Theories of Everything with Curt Jaimungal - David Bessis: What is Math? How Do You Learn It?

Episode Date: February 23, 2026

What is mathematics, really? Mathematician David Bessis joins me to argue that math isn't about numbers in a Platonic realm or a meaningless game of symbols—it's a cognitive technology for rewiring your brain. We explore why the official definitions of mathematics have been unresolved for 2,300 years, why understanding something means finding it obvious, and how the gap between a beginner and Terence Tao looks less like genetic destiny and more like compound interest on intuition. When asked what mathematics fundamentally is, his answer cuts through millennia of philosophy: it's what happens in your head when you pretend something is true until it feels real. LINKS MENTIONED: Papers, books, websites: - https://davidbessis.substack.com/ - https://scholar.google.com/citations?user=YmJL9KwAAAAJ - https://amazon.com/dp/0300283288?tag=toe08-20 - https://davidbessis.substack.com/p/the-magic-of-mathematical-intuition - https://davidbessis.substack.com/p/weve-been-wrong-about-math-for-2300 - https://arxiv.org/abs/math/9404236 - https://amazon.com/dp/0387900926?tag=toe08-20 - https://davidbessis.substack.com/p/the-curious-case-of-broken-theorems - https://mathworld.wolfram.com/FermatsLastTheorem.html - https://xenaproject.wordpress.com/2024/12/11/fermats-last-theorem-how-its-going/ - https://gwern.net/doc/math/1979-hersh.pdf - https://en.wikipedia.org/wiki/Problem_of_universals - https://plato.stanford.edu/entries/set-theory/zf.html - https://imperialcollegelondon.github.io/FLT/blueprint/ - https://amazon.com/dp/1015393233?tag=toe08-20 - https://annals.math.princeton.edu/2015/181-3/p01 - https://arxiv.org/abs/math/0610778 - https://pi.math.cornell.edu/~bts82/events/homotopyF20/notes/bar-construction-typed.pdf - https://www.edge.org/conversation/reuben_hersh-reuben-hersh-1927-2020 - https://mathshistory.st-andrews.ac.uk/Biographies/Ramanujan/ - https://amazon.com/dp/1107536510?tag=toe08-20 - https://viennot.org/abjc-lectures.html - https://pdf.sciencedirectassets.com/272332/1-s2.0-S0021869306X06555/1-s2.0-S0021869305006150/main.pdf Videos: - https://youtu.be/tYgiVnQubyw - https://youtu.be/RX1tZv_Nv4Y - https://youtu.be/73IdQGgfxas - https://youtu.be/lhpRAWxvY5s - https://youtu.be/rJz_Badd43c - https://youtu.be/c8iFtaltX-s - https://youtu.be/81sPQGIWEfM - https://youtu.be/wbP0KjWm0pw - https://youtu.be/mTwvecBthpQ - https://youtu.be/_sTDSO74D8Q - https://youtu.be/DeTm4fSXpbM SOCIALS: - Twitter: https://twitter.com/TOEwithCurt - Discord Invite: https://discord.com/invite/kBcnfNVwqs Guests do not pay to appear. Theories of Everything receives revenue solely from viewer donations, platform ads, and clearly labelled sponsors; no guest or associated entity has ever given compensation, directly or through intermediaries. Learn more about your ad choices. Visit megaphone.fm/adchoices

I'd like us to focus on two questions, both of which you have a provocative response to. One is inspirational, the second is philosophical. The philosophical one is, what is math? The inspirational one is, why do you believe that IQ, which is traditionally thought of as a necessary component to doing math well? Why do you think that's overstated when it comes to understanding or contributing to breakthroughs? Let's tackle the conceptual ground first and get into how fear holds people back and practical steps later. What is math? That's a very difficult question. That's a question that is

officially too difficult in the sense that nobody seems to agree on the definition. So I have in one of my substack posts, I have a screenshot of Wikipedia for a couple of months ago. It's been reshuffled, but the screenshots, the content stayed the same. The definition of mathematics, according to Wikipedia, says that nobody is agreeing on the definition of mathematics, which is a big issue.

You're teaching something to millions of billions of people and you don't really know what this is. So I think people have been mistaken because they were trying to define mathematics without any reference to human beings. So there are two ways of doing that. One way is to say that math is about, you know, it's a science of numbers,

of shapes, of things that exist in the Platonician world of ideas. And there is the other approach that is to say that math is about logic, about proving things, about having axioms, theorems, and all that. Now, it seems like this is correct from far away, but when you practice math, you realize that there is an issue with both definitions. Basically, these objects that exist, well, it's not really clear what this means. It's one of the few scientific areas where people still refer to things that we cannot

touch. They are very abstract. So nobody seems to agree whether these things make any sense. And the fact that it's about proof is actually something that's really contradicted by the practice of match. You spend your time dreaming, daydreaming, coming with crazy ideas, testing your intuition. So my definition of math, but it's mine. It's not an official one, would be that it's a special technique that involves imagining things and pretending they really exist and pretending they have properties that absolutely is true.

And this thing is gradually changing your intuition and make you believe that these things actually exist. So the clattonition side, which is that you believe that these things are really existing, is a side effect of math. And the logic side is the core technique to produce that side effect. When you say that we don't know what we're observing,

we can't observe math, we can't touch math, even physically speaking, you're not touching the objects because there's some barrier between you. It looks like there's direct contact here, but you never actually have direct contact if you zoom into the atoms and so forth. So what is it precisely that you mean

that we don't touch math? Well, what you can touch is you can touch a practice of math. You can touch someone who's doing that and you can discuss with someone who is doing that. And yourself, you can do math in your brain. So my favorite example of that is, can you imagine a circle? can you?

Can you see it? Can you touch it? Can you make it bigger and small? So, you know, this is a very simple experiment that shows that there is something really weird going on in your brain. You feel that you see something, but this thing is not really there.

So you sense that you can touch it, but of course it's not a physical object somewhere. And its properties, you know, being infinitely thin, being perfectly round, these things do not exist. exist in real life. So what is going on in your brain when you do that? So if we stop doing metaphysics for a while and we forget about the traditional way of thinking about this question, which is, yes, circle exists in the kind of platonition way, but if we just focus on what is really

going on, where there is you with your brain, doing stuff in your brain, and this stuff seems to have some effect on your brain and just weird thing in your brain, what is this activity? or do you characterize it? This thing for sure it exists. I mean, this is a true activity. It's a physical one. And so there is that article that I absolutely love, and I think that anybody interested in this question should read it.

It's a paper by Bill First on the Great Geometer. And I think it's from 94. It's called On Proof and Progress in Mathematics. And it comes up with that stupidly circled a definition of mathematics, saying that mathematics is what mathematicians do. But I think it's right in a very profound way that is probably the best way to understand

mathematics is to deflate the ontology, to stop trying to do crazy metaphysic that we don't really understand. Just focus on what we are really doing. What are we really doing when we do math? This is quite interesting. There must be some

through line though, because I imagine if in 200 years from now mathematicians or people who call themselves mathematicians all of a sudden are working on installing power lines and building homes, for some reason, it evolved to that. We would say you're no longer practicing. You're using the same word, but the word has changed, and you're not referring to the same object in a sense that people were referring to back then. You're completely right. And the actual definition by first from this article, I think,

is there is a seed in it that is, it contains a study of numbers and of shapes, you know. So you start with what we agree on being mathematical objects, even though we cannot really define them. But what we can do is we can define the fact that we're thinking about problems with numbers and shapes, because these things we're familiar with. So of course, yeah, it has to include arithmetic, it has to include basic geometry.

And then there's a bunch of other stuff coming with that. And this is the activity called mathematics. And first time even say that the job of a mathematician, is to try to improve the understanding of mathematics by humans, in the sense that we don't get quite perfect understanding of these things, but we try to make sense of them, we try to understand them, we try to study them, and doing that we do a bunch of things, and this is this entire field that we call mathematics.

So when a mathematician is thinking about circles and spheres and so forth, these are, let's call them the objects of math, And then there's the practice of math, publishing papers, speaking to people, a community. In physics, it's similar in that you have a model of an electron, but the difference is that we think the electron is something out there. Now, in math, what are you modeling? So this is the question.

We don't really know. I don't understand physics, to be completely honest with you. So it's not a perfect comparison point for me, because I always struggle with what is an electron. I've never seen one. but at least in mathematics you define the stuff you're working on so you give a definition

and that definition is basically playing Lego with objects that you already agree that are existing so the standard way to play Lego nowadays is to start with sets even though we don't really know what is a set but we have a fairly stable naive intuition of what is a set and from that we can build more and more complicated objects

so in a way the formal side of mathematics is basically the construction game that you're playing when you do math. What's interesting is if you study logic and formal logic

and formal mathematics, you have to accept at some point that what is true is formal statements that have absolutely no meaning in a sense. So there is a dissociation between the activity of mechanical proof and the activity of meaning-making. And when you do math, you actually try to align the two things.

And this is what makes mathematics interesting. So I think that these two aspects of mathematics are equally important. And actually, mathematics is this back and forth between having a meaningless definition, but it's just a synthetic game of combining axioms and deductions, and extrapolating meaning, and meaning is a human phenomenon. And this is where you cannot really eliminate the human in the mathematical practice. Even if you can formalize proofs,

the fact that the proof is a proof of the theorem that has a meaning is what matters to you. Okay, so just two minutes ago or so, you mentioned something about Lego, and we assume that these objects exist. But then it sounded like you're saying, well, these are just also the rules that we agree on, almost like chess. No one would say that the rules of chest exist outside of...

or that they're referring to something that exists. I mean, the rules of chess exist. But what I want to know is, what is this alignment of a human notion of meaning-making? And then the chicken scratch, that is just something that we agree, you have these symbols here, and then you can manipulate it and so-and-so,

and A can imply B. And so if you have A, then you could infer B and say B is true. So help me understand that, because when people say, well, this is a human enterprise, there's the tendency to think, well, then it's arbitrary. And it's a tendency that is very profoundly ingrained in our culture, but it's wrong. It's not because it's a human enterprise that is arbitrary.

And it's not at all about social deconstructivism of mathematics. It's not at all about that. It's just about acknowledging that there is something in mathematics that cannot be explained by formal logic and can only make sense if you view mathematics as a cognitive theory, as a connective practice. So let's pretend for a moment. that we endorse the formalist view on mathematics.

That is that, you know, mathematics is just a meaningless game of symbols, and these things, because they're not human, because they're very stable, because they're machine-provable, they survive humans, and they're kind of eternal in their own way, and there's no human involved in doing that.

Now, the problem is when you look at how the formal theory of mathematics is built, you start from the axomes of, say Zemolo-Frank. But why do you choose that? What do they mean? Why do you study this set of actions? So if you open a book about said theory, there's an interesting thing.

And actually, this is really what got me into that story. Some while ago, I was teaching a class on the foundation of mathematics. And I was trying to do what I always did when I was teaching mathematics first, tried to define the objects. And I realized something that I knew that I had not fully, realized before. In set theory, when you say for all X, you don't say for all

X a set, you say for all X. And when you have the axioms, let's say the axiom of extensionality that says for all X, 4O Y, blah, blah, blah. X and Y do not refer to anything. And actually, for all, doesn't mean for all.

It's just a symbol. So you have a set of strings to which if you stick to the formal theory, are absolutely meaningless. This is already annoying. Why would you study that? Well, because apparently it seems to be consistent

even though we cannot put it. Well, it's not a good reason. There are many sets of actions that are consistent that we do not study. We study it because we attach meaning to that. And meaning is a cognitive phenomenon. It's not in theory. There is nowhere in Zamero-Frank's theory

an explanation of what a set is supposed to be and the word set, the word set only appears on the book cover. It's never mentioned inside. There's no description of what is a set. There's no properties of set. They're just statements that are just meaningless. Now, you could say, okay,

we project meaning, but this is an optional thing we do. But this is not the case either. And actually, I think there is a very, strong technical argument for that, which is the way mathematics behave as a scientific field. If it was a meaningless game of syntax, if we were to introduce a bug in the system,

the system would collapse. If you have a math paper, they never really written in the full formal way. have human hand-waving blended with formalism. This is the way math papers are written. But sometimes, because of that, there is a bug in a math paper.

There is a theorem that is not true. And there are other papers citing that paper. And they use a theorem that is not true to prove other theorems and can go on for 50 years. It happened in the past that this thing

went on for 50 years without anyone noticing there was a bug. Now, if it was purely a formal theory, there would be absolutely no reason why mathematics would not collapse from that. When Andrew Wiles came up with his proof of thermos last theorem, the first proof he came up with was wrong. And then he fixed it. But if it's just, you know, assembling something that has no meaning,

there would be no reason why a false proof could do. be fixed. What does it mean to fix a proof? Every mathematician agrees that, you know, this is what Wiles did. He had a proof with a bug, and then he found a fix to his bug. So if you want to make sense of just this very simple objective event that happened in the history of mathematics, you have to agree that there is meaning. Because if there's no meaning, there's no way to fix a theorem.

fixing a theorem is coming up with a theorem that is close to the one you thought was correct. Or maybe having a lemma that is close to a lemma that you use that is wrong, but maybe there's a lemma that's close to it that is correct. But being close to is absolutely impossible to define in the pure formal approach to mathematics. It's not that it's incorrect.

It's not that it's an incorrect notion. It's a meaningless notion. You have no proximity between two formal statements that have no meaning. There must be some survivorship bias here, because I imagine that there's a variety of theorems for which someone said, all we need to do is fix this.

We found an error or a bug or what have you. And then in the process of attempting to fix it, you realize the theorem was false. And the ones that have survived, we say, well, that points to evidence that we as people are capturing something that the formal systems have not yet caught up to. Do you also see that or no?

actually, strikingly not. I mean, of course, you will always find examples of proof that are wrong. It just flattered wrong. This happens, you know. And probably the proof that Therma had in mind when he said, you know, I have a very beautiful proof that is too long for the margin. Probably this proof was unfixable. That's good.

We have good reason to do that. But to think that. But if you look at the published literature that is full of bugs, it's very rare that these bugs are unfixable and it's very rare that the consequences are really destructive. Yeah, of course, maybe a bunch of papers are citing this paper and they citing

a lemma that is not valid the way it's used, but you know what? It doesn't propagate usually. So there's something really strong and I think it has to do with that it's done by humans. I think if it was a machine trying to greedily learn and prove things, it would lead to disastrous consequences. But human beings, they write math papers when they understand what they're talking about.

So that's why actually they don't lose their sleep, you know. We all know that the papers we publish are full of bugs, but that's okay because the bugs are fixable. When I'm wrestling with a guest's argument about, say, the hard problem of consciousness or quantum foundations, I refuse to let even a scintilla of confusion remain unexamined. Claude is my thinking partner here. Actually, they just released something major,

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Sign up for Claude today and get 50% off Claude Pro when you use my link, clod.aI slash theories of everything, all one word. Wouldn't the Platonist just say, look, I have a simpler explanation. Fermat's last theorem is true because it exists. It's true. It's just our description of it was false, or our description of the proof of it was false. Okay, so you, this is platonition view.

You know, this is platonism. You pretend that there is something that is the canonical theorem that we're discussing, and this thing could have had an error of transcription when we wrote it down, that could be about the statement that could be about the proof. Now, this idea that these things exist

is very close to what I'm talking about, except that I don't think that we have to assume that this thing exists ontologically to agree that they exist in our intuition. There is another example of that. That is a more recent example that is, you know, there is a team with Kevin Buzard

who was trying to formalize the proof of Fermat's last year. And they started the project, I think, about 18 months ago. And in a few months into the project, they run into an issue with one technical theory they were using that's called

crystalline chormology. That is a very technical thing invented by Grotanig and one of his students. And they found a bug in a lemma that's been used for, I don't know many years, but for a very long time, that goes back to the foundational paper

of Chrysaline Comology. And they realized that actually the lemma was wrong. So in a sense, and that's interesting, there's a blog post by Kevin Bozzer describing his feeling, he being a formalist, officially a formalist. He says that, you know, the minute we found, we realized that this lemma was, force, the whole theory of crystalline comology collapsed, vanished, disappeared, disintegrated

into nothingness. And, well, that would be the official formalist standpoint. But on the other end, he says that in his head, he knew that it was fixable. Because he knew that crystalline chomology had been used for maybe 30, 40 years without, you know, people running into issues. And he says that, you know, if there was a massive bug in the theory, and if the theory had really no reason to exist because it was flawed, then we would not have been able to use it without running into issue for 50 years.

And this is, so you can interpret that, as you suggested, as the idea that there is somewhere a theory that's called crystalline comology that exists in a, kind of perfect world. But there is a different explanation, which I think is more interesting and probably more modern in its approach. That is to say that when we understand

Chrysaline Kognology, what does it mean for a human to understand something? It means to align a number of mental representations of the thing we're talking about, but also other thing for past experience and trying to articulate that

into a coherent world view, and this is, I'm really talking about a neural phenomenon. I'm not talking about some magical stuff. I'm not a spiritualist. I'm just describing that in your brain, this thing makes sense, and it's coherent, and it's meaningful, and it gives you a very pleasant feeling of harmony,

which is, again, a physical phenomenon. It's not a magical one. You feel that things are in place. where this thing say something about the theory there is that formal layer at the bottom and you're kind of building that's meaning on top of that.

And so what is mathematics if you look at it from this perspective it's a very unique technique of relying on what I call the game of truth. The game of truth is pretending

that you can write statements that are absolutely true, and you can make definitions that are absolutely rigorous, and you can play Lego with that, and make formal deduction of new things that will be absolutely true, which things, you know, when you talk about normal language, this never happens like that in real life. Language is not, does not behave like that. But when you do math, you pretend that there is that kind of alternate world of mathematics, where this game of truth is describing reality of this alternate world. Now, when you do that,

it has a certain effect on your brain. You're basically learning on a new set of synthetic images and ideas that you create playing that game. And this thing seems to have a very powerful effect on consolidating your intuition.

And this is what makes you both happy and also probably a bit smarter when you understand some mathematical concept, because you worldview is kind of compacting and becoming more luminous in a way. And this is the actual phenomenon that is going on. It's a, so it's a property of, uh, uh, neural systems exposed to that game of truth. And there is something that is working here. It's a, it's basically a machine learning theorem that says that it should say, I don't know how to write it properly, but there

should be a theorem that says when you play that game with a human brain, and probably with, with, with some sort of, you could play the same game with artificial intelligence probably, then it creates better representation that are very powerful and help you make sense of the world. What's the name of this framework of yours or this worldview? So there's Platonism, which was referenced earlier about mathematical facts exist. They're just true. They're timeless. Then there's a formalism which says that these mathematical facts are rules of a game. we could change the rules and we could equally explore arbitrary rules and they just don't

correspond to something at some human level, but that's fine because we just decided that these rules are the ones we're going to explore. What's the name of yours? So I name it conceptualism, but I should put a disclaimer that, you know, I'm trying to articulate something that has been in the air for a while. So if you read first on paper from 90s, 94, you get these ideas, but they're not pushed to the same degree of assertiveness, because of course, that was 30 years ago. If you go back to 1979, there is a beautiful paper

by Ruben Ursch that calls some proposals for reviving the philosophy of mathematics that contains some of the same ideas. So it's not just my ideas. It's a train that is, I think, you know, in a way in private conversation, even though many mathematicians would call themselves platonition if you push them a little bit

they will say okay maybe this is what I mean I mean I perceive them as existing doesn't mean they really exist same thing for the people who call themselves formalists when you push them they agree that like Kevin Buzhardt said in his blog post he said okay Christianic homology had disappeared

but in my mind it was still there somewhere so he agrees that there is an intuitive side to mathematics So it's a latent consensus that I think is very, not everybody agrees with that, but I would say that a good chunk of a current mathematical community agrees with the core, maybe not with the summary, but with the core tenets of this way of framing it.

So I called it conceptualism because I had to try to decide to put a name on it, and conceptualism refers to the, debate on Universal that is a very interesting medieval debate the story of William of Orcam and before him Abelard and all these people

who run into trouble for basically challenging the dominant Platonician worldview that I would say that all civilization civilization is primarily

believing that concept are real things and they really exist. There is there was and there is still this opposite view, that is concepts they don't really exist. It's just convention.

That's called nominalism. You basically declare that everything is an arbitrary game of social convention. And conceptualism is some sort of middle ground, but it's not, middle ground is usually a weak thing. It's a thing that you, because you don't want to fully agree with the two extremes, so you declare that you're in the middle.

Actually, conceptualism says something that is, additional to these things. It says that, you know, the abstractions that we're dealing when we do mathematics or when we describe the world, when you use the concept of an electron of things like that,

they're not existing in reality, in external reality. They're not mere convention of language, but they are produced by our cognition and they're by products of the way we structure or understanding of the world. So they, basically, they exist in our brain.

And I do think that's my interpretation, but I know that some people might disagree with that, but I do think that the things we've been seeing with, you know, the way neural networks, artificial neural networks work, it kind of validates the conceptualist approach that says, you know, basically, when you pile up layers upon layers upon layers of neurons,

then you will see that, you would get higher and higher level of features that are generated by the different layers of these systems and basically the neurons specialize on being excited when you encounter certain concepts. And this is really the, I would say, it's a conjectural interpretation. In the brain, it's not exactly one-on-one.

It's not one neuron-equal one concept. But there's something to it. I think it's a good metaphorical summary to what is the conceptual standpoint. It says that our brain fabricates abstractions in the form of concepts. How do we know when we found the right axiomatic system?

When we don't have issues with it. How do you know that you have the right operating system for your software? You can have two problems. One problem is you have a bug just doesn't work, it's broken, and you know that the first attempts to formalized set theory run into big issues

and Zemolo-Frank's solution was basically a fix of prior approaches to that. And the second thing that, second issue you can have is there are things you want to express that you cannot express in this set of actions. And actually, it's interesting to note that modern mathematicians, everybody's kind of believing that we're using Zemoro Franco,

but we do think that are not really fully within Zemoro Franklin. I don't think that category theory is well captured by a foundational set theory. I think it's, I don't know exactly how it works. I actually, to be honest, I'm a practitioner of category three, even though I don't really, I'm not completely sure that it's well-founded from a logical standpoint. interesting. And that's no big deal because it still works.

Wait, just a moment. So potentially something could be not well-founded but still work. So I don't really know. That's what I say. Maybe it's well-founded, maybe it's not.

But I don't think it's captured by said theory. I think it's something different and I'm not good at logic. And as a mathematician, you have to give up some, you cannot control everything in what you're doing.

There are some aspects of mathematics that may be a bit too abstract for you and you have to just basically accept that it just works, you know. Okay, let's rewind prior to category theory. Let's just pretend we're 100 years into the past. What I'm curious about with this question

of how do we know when we found the correct axiomatic system is in physics there are attempts to make axioms of axiomatic systems of physical systems like axiomatic quantum field theory or the direct von Neumann axioms of quantum mechanics or what have you. And then we test whether or not these are correct if they give us accurate predictions, if the

data, sorry, if the model matches the data. And then we'll say if it doesn't, then our axioms are incorrect. Now what's interesting is in math, let's imagine, so this is one idea of how it may work. We have a set of theorems. Let's say, let's enumerate them, one to 500. Maybe one is the Pythagorean theorem, and number 500 is Fermat's last theorem, and number 250 is the intermediate value theorem, what have you? Okay, what are axioms that can tell me that all of these are true? Because they already have deemed them to be correct. Okay, so this is prior to the development of set theory is what I'm saying. So, you know, there is an example of axiom having issues or not. Is the Euclid axiom, you know, is there, is it correct to assume the way it's been done for, basically, to

to millennia, that's, you know, through a point that's outside the line, there is a single parallel going through it. Well, if you do plain geometry, that looks like it's true, so you have to, you want to take it as an actual. But do you have to do it?

So, again, the two things are whether or not your system is consistent. If your system is not consistent, then you're in big trouble. You should throw away the XOmetics theory, because it just doesn't work. it proves everything, including the contradiction of everything. But if it's not contradictory, if it's consistent,

then it is a variable theory. The question is, what does it even mean? And you have an infinity of possible theory that you don't really know what they mean. So what happened with this axiom of having a single line going through an external point that is parallel to line, that you can decide that it's false,

and you can, and this is again the meaning-making activity that comes in with, you can declare that you have zero parallel going through that point, or you have an infinity of one of parallels, and in both cases, you have an interesting geometry, and you can interpret it as elliptic geometry or hyperbolic geometry, and it's actually very fruitful. So it's just describing something else,

that was not the intention of the people who wrote the original set of axioms. But it's actually very interesting. It's a different object. There is a general principle that says, you know, that whenever you have a consistent set of axiom, well, it means something in a certain model. The question is, what and what are you going to do?

So you can always extend that's one aspect of a special mental game called mathematics is you can always always expand the mathematical reality by new things as long as these things are not inconsistent. So, and in itself, this is, you can view it as a very entire platonition mechanism because basically anything that is formally correct can be made sense of, if you find a way to make sense of it. Okay, let me amend my question.

Let me say it like, rather than there being one to 500 theorems, let me say one to 500 statements that we think to be true, because by saying the word theorem implies that you've already found a proof to it, but let's just say we believe these to be true. We have some foundational system, maybe it's based on axioms, maybe it's not, but we have some way of proving, and we've convinced ourselves that these 500 are true. Now we're trying to find the consistent axiomatic system.

So let's condition on consistent axiomatic systems, not just all. Now we're trying to find which one is the correct one for us as mathematicians to explore. Now, what I want to know is I imagine that an axiomatic system, there's a case here, one that all 500, we found one. We found an axiomatic system that shows the proof of all 500, but it also shows a 501 statement that we believe to be false. Do we then say we're wrong as mathematicians and that the 501th one is correct?

Or do we say we have not found the correct axiomatic system yet? Okay, so it's a very tricky question because we have to there are some words that you're using that are not completely obvious to interpret. The word true is very difficult to interpret what you're saying. Let me just make it clear then. Okay, because some people would say

the Benak-Tarsky paradox is evidence that the axiom of choice should no longer be there because it produces the Benak-Tarsky paradox and we don't approve of this paradox. But you could also bite the bullet and say, sorry, there is some unintuitive result, and you just have to accept it.

So one is to say, we're going to abandon the axiom of choice because it leads to something that we don't believe to be true. Maybe true. Again, you have some issues with true and many people do, so we can explore that.

That's one route, or the other route is we can say that, well, we're just cowards and that's actually the truth. The Benak-Tarski paradox is not a paradox at all. The Loneheim-Skolan paradox is not a paradox. It's unintuitive, but you just have to accept it.

This is exactly the same story as the Euclid axiome. I think the Banar-Tarski's paradox is exactly like the axiom that there is a single parent. And it was shrugging to people that it could be otherwise. And in a way, Banar-Tar-Tarski might be saying

like maybe it's a hyperbology geometry and you're not happy because that was against your intuition. But again, I did not mention the issue with truth as a way to dodge the question. I would say it was rather to focus on the right what's really important here

because we tend to think that formal logic is about truth but formal logic is not about truth formal logic is about studying deduction systems and coming from a number of statements that you

suppose you know and getting to other statements. So in your example do we have a bunch of reasons that we believe to be true and if we believe the 500 of them to be true can we find

an axiome system? Yeah, just take the 500 results that you believe are true and make that your excellent system. Okay. That two things can occur. It can be inconsistent. In that case, you're in trouble because you're asking for something impossible. You cannot have these 500 things

being true together in the way you want. So you were wrong. But usually it's not supposed to happen. But then you have that 501 theorem that surprises you because you were expecting something else to be true. okay now there is a question is

if you're taken so let's say that your 501 theorem is and you get Banartarski and you get Banartarski and you're not happy because you don't like it okay does it follow from

the 500 axiom you took where you have no choice you have to accept it does it follow from another set of axiom that you try to find to get to those 500 here. But could you have taken another one and would it have been okay? Maybe. And the way to answer that question

would be to take the opposite of Banartarski to say that it's false and to add it to your list of 500 and ask whether this is consistent. So basically whenever you can take a bunch of results and make them work together, then you have a viable model of mathematics that you can function with.

The question is, all the models basically, when you get to a certain degree of sophistication of expressivity, they always contain weird things. You can say, I don't like Banartarski, but would you like a violation of the act, of choice?

I don't really think you would like that either. So you have to accept that some things, seem unintuitive and seem an intuitive is a very important way

and now we're getting into psychology and this is why you cannot get rid of psychology because an intuitive is not a permanent status

I'm not an expert in the Banartaski theorem but I don't find it disturbing to me

I don't know, I would not be able to prove it like right now because it's been a long time I thought about it, but last time I thought about it, it seemed okay, I didn't see any issue with that. Why? Because my intuition has been rewired by my practice of mathematics, and actually, I'm okay with it. I don't have any issue with that. I don't view it as a paradox. Now, some mathematicians and computer scientists who I've spoken to on this show, and I won't name names, but some of them are finiteists,

meaning that they don't believe infinity should be a part of math. They find girdles incompleteness theorem to be, well, the first incompleteness theorem to be unsettling, and their route is to say that our axioms that we're not supposed to capture all of piano arithmetic,

it's just supposed to go up to some finite number F, and yes, you have some potential infinity, but it's not an actual infinity. In those cases, it seems like what's happening is their axiomatic system proved something, which they find so horrendous that they abandoned the axioms. And it's unclear to me, when do we know that something that we're trying to capture with our intuition has been correctly

captured by something explicit? So intuition's more fuzzy, and then we have something explicit. We're trying to capture this fuzziness. But sometimes when we do so, we betray the intuition. And we don't know, is it because our intuition was false, or is it because our axioms are false and our intuition is correct. We don't know which one to overturn in. I don't know how that decision is made in generality, not just in math, because this also applies to, let's just say life. This is a standard example I say I think about.

We can imagine that a thousand years ago, or less we can imagine 4,000 years ago, the word life if it was around, referred to something like the sun was alive. You are alive, frogs are alive, Plants are alive, but the dirt is not alive. And then we're told, well, life has something to do with variation, heritability, differential fitness, and speciation. And then we say, well, yes, okay, by our definition of life, our explicit one now, you are alive, the frog is alive.

The dirt is alive, because the dirt has some microbes in some sense. The sun is not alive. But the whole point of the definition of life was to capture what we meant. So in that case, we just said, well, we were wrong calling the sun alive. But it's unclear to me, when do we say that this explicit axioms or our explicit description is correct versus not correct? So this question of correct or not correct, I think it's more relevant to the non-mathematical world. Because again, in mathematics, whenever it's consistent,

you have an interesting theory to build. and actually think that the people who just want to do finite constructive mathematics are doing great stuff. I love that kind of math. But on the other end, the people who, you know, use infinity in ways, they approve theorem that are meaningful.

So, and you could say that, you know, doing axiomatic set theory is just affinity theory of sentences expressed in the language of sets, set theory, and it just happened that some people are happy with the meaning they project onto it. So in mathematics, you have that priority of domains of research and objects.

You study and phenomena you can study. And it's why actually you don't have that phenomenon in mathematics of having schools where people say life is like that and all the people who say life is, some people who say, for example, I think a while ago there were people disagreeing with the fact that humans were animals, you know, probably. I think it's becoming marginal, but it used to be like that. It's not like in mathematics some people are arguing that humans are animals

and humans are not animals. It's that some people work in theories where humans are animals and some people who work in the other theory and they prove things and it's always interesting and they can exchange and discuss and they're happy to have different taste.

It's a matter of taste. Now, in the real world, you have that issue, definitions, what you say, what you describe as axiom, is basically the definition we're using. I have to be recalibrated from time to time

because we want to talk about that weird thing that is called reality and we want to have some stable and it just doesn't work. It's not really stable. So we have to recalibrate it from time to time. And it operates really differently. In mathematics, you don't recalibrate things.

You say, okay, I'm going to do a slightly different theory. It's not going to replace the prior theory. It's going to save things that are a bit different, but I like them better. They kind of better describe what I want to express. But it's always additional. You never destroy anything you've done in the past. We still, we can open a math book from 300 years ago

and still we're happy to read it, you know? Maybe some from that, it's okay. What is the difference between the math that, say, lean is doing a formal system, a program, seems to be doing math, and a mathematician? Which one is actually doing math? I think the mathematician who is using lean to do math is actually doing math. Lean is, I'm pro-Line.

I'm not against lean. I think it's a great idea. And there are serious issues with the way mathematics is written and done. This issue comes from real problems that people are not doing it on purpose the wrong way. when you write a math paper, you're trying to express something and you're trying to prove a theorem. You're not trying to prove a theorem

to a machine. You're trying to prove a theorem that would be accepted as a theorem by your fellow mathematicians. And you do that not by producing what is the official definition of a proof

that is a sequence of derivation within an axiomatic So, you know, these things only exist in the dictionary. It's only in the dictionary that people pretend that mathematics is a game of formal deduction. When you write a natural math paper, it's not what you're doing. You're basically waving hands. You're explaining in a kind of hybrid language that is blending some plain language phrases and sentences and some formulas that are more or less rigorously defined.

Most of the time, the definitions are correct. The definitions are the most important part, because this is the only solid ground you find in the newspapers, is the definitions. Many mathematicians will tell you that they look at the definitions and the proofs, you know, that's another story, the proofs. But the definitions and the core statements should be at least rigorously.

done. Now if you you do that because if you were to write an actual formal derivation it would take possibly, I don't know, possibly millions of pages if you were doing that. Yes, yes. The actual Zemarrenkel

framework. And nobody actually does that. And there is an example of that, which is our Russell and White said prove that one plus one is equal to two. It comes after hundreds of pages. So if you just want to prove a basic results, you need hundreds of pages, just imagine if you want to prove

the classification of finance impor groups. So lean and that approach is to say that, okay, we may continue to write papers like that, but we should

do the missing part that is to make sure that the proof is actually a real proof in the sense of formal mathematics. So there is an effort to try to build the

interface between the actual math publications and to offer some interface to a formal system where some proof assistant, because humans cannot rewrite them, but if you have correct UX, then possibly you can build a complete derivation of that. And I suspect this is how mathematics will be done in a, in maybe, I don't know, I won't risk to propose an exact time frame but I think we will get to there because there are issues with papers are published and we're

not only, we're not really sure if they're 100% correct, but this also takes a lot of human effort to make sure that it has a good chance of being correct. So this is very disruptive to the actual functioning of academia. I have a personal

example of that. One of my papers took one of my papers took almost seven years to get accepted and that was It was just a nightmare. You know, three referees gave up because the paper was too complicated. For good reasons, because it's difficult, but for bad reasons, because I've been sloppy at places, which is what happens when you write a very long, difficult paper. And that's a very, that's inhuman, you know, to live with a referring process that takes eight years and years.

Yes. Because everybody was kind of agreeing that if the paper was correct, it should be published. But the question is, was it correct or not? So if there have been something like Lean, and what I would dream to be the future of AI-assisted mathematics, that would be a system that basically fills the dots, you know, from the human proof to formalized proof, because this should be a pretty mechanical thing to do.

Then I should have been able to press a button and say, okay, my paper is correct. How would I have been very happy with that? Okay, many threads. I'll pick on two of them. You mentioned definitions earlier, and a mathematician I was speaking to actually from the University of Toronto right here is Dr. Barnatin. I asked him something like, what's your favorite proof? Which is just a standard question that many high schoolers have. Okay, the square root of two is irrational or something like that, whatever. He said, Kurt, what I find more interesting is definitions rather than proofs. We have an emphasis in undergrad for theorem proof, even in graduate studies, but as you do research,

it's more what are the right concepts. He said he found that to be more interesting. He also said, and I want to hear your take on this, Jor also said that we didn't prove, it didn't take 100 or 200 pages to prove 1 plus 1 equals 2. We didn't validate that 1 plus 1 equals 2. Instead, we validated that the axiomatic system was correct. Because had that axiomatic system produced something different, we would have thrown it out. But yet, it could have been a perfectly valid system in and of its own that aliens maybe want to investigate and maybe it's consistent. Although something I don't know about, and perhaps you know about, is there are different logics like periconsistent logic. So I don't know if an axiomatic system producing an inconsist

actually breaks it from a formalist standpoint. But that's a whole other rabbit hole we can get to on another podcast. So anyhow, do you agree with withdraw there that we often say, as popularizers of math, not science, popularizers of math, that it took 150 pages to actually prove one plus one equals two. We say that, but isn't it more correct to say it took 150 pages to show that this formal system is consistent with what we know, and then that gives this formal system more credence. Yeah, I think that's completely correct. It's not just about the formal system being correct, is about it having the expressivity we want. And I think this is what they were trying to do, you know, trying to recover a result that people were previously agreeing on. Nobody was disagreeing

that one plus one equals to two. And I completely agree with him on, you know, proof are the technique used by mathematicians, but they're not the reason why they do mathematics. They do mathematics because of the meaning they project onto that. And meaning is about perceiving these things. I mean, I prefer to think of mathematical objects and to think about mathematical reasoning.

I actually, when I was 17 or 18, I was happy with doing reasoning because I was maybe that was kind of new for me to be able to write down actual proof and that. And when I grew up as a mathematician, I started reasoning less and less and digitalizing more and more. and to me understanding something is about being able to relate you know kind of visualize it visualize it's it doesn't have to be vision but it's about feeling things yes and my my favorite things in mathematics are not proof there are objects so the definitions are the gateway to the objects but just the fact that certain objects exist I found it really beautiful. The fact that there is a monster and I mentioned 196

883 is beautiful. I mean that this subject is existing and the proofs are the gateway like the definition. They're gateways to creating mental representation but the reason

is the reason why we're doing that is to get this representation. It's like having a treadmill. I mean nobody loves trademines. people like to be in good shape, you know, it's the same thing. You and I met on Substack. I'll place a link to your Substack on screen over here. People can visit it.

And I'll also place a link to mine in case people are interested as well. What many viewers may not know is that you had a permanent position at CRNS. Or CNRS, sorry, if I'm not mistaken. Yes. you essentially, if I'm not mistaken, had tenure for life. So it wasn't that you couldn't get a job or something like that. Many people leave academia because they couldn't get a job. They go to industry.

You left both industry and academia. Tell me about that. Tell me why. So it's actually even worth than that. I have a tenured permanent for life, research-only position, which is something that's up pretty crazy. So basically, I had no teaching. I could have spent my life doing whatever I wanted every day.

And the pay was not great. So that was an incentive to go to, you know, to not keep this one position permanently. But still, I had the option to keep it forever and I quit, which is very unusual. It's extremely unusual. It's unheard of. So there are very few people. Eric Hull is someone else who happens to be on Substack, who I'll be speaking to at some point as well, who was also on, I don't believe he was tenured, but he was on tenure track, and then he left that.

He left academia, and it's so rare because there's prestige associated with academia, there's safety associated with a guarantee of a job. In your case, the people who have tenure most of the time, the one part of their job they dislike are the administrative tasks that they have to engage in, but you didn't have to teach, you didn't have to do all this miscellaneous auxiliary work. So what is going on with you? So there are two sides to the question. One is what was my driver for doing that? And the second question is what made me brave enough to do it?

And I would start by the second one because the second one is very anchored in a very specific moment of my life. So when I was a PhD student, so when I was 18, 19, 20, I was on a very fast track of going to elite math institution, learning very fast, progressing very fast. Then when I was 21, 22, I ran into very serious personal issues

and I had to quit studying for two years and I abandoned my first attempt at doing a PhD in pure mathematics. So when I returned to, I restarted my PhD when I was 23 with a new address. advisor. And when I did that, it was very difficult for me. Very, very, very difficult. Because my brain at least stopped being able to do math for two years, and I had to re-learn math. And that reinforced something that was always present for me in my interest in mathematics, is to view it not for the sake of doing mathematics, but for the sake of understanding what was going on in my brain. And I viewed my mathematical career as an experiment into, okay, how far can I push this thing?

Can I really become a mathematician or not? That was an open question for me. I was very insecure by then. And so I did my PhD, even I had a two-year temporary position in the US and I was hired at C&RES and I got that permanent position. But when I was, I got my permanent position at the year I was turning 30. and I was still very insecure in my identity of, you know, am I, you know, it's ridiculous because in a way you have all the creditions you want.

You're a pro mathematician from any external perspective, but internally I was feeling like I was a fraud. Like many researchers do, I do think that actually people who do good research start feeling like their frauds, because that's a very good incentive to work hard and try to do great things. Now, I quit mathematics at the moment just right after proving a big theorem in my small domain.

So it's a big theorem for me, that's a small theorem at the grand scale of mathematics. But that was a fairly big one for me. And it resolved my inferiority complex. And I realized that from a pure ego perspective, I would not be getting much more by staying, mathematics. So it gave me the permission to be really sincere about what I wanted to do. Interesting. And I had just been giving that course. I discussed that course about the

foundation of mathematics. So that was a very unusual course. I gave it at the Ecole Normale Superior in Paris. That's a very prestigious math institution, but I was not teaching the math students. I was teaching the humanities students. And that was an optional course that was basically it was a not for credit course and I had a very small number of students were brave and crazy, you know, just trying to understand what is mathematics, you know,

they may have been heading to do a PhD in philosophy or stuff like that. And so I created that course that was a unique course of me trying to explain what is mathematics. And doing that, I realized that many of the things

I had thought about mathematics were wrong. I had stopped interrogating the foundation of mathematics because when you become a practitioner of mathematics you basically, you know, the stuff works, that's okay, you're okay with it,

you just prove theorem, that's your job. But when you take a step back and you realize it doesn't make any sense, I thought it was a very profound things because, I mean, seriously, this thing we're telling a story about mathematics

and we've been telling that story for literally over 2,000 years and this story is meaningless when you look at it closely with honest eyes. We're used to the story so we don't realize it's meaningless but the story about

Platonism and the story about formalism the two of them are completely crazy when you look at them carefully. So I realized that there was this thing going on about the back and forth between formalism and intuition

and that was something that was taking place in my brain. And what happened to me from 17 when I started to study mathematics until 35, when I decided I wanted to quit mathematics, is I transformed my brain in a very profound way.

I became able to see things that from my own perspective seemed genius level not that my research compares to the actual geniuses doing that, but if I was, you know, the theorem I proved right before quitting is a theorem that was incomparable with what I was capable of doing when I was only 30. I transformed my intuition, my cognition, in a way that was unthinkable to me within my prior system of thinking.

So it changed everything I thought I knew about mathematics, cognition, our ability to grow, and I always knew that I wanted to tell the story to the general public. And I realized that this interest in not proving theorems, but about understanding of mathematics works,

is a topic that is not, taken very seriously. It's not something you're supposed to do as a full-time job when you're a mathematician. You're supposed to prove theorems. And I was feeling uncomfortable in that, having that research on your position, not really wanting to prove any theorems.

I was exhausted because I had worked too hard proving theorems. And I wanted to do soft stuff, general public stuff, and I thought it was better to have a clean break from academia to do that. Searchlight Pictures presents in the blink of an eye on Hulu on Disney Plus, a sweeping science fiction drama spanning the Stone Age, the present day, and the distant future,

about the essence of what it means to be human, regardless of our place in history. The film is directed by Oscar-winning filmmaker Andrew Stanton and stars Rashida Jones, Kate McKinnon, and David Diggs. Stream in the blink of an eye now only on Hulu on Disney Plus. Sign up at Disneyplus.com. If I'm understanding correctly, it was some insecurity that made you want to prove yourself intellectually, maybe it yourself and also to others. You could have gone into some other domain, maybe chess playing. And let's imagine you became the best in the world at chess. Then you would have realized, okay, this isn't what I thought it would be or it doesn't feel, or it does feel that hole.

But now I get to examine, what do I actually want? Because the motivation for going into it was insecurity. Yeah, I think I was a student of mathematics until I was 35. And then I was trying to understand what was mathematics and what were my capabilities. So part of that insecurity is social and part of it was not really understanding the topic. And then at some point, because I had made progress

that was beyond doubt transformational for me, Then I realized that, okay, what I've been conjecturing about the way it functions, conjecturing about the way my brain works might actually be correct. So I might be onto something bigger. But this thing bigger is outside of mathematics. So it's maybe comparable to, you know, you've been hired as in the founding team of a startup and the startup becomes very successful. But it's not really your startup.

You know, it's not your business, not your idea. you might be successful in that, but there is this other idea that has been your idea from day one, but you did not really feel that you had the legitimacy to go into that direction, and then at some point you decide, okay, I'm going to leave this company,

I'm going to start my own company. That's basically an idea like that. But the problem was I had no money. So the first thing I did is I took a break. So I took a suspension from my job. I was not crazy to quit formally from day one. I could take a temporary break for a couple of years. So I took a break.

I found a job just randomly, asking a friend where the friend was a sister-in-law had a company. I basically joined a company for one year. I made enough savings to have a personal runway of maybe 18 to 24 months. So it was the first time in my life that I actually had that kind of runoff so I decided I would use that money to

write the book I was dreaming of writing about mathematics and I started doing that and I did that for maybe six or nine months and then I realized I was not capable of doing that and my money would run out before I completed the book. So

then I decided to start a company because I was reading a lot about machine learning so I I decided to start a company just to finance my writing. So I thought it would keep me busy for one or two years, and it kept me busy for 12 years. I became a founder and CEO of a Maltek startup using machine learning to apply to customer data,

first-party customer data for B2C companies. So very hands-on business, very concrete, very business-oriented. I'm going to place a link to your book on screen. it was a hit in the French-speaking world, and it's been recently released into the English-speaking world, and so people can get their hands on that. Now, self-doubt, diffidence, this could lead someone to pursue math,

but it can, for most people, prevent them from pursuing math or physics. They feel intimidated by it. There's some fear that's holding them back. What is your message to those people? Wow, that's intimidating because, you know, I know that these people are really suffering and math is in effect putting a ceiling on the options of many people. The message would be, first of all, and it's a very important one,

it's not hard-coded in your genome. There's a persistent belief that this is genetic. And of course, there is some genetic variability. I'm not a blank slate, this person. I do acknowledge it. There is some genetic diversity in human beings. And actually, some people are objectively, cognitively impaired, and they cannot do mathematics, let's say, high school level or whatever.

But this is a minority of people. And the situation you're describing is about people who are otherwise intelligent, people who can read books, for example, people who can read my sub-sac. So if you can read my subsac, you're smart enough to do math. I have 100% certain that your genetic makeup makes you capable of doing that.

Now, the thing is, it's a very special, mental practice of things you do in your brain to do math. and if you've never practiced it, if you never found how to do it, it's hard to guess what's the right thing to do. But of course, if people tell you that this is genetic, you will not even try.

So the core message is, yes, it's hard. Yes, it seems impossible because there's a certain practice that you need to learn. That is invisible. That's something that you should do in your brain. But you can learn to do. it. Of course, you will never get to the same level of people who started doing that when

they were two or three. But here you can make step up progress. What's some concept, whether it's mathematical or physics or otherwise, some abstract concept, some equation or theorem, that you struggle to wrap your mind around for, let's say, months, and then it clicked. What was that and what made it click? So there's an interesting one. It's a technical one. It's a group cohomology and the bar construction group cohomology.

So these are things that were invented in the 1940s and 50s by Eilenberg and McLean. It's basically to associate certain topological invariants associated with groups. So it's really pure math. But it's something I learned when I was in my early 20s,

when I was, I think, graduate students doing, I was doing growth theory, and I was supposed to know that stuff. That was really basic stuff in a way. And I never understood it. I mean, so what does it mean, so what does it mean, not understanding it?

I could read the definition. I could read the construction. I could read the theorem. I could even apply it, if needed, even though I never really applied because it didn't click. I did not really understand it. And I remember the very night I understood it when I was, I think, 33 or 34, I have to check the date, but I know exactly when it took place.

It took place when I was actually working on proving the big theorem I was mentioning, which is actually in a way of the theory about group cohology. So I was supposed to be a world expert in group cohology in a way, but I still did not understand. the basic definition of it. And one night I woke up and I realized that reinterpreting that thing that was

defined using pretty terrestrial mathematics, reinterpreting that into very abstract stuff involving category theory again made it trivial. And it's a very common phenomenon. There's a,

many people who studied under the growth andique were describing the fact that you know, he was starting to lecture saying, you know, it was basically a stairway to going to the heavens, you know, piling abstraction unto abstraction and to abstraction. At some

point, you just realize that you're reaching the ground. And that's exactly what happened. Like piling as abstraction and unabstraction, realizing that group homology should be interpreted using groupoid covering, the universal cover of a group

and the nerve of that as a category and because the geometric realization of the nerve is a function of two categories, then the bar construction is trivial. That was incredible. That was, you know, that moment where this thing, you spent 10 years,

a shame that you could not understand. And realizing that just increasing the temperature of your abstraction, making it like 100 times more abstract, made it one of the times more simple. But it was absolutely fabulous experience. Now, this podcast is watched by researchers in math, yes,

but also in physics, in computer science, and then philosophy, there's also a large portion that are just lay people. So for those who are not mathematicians, who have no idea what you mean when you say nerves and functors, to explain, let alone cohomology, let alone group cohomology, explain it so that they could understand. Wow, this is very difficult what you're asking, but I will try.

So a group is the structure of symmetry of an object. When you have a triangle, you can have mirror symmetry by going through one vertex and the middle of the opposite edge, and you can also have rotation. So you have a dihydro group, that's the structure of that. When you have N marbles, you can permute them, have a symmetric group,

basically, sorting things is a symmetric group. And basically, the thing about group homology is, the core idea is how do you associate a space to a group?

How do you go from having that thing that is very algebraic, that is the symmetries that you can compose, combine, multiply, you know, that you have a structure

with some kind of multiplication, like the Rubik's cube is a group theory object. Everybody can if you want to feel the complexity of a group, just take a Rubik's cube. That's a good example. And how do you associate a space

with a shape to a group? And the construction I was interested in was really about doing that. And now what's interesting is because when you have a space, then you have certain environments that are called comulgical environments, but the commulogy part is actually not the relevant one

to what I'm talking about. It's really, how do you go from a group to a space? And these things have been, this is really the central idea about category theories. You have mathematical objects of different natures and you can transport

and reinterpret results about groups as results about spaces and by the sign you do that and you're happy because it illuminates your understanding, certain results about group

that you can prove using the shape, of the spaces that you associate to them. And you can't just assign spaces willy-nilly, so there must be some condition. Yeah. And you want, of course, that to be functional in a way that if you have a morphism between two groups,

then you have a morphism between the two spaces, of course. It has to preserve the structure. David, why don't you tell me something that people misunderstand about you or misunderstand about what you're saying? That happens to everyone. Everyone's misunderstood at different points. But what I'm referring to is you're a public person,

you've written a book, you have a substack, there must be something that people say, ah, David, yes, I understand what you're saying. You are saying X. But then you say, no, no, no, I'm not saying X. I'm saying Y. So what are you constantly misunderstood about? So there's something that I find really annoying

is people keep saying that I blame teachers for the poor situation about mathematics. And I actually don't blame teachers. I think they're victims of that. And people say that, you know, people think that I'm making some social commentary about people's relationship with Matt. Actually, I'm making a statement about the nature of reality

and the functioning of our cognition. And I think that we are making fundamental mistakes about human cognition. And everybody is trapped in those mistakes. And I'm trying to entangle that. and trying to articulate a non-absurd way of understanding

what is going on in our brains. Not that I'm competent as a neuroscientist, but I just try to make sense of the actual practice of doing mathematics. And I'm always... I don't know how to react when I see... I've seen reviews on my book where people were saying,

he's criticizing school and teachers. No, I'm not doing it. but he's so much bigger in shoe than Matt. When we spoke over email, I had to push our interview forward a week, and then you quipped. Well, it doesn't make a difference. The foundations of math have been broken for 2,300 years.

Another week is okay. And of course, that's a joke, but what did you mean by that? Well, it's something embarrassing about what I'm trying to do, and every time I write an article on substack, and I have to press the published button, I think, okay, why am I discussing these things? I mean, these are subjects for cranks,

you know, trying to discuss the foundation of mathematics and the nature of mathematical cognition and where does cognitive inequality come from. And the reason why I'm actually stepping into that ridiculous territory, I mean, you're not supposed to do that. You're supposed to talk about normal stuff is because I do think we're seriously wrong. And it's a very bizarre feeling of feeling

that something that's fundamental as mathematics is misdefined in a very profound way and has been familiar. It's an embarrassing situation and we should get out of it. I'm very happy whenever I see that someone said the same thing before me

because that's reassuring to me. There is a short story. video interview of Ruben Hirsch. I'm a big fan of his 1979 article. I'm a bit disappointed by what he did after that. But about one of his books, there was an interview by the American Mathematical Society and it's available on YouTube.

And he's describing that situation. That is really a crazy situation. It's exactly the same one as what I had experienced giving that course to humanity students. He said that he was once teaching a course about the foundation of mathematics. and that was the first time he had a careful look at it

as a professional mathematician and he read about Russell and Hilbert and Brower and all their theories about mathematics was to be founded and he realized that none of that made any sense

and he said what can I tell to this poor students? I mean what? Hilbert was wrong and Russell was wrong and Brewer was wrong and we all mathematician failed. And that's crazy emotions to have. You're not supposed to stumble upon bugs that big in our culture.

But this is a very big one, very profound one. What if someone says, okay, look, the fact that we have had incorrect ideas about the foundations of math for 2300 years, actually, that's not a point in your favor. That's a point against trying to conceptualize math correctly, because if we've been so off for 2300 years and yet we've produced Growthindyke and Terrant Tao and and Hilbert and so forth

and neural nets and etc then that means that the foundations of math weren't terribly important to understand from a philosophical or cognitive standpoint anyhow so there's two different directions

growth and Dick did not need anyone to teach in mathematics in a way. And that's a phenomenon that's very typical of mathematics. If you ask the top mathematician, they will all tell you that, of course, yes, they learn very important thing at school and they're at this great teacher or maybe this parent or maybe this book they read that was very inspiring. But they also feel that there is something about their own mathematical practice that is something

they learn by themselves. And I don't think that anything I've been writing can help produce new Ramad Lujans and new Rotanic. I think they're generated by some random circumstances and very unique cognitive trajectories that cannot be produced by any advice, external advice. I think this thing materialized very early in infancy, and it has not. nothing to do with whatever I can say. But on the other end, there's something that's really broken with now. I mean, if you think about the transformation

that started 200 years ago, where primary school started to become something for everyone. That was not the case. 500 years ago, people were not learning to read. And then 200 years ago, gradually, country after country started to have a free public mandatory primary primary. education. That's basically about two things. Reading writing, mathematics. That's interesting,

that we basically succeeded at doing that for reading and writing. Of course, there are some people who struggle with reading. It's not perfect, but people, on average, people have some fluency in reading and writing. And that's a big success. We could not have the world we have today if we had not achieved that. I mean, the digital life, is a written knife by many aspects. Yes, we have TikTok, but you still need to enter your password, you know? So you still need to be able to read and write. Now, this other basic thing that is making people able to understand mathematics is a hit

and miss, you know. Yes, okay, people believe in numbers. They can manipulate even negative numbers that, you know, maybe 200 years ago where the M2 2 abstract for average people. We made some progress, but still, people seem to be blocked somewhere in their relationship with mathematics. And there is something fundamental that doesn't work, but we fail to onboard a massive chunk of a global population. And I do think is a serious issue, actually. If you think about the role, well, maybe AI, let's not go too much into that discussion about what

can do and replace about our own cognitive ability, but we're living in a technological society where mathematics is at the foundation of basically every sophisticated object that we're manipulating. And some people are apparently handicapped. I don't think they're really handicapped, but for all practical matters, they are mathematically impaired, and that's a big issue.

And I do think that, you know, when we tell people, mathematical objects they exist in a parallel universe, I can see them. You don't see them? There's something wrong with you. Or is it supposed to help them?

There's something that is fundamentally broken with the idea that I know that many professional mathematicians are platonists and I think they're misled when they say that. They want to say, yeah, I see them, I feel them, I want to believe

that these objects are real. And this is an important step. I think it's what I call methodological platonism. But it's very useful when you have someone who doesn't understand math at all to reassure this person

that they're not missing a secret connection with these magical entities. It's just that right now they don't have an intuition for them and mathematics is about building those intuitions.

Now, it's something, you know, you're transforming something that is magical, into something that is tangible. That is, it's a learning process. And there are certain things you can do in your head that will help you develop your own intuitions. And it's about your own neuroplasticity.

And when you learn them, this will be obvious to you. Now, that's more interesting, that's more tangible, that's more practical. But if we refuse to reject the idea that Platonism is incorrect, Then if we continue to be platonitions, there's no way we're going to be guiding people

into building their intuition. Speaking of having some connection to a secret realm of a sort or some obscure connection, the mathematician that shocks me the most of everyone that I've studied, even above physicists, above Einstein, above Hilbert, above Groton Dick, above Terry Tao, even, is Romantagin. there's something so odd about him.

I haven't encountered anyone like him before or like him after. I don't know what the heck was going on there. He said that he would sometimes see formulas and dreams and had to do with gods or meditative practices. He had little formal training. And there's specific conjectured formulas. It's not like the primes are proportional in frequency to this and that. It's no. You take the derivative with respect to n of the square root of n minus 1 over 24 times the exponential.

And these turned out to be true, many of them. What the heck is going on with Ramanogen? This is incredible. Honestly, this is incredible to everyone. And he wrote like hundreds of thousands and thousands of formulas. You just take one of them and you look at it and you say, oh, is it possible that someone dreamed that. Now, you have to accept that he did produce them. He was basically taking a piece of paper, writing them on the formula saying it's a theorem and not being able to explain anything about a possible proof,

about his thought process. Well, he explained some stuff. He said, okay, I was dreaming and then saw blood flowing from the ceiling and I saw a hand writing the formula on the screen of blood and I just took notes. okay well okay so

let's accept that it really existed because he really existed now what's the explanation you have basically three ways of explaining that and I'm going to tell you my preferred one

the first one is he was really inspired as he was claiming by his personal goddess it's a mystical explanation the second one is he was a mutant with superhuman capabilities

that are absolutely miraculous and his brain was a different machine from our brain. And the third one is something with his brain that enabled him to do that. Now, many people,

and it's crazy to find that. Many people reject the third explanation because they don't find it credible. But I find the first two personally hard to believe. I struggle to believe that he really had a personal goddess.

That's not my way of looking at the world. Now, whether or not it was genetically different, it could be, you know, exceptionally gifted from a genetic standpoint, but there's something wrong with it. If you look at genetic variability, we are all different. Like, let's say height. height is massively genetic and it's proven.

Okay, I don't know if it's 70 or 80%, but the variance, is in a country where there is enough food for everyone, the virus is mostly genetic. But we all have the same height, exactly, plus minus 20, 30%, but this is nearly exactly the same height. There's no difference of orders of magnetic.

Look at running 100 meter dash. Some people run faster, yeah. Hussein Bolt ran 1.5% faster than the second one. in the record setting race and 3% faster than the slowest person in that race. Men run faster than women, but the difference is about 10%.

It's nothing. And when you look at Romanogen, it looks like it's 1 million times better in terms of mathematical intuition, but a regular mathematician, it's incommensurable. And this is, the very nature of that points to something that is not genetic.

You usually don't get that much variability within a given species. It's not like that. It looks like something that is produced by cognitive transformation and neuroplasticity. Let me give you another example. Playing the violin. I don't know if you play the violin, but I don't. If I take one, I will be incapable of doing it. anything with it. And if I know

that if I train for towards a day for the next to 10 years, I would still be basically incapable of playing the violin. But I also know that it's not genetic and some people learn how to play the violin if they start early.

And of course, they look like a magician when you cannot play the violin. It looks impossible to do. And my feeling, it's the same kind of learning, except that it's takes place entirely within your brain. So this is, you could object

that everything I'm saying is very arbitrary and very external to the actual formulas of Ramanlton, but have a more concrete example to give you, which is if you take this perspective that he actually had a normal brain, but did very unusual things with his brain, then it opens an interesting mathematical question

which is how did you do it? It's a way to get to these formulas through thinking without making written computations and I had, so when I was

a student at the Econnor Superior, as a first year student we had a course by a mathematician named Xavier Vieno and you can find videos of him on YouTube and he has a website

with all his thighs and all of that and the course was absolutely fabulous he was basically constructing intuitive objects things like you know dominoes examinos trees forest

stacks things you know kind of combinator objects that you can play with and he was explaining how to create a dictionary between certain constructions made with those very concrete, very intuitive objects,

and increasingly complex algebraic formulas. So this field is called enumerative combinatorics, but he was pushing it to the limit. And it was a full semester course, very dense, but at the end, we actually proved one formula by Roman engine using completely visual

objects. And it doesn't mean that he did it like that. As always, in mathematics, you have many different avenues to proving a theorem. But it proves that for this basic formula, but just one example, it is possible

in a normal, maybe gifted, we were all gifted, but we still we were like maybe 30, 40 students in the room and we all find it obvious at the end. You know, that's a pretty strong phenomenon. it's possible for a brain of these 20-year-old students

to find a visually obvious way of proving the theory. So it's not just about ideology, but deciding whether it's nature or nurture or whatever. It's also about having a productive approach to removing mystification and trying to actually create things that were made possible because you believe that it's not

a magical stuff. So then you're going to find the solution. I think it's, when you do mathematics, there are a lot of things that look like magic. Basically everything that's been proven that you don't understand looks like magic. But you have to say, okay, it's not magic. There must be a way to find it obvious.

Because someone found it obvious, otherwise they would never have been able to prove it. Yes, one of your most popular substack pulse is about how there's a Pareto distribution when it comes to what looks like talent, but we can just call it performance or something like that. And that's not explained by a normal distribution of IQ. Again, we have to bracket Ramanogen because that, to me, looks like magic still,

even hearing your explanation. So everyone else outside of Ramanjanjan, even if mathematical ability is normally distributed, and there's some evidence for that, but let's just assume even if that's the case, what you're saying is that ability is not the same as performance, and performance is something like ability times effort, times opportunity, times accumulated advantage.

I would say ability is a complex stuff. I mean, I do think it's not just, it goes beyond performance. There's something about the, the way, when you do math, you have the feeling that some people are stronger than knowledge. It's not a very well-defined notion, but you do have a feeling. When you speak with Terry Tao, you're scared to death because it's obviously much stronger than you are. okay. Pierre Deligne is

obviously stronger than I am. I was working on related topics and I spoke with him a couple of times and I was very intimidated because he was so much smarter than I was. Now, this thing goes beyond the actual productivity. It's not about the number of papers they wrote.

It's not about the theory and they proved it's just about, you know, you speak to them and they look at you and you see in their eyes that I understand everything what you're saying. I think they're going to reply something to you that is exactly what you've been trying to find

for the past six months and you could not find. And they just replied to you on the fly. This is mind-blowing. And so it's not quantified, but if you had to quantify it, so I was discussing with my friend Hugo Duminico Pan who got the Fields Medal a couple of years ago

and he was saying that he's in his own subjective, it's very subjective. You know, subjective perception, Terry Tao was at least 10x faster than he was in terms of understanding new mathematical concepts. And a lot of the work of mathematical is making stuff, making sense of new stuff. So this uncalibrated subjective scale of being strong at math doesn't look like a Gaussian distribution. It looks like a parietal distribution where you have orders of magnitude,

difference of wealth. it's not Elon Musk is not like 2x richer than you are I don't know many times richer than you are it's incommensurable we have a perception

of people's strength mathematics being distributed in the same kind of distribution as well now that does not negate the fact that there is some genetic inequality between people

that by default should be expected to be Gaussian and it's one of the inputs to that for sure. And there are many many algorithms that

can start from a Gaussian input and turn a parietal distribution, but they tend to be noisy. And noise is the important thing. And now this is related to, of course, the experience

of progressing in mathematics, where it really feels like it's a capitalization process where you stack abstractions on top of abstractions. And it's not just about doing it on paper, it's about

doing it in your head, about using your prior abstraction as a scaffold to create new intuitions. So you build intuitions on top of intuitions. And when you do that, over the sustained period of times, I'm talking about

the practice of doing that, you know, several hours a day, basically every day. Some people work on, maybe not every day, but most days, for 10, 20, 30, 40 years of your life. And what is the effect of doing that on your brain? Well, this looks like a capitalization process. And what's important in a capitalization process is that it's not fully deterministic. Because there are so many things, you know, maybe you will have ups and downs in your life,

maybe and so on and so forth. So there's an interesting good news for everyone here is for the very same reason why the gap is massive. It means for the same reason, it means that you can make stellar progress.

It's like with wealth, of course. Maybe you're poor. But if you find a better job and maybe start to manage your finance to be better, you're not going to be absolutely wealthy, but you're going to to run, you know,

go to a safer place in terms of your personal wave. There's something like that about mathematics. You cannot become Terry Tao or Roman Jung. If you decide today, you cannot work so hard

that you will get there. I don't believe that. But still you can become maybe 10x or 100x or maybe one for the next better than you are. Because then maybe there's like 12 orders of magnitudes and maybe there's still two or three

that are still accessible from where you are right now. Terry famously said, Teres Tao famously said that he has little understanding of topology, something like that. But then what's funny is that we then think, yeah, but your bar for what you consider to be understanding topology is far higher than most people. Most graduate students who take a few courses in topology would say, I have a grasp of topology. Terry Tao could probably write those textbooks and then still say, I don't, but I don't understand topology.

So what is understanding? What does it mean? You talked about it earlier when talking about your K-Py proof or the group cohomology. Yeah, I have a very simple definition. Really understanding something in mathematics is finding it obvious.

You find obvious that 2 plus 2 is equal to 4. Your goal should be that things in mathematics should be that obvious to you. that's a very high bar, but the thing is setting it like that forces you to ruminate until you actually manage to see them like that. And this process of

I did not understand group chromology, even though I understood the definition, but it was like, you know, it was like a stone coming from outer space, and I did not really realize why there was this stone that was having bizarre properties that was just in photo me.

I could not understand. understand where it was coming from. And then one day, I found a way to find it obvious. It was a very abstract way, but still, it was very obvious. And I still find it obvious. You know, I remember.

It's a trivial. It's just because the geometric realization of the nerve of the categories, the functor of two categories, and you apply it to the universal... Cover groupoid? Cover groupoid. You do the, you know, it's a Geroa theory for groupoids. It's trivial.

I'm saying it in words. I'm sure that among your listeners, there's absolutely nonsense to 99.99% but maybe they have one or two guys listening to me saying, oh, wow, this makes a lot of sense. It's so obvious. Once you, it's like, you know, riding a bike. Once you learn that you cannot unlearn it.

Good math, math that you really understand, successful math, is math that feels transparent to you. It's like riding a bike. You cannot unlearn it the same way you cannot unlearn 2 plus 2 equal 4. I have one more example of that. Something that you take for granted that every kid in Western countries

take for granted. You have a map or a piece of paper and there is X and Y coordinates. Okay? This is crazy. Like, how could you unlearn that? At a point in a plane

is defined by two coordinates, X and Y. Just imagine the kind of general lobotomy that would take place if we remove that from the things we find

of us. It's impossible to go back. The entire human race has accessed some kind of spiritual elevation where it's obvious to everyone that appoint in a planet as X and Y coordinates.

Now, go back 400 years ago. This is actually what Descartes invented. Before him, there were two separate fields of mathematics. There was plain geometry. It's about geometry. Ruler and

compass. And compass. And there was algebra, formulas, numbers, equations, and they were like they were not talking to each other. You just found a cognitive bridge

between these two things. Actually, it's the first example of category theory, and you have back and forth between algebra and geometry, but that's another story. You transform the entire human civilization by connecting two things that were unrelated before. And after you did it, it feels so obvious that people,

they don't even think it's advanced, not, but it was cutting edge 400 years ago. And this is another argument, against the hereditary case about mathematics. I mean, look, this was like hardcore research 400 years ago. Now it's all just to everyone. We did not mutate that much in the past 400 years to turn something that is cutting edge into something

that is obvious to everyone. So there is some global elevation of mathematical cognition. And it doesn't require any thinking to you to just think that, you know, you have X and Y. It's just obvious. And this is what you want. this is the way you should understand math. Now, speaking of obvious, when you were giving a seminar and Sarah was there,

Sarah said, I didn't understand anything you said, or something like that. So it wasn't obvious to him. What changed about how you present, what occurred in your mind after that? So it's an incredible story. I mean, that was one of the weakest math talks I ever gave, because I wanted to prove the theorem back then, but I couldn't really prove it. So instead of having a talk where I was presenting some big fancy new theorem,

I was just kind of reviewing the basics of the things I'd been working on for years, making them very easy. And there was Ceres sitting there at the front, looking at me for the whole talk for 90 minutes. It's a very long time. And then, at the end, he walked

and said to me, you're going to have to repeat everything to explain that to me because I did not understand a word. and to be honest, it was right. I think after that I realized that my talk was crap, not because I had given a bad talk, but because there were miracles.

So my thought talk was rigorous, but it was miraculous in many ways. There were things that were occurring that I could not really explain why they were working. And it's actually related to the stuff that later enabled me to prove a big threat I mentioned. So he was

spotting not a logical flow in my argument, but some intuition gap. There are things that were not quite right. They should not be like that. There was some missing ingredient in my description. Now, what I found interesting is not just his critics, it's criticism of my own

talk, but the way he did it. I thought, how can you dare to do that? it's not just that it's rude in a way, that it's, that it's exposing himself to saying, I don't understand a word. Why is it doing that?

So my first reaction was to think, okay, you can afford to do that when you receive a field's medal at 27 and you became the first person to receive the other prize. And of course, you can say, there's just provocation, you know. He does whatever he wants. But then I thought, you know,

maybe the fact that he's doing, that has been become the mathematician he is. And I realized that I was not capable of doing that, not because I was not smart enough. Obviously, you don't need to be smart to say that, but because I was not brave enough or I was not secure enough to expose the fact that I did not understand anything. So I tried to replicate the technique to see what happened. So when you go to a mouth conference, usually you're in a small, group of maybe 40, 50 people coming from all over the world, working on your domain.

But within your domain, there are like subdomains and tiny subdomain. So maybe when you have sitting at the dinner, because it's, you know, these conferences, usually the last one week, it's a very remote place and you just live together for one week. And at dinner, you're sitting maybe with a grad student or maybe with a postdoc. And you discuss, okay, what are you working on? And that person will start to explain something to you. usually you don't understand a word

because it's not these objects they're mentioning are not objects you're working on every day maybe you learned about that a long time ago and maybe you did not really understand like the group comedy thing like you're supposed to know it but you don't really know it which is a very

embarrassing situation because when you're a pro mathematician you're supposed to know stuff you're not supposed to not know stuff so maybe you ask a couple of questions and you don't understand the replies and after that you use this shut up and you change the topic

you just discuss about something and you make jokes people end up making jokes and you know discussing anything but mathematics now what I did after the after Serre did that very provocative thing is I tried to replicate that

so I was discussing at a conference with with a guy and he asked him what he was working on and he explained and I did not understand and I took him

apart and said okay please explain to me start you know my brain is damaged that's what I told him I have brain damage and I have attention deficit disorder

it was a private one-on-on conversation so I know that it's not fun to play with these expressions when you're making public statements but I'm just relating a private conversation I had with that guy a long time.

And he smiled when I said that. And I just assumed that I cannot focus my intention for more than 20 seconds. And please explain in a very simple way. And what I realized is that this created two subtle but very important changes in the dynamics of the relations. The first thing is I had given myself an excuse to, ask all the questions I want. Because the premise is I'm stupid, I'm slow. And if I declare that up front, then I will not be embarrassed.

Because usually when you don't say that, you ask a question, but then you don't understand the answer. And that's very embarrassing. And then you maybe you ask a second time and you still don't understand it. And that's even more embarrassing. And then you stop. But if you say, my brain doesn't function, please help me. Then it's fun, you know.

And you can ask as many stupid questions. you want. And this is what it created for me. But the other thing it created is it changed his own attitude. It's basically when you walk in a foreign country and you walk in a restaurant, they serve you of a tourist menu.

Because they think it's what you walk. And this guy, so I have something with my memories, because I wrote down that story, it kind of interferes with my memory. So I think he was a postdoc or a grad student, but I'm not entirely sure because once I written

it in my book I can forget the details but you see my brain is not working so fine but this person had in that kind of implicit pecking order of mathematician he had lower status than mine because I was a permanent CNRS researcher which is a fairly prestigious

position so until then he was serving me the tourist menu he was trying to look impressive so he was explaining things by the book and not by the easy book, by the really look,

look how impressive I am doing this kind of very abstract mathematics. And after I told him, my brain was damaged. Then he shifted. He served me the menus, the menu for the locals, the menu for himself.

I was asking for, not how he was talking about his research topic for a research proposal where it has to look very important and glamorous and all that. I was asking for the way he was making sense of it for himself.

And he was very happy to serve me that menu because that was much more personal, much more intimate, and much more simple. Of course, he was starting from very basic examples. And I realized that there was something magical in this tactic of pretending you're very naive and you ask stupid questions. Interestingly, years later,

when I was, I had become, a startup founder and CEO. Once I was in Silicon Valley and I was discussing with some guy who was a pretty high VP at a very big software company. And

he was someone I was having regular discussions with, but it struck me that once he asked me for something about machine learning, about the way we were doing machine learning and our company and I started to explain things in a very conceptual way. And he said, stop, stop, stop.

Excuse me, but I'm going to ask a stupid question because that's the way I understand it. And it struck me that it's the same exact pattern, the one I noticed with top mathematicians. At Medcan, we know that life's greatest moments are built on a foundation of good health, from the big milestones to the quiet winds. That's why our annual health assessment offers a physician-led, full-body checkup that provides a clear picture of your health today, and may uncover early signs of conditions like heart disease and cancer. The healthier you means more moments to cherish.

Take control of your well-being and book an assessment today. Medcan. Live well for life. Visit medcan.com slash moments to get started. Super interesting. Okay, so now let's assume some younger mathematicians are watching. Okay, and they've heard this story, but they want to know how can they apply it for them? yes, there is the specific tactic of admitting ignorance or

or playing dumb, which may or may not be playing, but it doesn't make a difference. Okay, so they're watching. So they heard Sarah speak to you. Sarah confesses his ignorance or his lack of understanding, but there's a hierarchy there. There's a dynamic. Now, you

to that other person, you could afford to say, look, I'm not understanding, I have ADHD, blah, blah, blah, I have brain image, you get the idea. You could afford to say that because of the dynamic. Now, there must be a productive way to say you don't understand. You gave the example where you were at a bar and then you say, look, I don't understand this. And then the person explains it.

Then you say, I still don't understand. The person explains it. And then at some point you just give up. What is the answer? To not give up? What is to ask differently or to confess foolishness? What is it? So, first of all, I'm describing reality.

I'm not necessarily having solutions for everyone on every single day. But what it says is there are social feedback loops that will amplify. And that's, you know, when you look at a parieto distribution, it must have feedback loops. Otherwise, you don't get parietal distributions. So I just mentioned an explicit feedback loop that is a social one. First of all, at some point, if you continue to, if you persist, if you make progress, some of these natural easy feedback loops will activate. But probably you're not very.

there yet. So message one is be patient, things will get better. Your rate of making progress in mathematics will actually increase over time. The second practical advice is clearly you cannot go to Jean-Pier-Serre and ask him to

repeat ten times the answer to a stupid question. It will not work. You cannot afford to do that. So maybe just find someone you can speak with. And I do think that peer learning is super useful in mathematics.

If you look at our mathematician learned stuff, they learn stuff at the blackboard. But what do they do at the blackboard? It's just usually two or three people speaking. It's with a piece of talk and trying to make sense of something that is too hard for each of them.

But with this back and forth, being very naive, being very open, you can make progress. And I do recommend to all students of mathematics to find the right peers. People who are not here to, you know, to look up, to look down on you,

sorry, to look down on you. And they actually are willing to be at your level and accept that you're their peer. And you, the two of you will make progress together. I think it's, it's the best way to, to navigate mathematical studies. When I was a PhD students, in my second attempt, finally successful attempt at doing a PhD in pure mathematics, I was very shy, very insecure,

and I could not really speak with my advisor because I was too scared. But there was this prior student who was still in the same research team. It was a bit older than me and much more advanced. And with him,

I felt comfortable asking stupid questions. And it was very kind. It was very nice. Maybe if a person next door is an actual asshole and you should not do that. I don't know. But try to look for someone with whom you can have that relationship. I recall you said, and it could be in your book, I don't recall the source,

but you said that textbooks are meant to more be like reference manuals than primary sources. And that peer-to-peer, when a mathematician or someone is teaching something to someone else, They're often teaching these implicit metacognitive tools about how to think about this and that. Exactly. And it's beyond that it's also dynamic in the sense that when you understand something, you find it obvious. And when you find it obvious, usually you have many angles to look at it. So if we were to continue the conversation, and if you were a mathematician, and we were to continue the conversation about the thing I said about group cohomology,

which I said in a very provocative, short way that nearly nobody in the room could understand. But if we were to continue, I would not know how to continue if you did not share with me what you already understand and what you don't understand. And then, because this thing that is now obvious to me,

it has many facets, and I can choose the right angle depending on what you're telling me. So it's the back and forth. And this is one in one discussion. is it's not just that it's more informal, is that it's more personalized. And great mathematical teaching is personalized.

You can go a million times faster when you personalize it. Right, right. Okay, so suppose you're lost and you say, save me, you can't have someone save you unless they know the general location where you are. Yeah, I need to give you my hand, but where are you?

So you need to scream in the dark and I will try to find you. Right, right. And this is really about that. So if you do that, you can... And interestingly, there's another phenomenon, which is... And I experience that with my kids.

I have two boys. One is six, and the other one is two. The one with six, I have some experience now. I had some experience now teaching in mathematics. But in a very casual way, never fall, just trying casually to discuss things. And what I realized when I discussed with him is, maybe I started out with the idea

that I would teach him concept A or result B. But maybe after adjusting for where he's actually situated, that's concept C that I would teach him. And I would never get to concept A

because that's not the right one for him. And very often, when you start a conversation with someone trying to explain some, let's say, you want me to explain why the bar construction is trivial if you look at it as in terms of a function of two categories. Then we just have to explain to you what is a category

and what is a factor. And maybe that would take the full hour. And that's fine because you will have made progress. What is really important is not to get stuck where you are. It's to make progress. And maybe it will be for our next conversation that we'll explain the theory. And you have to be very practical, very humble about

about what you're going to learn. When I discussed with that guy at the conference where he was explaining me his research topic, he actually never reached the actual moment where he was discussing what was his actual contribution to mathematics. He just explained to me some preliminary that I should have learned when I was a graduate student and I never understood.

But I was fine. I was happy when I exited the competition. So there's so much to... It's like, you know, you can have Hussein Bolt as a coach. He will not make you run the same speed, but maybe he will run faster than you were running before he coached you. You know, that's what you want. You want people to be happy and make progress.

That's a very humble, but universal situation. Ah, okay, so the lesson that I'm taking from this is that it's often said that the teacher needs to meet the student where they are, and that's true, but here, because I was asking you to give advice to the mathematicians and the researchers who are watching, It's also your responsibility, if you truly want to make progress, to let the teacher know where you are,

to not pretend you're in some place that you're not. Yeah. And you should not have exaggerated ambitions in terms of understanding. And a really good math teacher is someone who meets you where you are and makes you, you know, many mathematical, So, math, think of

mathematics as rock climbing. It's a very common metaphor. It really expresses something. Maybe you cannot climb that wall

right now, but you can climb the small and the very good math should be very honest about that.

You mentioned draw by not on saying that his favorite proof is a proof of square root of two being irrational?

I know why he says that. I was just saying that's an example of a common. Okay. Oh, okay.

But for me, it's my favorite proof. because this is a proof where I can explain what a proof is. And I can explain that to a child. And that's fine. You just explain what is a proof. And you explain something fundamental to someone, but it's very naive, it's very simple.

I cannot explain how the proof of very complex theorem works, but I can explain how this one works to everyone. So you could request your friend to provide you more and more information, But here, suppose someone was to come to you and say, teach me about the bar construction. You say, okay, in order to do that, I have to teach you about a functor. In order to do that, I have to teach you about categories,

which requires an entire framework, category theory, the introduction of, and may take an hour or two hours. I imagine someone would feel like, I'm just going to waste your time. Why don't you just tell me the topics that I need as a prerequisite, and then I'll go and learn that on my own.

So what's your answer to that? So there are two things. You want to understand of the toaster is working, are the inner workings of the toaster or you want to understand how to use the toaster. So another important thing is most of the time when someone comes up with a question is because they're stuck somewhere,

they want to prove a theorem, but in the theorem they need to use some technology somewhere and they don't know how to use the technology. So here you just keep it simple. You say, okay, no, no, no. Let me show you. This is where you put the bread, this is the button you're pressed,

and after two minutes you're done. So you just explain the interface of the theorem. And then maybe, because it's a bit frustrating, you say a few and waving make a few end-waving remarks about how it works internally, but you're not really serious.

Now, if the question is, how does group cohology really works? Why is this construction working? Then you go to the other direction. It's really two different demands. You have to accept, but some people don't accept it. I don't think as someone like

Growth and Dick ever accepted being presented with the technology that you do not understand. And that's a very unique and very powerful approach to being a mathematician to reject everything you don't understand. But there are other people

who are very happy to manipulate notions they don't fully understand and they can leap through entire territory is very fast, basically standing on the shoulders of giants

and being happy with it. what's really important is to understand the human ask that someone is someone coming to you with a question. And it can be very different depending on the context and what they're trying to do. I have a question about hand-waving. So suppose we went back a thousand years and then picture proofs were accepted.

Now we would say that there's something hand-wavy about that, meaning unrigorous or gestural or notional. And then someone would say, well, the way that we thought of continuity was hand-waving, but the rigorous is epsilon delta. But then we'd say when you'd take a real analysis course or you'd take a certain real analysis course, like the one at the University of Toronto, it's called MAT-157s, my favorite course, actually. They make you painstakingly prove with modus ponens every single line of certain proofs. And then you'd say, okay, well, that was the truly rigorous one.

But then you can imagine someone saying, no, no, that's not even true rigor. lean, putting it into lean and getting a computer to say it's true is the true rigor. Do you think we'll ever get more rigorous than that? Do you think we've bottomed out at the most rigor right there? I'm not sure. And again, my definition of mathematics is that it's this back and forth process between understanding and formalizing what we understand. And there's a never-ending back and forth.

And in a way, Russell and Whitehead, we prove that 1 plus 1 equals 2 in a very different way. So you could say that they tested the equation but they also consolidated in a different way. So

there but it doesn't mean that it's going to be productive to always question the rigor of stuff. I think you should have a good reason to question it. And you mentioned

that 2,000 years ago some picture proof were accepted well, you know, if you read some actual papers by first one, it looks like they still accept it today, you know? And that's fine because there's some stuff that is easily drawn but very hard to

prove. I have, in one of our papers I have a lemma where I give a proof that is one picture and the text says a picture is worth a thousand words and that's all. End of proof. Interesting. It was accepted. It was published. David, what's the most inspirational advice you've received?

Oh, it's a very practical one from my PhD advisor. When I was at Yale, I had that very nice temporary two-year position, and I had proven the CRM that was not very hard, not very profound, but I had found a way to write the paper in a way that, you know, it's like when you write a good piece where you found a good angle that makes it very sexy. But it was not very profound in a way,

but it was referring to questions that were asked by fancy paper. So I was thinking about you know, submitting it to a very good journal, but I was not sure whether it would get accepted. And he said,

go for it. I don't know. Maybe not. It's not clear. You know, there's pros and cons, I don't know, but don't self-sensor yourself, you know. You will see. Now, so, and, that was the first time I accepted taking some risk for my ego, because usually

until then I was very careful. I was so insecure that I was careful because I was afraid of being rejected or, you know, people who are saying your paper is crap, you're going to reject it. So that's difficult. And I submitted it and it was accepted very quickly. permanent serialized job thanks to the paper. And the lesson was if I don't fail often enough,

it's because I'm not trying hard enough. And I should do things, even if I'm not sure they're going to work. And that was very helpful for the rest of my life. I very much like that. There's some percentage you can put to it. That if you're not failing 10% of the time, then it means you're not trying hard enough.

But anyhow, thank you. Thank you for spending so long. me. Thanks a lot. That was a wonderful conversation. It was very interesting. So I know your style and I know that you have this thing about being very patient about asking, you know, the tricky questions and spending the time on them. So I played the game. I hope you're happy with the result. Hi there. Kurt here. If you'd like more content from theories of everything and the very best listening experience, then be sure to check out my substack at kurtjymongle.org. Some of the top perks are that every week you get brand new episodes ahead of time, you also get bonus written content

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