Well, I've no idea. It just seems to be a content-free sentence. In your psychology, in your mindset, nothing would be changed if you found out that you were the only conscious person or that you were living in a dream? I don't know how I could tell the difference between being the only conscious person or not being the only conscious person. I mean, I assume other people are conscious just by analogy with myself, but I can't imagine any experiment that could actually prove beyond doubt that someone else was really conscious rather than just a computer program running inside someone's brain.

Starting point is 00:46:12 Again, it's this problem of what is free will. I can't think of any experiment that can tell the difference between having free will and not having free will. Okay. How about this? will. Okay. How about this? I imagine that there is no experiment that can test whether you love your wife or your wife loves you. Yet I imagine also that the answer to that matters to you. So in a similar, well, does it first of all? Well, it's another of these terms that's really hard to define. I mean, I think this is another question about my internal state of mind, which as I've noticed in several of your other questions, I have a very hard time figuring out what my

Starting point is 00:46:50 internal state of mind actually is. Is that because you're not introspective? Do you tend to think about yourself or do you just tend to think outside yourself? I do sometimes think about myself and the conclusion I tend to come to is that I have absolutely no idea what is going on. I mean, it's a problem that completely baffles me. I mean, I have no idea what is going on in my own mind, let alone what is going on in other people's minds. So do you have any thoughts about equivocal questions about the hard problem of consciousness? You have to remind me, which is the hard problem i vaguely remember penrose talking about this but i've

Starting point is 00:47:31 forgotten which is the easy problem which is the hard problem the easy problem are the correlates so for example anger is associated with so and so part of the brain and a certain circuit the hard problem is how is it at all that consciousness that is experience can arise from material only? Yeah, this is like all these questions in Douglas Hofstadter's books. And again, I've no idea. We have all these experiments,

Starting point is 00:48:00 like suppose you simulate the brain by taking a trillion small computer chips and wiring them up exactly as the neurons are wired up in the brain so that it behaves exactly like a human brain. Is that conscious? And then you could go a step further and you could write a complete description of the human brain and what it's doing on a very big piece of paper. And is that piece of paper conscious and you know if if you you know erase bits of this piece of paper according to certain rules thus simulating the action of the brain are is you know is is that piece of paper with all these things being erased repeatedly conscious because it's you know in some sense it's indistinguishable from a human brain. And on the other hand, it seems utterly ridiculous to claim it's conscious. I have no idea how to resolve this problem. What are some of your favorite mathematicians

Starting point is 00:48:55 and physicists alive? Alive? Oh, well, I think the one I like reading most is probably J.P. Serre. I mean, I don't work in his area, but he's so good at writing that I read almost, you know, I try and read almost everything he's written. He's a number theorist? He's a number theorist. He's also an algebraic geometer. He's also an algebraic topologist. And he's also done quite a few other things. I mean, he's sort of one of the closest things we have to a universal mathematician. What about Terry Tao? Yeah, again, I've been collecting his books and they're on my big pile of books I really ought to get around to reading someday. And yeah, that's interesting because he thinks in such a different way from what I do. I mean, he's an analyst and, well,

Starting point is 00:49:53 again, he's not just an analyst. He also does lots of things in number theory and so on. But he's obviously thinking much more in terms of inequalities and error estimates and all the other things that analysts do than I do. So I'd say I'm much more towards the algebraic side where I do equalities and then sort of get a bit nervous whenever anybody writes down an inequality sign. What about physicists now? I don't really have favorite physicists. I mean, I enjoy reading Richard Feynman, but that's largely because of his popular books rather than because of his physics. Who else? I read, you know, Steven Weinberg has some rather nice books on quantum field theory.

Starting point is 00:50:44 Yeah, I don't spend so much time on physics, really. What are your thoughts on Ed Witten? Have you collaborated with him or met him? I've met him and he's simply terrifying. He produces fantastically brilliant papers, about one a month or something. And I don't know, I have no idea how he has the energy to do this. I don't know. I have no idea how he has the energy to do this. And, you know, it's a stream of endless good ideas that you know, he really had almost had nothing to do with before that. So, you know, he was just sort of revolutionizing the area of mathematics just as a casual side effect of something else he was doing.

Starting point is 00:51:42 Let's get to the monster group. What are some of the applications of the monster group to physics? There aren't any. Are there conjectures? Well, there's been some speculation. I mean, Ed Witten, for example, has some speculation about a connection between the monster and some sort of gravity theory, and that the mathematics behind the monster is quite similar to a lot of mathematics that turns up in string theory so you can speculate that there's a connection but no one has ever found any sign that the monster has anything to do with you know real everyday four-dimensional physics there are some people who try to build physics off of the e8 and what has a dimension in 8 has an echo in 24 and then 24 somehow associated with

Starting point is 00:52:26 the monster so I'm wondering can one base physics off of the monster group? I would I would write that off as a really crazy idea I'm not saying it's wrong I'm just saying it's really crazy I mean it'd be really nice if the monster group did turn up in physics somewhere. But as I said, there's no realistic sign of this yet, apart from the fact that both the monster and string theory use similar ideas. For people watching, just so you know, Professor Portraits has a great YouTube series on teaching graduate level mathematics. And I recommend you check that out. Do you enjoy teaching? Or is it just your connotation? It's fun making these videos, except it's also kind of stressful because I mean, it's, I haven't realized just how many errors I made in lectures. You know,

Starting point is 00:53:24 you're lecturing and you don't notice anything wrong. And then suddenly when you put up the YouTube, people are pointing out all these embarrassingly stupid misprints you made. And, you know, I guess I should just feel relieved that I'm not a, you know, I sometimes think, well, it's thank goodness I'm not a doctor or a cop or something, because, you know, if a doctor or a cop makes a mistake, you know, there's someone dead and there are headlines and committees of inquiry. And when I make an embarrassing mistake, I get this polite note from a YouTube comment. So maybe I shouldn't complain too much about it. You mentioned that innovations coming from outside the field of academia, for math in particular, is rare, unlike the past, which was, you gave some examples for Matt and Kaylee, who were lawyers. And I'm wondering, is this still true today? As far as I know, you mentioned one of the reasons is that it's easier to get a position in academia.

Starting point is 00:54:18 But I don't know if that's true today. Well. Unless you're a postdoc. true today? Well, unless you're a postdoc. It's very much true compared to now, compared to a few hundred years ago when there basically weren't any positions for mathematicians in academia, maybe one or two in the world. The other big problem is all the easy stuff in mathematics and physics has been done. So people like Fermat, when they started doing mathematics, there were whole loads of things they could do, which, well, they were difficult for the time, but nowadays they count as easy. Whereas if someone was starting from outside trying to do mathematics now, it'd be incredibly difficult because anything you can think of in less than several years' study, someone has already thought of it. So it's partly that there are, I mean, people coming from outside doing something big in mathematics has become very rare because it's so difficult to find any problem that an outsider can do.

Starting point is 00:55:27 problem that an outsider can do. What advice do you have for outsiders who want to contribute to math or at least study it at the graduate PhD level, but they're on their own? It's the pandemic. They don't have peers. Much like yourself, you don't collaborate. So what advice do you have for them? If you really want to contribute something new, you need to find a relatively new area in mathematics. So, I mean, every now and then, someone has a new idea that breaks open a new area. And then, you know, it's like, you know, there's this dead whale theory. You know, if you're at the bottom of an ocean,

Starting point is 00:55:57 there's usually nothing to eat. But every now and then, there's a dead whale comes down and everybody goes frantic eating it. And every now and then, there's a really new area in mathematics. And if you can spot one of those and get in on it, then there's a lot to do. A typical example, a few years ago with things like fractals and chaos became really big things. And there was quite a lot to do.

Starting point is 00:56:20 It's very hard to suggest new areas because if I knew about them, then I'd probably already be working in it or something. Yeah, there's one thing I was thinking of that might be possible to contribute, that none, people who are not mathematicians, might be able to contribute to, and that's artificial intelligence doing mathematics. I mean, you know, so Google came up with this artificial intelligence program that could play Go or chess very well. And I'm pretty sure the people at Google programming this were not expert Go or chess players.

Starting point is 00:56:58 They were, what they were really good at was building an artificial intelligence system. And sooner or later later artificial intelligence is probably going to get good enough to do serious mathematics so that might be one thing to look at if you're an outsider you know get into artificial intelligence doing mathematics I think there are not so many pure mathematicians looking at that now. And in a few years, it might take off quite suddenly. So in some sense, let's say 30, 40 years from now, the computers will be better than us at doing mathematics. So why don't you be the puppet master behind the computer if you want to help the innovation? Yeah, exactly.

Starting point is 00:57:39 At some point, did you ever feel like math and physics wasn't for you, either because of its difficulty or for some other reason? No, not really. It never seriously occurred to me to do anything other than mathematics. I mean, maybe when I was six, I thought it'd be fun to be a fighter pilot or something. a fighter pilot or something. But apart from that, I never seriously consider anything other than mathematics or possibly computing

Starting point is 00:58:09 as an emergency backup. What are your thoughts on the many worlds interpretation? Well, like I said, you have all these phrases like maybe there are many worlds, maybe we're a computer simulation, maybe we're a cellular automaton, they don't seem to mean anything.

Starting point is 00:58:33 So what's the difference between a many worlds interpretation and a non-many worlds interpretation? Well, there doesn't seem to be any difference. Okay, I'm going to get to some audience questions from reddit i think that modern geometry is overly complicated as in if you look at john lee's introduction to smooth manifolds you'll find that it has plenty of little lemmas and theorem theorems that say very little the content is little but they're elaborate proofs and i'm wondering if this inadequacy is due to basing the foundations of math on set theory, leading to very cumbersome frameworks for geometry. Do you sometimes feel that there should be a way to base mathematics off of some alternate framework, some topological geometric consideration that's less cumbersome?

Starting point is 00:59:21 Well, there have been endless attempts to do this. I mean, what you have to remember is that set theory didn't become the, well, practical foundations of mathematics just because someone decreed it. What happened early in the 20th century, people were trying dozens and dozens and dozens of different attempts to have foundations of mathematics. I mean, the Principia Mathematica of Russell and Whitehead, Lambda Calculus, and Quine, or however you pronounce his name, came up with another foundation of mathematics. And the reason we use set theory, because it was won out as being the most convenient one. And people are trying all sorts of things these days based on type theory or category theory. And again, none of them have really taken over from set theory. So set theory is sort of won by a sort of evolutionary process.

Starting point is 01:00:20 People keep trying to modify things and taking the fittest one that survives best. And set theory is the one that has won out. So when you say set theory is clumsy and there are lots of problems to it, yes that's true, but the trouble is everything else has even worse problems. I figure that when computers start doing mathematics that it may be that they won't use set theory because some forms of type theory or something a little bit more computational than set theory may be a bit better. And in fact, there are about a dozen or so different attempts to find a good foundation for computers doing mathematics. And most of them don't seem to use set theory directly. But again, what we're seeing is an evolutionary process.

Starting point is 01:01:05 People are going to try out all these different approaches and probably one of them will turn out to be better than the others and take over. I find that when I talk to category theorists, they say the foundation should be category. Oh, that's nonsense. Category theory is a useful bookkeeping device, but the idea that it's the ultimate foundation is just silly i mean it's like you know finite simple group theory saying the ultimate foundation should be finite simple group theory or whatever and the problem is category is just not a primitive concept um you can write down axioms for category but there's you know it's you know i mean maybe you want to base mathematics on say an elementaryos, which is a special sort of category. Well, fine, you can develop mathematics inside an elementary topos.

Starting point is 01:01:51 That's no problem. The trouble is this gives you no explanation for why the axioms for an elementary topos should be consistent. Set theory at least makes a sort of gesture about trying to explain why the axioms of set theory are consistent. I mean, I don't think it's a very successful explanation, but at least they sort of make a, you know, they make a sort of token attempt. Trying to claim the foundations of mathematics are category theory, there's just no attempt to explain why the axiom should be true or consistent or whatever. This is a similar question. I have always been interested in seeing whether we could develop an all-encompassing framework despite decades of seeing a proliferation of different set theories with all sorts of independent results. I know Hugh Woodland or Hugh Wooden has been working on Ultimate L, some Ultimate L research program.

Starting point is 01:02:47 While others say pluralism is here to stay, people like Joel, David, Hamkins, who is a mathematical Platonist. What are your thoughts on this subject? Do you think mathematicians like Hugh Wooden and Peter Kullner may find success in developing an ultimate canonical model of set theory, or would you more closely align yourself with the multiverse view of set theory that Hamkins advocates for? I think, well, my views on this change every month or every year or so. So the current views is that my impression is very much that there are many different models of set theory and there's no ultimate model. In some sense, it's a consequence of Gödel's incompleteness. There's no ultimate model of all sets. The best we can do is have a hierarchy of increasingly powerful models or

Starting point is 01:03:38 systems of set theory. So, you know, you can start off with, say, Zermelo set theory might be, you know, somewhere near the bottom. And then above that, you can have off with, say, Zermelo set theory might be, you know, somewhere near the bottom. And then above that, you can have Zermelo-Frenkel set theory, which is a more powerful model. And then you can start adding in your favorite large cardinal axioms. And there just doesn't seem to be an ultimate model. And I think it's unlikely that it will be. to be an ultimate model. And I think it's unlikely that it will be. In fact, if there was an ultimate model, then you could just, you know, hit it with Gödel's theorem and point out there's an even bigger model that proves the consistency of your so-called ultimate model.

Starting point is 01:04:16 You may be able to find a sort of ultimate convenient model that can do everything we can think of at the moment. But I mean, in some sense, that's what Zermelo-Fraenkel set theory is. I don't think anybody claims that Zermelo-Fraenkel set theory is an ultimate model, but it's enough to do almost everything we need to in mathematics. So it's a very convenient one. You said that your views on this change month by month or year by year. So what was it before and why did you change it i don't know i sometimes thought that you know you can it can oscillate between finitest mathematics where you deny everything that can't be expressed in terms of

Starting point is 01:04:56 computation and you can go from there all the way to platonism where you claim that there is a single ultimate universe of all sets, and there are all sorts of intermediate points. And I sort of float up and down this depending on, I don't know, depending on my mood or what I've been thinking about recently. So recently or currently, what do you think of finite? You called it finite, as far as I know, it's called intuitionist logic, the computational model. That has a lot going for it. Basically, especially if you're trying to do mathematics on a computer, because of course, a computational based foundations of

Starting point is 01:05:34 mathematics are particularly convenient if you're actually trying to do computations. I mean, you know, computers can't directly manipulate sets in the way that they can directly manipulate integers. The only trouble with the finitistic foundations of mathematics is they're very restrictive and it's really hard to do most mathematics in them. I mean, it's sort of what the analogy is like, you know, it's one of these religious sects. I think someone once said it's a bit like the Shakers who built really sturdy barns but didn't have offspring. And Finiteist mathematics is like that. It's really solid and an absolutely solid foundation, but it's difficult to do anything productive with it. I mean, if you look at any, almost any piece of mathematics, for instance, if you look at, say, Wiles' proof of Fermat's last theorem, okay, Fermat's last theorem makes perfect good sense

Starting point is 01:06:36 to a constructive, finitist mathematician, but trying to encode Wiles' proof in finitist terms, well, it would probably be possible, but it would be a real nightmare to do. I mean, it would take years and years of really boring, tedious work, even if you could actually do it. Arian Raj asks, can people after 40s learn mathematics up to an advanced level? Now, let's break that down. I don't believe he's asking, can they research? Can they contribute to research? I think this person wants to know, can I learn enough that I can, let's say, understand one of your papers if I'm starting at 40? I'm sure you could if you're willing to spend enough time on it. You have to remember that

Starting point is 01:07:19 when people in their 20s are learning mathematics as grad students or whatever, they're spending, you know, eight or 12 hours a day every day for years learning it. And if you're prepared to put that amount of time into it in your 40s, my guess would be you probably could. The problem is, you know, by the time people are in their 40s, they tend to have jobs and kids and all sorts of other distractions. So it's very much harder to put in the amount of time necessary. Boris Costello wants to know your take on the unreasonable effectiveness of mathematics in the natural sciences. This has really baffled me. I mean, what was my take on it? Well, that just sums it. I mean, so the problem is there seems to be no

Starting point is 01:08:02 reason at all why the universe should make any sense. I mean, you imagine what people thought about the way the universe worked, you know, two or three thousand years ago, you know, things just happened for no reason. Trees just grew because that's what trees did, or maybe there was some god making them grow or something. And we've discovered since then that nearly everything seems to be ultimately controlled by mathematics. And I've no idea why or no idea why this should be true. You know, it's possible you could have a universe that just sort of was like a universe that sprang from the mind of a fantasy author, where things just worked by magic for no particular reason. But we seem to be in one where it runs on mathematics.

Starting point is 01:08:46 Rajesh V wants to know, are you puzzled by most people's inability to understand math, or do you see a reason behind it? I'm not puzzled by most people's inability to understand math. What really puzzles me is the fact that anyone can understand math at all. I mean, from a point of view, if you're an evolutionary psychologist, you can pretty much prove that no human could possibly do mathematics. Because, you know, think about where we evolved. We were running around on the, probably the African plains being chased by leopards or tigers or something. And if you get someone who, you know, stops to wonder about large cardinals, well, you know, 10 minutes later, they're going

Starting point is 01:09:25 to be passing through the digestive system of some large carnivore. I mean, the mathematics gene should have been wiped out by high evolutionary pressure. So, you know, why do humans get interested in mathematics? This is what's difficult to explain. What's the longest time you've thought about a problem before finding a solution to it? Well, it's either, I think, let me see, moonshine conjectures were probably about 10 years. I might be able to give you about 40 years, only I don't know whether I've actually solved the problem yet. So somewhere between 10 and 40 years, depending on whether the problem I'm currently working on works out or not. Is the problem you're currently working on a monumentous one?

Starting point is 01:10:17 Is it one that people would recognize if you were to say it by name? If I were to say it my name, everyone would think I'd gone completely mad, so I'm not going to say anything more about it. It's about God, isn't it? I said mad, I didn't say crazy. What do you think of Ramanujan,

Starting point is 01:10:37 a self-taught math genius? This is just, I'm just overwhelmed by it, because he did something that I thought would have been almost impossible, that he managed by his own unaided efforts to turn himself into one of the greatest mathematicians. I mean, you look at what he was growing up in. He was growing up in extreme poverty. And I mean, even surviving in those conditions would have been as much as most people could do. And to do this phenomenal mathematics, I just cannot imagine how he did it.

Starting point is 01:11:15 I mean, you know, I mean, Ramanujan himself said that, you know, how did he get his mathematical formulas? And he said he got them in dreams from some goddess or something and i can't actually think of a better explanation i mean it's it's it's almost miraculous that he managed to do this right some of the formulas are so arcane and so esoteric that you wonder how can you come up with it without deriving them and he would just they would just pop into his head and they would turn out to be true well i i don't know i i think i think he was fudging a bit then i i think it was um i mean you know some mathematicians have this a bit of a desire to keep their methods to themselves a bit and maybe maybe romanogen was doing that to some extent um but i i really don't know i mean, I've looked at some of his papers and books, and I just cannot imagine how he thought of them.

Starting point is 01:12:10 I mean, some of his formulas you can prove by using techniques that Ramanujan didn't know about. I mean, it's kind of notorious that Ramanujan never knew complex analysis, which is a very powerful method for evaluating integrals. But somehow he never seemed to feel the lack of it. He was able to do by real variable methods what most of us need complex variable methods to do. So somehow he just got really, really, really good at real variable methods in a way that almost nobody since has managed to equal. Bob Bobbity says, thanks for the Q&A. I'm just wondering if you plan on continuing your number theory series? Yeah. If you're talking about my YouTube channel, well, I'm not really planning to, but I'm not not to it's it's just sort of topics just turn up at

Starting point is 01:13:06 random depending on what i feel like and what people vote for and so on so probably if i keep going number theory will probably continue i requested to you over email for you to start a series on quantum field theory what are the chances of that uh not all that high because I had a sort of bad experience with it. Like I tried to understand it and invested a few years and felt I'd really, it's like really hitting a concrete wall. When you read quantum field theory books, virtually every one of them tackle the problem differently, unlike mechanics or quantum mechanics even. I would be so curious to see how you tackled it. Well, yeah. So when you say differently, I think you're presuming, meaning there are lots of different approaches, like you can do it via operator algebras or via perturbation theory or by trying to make sense of

Starting point is 01:14:03 infinite dimensional integrals or by doing topological quantum field theories or whatever so so there are all these different approaches and what they all have in common is that none of them really work um so i was looking at all of them and couldn't really make much progress on any of them so i mean the the the one that people have that's been most successful is the perturbation theory approach. I mean, this is sort of what physicists actually use to calculate numbers that they check in their experiments. But again, you know, perturbation theory produces these divergent infinite series. And to make sense of them, you cheat.

Starting point is 01:14:46 You just take the first few terms, which makes no mathematical sense. And I was never able to figure this out. So you see an arbitrariness as to when they choose to cut off? Well, maybe physicists who know more about this would say it isn't arbitrary but as far as i can figure out that there didn't seem to be any very good reason for stopping at some point rather than another i mean i guess you can stop when the terms start getting big again but that's sort of mathematically very unnatural ja524309 that username, says, I'm curious why you think memorizing proofs is useless or less useful than doing calculations.

Starting point is 01:15:31 Could you elaborate? I should say, I agree with some of your other remarks, e.g. there not being a best proof of any particular theorem. That was when you were asked about the book. I was watching a talk by Barry Mazur the other day, and he spoke also how different proofs can sort of allude to different possible generalizations of the same problem, which may or may not necessarily themselves be able, which they may not be able to unify. Yeah. Sorry, that was. Yeah, let me just read the first part because I believe the question is...

Starting point is 01:16:06 I got confused by having two questions. Oh, I'm confused. I'm confused reading it. Okay, I'm curious why you think memorizing proofs is less useful than doing calculations. Let's do that part first. I think this is one of the things that probably varies from person to person. So people have very different styles for learning things. And maybe for some people, memorizing proofs is a useful thing. I'm just saying for me personally, I never found memorizing

Starting point is 01:16:32 proofs to be very useful. That's mainly because my memory isn't as good as some people's. I mean, I've noticed that some people have incredible memories. I mean, you know, they can sort of read a book and basically recite it back word for word. So von Neumann, for example, could do that. And maybe for people like that, they can memorize proofs. But for me, at least, I mean, I could memorize a proof if I put a lot of effort into it, but it would be, you know, I mean, it's, there's what economists call an opportunity cost. My time is actually more productively spent doing something else because I seem to learn.

Starting point is 01:17:11 I seem to understand things better by actually doing calculations with examples. So so so I guess my answer only applies to me and doesn't necessarily apply to other people. You mentioned learning styles. Okay, so what are some of the different learning styles and how does one figure out their own, the one that fits for them? Well, for me, learning is very much, I have to, my learning style is, as I said, I need to do calculations and I need to read things over and over again because I'm rather bad at learning things. Other people's learning styles,

Starting point is 01:17:46 I don't actually know much about because all I can notice is that they're obviously different from mine in some ways. A really obvious one is I'm terrible at learning languages, whereas other people can learn languages really easily. So they must be doing something different. Do you have a philosophy of mathematics? And if so, what is it? Philosophy of mathematics. Well, this is like the question about what the right foundation

Starting point is 01:18:14 should be. And as I said, there are several underlying philosophies. So there's finitism, where you think everything should be computation, and Platonism, where you think everything exists in some mystical realm. And there's formalism, where it's just a game you play with rules. And I'd say I sort of kind of oscillate between them, or maybe amalgamate them all. I mean, they all have some good points and some bad points. So I don't really have a single philosophy. I just sort of kind of amalgamate all philosophies and pick and choose whichever is most convenient at the time. Now, there seems to be something a little bit off about this. It's like trying to belong to several different religions all at once.

Starting point is 01:19:05 But somehow for mathematical philosophies, maybe you can get away with it. What do you mean you pick and choose which one's useful? What would be an example of one that's useful for you? Well, I mean, I think the different philosophy wouldn't really be useful for actually doing math research. The different philosophies would be useful for trying to understand what the foundations of mathematics and the universe are. So if you think the universe should be based on computation, then maybe you try working with a finite artistic computational model of mathematics. If you're trying to understand some deep,

Starting point is 01:19:47 complicated theorem, then it's very difficult to prove these theorems in a finitistic model, so you may have to sort of switch to a more platonic model where there is, you know, you assume the existence of very strong constructions like the power set of a set. I mean, it's very difficult developingions like the power set of a set. I mean, it's very difficult developing things like general topology, for example, in a finitistic universe. How does one know, by the way, this question is from Bag Biggered, how does one know when they're on the right path in finding a solution? I've tried substantial unsolved problems and I end up making progress, but then I feel

Starting point is 01:20:24 like I'm chasing my own tail. This is one of the hardest problems in research. And if you could solve this problem, you'd become the world's greatest mathematician. It's sort of like asking, how do you time the stock market? Nobody knows how to do it. And if you could, you'd be really rich. to do it and if you if you could you'd be really rich um so so it's this question of knowing what the right solution to a problem is the right way to solve a problem is and you know i mean this is in some sense the hardest part of research trying to trying to figure out whether you're on

Starting point is 01:20:59 track or not um i heard a story about that. There was this mathematician called Nagata, who was, you know, half his work was proving these amazing theorems and half his work was proving these stunning counterexamples to theorems. And the story I heard was that, you know, every morning he would try and prove a theorem and every afternoon he would try and find a counterexample. So one or the other would work out. So he had both bases covered in some sense. He didn't need to know when he was on the right track because he was trying all possible tracks simultaneously and just found out which one worked by trying everything. Teham asks, can you explain Langland's program and does it have any connection to physics? And does it have any connection to physics? No, yes. Langland's program, if you had to say it in one sentence, it would say that automorphic forms on groups have something to do with representations of Galois groups, which sounds like this obscure technical stuff. But rather amazingly,

Starting point is 01:22:05 an awful lot of problems in mathematics turn out if you go very deep into them to be related to the Langlands program. For example, the classic example of this is Weiler's proof of Fermat's last theorem was essentially using this idea. Does it have anything to do with physics? Well, some of the more recent work on geometric

Starting point is 01:22:27 langlands is using ideas from quantum field theory so i can give you a definite maybe on that one there are certainly some ideas that turn up in both langlands program and physics whether or not there's any more direct connection i don't know but it wouldn't surprise me langlands program seems to, it seems to be one of these things that turns up almost everywhere if you go really deeply into something. You mentioned that you're extremely interested in learning more about it. I've been so for decades and there's just so much to learn. I mean, I mean, Langlands program seems to be the ultimate black hole for what you need to learn in order to understand it.

Starting point is 01:23:09 There are several very deep subjects like representation theory and algebraic geometry, which are themselves notorious for how much you have to learn to get into them. And Langland's programme sort of subsums all of these. You need to know algebraic geometry and representation theory and a lot of hard analysis. So it's more or less, it's almost like it's every hard area of mathematics all combined into one. Well, you do have a course on representation theory and on algebraic geometry. So you should have some advantage there, no? These are just scratching the surface of what you need to know. The representation theory I've put

Starting point is 01:23:49 on is just finite groups in characteristic zero, which is the easiest example. It's not quite trivial, but it's the really easy part of representation theory. Langlands program, the groups become infinite groups, the representations become infinite dimensional, everything becomes far, far more complicated. Just Us Perthes asks, how can a person who's self-studying deal with gaps in knowledge? When I get stuck on a new concept, I'm often unsure exactly what it is that's preventing me from understanding it. concept, I'm often unsure exactly what it is that's preventing me from understanding it. And I, in other words, I don't know what I don't know. Well, one thing you can do these days is there are lots of math discussion groups on things like Reddit, and you can just try asking. I mean,

Starting point is 01:24:36 you may have to poke around a bit to find the right sort of math discussion group that will answer whatever level of question you're asking. Some groups, I think there's a, there's, if you search for math overflow and math stack exchange, these where you can ask questions and you'll quite often get serious answers from mathematicians about them. Okay, for the people watching, please ask any further questions, because I'm going to end this at some point soon. But make sure your question is asked in such a way that if I read it verbatim, it makes sense, because I don't like having to parse them in real time. And just one question at a time, because my short term memory seems to be getting worse and worse. Yes, just

Starting point is 01:25:19 keep it at the length of one sentence, one short, one 14-word sentence. Okay. What have you been reading recently? Let's say nonfiction and fiction. Gosh. My mind is, for some reason, completely blanking out on this. I mean, I can give a boring answer of all the books I was reading for my YouTube videos. You tend to look at older books, at least you reference them plenty. For example, you referenced a book that was 100 years old, where the calculation, where the graphics look

Starting point is 01:26:00 like it's a modern graphic, but it was by draftsmen. Yeah, that book is absolutely fantastic because it's written in the days before computers, and the graphics there just blow away most of the graphics people produce by computer these days. Somehow their pictures have a sort of soul in them whereas if you if you just get a computer to draw a graph of function it's just you know the aesthetic quality is not there yeah yeah the heart isn't in it exactly i mean that the you know no one has no one has sweated blood to do the picture you're just not interested and i can't imagine how they did it. I mean, there are just pages and pages of numerical data they must have all done by hand. And there are pages and pages of graphs that must have been painstakingly drawn and calculated. I mean, the amount of effort they put into that

Starting point is 01:26:57 is just extraordinary. How are you finding YouTube, by the way? I imagine it's new for you. How are you finding YouTube, by the way? I imagine it's new for you. It's interesting and very stressful. As I said, I get really, really bugged out by making mistakes on my videos. It's just so embarrassing having these. Well, what I was thinking is, what it reminds me of, you know you get these books written, you know, the million stupidest things ever said by Professor Watts, by President Watts's name and things like that. I suddenly have an awful lot more sympathy for President Watts's name who blurts out

Starting point is 01:27:36 something stupid whenever he opens his mouth because I've noticed in my YouTube videos, I'm continually making really, really dumb mistakes. I mean, it seems to be almost impossible not to. You can prepare and check everything and it doesn't matter. When you're actually talking, the stress causes you to say really dumb things. You get stressed out when you're making your YouTube videos? It's somehow very, yeah, it's very stressful. And when you're stressed, your mind just closes down and you make the dumbest mistakes. And you get to the end of the video and you wonder how you could have said something so stupid. What is it about it that you like? So you just mentioned a negative aspect. As I said, research consists most of the time of getting nowhere.

Starting point is 01:28:26 And if you're making YouTube videos, you're at least doing something. And I was rather surprised people actually seemed to like watching math lectures on YouTube. I mean, yeah, I mean, I started putting these videos because they were my, you know, lectures for courses I was supposed to be teaching. And I had to make some videos for them. and I had to make some videos for them. And it sort of didn't really occur to me that people would actually watch them for fun because I can't stand watching lectures. But other people obviously learn differently. How much time do you spend thinking

Starting point is 01:28:56 about your own particular background and knowledge and skills and relating it to previous knowledge versus spending time thinking without relating? relating it to previous knowledge versus spending time thinking without relating? That's an abstract question. I'm not quite sure. I would say, I'll try and answer it. I would say that nearly everything I do, I'm trying to relate it to other things I know. I mean, a lot of what research consists of is making unexpected connections between things. So if you're working in a completely new area

Starting point is 01:29:34 that has no connections to anything you know in the past, it's going to be really, really difficult because it's going to take a lot of time to build up enough knowledge to start making connections. to build up enough knowledge to start making connections. So you really want to be relating everything you do to, hopefully, to something you've done earlier. I mean, you're trying to build up a big, in-connected web, not have a lot of series of discrete domains of information that have nothing to do with each other. Neon Dagger asks, math people often talk about being drawn in by a wow subject that is hidden from them earlier on. Is it important to wow students to draw them into math? If you're trying to encourage people to do mathematics, this would be one way of doing

Starting point is 01:30:19 it. But it's, again, one of these things where different people have different approaches, that some people are drawn in by this wow thing and others are drawn in because they just like working steadily for... Your screen has gone very weird. Okay, one second. There we go. There we go. There we go. Do you remember what you were saying? As I said, I'm under stress, which makes my short-term memory even worse than usual.

Starting point is 01:30:56 Do you feel stressed being interviewed right now? Very much so. I mean, I've noticed this, that I'm having great difficulty thinking about questions that people ask. I have the same phenomenon in lectures. People ask me a question in a lecture, and I've no idea what the answer is. And then I get out of the lecture, and it's completely obvious. So somehow, yeah, being interviewed or talking, whatever, the higher parts of my brain simply close down. How often are you interviewed?

Starting point is 01:31:33 I think, well, not very. There seem to be one or two people asking me for, you know, they're undergraduates doing projects where they have to interview a professor or something. So I've had maybe two or three interviews. Your screen has gone weird again. Okay, okay. Let me just switch cameras then completely. I have a feeling it's just overheating.

Starting point is 01:31:56 Yeah, I was looking, just so you know, I was looking for interviews of you, other people doing podcasts with you, and I couldn't find any. Okay, what's up with Golbach's conjecture other people doing podcasts with you and I couldn't find any. Okay. What's up with Golbach's conjecture and why is it hard to crack? I don't really know why it's hard to crack. It's people have got very close.

Starting point is 01:32:19 I think there was a, I think this isn't my area. so I may have misremembered. I think there's a Chinese mathematician, maybe Chen or something, who managed to get very close. So Goldbach's conjecture says that every even number is the sum of two primes. every even number is the sum of two primes. And you can get quite close to that, that with a lot of work, you can prove that every even number is the sum of a prime and a product of two primes.

Starting point is 01:32:52 And for some reason, there's some horrible obstruction to going any further than that. And I've never managed to understand what the obstruction is. I think Terry Tao, who you mentioned, I once saw a discussion by him of what this obstruction is. So he might be a better person to ask. And again, we can prove that every odd number is the

Starting point is 01:33:14 sum of three primes. So we've got three primes instead of two primes. So we've got very close to it. But again, there seems to be this weird obstruction to pushing that last little step. So, as I said, it's not my area, so I don't know exactly what this obstruction is. L. Ham Ramanhi asks, what advice do you have for someone in their 20s trying to learn math? I know variations on this question I've been asked before already. Well, go for it. I mean, it depends very much on how much time you can spare to put into it. I mean, people in their 20s are quite often trying to have their career starting up, so may not actually have a whole lot of spare time. So I'm probably just repeating the answer I gave earlier, but my advice would be to find something you're really interested in.

Starting point is 01:34:08 And my usual suggestion for that is something like modular forms, which is that that's the area I found particularly striking. I just wanted to conclude by saying thank you. I appreciate you being generous with your time. I watch your videos. I gain from them. And I'm pretty sure plenty of the audience from this channel will gain from it as well. Yeah, well, I think having interviews like this is really good practice for me since as you know, as I'm sure you observed, I'm rather nervous, not very good at them. Well, just so you know, I'm extremely nervous as well. And partly one of the reasons I have to go urinate is because I urinate when I'm nervous. It didn't show.

Starting point is 01:34:53 I was thinking how calm and professional you looked. No, thank you. I'm absolutely not professional. I appreciate that you would even give me a modicum of that. I'm extremely introverted and i have social anxiety it's just that i have to trade myself to overcome it to some degree or is everybody faking it or there's some people who are really extroverts no there are plenty of people who are introverts just so you know it was somewhat ill-advised on my part to speak to you because usually I have, I ask a professor, can I speak with you

Starting point is 01:35:28 six to eight weeks from now? And then they'll say yes or no. And then I have six to eight weeks to look up their corpus of work and make notes and, and ask them somewhat challenging questions or technical questions for furthering my understanding, basically trying to get in their head, not trying to disprove or have a contentious interview at all, basically trying to get in their head, not trying to disprove or have a contentious interview at all, just trying to understand. And for this one, I messaged you on Monday. I believe you said yes on Tuesday.

Starting point is 01:35:53 And then I had one full day only this week to go through your entire work. And I could barely get through the, I can barely get through the overview of the proof for the monstrous moonshine conjecture. Well I think you're way ahead of most people on that. The surprise is not that it's hard to understand, the surprise is that we can understand it at all. I'm much more algebraic, far more algebraic than I am geometric, and I dislike when people show group theory at first

Starting point is 01:36:21 with the Rubik's cube or with the triangle, because to me, I'm just thinking they're far more symmetries than what you've just shown. So someone will say the triangle has six. It may not, it may have eight. I don't know, but whatever. Then I'm thinking, yeah, but isn't there also the symmetry of, let's say you invert the thickness of the line and then bring it back. So is that not a symmetry or is there not a symmetry of inverting the colors and then bringing that, like, why are you choosing that? that just just show me the axioms algebraically i understand that yeah well yeah um yeah but i think that it's it's the usual problem that you you can't give a presentation of group theory that satisfies everybody because people's learning styles are so different some want to see the axoms, some want to see examples and so on. Were the questions okay? Yeah, except the ones where I kind of got stressed out and couldn't think what to say.

Starting point is 01:37:16 I think there's one weird one. You asked me what I had been reading recently. My brain just completely froze. Yeah, I find that happens for me as well. I was actually thinking about that. I'd read somewhere where someone was saying, well, they'd asked people what they'd been reading recently, and some people were unable to answer with a single book as an example of how illiterate people were.

Starting point is 01:37:41 And I used to think to myself, well, exactly the same is happening with me. I just can't think of anything I've written in the past, anything I've read in the past day. I just couldn't think of anything at all, even though I've looked at several books. Yeah, I'm just hoping I didn't say anything that's going to get me condemned on the internet. You know, there seem to be too many cases of people who inadvertently say something harmless, which turns out to be a code word for some horrible crime. See many mathematicians, physicists too.

Starting point is 01:38:11 They have a difficult time with answering questions that have to do with that are ambiguous. And yeah, well, me too. I mean, you know, whenever there was a, freezing up every now and then with one of these more general questions. Right. And I would freeze up too. But the difference between me is that I have an artistic side. That's the filmmaking side. That's the standup comedy side. So in a sense, that's almost like you deal with ambiguity

Starting point is 01:38:40 as your regular male you. So for me, I don't have too much of a problem with it. And I find that much of the, I don't know if you'll agree with this, but I've heard this said that much of the solution comes from specifying the problem. And so I wonder how much of, let's say, is an ant conscious or what is consciousness and so on, so on. How much of it is it, the solution is just, well, how do we define it properly? Yeah, well, and this seems to be the cause of half the problems in philosophy. They're arguing about something that simply hasn't been properly defined. Possibly you could say the same about religion too.

Starting point is 01:39:24 possibly you could say the same about religion too yeah and then some will even say that actually it's the opposite the problem in religion is that people are trying to define it and it should be left implicit possibly yes i mean um it's like me backing out of your question about god by saying i didn't know how it was defined that's everyone that's every well that's not everyone but that's most mathematicians most physicists and that would be me prior to two years ago yeah well mathematicians were actually trained to have things well defined before you start discussing them of course. But... Sorry, I wanted to speculate around that. Because I wonder how much of it is, it's not that they don't like that it's somewhat meaningless, quote unquote, but that because it's

Starting point is 01:40:17 meaningless, they're afraid that what they will say will then be used against them in some foolish, like, oh, you're imbecilic for saying so and so and it's more about the fear of commenting on what's vague and saying something foolish than it is about commenting on it at all and the reason why i say that is because in our everyday life like i mentioned to you we do use the word game and chair and i but those are actually ill-defined like what is a chair maybe it's something to sit on, but then you get stumps. And then, but what is I? The more you look at I, what defines I? And you know, there's various.

Starting point is 01:40:51 Well, I'd say everything is ill-defined, but some things are more ill-defined than others. So chair, I will normally know whether something is a chair or not. Whereas something like God, I've just no idea. It's much, I mean, chair and God may both be ill-defined, but God is a lot more ill-defined than chair. Before we were talking about Ramanujan, and you mentioned that when he said that he got his ideas from his prayers, his visions, meditations, and so that probably he didn't. He had a secret mechanism that he wasn't revealing. And I'm curious, do you

Starting point is 01:41:28 have a secret technique to mathematics that you don't reveal? No. And I don't know about Ramanujan either. I'm just sort of guessing that maybe he was giving an answer to a slightly silly question. I mean, as I said, it's just very difficult to analyze your own mind and figure out where ideas are coming from. What do you think would be the implications if Ramanujan did indeed get his formulas inexplicably from his visions and so on,

Starting point is 01:42:00 from a goddess or a god? Well, I think that would just be his explanation of, explain what's going on in his subconscious brain. I mean here we call it subconscious, other cultures might call it a vision from a goddess. I mean they're both ways of saying we have no idea what's going on. You mentioned before that modular forms are a field that you find to be the most fascinating or provocative or stimulating. I don't recall which one of those adjectives you used, but why is that the case? Well, so what modular forms do is they seem to turn up in all sorts of areas of mathematics that appear to have nothing to do with modular forms.

Starting point is 01:42:48 Obvious one I used quite a lot was they turn up in the theory of the monster simple group and simple groups seem to have nothing to do with them. Marina Vyakovska used them to solve the sphere packing problem in high dimensions and again that has no obvious relation to modular forms Andrew Wiles used them to prove Fermat's last theorem which there's no obvious reason why they should turn up so they're turning up in completely unexpectedly in all these completely separate mathematical areas and whenever they turn up all these utterly bizarre coincidences start happening I mean in order for modular forms to solve a problem you need a whole chain of bizarre numerical coincidences and these numerical bizarre coincidences often seem to happen so you know there's something very weird going on.

Starting point is 01:43:46 As to why modular forms turn up in all these places, I just have no idea. It's just one of these bizarre mysteries. Is one of those connections, those coincidences, the 1,9,6,8,8,3? Yeah, exactly. So there's this, that's a typical example of this bizarre numerical coincidence. You take a representation of the monster group, and it happens to be almost the same as a coefficient of the elliptic modular function. another mathematician. I recall you saying that you believe mathematics is discovered, and he says it's invented. And I don't know which mathematician this is, so don't worry, this is an anonymous mathematician. He says, mathematics is created because it's a language used by humans so that we can understand and make sense of the world around us. For mathematics to

Starting point is 01:44:36 be discovered, you would assume that the way humans perceive the world is absolute truth, and in some way establish humans as the universal standard of everything. I might be rambling here, but I get the feeling that if you choose to see mathematics as something that was discovered by mankind, you are taking a human-centric and deterministic view on the universe. We only created it to make sense of things, just like we created the spoken word to communicate. So what are your thoughts on that? Well, whoever said that is indeed rambling, as he suspects. The point about mathematics is suppose there's another civilization on, say, Alpha Centauri.

Starting point is 01:45:16 And they will have the same mathematics that we do. I mean, obviously, they won't use the same notation or terminology in the same way that, you know, English and French people do the same mathematics, but they do it in a different language. So they will they will have the same mathematical theorems expressed in their own weird language and they will have found the monster simple group and they will have found modular forms and they will have found everything else. So it's not a human centric way of viewing the world, it's a universal way of viewing the world. I mean, at least that's what I think. Maybe it's me who's rambling, but I would say it's discovered, not invented, because any other advanced civilization will have an equivalent mathematics. Do you think that we'd share an idea of what the few key theorems are? Or important theorems? Or for example, like the book. Do you think we share the book?

Starting point is 01:46:09 It's a very speculative question that this is in some sense meaningless, but in spite of that I will say yes, I would guess that other advanced civilizations will on the whole share the same key theorems. Maybe not exactly, but... Let me be more explicit as to what I mean. Let me be more clear. What I meant was that they would have the same idea as that this theorem is a key theorem. Not that they would share the same theorems. It might just be one line as a lemma, but we consider it to be a holy grail, and they would also consider it to be something of profound importance. I think there'd be a lot of overlap I mean I mean even between

Starting point is 01:46:45 different I mean there are different mathematical um I mean before the world sort of became unified there were different mathematical um cultures like the Greeks and Indians and Japanese all had slightly different approaches to mathematics and they they weren't exactly the same on what they agreed were the central theorems or approaches to the subject, but there was a lot of overlap. Yeah, I recall someone's estimation for pi was four. I don't recall which time or which spatial location. Yes, there have been some very weird estimates for pi. Okay, so Tori Ko says, what does he enjoy or find meaningful about math? Well, I like it because it's really describing how the universe works in some very fundamental way. I mean, as I think I mentioned earlier, the universe seems to be running on mathematics in some very deep

Starting point is 01:47:45 sense that, you know, physics like the standard model or general relativity is really just an application of mathematics. So when you're discovering something new in mathematics, you can think of it as being discovering something new about the way the universe is working. So to me, when you say that, if I didn't have any context as to who you were, I would have thought you were either a theoretical physicist or a physicist in some capacity. Why is it, do you think that math somehow has a more direct access to the mechanics of the universe than physics? No, I mean, mathematics has access to the way the universe works through theoretical physics. I mean, theoretical physics is the interface between mathematics and the actual physical world in some sense. I mean, not all

Starting point is 01:48:37 mathematics seems to have a direct relation to the physical world. I mean no one's found the monster simple group in our four-dimensional universe yet. But there's a lot of overlap. I mean it's quite hard to find an area of mathematics which has nothing whatsoever to do with the physical world. What type of mathematician would you consider yourself? A pure mathematician? If you have to divide mathematicians up like that, then I guess I'd count as pure rather than rather than applied or a physicist. I mean, as I said, I sort of tried doing theoretical physics a bit and that didn't work out so well. Is that the only reason why you abandoned theoretical physics in favor of pure mathematics?

Starting point is 01:49:26 Because you just had a difficult time? Yeah, it seems to be a perfectly good reason for abandoning something. Yeah, Harish Chandra apparently did the same. He started off as a theoretical physicist and switched to mathematics because he found it too difficult. mathematics because he found it too difficult. Okay, Terry Cole, the same person says, what does he think about biology, psychology, philosophy? Do you think that you would gain as much of a sense of understanding by exploring those fields? Well, again, I have the caveat that I don't actually know a lot about these fields. Biology, I tried reading some biology textbooks and they're incredibly scary and you read a biology book on the way that a cell works for example and it's perfectly obvious that a cell can't possibly work in the way it describes it. It's just so complicated that the way

Starting point is 01:50:27 cells work is so complicated and bizarre it's very hard to believe it exists. I'm sort of worried that if I read a biology book, you know, the universe will suddenly catch up with this and I'll drop dead. So I think biology is a bit dangerous to study because it seems so incredible. Also just so much of it, I I mean, it hadn't quite dawned on me just how horrifically complicated cells are. Timberfin says, is nature fundamentally mathematical or something else? Now, I know you touched on this, but again, for Timberfin. Well, depends what you mean by fundamentally. I would say if you go deep enough into anything, you eventually get to mathematics. I mean, you might start with trying to understand nature

Starting point is 01:51:11 via biology. And if you look really deep into biology, you find it's really biochemistry. And if you're trying to work out how all these biochemical molecules work, you discover it's really quantum mechanics, which is physics. And if you look at how physics works you suddenly find you're down to mathematics. So mathematics is sort of at the core of everything if you go deep enough. Bernardo Fitzpatrick asks, I would like to know, does the professor know of Italian mathematician Dr. Olivia Carmelo, and if so, what does he think of her use of Alexander Grothendieck topos as a unifying theory in mathematics, and her interpretation of topos or topoi as bridges in order to facilitate this unification? Well, I don't remember the name offhand, but topos have indeed been used as a unifying thing in mathematics. So topos is this

Starting point is 01:52:07 construction that Alexander Grothendieck came up with and it was shortly afterwards realized by Lorvier I think that a topos is really a sort of almost a sort of intuitionistic model of set theory or a weak version of set theory. So you can in fact do most mathematics inside a suitable topos. And in fact this is almost one theme of Grothendieck's algebraic geometry that anything you can do in mathematics you should try and do in an arbitrary topos and that gives you a sort of way of generalizing lots of theorems. Does he have an opinion on Grothendieck? I'm just overwhelmed by him. I mean, I just, I mean, there are a lot of great mathematicians, I sort of think, well, you know, I could have proved that guy's theorem if I'd just been a

Starting point is 01:53:00 bit smarter or worked a bit harder. And then there are other mathematicians, I look at their work, been a bit smarter or worked a bit harder and then there are other mathematicians I look at their work and I realize there's just no way I could possibly have done that no matter how hard I worked and you know Grothendieck's one of them and you know there are a few others like John Thompson's work on finite groups I could just never have done and Andrew Wiles's work on Fermat's last theorem I just couldn't possibly have done so So he's one of the ones who just, I have to admit, they're just way out of reach of anything I could ever have done. Have you met Wiles? Very briefly, once or twice.

Starting point is 01:53:43 Okay. This comes from Peridot. Many people decide that they're more analytical than algebraic, or vice versa. Can he expound a bit on why he personally finds inequalities unappealing as opposed to equalities? This goes back to our previous conversation. They're not unappealing. I like inequalities. It's just that, you know, I just don't, what's the word? They're too messy. I just haven't got them completely internalized. I don't have a good feeling for them. The reason for this, I don't know. I think I've just, it's probably just a matter of practice. I've spent a lot more time working, thinking about algebra and equalities and so on.

Starting point is 01:54:18 And presumably, if I'd spend all that time thinking about inequalities, I would be the other way around. time thinking about inequalities, I would be the other way around. I mean, I think the difference between mathematicians who are more algebra and mathematicians who are more analysis is very much what they spent most of their time practicing. And the last one is from the same person, Peridot. He says, for the ABC conjecture question, can you please allow him to be as super detailed as he possibly can be? And if he says he doesn't know much about it, ask him why he thinks it hasn't triggered his curiosity. I'm assuming there was a previous ABC conjecture question, but I can only conjecture as to

Starting point is 01:54:56 what that conjecture question was about. So let's say his last sentence says, I am consistently curious about the process of how mathematicians decide what is and isn't worth their time from a mathematical perspective. So I assume he's wondering, why haven't you tackled the ABC conjecture? And why do you choose this topic rather than this one? Other than pure interest. I haven't tackled the ABC conjecture because it's just too difficult. I mean, you have to select problems that there's some hope of making progress in.

Starting point is 01:55:23 If you're asking about my opinions of Mochizuki's work, well, I just have no idea. I think I said earlier, I tried to understand it for a few hours, and it became obvious to me that it would take me many years of study to get a vague idea of what was going on. So Mochizuki is another one of these guys who look at his work and there's just no way I would have been able to do that. I mean, it's a bit controversial about whether it's right or wrong at the moment, of course, but whether it's right or wrong, it's still an incredible, Mochizuki's work is still an incredible achievement. And how do you decide what's worth your time and what's not from a mathematical perspective?

Starting point is 01:56:15 If it works, it's worth my time and if it doesn't work, it's not worth my time. You can't tell in advance, you just have to, if there's a problem or area, you start working on it. And sometimes it just doesn't seem to be working out. And sometimes it does. And deciding what is worth your time is actually one of the hardest problems in mathematics, because you may have spent several months working on a problem. Maybe you're just about to make a breakthrough. And if you only spent another week on it, then you would crack it. But you can't tell. Maybe, you know, maybe it's a week of the biggest problems in research is is figuring out um which which approaches are likely to work how many weeks or months do you usually give on a problem before you decide it's solvable or not by you well i never really give up on problems what what what you do is you you work on problems for a bit and

Starting point is 01:57:25 then you get a bit depressed because you're getting nowhere and then you put it aside but always come back to it later after a few months or years when when you I mean what what generally happens is I I I forget why my approach didn't work and and a few years later I think oh why doesn't this approach work and I try it again and then I then I discover why it doesn't work and a few years later I think oh why doesn't this approach work and I try it again and then I discover why it doesn't work again but every now and then you make a little bit of extra progress and sometimes after coming back to the problem a dozen times you finally manage to solve it and there are plenty of other problems I've come back to many times and I'm still stuck on them. How come you forgot that you tried a certain technique and it didn't work? Because

Starting point is 01:58:09 I would imagine you take notes and you would have access to those notes. Of course I don't take notes. Taking notes requires being organized. I'm very bad at keeping notes of what I do. I mean, scrapbooks of your work, no? Well, I tried keeping a scrapbook of some of my work and I looked at it a couple of years later. It was completely illegible and I couldn't figure out what on earth I was talking about because I'd been too lazy to write up proper notes. And, yeah, it may not be a bad idea to sort of forget what the problem was and come back to it, because sometimes when you come back to it, you suddenly realize that the thing you you were stuck on it wasn't really valid and there's a way around it of course most of the time it was valid but whatever you know i had a note and it's just a bullet point

Starting point is 01:58:56 it said no ghost theorem and it reminded me i wanted to ask you if you can explain to the audience what the no ghost theorem is and string theory and then its connection to the moonshine conjecture. And you're allowed to go into technical detail if you feel like the audience is going to lose you or even me then that's fine. This I could probably explain if you gave me a semester course but here goes. First of all the name no-ghost theorem is a sort of pun or word play. So in theoretical physics there there are things called no-go theorems, which mean something doesn't work. I mean, you can't, there are various no-go hidden variable theorems. The no-go theorem is a sort of play on words. And what a ghost is in string theory is a name for negative norm vector in a Hilbert space. and this is extremely disturbing because Hilbert spaces

Starting point is 01:59:45 aren't allowed to have negative norm vectors so if you've got a negative norm vector that rather messes everything up. So in order to get quantum mechanics working you've got to make sure there are no negative norm vectors, in other words there are no ghosts because negative norm vectors are called ghosts. So when you take a string and try and quantize it, you find you've got this big space and sometimes it has negative norm vectors in, which is really tiresome. And the no ghost theorem proved by,

Starting point is 02:00:17 I think it was Brouwer and Goddard and Thorne, says that if the string happens to be living in 26 dimensions, then everything is nice and you don't get any ghosts. And if it's in 27 dimensions, you do get ghosts. And it's really bizarre this number 26 turns up. I mean, you can imagine something special happening in two dimensions or maybe three dimensions, but 26 dimensions is just ridiculous. So that's why string theory works in 26 dimensions it's because of this no ghost theorem and then you get super string theory where the critical dimension is 10 rather than 26 and that's still a bit tiresome because we

Starting point is 02:01:01 want to bring it down to four dimensions and there are various ways you can try and get it down to four dimensions but none of them seems to be totally satisfactory yet and 26 dimensions is really nice because it ties up with the monster and the leech lattice so the leech lattice lives in 24 dimensions and it turns out that if you've got the leech lattice as a very natural construction You sort of embedded in Lorentzian space and that involves adding two to the dimension so the 26 dimensions of a no-go theorem and string theory is is Very close related to the 24 dimensions that the leech lattice lives in. And the leech lattice is connected to the monster? Yeah, so actually that's another of these bizarre coincidences that I mentioned

Starting point is 02:01:50 that always appears whenever modular functions and forms turn up, because the modular forms are working, doing very special things in 24 dimensions. Why is it that in 26 dimensions there's no ghosts but there are other flavors of string theory so do they have ghosts in other dimensions? Yeah yeah yeah so 26 dimensions is is what happens if you take the simplest possible version of string theory. As you say there are lots of other flavors in string theory and these tend to have different critical dimensions so that the next simplest is super strings and 10 dimensions and you you can sort of fudge around a bit and find things going on in 18 or 14 or 6 dimensions but these are all getting a bit more complicated

Starting point is 02:02:37 26 is the is the simplest case and lower dimensions get more and more complicated are there higher dimensions than 26 or is it just proved that that's the cutoff anyone has case and lower dimensions get more and more complicated. Are there higher dimensions than 26 or is it just proof that that's the cutoff? Not that anyone has managed to figure it out. For the ghosts, I mean. 26 seems to be an upper bound for all this nice stuff going on as far as anyone can tell. Okay, man. So what's your plans for next? Where can people find out more about you?

Starting point is 02:03:04 Where can people find out more about me? Where can people find out more about me? If I ever find out, I'll close it down. I don't know, Wikipedia, maybe. I don't know. Your YouTube page. I'll link it in the description. Yeah. What's next for you? What's next for me? I'm trying to get this problem that I've been working on a long time to see if it either works or doesn't. And this, well, I hope it will be solved soon, but I've been in this position of thinking I've solved at least a dozen times in the past and every single time it's failed. So I don't know. Professor, thank you so much. This has been a pleasure. So I don't know. Professor, thank you so much.

Starting point is 02:03:44 This has been a pleasure. Oh, yeah, well, thanks for the interview. It's been an instructive experience. And I hope you're going to edit out all the most embarrassing things I've said. It's actually only going to be a compilation of the embarrassing things you've said. Yes, I should have.

Starting point is 02:03:58 Yeah, it's like when you have a newspaper interview. You say one dumb thing in it, and that's the one thing that gets published by the newspaper.

Theories of Everything with Curt Jaimungal - Richard Borcherds (Fields Medalist) on the Monster Group, String Theory, Self Studying and Moonshine

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