How natural is our mathematics? (contd.)

0:00 He then goes on to three things that happened with equilibrium in statistical mechanics. This was the first one. Do people know this? The Li-Yang Circle Theorem. I think if you take the Ising model and you do your stuff, then you end up with some classical complex polynomials. And because of what do you know about phase transitions and all that sort of stuff, you know those complex polynomials have to have zeros on the unit circle. And, you know, they labored away for years, apparently, well, this was the story I originally heard. And then they came up with the proof that they kind of knew physically it had to be the case to Sparman. So they grew up to say, ah, here's a great case. the mathematical result we never would have gotten this had it not been for Laurie's physics. Something he didn't quite tell us something I've probably noticed recently. So in 51-52 Li and Yang were developing his theory and Mark Katz became aware of their conjecture which was later to become the Li-Yang circle theorem and he brought to his mind a theorem of polyas remarks on the integral representation of Riemann's to function. And he realized that a slight modification of Polya's proof could be used to prove a special case of this conjecture, or this to their attention. So Li and Yang were then able to get out for reasoning, and within a couple of weeks, produced proof. Captain says, I recall Professor Yang telling me at the time that Hylsats spy, Polya, was one essential ingredient in their proof. So it wasn't, so we had a bit of a story about how this could have came almost out of nowhere. It came out of a business system. So this is trying to construct a proof of this result. But it actually turns out there's actually some pretty strong connection going on with some work going on in number theory now, the real zeta function. We'd be seeing statistical mechanics in number theory very closely attached to each other all the way through this. Okay, so that's possibly slightly played to that one. He then goes on to say, so would these only physically natural notions of a partition function These physical techniques, we then apply it to other areas of mathematics. Differential dynamics. So we can use it, for example, on the theory of hyperbolic manifolds, differential geometry.

2:30 And then we can end up with results like prime factorization for lengths of close to GD6. Compact manifolds of negative curvature. So again, an important step in mathematics, but we wouldn't have got there if we hadn't had. Okay, but I mean, well, given I've slightly deflated the idea that partition functions are unnatural mathematically, also, I mean, in a sense, the more that sort of thing is going to happen, the less likely you are to think that a mathematician couldn't have arrived at it, is that flexible a tool that you can shift around well over the place. You are then going to perhaps question whether it was so mathematically unthinkable. He finally goes on to this case, which is a slightly curious one because, in a way, this is a perfect case of simultaneous development. Some of you probably know this, the KMS, Kubo-Martin-Shringer, and the mathematicians, Tomita and Takisaki, he sort of did more or less the same thing, using the work of one-on-one-man algebrism. Okay, he's partly, I don't know why, quite. I mean, he doesn't make too much of it in a way, because he's kind of, the mathematicians have always stayed up for the same thing. I suppose he could say, well, this isn't very good maths that's come out of that. We wouldn't have even got to this stage of even formulating it if it hadn't been just a lot of mechanics. But again, the more the mathematicians are able to do it, the more you're going to think, could have got there anyway. Yeah, it's just a little hint for those who want to win a million. This is the big game, although this is Colin's game. So you're going to dream up a quantum dynamical system whose partition function is the real zeta function with spontaneous symmetry breaking over the pole, such as the edels, which are reversible adels, call the symmetry group in this form group. They form the symmetry group for the set of KMS states Okay, so just to finish up, what was Ruel going to make of this? Well, so he's saying physical naturalness is often quite different from mathematical naturalness. The intrusion of physics, therefore, changes the historical development of mathematics. This is relatively modest in the 20th century. Also, the influence of historical actions should not be overrated. Some concepts, like those of natural animals or groups, would have to appear sooner or later in the development of human mathematics.

5:00 So he accepts it as quite a chunk that would have happened anyway. But external circumstances might have been different and led to different mathematics. I further think that whatever...so now in his game, once he's allowed a certain chunk of mathematical naturalness, he's now going to bring in the mind. I further think that whatever naturality mathematics does have is not due so much to logical necessity. It's just the peculiar nature of the human mind. I mean, the way in which our logical thinking is linked with visual intuition and tied to illogical, natural languages. I mean, our idiosyncratic likeness for short formulations, which we call elegant. So, the aesthetics is taking off in a very psychological way. And for somewhat repetitive ways of arguing, which we describe as natural. This is not to say that mathematics is an arbitrary construct. Of course not. It is a structured subject, and in some sense, it is nothing but a structure. That structure was not made for us to understand, yet the human mind can grasp it, that's what makes mathematics so fascinating. We like to think of the discovery of mathematical structures as walking along a path laid out by the gods. But, as Antonio Mercado wrote, maybe there is no path. So, yeah, you get the sense of the idea that the mathematician has that they are covering, you know, this rich structural world, and really it's you that's making the path in your activity. Okay, well let's wrap things up now with some conclusions. I think when you read these people, people like Juan Eumann and Ruel, they're not really that consistent. They're something, they have kind of a philosophical training. Yeah, there's something always rather fascinating about them. They tend to weave their way around, around the corners of that box that I put up. On the other hand, I think they make wonderful places. sense of what's at stake. I think they're good places to build from. Also, another thing we're going to have to have a lot of people coming in, if people want to get rid of this, we're going to have people coming in and working on this, zooka-zagging through many different ways. If you do a local study like, of one particular feature, like Ruel is doing here, I think you can kind of drag it anywhere you'd like, in some sense. You know, he took it very much for physics, it was instrumental in it. But, you know, I tried to pull it back

7:30 I think you need a much larger picture to get a better grip on the right way of stretching things over that square. Thank you. history of mathematics and the history of physics, I think one thing that would be, that would stand out, would be that both disciplines tend towards greater generality. Both disciplines tend towards greater... What were the two disciplines? Mathematics and physics, yes. Both disciplines tend towards greater generality, and they might sort of conjecture from that. Well, greater generality means theoretical economy, tending towards greater theoretical economy. Fewer and fewer concepts leading to lots more structure. So they might conclude that what's natural to the human mind is this tendency towards theoretical economy. Well, I can't really put myself in the mind of the Martian, but, um, I mean, some people look at, you know, well, this is sort of a royal sport, you know, in a slightly negative sense, you know, the human is, he's got that thing they like doing, they like things to be, yeah, a little pest you in the way you're saying, and that's just, that's the way they are, and he's rather, he imagines that, I mean, computers will be the thing that provides the counterfactual force, if you like, you're not, probably not Martians, because people but when we start using computers well then we're going to have very different mathematics because I mean, you know, elements and brevity is not of importance to the computer that's just thinking I would take it to disagree well, theoretical economy is also an aesthetic kind of quality yeah, yeah but is it just purely

10:00 human relatives is it just something about strange creatures that we are Well, I mean, I didn't quite get, you know, the explication of naturalness. It eventually does come down to the mind, doesn't it? Well, I mean, I wasn't actually putting forth my own. No, no. I was just sort of suggesting there are different ways of looking at it. Well, some people, yeah, I want to say it's just down to the mind, but others. When you got through to that real crystal clarity, Is that really just the way we humans are looking at things? Balls in boxes. Well, what alternative is that? Well, some might imagine they are recovering. They are getting to the bottom of how things are. In the fairly sparse world of what's called the structures? that seems to be a mystical right Pythagoras had a mystical approach to numbers they all had to be rational and then when one of his apparently when one of his students discovered the irrational numbers he had him murdered because we don't go into such things anymore Yeah, is there a sort of fairly straightforward analogy here between what counts as natural in mathematics, as we might find Gauss thinking up, and what counts as natural in physics, which tends to be things like, I mean, I guess I'm really just saying a really simplistic version of the whole, you know, sets versus categories thing. But in physics, you end up with things like sort of, you know, entities and properties of entities. And actually, on an even more simplistic level, you just end up sort of like thing and noun and property of thing, adjective, and things, doing things to each other, verbs, and it's all really simplistically there in the structure of our language. let alone in the structure of our thought, but is that, if you're going to sort of question

12:30 the naturalness in mathematics, well I'm not sure whether people do question what I'm claiming is analogous in, in, in physics, I mean. That's only a way out, isn't it, it's just, you see, the common element, the common cause is in some of the mind. It seems to drive you towards just being very psychological about it. You sort of think, well, the way I am as a human being means that certain math is going to be natural and there's no point trying to think about things in any other respect than looking for existent entities all the time because that's just what's natural to my way of thought. Yeah, that's something, you know, to look for common causes, and that will be one way out. But it's possibly eating some people in a way they don't want to do something which can't impact you. It's the way we view the world, categorize the world. But yeah, it's a valid option. Of course there are many questions, Can I just ask you to follow this one from your closing remark that case studies could be pushed in any direction. I can imagine that's true, but do you have privately in your pocket that you're willing to say out loud now some view about that square and where the preponderance of case studies would be? If somebody could do 200 case studies, without the violence to the history and the philosophy, where would the preponderance of evidence come? I can't, I do, I vacillate between, what else do I do? Between, or probably that last solution of the, I feel you can't finish the solution, but then there's something, something that does really feel like you're carving out, because you start

15:00 that don't feel torn out, but then you end up somewhere, and then light is shone on it. There's enough spookiness that you can't ever smooth it out. It feels like, you know, if you push it off one way, it crops up there. If you try and get rid of it there, it comes back over here. So there's some sense in which I don't want to go along that line, thinking it's all just... little exchange to the previous question didn't seem didn't as i understood it relate very closely to the bigger question or to them or to the idea that most of mathematics is well like being driven by physics it's much more about mind versus mathematics rather than the empirical world versus mathematics yeah if you take a little bit like it's by the mind Yeah, I mean, to try and get some grip on this various issues, we need a clear taxonomy was the big question, which I took it to be lined up with the left-hand column of the world and mathematical nature, where mathematical nature meant intrinsic to pure mathematics, the idea of autonomy and taxonomies that were internal to it. And then, on the other hand, we have a mind that's societal, setting aside society. Kantianism is all about the mind, not about, I think, the world. The mind contributes, the mind is the underpinning of the apparently autonomous taxonomies and mathiluses and mathematics. So, I mean, in that simple mind, trying to make a concrete set-up of either the physics or the mind, they are two claimants to being the main determinants of mathematical nature. Then on the third position would be a kind of gungo-Platonism, that no, mathematics has its own autonomous naturalism taxonomism. Okay, so that's three positions. Leitism, physicalism, you can call it, and mentalism. Now he does 200 case studies. Which one are you going to know? Yeah, I want to know the answer.

17:30 I really don't know. I also, I, I, moving, I can't help myself. My name's Jack, I mean, I don't even know what the sermons would look like. I hope you should be clear as to what sort of evidence would have led you to go with you rather than your mother. Well, Rubella was presenting a certain type of evidence. Like, you know, I think that wasn't very good evidence in many ways. Well, I really didn't understand very well what Rubella was saying, actually, and what that evidence was supposed to imply. Oh, okay. So, I'm sorry, I'm going to gain absolute advice. So, physicists, in their encounter with the world, have adopted this thing, equilibrium statistical mechanics. A very important feature of that is the equilibrium state. When you cash that out in mathematical terms, it leads to a mathematical concept. That's the sort of concept you would never have gone to, purely through internal mathematical considerations. Yet, it starts leading into new mathematical results, we start applying it in different mathematical fields. It's effective the course of mathematics. Mathematics would never have gone along that byway, had it not been for the business thinking in terms of... But are you wondering what I'm saying? That claim seems to be entirely plausible and actually innocuous. and I'm not very interested in that research in physics is going to lead to certain branches of mathematics and otherwise it seems to be not a problem. I mean, I'm not really seeing what I should make out of that. So something further, for example, and what's more, those branches of mathematics are fundamental to mathematics. Part of the core, yeah. Because that further came, but I think I'd like to understand that better. would be evidence? Is it that the evidence would be case study after case study, okay, one locates a new piece of mathematics, and now there's some further piece of data, I guess, which says that this piece of mathematics is fundamentally mathematics. In the, I don't know, citations, it's cited by more, you know, is that the sort of data that would then support to the world's thesis? Yeah. Right. Great. And the absence of data like that when it's brought to neutral between the mind and the patients.

20:00 Okay. I'm just doing it more. You're interrupting Jeremy's credit. Well, yes. My concern was taking it away from the physics of it somehow in itself. Although, I mean, of course, you can come back to the physics. anyway. I'm trying to deflate some of his evidence. He hasn't taken, you know, his name was very, very bad. He's done the prosecution, he hasn't done it. What counter-servidence between the mind and the evidence? Well, yes, that's going to be a toughie, doesn't it? Yeah, possibly the evidence. What sort of coincidences you're willing to countenance with a common structure to the two will happen to be a coincidence? That leads people along the it's out of their line. Things like where did that really come out of? Why is that now our best hope to read that process? It's a bit surprising when it's a bit. I mean, my question, I think, relates to the discussion that's just been going by now. I mean, I wanted to go back to your opening remarks that philosophers, I don't know if there's other species, I'm not sure, not conditionally. We look back on this period in amazement at the lack of dialogue between philosophy of mathematics and philosophy of physics. I want to be clearer than I am what questions you think they will be amazed weren't being answered by this lack of dialogue.

22:30 Well, I suppose that's a slightly presumed that for us people go along a direction that wants to come to terms a bit more with the way things are done, a little bit more interesting. mathematical practice and post I mean it isn't there's a picture of something presented the 20th century the way you get a divorce in the 30s between mathematics and physics and the quantum bill theory to have oral and then they go along about their own business and then back in 75 they reunite again and finally do what each other needed and that's what about this coming is precisely a product of that kind of reunification. And now they're really helping each other in amazing ways. Physicists are coming up with mathematical conjectures. Mathematicians are not math. if you're interested in the way mathematical physics is done, then you had better know that they have learnt a better way of going on. And that better way of going on isn't just and it's interpreted in their one way. They are now very interested in going into the mathematical network and seeing how that belongs to its own mathematical network, because they know they're going to learn from those interconnections. I mean, Gene Bigner, whom you cited as being too hard on physicists, I take it, Yeah. Wanted to know how it was that mathematics worked for physicists. Yeah. Well, that's the question, right? And that's a particular question, and maybe these case studies would help, I don't know. But I'm still not clear myself how... I mean, I take it you are wanting to claim that philosophers of mathematics of whatever persuasion or whatever, just by being philosophers of mathematics, they ought to be interested in what goes on in physics. And I'm wanting to know what the argument is. Well, let's see, you know, we all have all our resources. I mean, whenever I say, let's slightly shift direction, everyone says, why do you say everybody's got to do this. I'm just saying there should be some attention being paid to the interface with physics. You know, how about this? I mean, chapter four of my book?

25:00 Right, okay. Where I start to bring in brain-based ideas into mathematics. But it's absolutely clear now that a lot of the evidence for mathematical propositions So is the claim, I mean, one sort of claim would be that it provides data about the extent of mathematics and the mathematics that needs to be understood, so, you know, if you're learn to be a philosopher of mathematics, you need to know more mathematics than you learn by doing A-level maths or something, which sounds a reasonable position to take, but I'm still not getting a clear feeling, sense of why it is that the mathematics, I mean why, for example, if you want to understand the nature of mathematics, would it be more important to study work that's going on in physics or mathematical physics than, say, to understand, which would take a long time, Weyl's proof of Fermat's last theorem, where one has a case of a result that, you know, brought together very diverse areas of mathematics and in some sense it would seem one could see what is essential to a creating process in mathematics, and that might be, you know, that seems a good thing to think about if you're trying to understand the nature of mathematics. I mean, I'm not clear why looking at the application in physics or the development in physics is going to be particularly crucial to philosophy in mathematics. Well, I certainly agree that people think that looking wilds would be a good thing for somebody to do in the future. I really generally want to know, get some kind of sense of the impact, can't we want to know, the impact of physics on mathematics? Do you think it's not a, you'd think that's a purely historical question, to want to know the shifting...

27:30 Well, on the face of it, yes. I mean, I'm wondering what philosophical questions would be answered. I mean, maybe you still think that there's something that leads you to some normative notions of things having worked well. Encounters with physics is to be promoted. Things have only worked very well when, as Neumann warns us, we make sure our mathematics doesn't just dilute itself and dribble off into a thousand channels. There's a normative side to it. we should make an encounter or for this to do you can have a slight moment of development or do you still want to have a struggle well that's moment of in a sense that involves good practice in mathematics what would be how should we allocate resources in mathematics where should our appointments be made these are all crucial questions in the development of mathematics but I'm not that's what we would call the philosophical question. I'm not saying it's not, but I'm not getting, I don't think on the face of it it is, so I think it needs an argument to say what is philosophical in that kind of enterprise. Well, we've met at philosophical games, I mean, people have very different notions, don't they? I mean, I can't quite possibly see some of her work and suggest that it does have a normative side to what this issue would be doing. That's all part of her philosophical package of what she's doing in philosophy of physics. She has some notions of, we've been led astray with a sort of faulty philosophical idea of what physics is, to devote too many resources to fundamental physics. a less than ideal representation of the relationship between mathematics and physics you might say have the same effect in simple terms I can ask you a question in Chinese you can go away and try and learn Chinese and try and give me an answer or you can translate it into an English question and give me the English question English answer to the English question and then translate that that's I think the sort of simplistic view of it and the philosophical question is why is there a common structure to the two languages

30:00 that's a philosophical question as opposed to something really practical as well, would you prefer to go away and learn Chinese and I don't know how to speak Chinese and then answer my question sort of in Chinese or would you prefer to translate this into this alternative I mean that's that's just how I see it So why are the mathematicians and physicists encountering practically the same thing? Why do they need the same thing? Isn't that a pleasant question? Does anyone ask this question? There's three people in the queue. Anyone else? I just wanted to check something. I was just a bit surprised that you didn't respond by saying, well, there is evidence of a phenomenon, which is a philosophical experience. One could then argue that, well, is there a phenomenon there or not? You mean the goodness in history? Well, I think the way that we were hearing it wasn't that, I like the way I paraphrase it. That's what makes sense to me, that we seem to be finding that the source of mathematics that comes out of physics is very deep mathematics. Just being within mathematics. The mathematics community is deep. It's a strange phenomenon that seems to call for an explanation. It's not actually a bigness question at all. It's something like a whole other way around. Yes, or it seems to be parallel. No, it's not playing the game desperately putting the mathematics first or the physics first. What I find myself more obvious from the requiring explanation, and raising philosophical questions, is why should the investigations of physics to produce, you're particularly good at producing deep mathematical results. I find the alternative view, you know, why should deep mathematics have applications in physics to be less troublesome to me personally? Would that sort of at least be a fantastic case for why one should be interested in the relationship in physics about that? It doesn't strike me as, it seems to me some story has to be told about that. I mean, The fact that, you know, very deep mathematics, the whole development of calculus giving rise to 19th century development of mathematical analysis arises from concern with the physical world.

32:30 Now, I mean, maybe that's a deep question that has to be looked at philosophically in an important way. I mean, but, well, maybe this is the very sort of thing that one should be doing, but it seems to me that, all right, then one can say, you know, it's very hard for us to have, to find coherent structures to investigate Abognizio, and so abstracting them from the physical world around us is our best chance of establishing something that's deep in the sense of being rich enough to end up having a highly developed mathematics. You mentioned the case of chaotic numbers. They certainly arose not in physics at all. they were purely let's do something analogous this business about metrics on spaces that we already were doing the theory of reels getting reels as a completion of the rationals and then 80 or 90 years later that turned out to be significant for physics isn't clear that it goes particularly one way rather than the other. That it happens in both directions is very striking, but I'm still not clear that there's a great philosophical issue that's different from simply the fact that mathematics exist, which is remarkable. Well, I certainly agree with the two-way. I think it's part of what I'm saying about And that's an analogy, as you say, to get it done with your bio-scouts as well. Expansion, if you're serious. But if there's a mystery, I mean, there is something, you know, there is a mystery there. You know, you've given a sort of explanation where there's a beginning, but I don't mean too many people even beginning out on an explanation like you've just done. wouldn't it be helped by just to get the balance right by a few we need to get a bit more feel of the field before we can start really talking about it it's all very well coming out with our capitalist cake measurements was one of the early

35:00 the early ideas of how much beer in the cake I mean I'm quite worried about that example I mean it was hardly one of the keys after a dull mathematics oh yes well it's just to return to the the concept of naturalness to see whether you think i'm sort on the right lines with the example of the jewel of the jewel which is the first one yeah which i i thought seemed to point the right way and it Thank you.