How natural is our mathematics?

0:00 Well, okay, so it gives me great pleasure to be able to go through one of our own, and we came to talk on mathematics, and that's why I'm not sure what we have. Should we be more excited? It's great to be invited here as a philosopher of mathematics to a philosophy of physics seminar. I'm rather hoping this is a pointer to the future. I think our ancestors when they look back at this phase will be rather amazed at the lack of dialogue between philosophy of mathematics and philosophy of physics at this period. In some ways, mathematics gets rather trapped. I was just thinking if I were clever enough to now leave PowerPoint, I might have logic as the hard place and physics as the rock. Mathematics trap between the rock and the hard place in philosophical terms. As you know, Edward, work's been done trying to reduce mathematics to logic as far as possible. And when this isn't possible, when there's some sort of excess of mathematics that doesn't seem to be reducible to logic, people see this as great worry, great concern, and they then try and push it to the right, into the physics, specifically in one solution in the philosophy of mathematics, to see this excess as something in the physics. but also from the point of view of the philosophy of science again I guess they've got enough worries of their own philosophers of physics have enough worries thinking about more theories and the way they relate to the world possibly not enough attention is paid to the role of mathematics in the physics that's a theory I have something I've tried to do I've just had a book that's just come out something I've tried to do is to look a little bit into some sort of depth into the inner life of mathematics is really quite rich if you go and look there. Something I haven't really looked at too much is the interface with physics, which of course is going to be important for a more satisfactory philosophy of mathematics.

2:30 So I thought this would be a good place to make a bit more of an effort to see what philosophers of mathematics and philosophers of physics might talk about. So the title of my talk is coming in response to an article by David Ruel. Some people know this, he's somebody behind the bifurcation theory, chaos, turbulence. He wrote this article in the Bulletin of the American Mathematical Society, Is Our Mathematics Natural? The Case of Equilibrium Statistical Mechanics. Back in 1988. I've changed from his is, I've changed to how. because I don't like these questions in philosophy. I prefer how questions are a bit more subtle in some ways. If things are going to be sort of yes or no type answers, then possibly they weren't the right sort of questions. And curiously, I think this reflects a theme which will crop up through the paper, that mathematicians are also, in a sense, going from is to how at the moment. I'll make that a little clearer as we go on. Something that Ruel is touching on in this paper is the relationship, the credit, should we say, the credit to be apportioned. In the great enterprises of mathematics and physics and mathematical physics, what credit should be apportioned to each party in these successes? And very often it strikes me when people try and do this, they get caught in one extreme end of the range or the other extreme end of the range. I mean when you read Wigner on the subject it sounds like all the credit goes to mathematics, the unreasonable effectiveness of mathematics. The mathematician gets quite a boost from reading that paper. And he's almost a little suspicious of the physicist. It's all a little bit too easy for the physicist if you read that paper. It's like you're coming along with a bunch of keys and you just pick any one and it opens the door and in you go. Another person, Mark Steiner, has recently been writing on the applicability of mathematics as a problem. Again, mathematics does a heck of a lot of work, the work in terms of applications. He's led to some notion of a user-friendly universe. The people on the other side of the coin tend to react rather strongly to the bigness discussion of this problem. and something you pick up from science literature as well as philosophy of science literature

5:00 it's almost as though what's the mathematician's role that's just to pump our structures they're rather indiscriminating they just chuck out these structures and it's up to us, we scientists, to sit through them can we make any sense of these things or is it merely mathematics and you get that word merely tagged on and then there's another well, and Feynman does it sometimes can recreate it whatever we need anyway. We really don't need you. Whenever we need something, we can recreate it. And then sometimes you're not even up to the job, you mathematicians. In fact, we have to do the work ourselves, and then we can help you out. And it's those kind of attitudes that have led somebody, a computer scientist called a metropolis, to try and lure the mathematician away. Okay. Here we go. The relationship to computer science. The relationship to computer science and mathematics scarcely resembles that which exists between physics and mathematics. The latter may better be described as an unsuccessful marriage with no possibility of divorce. Physicists internalize whatever mathematics they require and eventually claim priority for whatever mathematical theory they become acquainted with. The mathematician see to it that every physical theory sooner or later is free to more shackles of reality, deliberating in the thin air of pure reason. We mark computer science is a very different move. It turns to mathematics in much the same way that engineering always has to borrow us freely from already existing mathematics to earn it for altogether different purposes or more likely for no purpose at all. Computer scientists raid the coffers of mathematical logic, probability, statistics, their algorithms and even geometry. But far from resenting the raid, each of the disciplines is buoyed by the incursion. So, quite an allure for the mathematician if you can break away from your wedded vows to physics. And it would be interesting, I think, if people start looking at this in a bit more detail, what computer science makes with mathematics. I mean, something that would be interesting is this boundary that already was a little vague, a big use in some ways between mathematics and logic. Is there a true logic that isn't a mathematical logic? this becomes muddied even further I mean some of the elements of philosophical logic, things like temporal logic where do they find their use precisely in computer science model checking the great market if the philosophical logician wants to sell their product for something useful

7:30 these are the guys that could make the case for them but it's not always the case that the computer scientist finds what they want, rather like the physicist was saying, you know, we don't always get what we want from the mathematician. Here's a local computer scientist, Samson Abramsky, saying, you know, we don't, to start off with things were good. Notions of function, set an algorithm were available off the shelf from mathematics and logic for use of computer science. By contrast, there's no pre-existing theory of criticism's interaction information flow in which the second generation models can build. He said he's got to go about constructing some new mathematics. Often with some little equal from physics as well, it's all interesting, a lot of interconnections going on. Oh, yeah, I can come back to the how-weather thought here as well, because there's one side of what Abramsky's doing that is around the notion of it's not just whether a program can do something. We want to know how a program does something. And if this program does some task Are they the same kind of how, or are they different hows? So there's a whether to how dimension going on there. Might you say what second generation Well I suppose we're going off from the original people working in languages like Lisp and some of the early languages that were pretty closely tied to the logistic of the thirties, churches, Turing's and so on. You know, now we've got to know how is the internet going to work, how do we know a language that's capable of sorting out those problems out. A lot of interaction between computers, it's not just single boxes doing something on their own now. Massively interconnected. just to end with this little section with a more balanced response perhaps from a mathematical physicist at this time I don't know if any of you come across he works with string theory so he's a nice reconciliation It's clear to many physicists and mathematicians that we're going to continue we're seeing many exciting domains of interaction between physics and mathematics. So much so that some researchers are even predicting a merger between the fields. It's a very good place to give the vast instances of the means

10:00 that mathematicians and physicists have about a given subject. And it's not a good thing for the merger to happen. They benefit from each other's so much precisely because they have such clearly distinct views about a given subject. It seems a bit like marriage counselling suggestions. You know, let's value what's good about each other individually and what you can do together. Okay, well, let's turn now to Ruel and his notions. He's going to focus on the idea of a naturalist, mathematical naturalist. And what are some rather hefty terms involved here? of the world, the mind, society, mathematical nature. His task is going to be, mathematicians see themselves as dealing with a certain core bit of mathematical nature, which I'll talk about as we go on. But they're a little bit wrong in a way because the world is very important and the world, our encounters with the world really do change the course of history, the course that the mathematics takes. I mean, some of you may think through the 20th century this idea is almost dropping out a lot of mathematics develops I'm going to say steam but he's saying there's some really critical ideas in mathematics that would come had we not had them engaging with the world and then he's also going to bring in the mind in some way that we'll look at it later I'm not going to argue to any strong contusion about how you should put these together There's enough possible combinations for anyone in this room to have a different point of view about how you should weave your way through that. I thought I'd give you one perhaps I hadn't even thought of, but it crops up one of the terms that we need later so it's not a bad opportunity to bring that in. This is Yuri Manin, one of these endless stream of brilliant Russian mathematicians that have come out and have been released since the Cold War. is a curious nation. So there's, do people know about pihanic numbers? Do you answer, I mean, from a mathematician's point of view, if you're dealing with the rational numbers, you can put, between any two rationals,

12:30 you can put various distances between them. There's a sort of very, the distance that most of you would put between them, which is just to subtract them, I think. And then when you try and complete the rationals under that kind of measure, you end up with the reals. That's fairly well-known. But there's mathematicians that came up in the late 19th century, came up with other types of measures, other kinds of distances you can put on between rationals. And when you complete according to those distances, you get these periodic fields for every prime, p. And this is really, you know, a lovely core bit of mathematics. This is the sort of thing we would say the mathematicians are holding to be core piece of mathematics. So the real numbers are seen as competing at the prime at infinity. The other piatic fields are competing at the other prime, the actual prime at prime. And what you do is you put them all together, you see them as one, you multiply up these fields into what are called adels. So you inject the rationals into all these fields at the same time, all their completions. And you can do some really good things in number theory if you do that. And you say, it's kind of a curious idea, because the mind is a material body, a material, well, it's a portal, I suppose, by some material nature of our minds, of our brains constructed out of massive particles it makes us look, it makes us turn to the one, that one single rather, you know, arbitrarily it makes us turn to one completion, I mean the real completion. But we could put it just as perfectly as well, have projected out onto the other, the other sorts of completions, the other chaotic completions. A weird idea, a rather interesting sort of twist of your way between mind, will, and mathematical nature. What is the suggestion? If we weren't built as we are, as for particles or something, that would be a reason to spiritually... What is the suggestion? Well, I suppose, yeah, if we were made out of law... Does he mean very large numbers of particles? Is the point that because you get our normal way of doing things when you sort of take P to infinity, it's something, it's to do with very complex or something, or very large numbers of things involved, I don't know. No, it's not a large number, no, it's to do with having maths, is it? Right, okay. But when we push back, there's certain ideas, there's quite a lot.

15:00 If you put Pierre and theoretical physics in on your search engine, you'll come up with whole slews of papers on this. Okay. That ultimately we're going to need all this adelic sort of mathematics if you really want to get at the structure of the universe. But then somehow, you know, some symmetry has been broken, if you like, it's come out in terms of us looking at the real material nature of all these things. I mean, I just want to pop that in as if there are some very weird ways you can loop through these corners of this square. Okay, clearly a daunting task, trying to put pieces together. I ought to say a little bit about what goes on in this corner here, the mathematical nature. I mean, certainly not, when you say mathematicians believe in this mathematical course, mathematical nature, he's certainly not meaning, you know, the space of all possible mathematical theories, it's not like that. It's something sparser than that. There's some very important, you know, people know, people have a sense of what's the good stuff. the sort of people that when it feels metal, they're always going to be working on that good stuff I mean there are going to be disagreements between them, your vision of what's in the core, but there's going to be some elements that everyone's going to agree should be there just about anything that derives from Gauss will be in the core this is just amazing he touches there's a sense of continuity many of our ideas should back two gams, a man with a great touch. There's also an idea of a sort of iceberg field that we're sort of perceiving little pieces of this space. If only we could just get behind, get the right point of view, we'd actually understand how it all fitted together properly. So there's a great sense of sort of spooky connections that I just sort of tell you about. I can just find that link to that. We came from a really different angle than the other. Now, of course, you can respond to this. Some philosophers would respond to this in saying, well, you know, all there is is sort of logic and some space of all possible definitions in set theory. Anything else about the things that mathematicians study, it's either because that's what physicists of need say,

17:30 or else it's some sort of socio-psychological sort of practice that would have driven them to find this interesting rather than that people in the strong program have even gone as far as to take the logic out as well it's all over there, there's nothing there at all there's a bit of a backlash that wants to say one need to take this socio-psychological stuff in a negative sense, one can take it in quite a positive sense in the sense that if you've got a well-arranged society then things that emerge through the deliberations of that society are, you know, that is what rationality is about. So there's another way through there. Robert himself, he starts off by becoming engaged in sort of counter-factions, I don't mean, one way of doing it. You know, there's the obvious temptation, you know, if we met the little green men from Mars, what would their mathematics be like? You know, it wouldn't be wonderful if you could have access to another civilization. Politicians talk as though you would expect them to have very similar mathematics, I think, which is the same, similar level of civilisation. But of course, I mean, it's just the game you can play. He then goes on to the thought, well, okay, something is happening, computers are coming in, they're going to do mathematics differently. This will show us that what we held to be this core mathematical nature is rather heavily reliant on the way we were constructed. But he's saying that's still pretty early days, too. And it's true, computers have really come into mathematics in a very minor way, it's surprising, much less than physics. So what he's going to do instead is turn to Exos and history of mathematics through the 20th century to see if he can help himself answer this question. So let's have a think about natural, naturalness. in chapter 9 of my book I have a chapter that talks about naturalness and it's something that my mathematicians just use it all the time if you go through any bit of exposition this is just the natural thing to do it seems a little bit strange in a way like protesting too much now of course I mean if you're thinking where does the term appear you're perhaps the first thing that comes to mind would be natural numbers,

20:00 probably connected to the world, isn't it? You might also know about natural deduction, which is a way of, there's an idea of a way of getting more, getting a bit of grip on the way mathematicians reason. Natural deduction is a way of construing logic in a way that's closer to where mathematicians think, so they're going more at the mind, should we say. The three senses that come out from this chapter of mine Point in this way Certainly first off, particularly the way it's used in terms In mathematical terms It has quite a technical sense Freedom from some arbitrary choice Something that's going to be called natural If you didn't have to make an arbitrary mathematical choice A bit like some coordinate free type to it. Which, I mean, there would be a very clear case of that. A natural equation, this is a technical term, is an equation which specifies the curve independent of any choice of coordinates or parameterization. All of those, even natural logarithm, although I didn't know the curve, and it crops up for a 1668. But it's clearly linked to the idea that it's to do with, you don't need any nasty number in there, one or one will do, as your number, the hyperbole. So it has that fairly technical sense. Used in exposition, though, tends to have this meaning as the obvious next step. It's non-contrived. You're just forced to make it. It's forced upon you. Okay? This is the sense you're getting that the matter still dealing with the nature. It's very resistant to what they're dealing with. Every bit of resistance a little bit more freedom, but almost as resistant as that confronted by the scientists. You get this kind of playful side of it as well, but I'll do this. Here's some of you discussing. In some sense, although, if you look at the space of groups, how many trillion there are, but the vast majority of them are of a certain type. Still, it's pretty much shows that most of them are in nature. In the broad sense, not simply of chemistry and physics, but of number theory, biology, common plurics. They do this quite a bit. Are close either to simple groups, which are things that are opposite age from these type of groups. Autogrups, which arise naturally. He focuses on this one, and he slips this one in.

22:30 I put the emphasis on that. He slips back into naturally, which arise naturally in the study of simple groups. things that you're kind of forced to look at. But there's a third notion and that's something I want to talk about a bit later that really there is some notion of simplicity in the natural equal simple. The idea that if you really work over a piece of mathematics long enough, you can extract some really simple essence of it. Here's a protection from a mathematician. The quest for ultimate triviality is characteristic of the mathematical enterprise. I'll give you a sense of this later on. Curious idea. Okay. Just a slight interlude on von Neumann as we're talking about mathematical physicists. In 1947, one of them wrote an article called The Mathematician, and you've never seen that. Often you probably will have heard quotations from it, it's nothing right. So he's trying to get it probably into one of those sorts of books that try and ask all the professionals to give a sense of what they're doing. So, The Mathematician, one of them is asked to be The Mathematician. he makes he's making various comparisons with theoretical physics they share quite a lot in some sense the criteria used to judge the theory often very much of an aesthetical nature it doesn't look very similar in that sense but of course theoretical physics does have another side to it there are things like certain sort of hard concrete problems that take their attention in a certain way but an awful lot of the work and the judgement processes of the theoretical physicists are very similar to the mathematician. Okay, that is about a very aesthetical nature. Okay, and he elaborates the sorts of criteria used in mathematics. These criteria are clearly those of creative art, and the existence of some underlying empirical worldly motif in the background, often in a very remote background, overgrown by aestheticizing developments and followed into a multitude of labyrinthine variants. Well, this is much more akin to the atmosphere of art, pure and simple, than to the new empirical sciences. Okay, so then he's heading on, then he's going to get, so, as a mathematical participant travels far from its empirical source,

25:00 or still more of a second or third generation, and indirectly inspired by ideas coming from reality, to be set with very, very, very dangers. This you may have heard. It becomes more and more purely aestheticizing, more and more purely art for art's sake. this need not be bad if the field is surrounded by correlated subjects which still have close empirical connections or if the discipline is under the influence of men with an exceptionally well developed taste but there's grave danger so interesting questions there what does one want to make between this aesthetic and this good it's not just this aesthetic So, I mean, are some forms of aesthetical taste associated with better ability to come up with ideas that are going to be useful in the future? As I say, Gauss is seen as someone he knows what to look at, he's got a wonderful touch, and read them again. I mean a T.O. is often held as one of the Britons the man with the best taste in Britain I was hearing recently that when he was in Oxford he was stopping people studying poinsett theory poinsett topology that's just bad taste you should study algebraic topology and well, you know, lo and behold a few years down the line, he does find its uses in field theory, in quantum field theory somewhere where tiers were and he claims, you know, they had to teach me this significance for Fourier transform I don't know, you know, just as a mathematics it's not clear whether von Neumann there is presumably he's counting himself in this and that was an exceptional case another elitist, British elitist G.H. Hardy, perhaps more differently with him. It appears that Von Neumann came to Cambridge in 1936 to talk to the mathematics department. And I'd love to know what he talked about. I've got an idea what he talked about, but I can't remember what he was talking about. But at the end of it, Hardy famously gets up and says, obviously a very intelligent young man, what's that mathematics?

27:30 That's not a taste that Hardy shared. A few years later, back to Reuter again, he can be saying, many of us remember the feeling of ecstasy we experienced when we first read von Neumann's series of papers on rings of operators in Hilbert's face. It's a paradise from which no one will ever dislodge us. That's Hilbert's statement. It's just like an allusion to Hilbert talking about Kantor's statement. But there's always this worry, and then Reuter also tells of Staphon that when von Neumann over in the States and away from his beloved middle Europe, along with another emigrant, Stanislav Ula, he tells us that Stan soon found a way to von Neumann were cheerful about being in this environment in Los Angeles. Stan soon found a way to cheer up his brooding friend. He began to make fun of von Neumann's accomplishments. He would mercilessly ridicule continuous geometries, building space, and rings of operators. Like everyone who works without distractions, von Neumann needed with constant reassurance against deep-seated and recurring self-doubts. Rather than feeling offended, von Neumann would burst out in a laughter of relief. Okay, so this is... It's nervousness. What happens if he doesn't... If he's just empty, Baroque construction. Now, I mean, interestingly, in the case of von Neumann there, I mean, Hardy was famous for being the one, you know, his big goal in life was to solve the Riemann finances. And what's the best effort now these days to solve the Riemann hypothesis? One of them comes off of Alan Kohn, a mathematician, who's using type 3 von Leumann algebras, those rings of operators, using a, back to those Adele's of Magnum that I mentioned right at the beginning, he's using that in some kind of packages to solve the Riemann hypothesis. Also, the type 2, the von Leumann algebras came to different graphs as type 1, type 2, type 3. And the type two have paid off as well, yeah? They've played a big role in this linking of Knott's theory, Yann-Baxter equation, constant groups, cluster of ideas off there. Okay, so, there's a play. Right, I mean, how does one judge have one engaged when one's moving in the right direction? It's naturalness. I mean, of course,

30:00 so when you make these connections later on down the line, internally for a while and suddenly the amazing connections are made. But Ruel comes down a little bit hard on this. He says, well, you know, it's not that hard to do in some sense. It's like we live in a very high dimensional space so everything is near everything else. You can pack lots of things in very close together in a high dimensional space. So it couldn't be laid out with mathematics and we shouldn't really be so very surprised that things are related. I mean, the feeling back on the other side, though, is it really isn't that easy to come up with these really amazing connections. It's not easy. So, the world's game plan is to take something. It's going to take statistical mechanics, equilibrium statistical mechanics, show us that ideas in that which are natural physically, i.e. the partition function and equilibrium space, things that are not mathematically natural, they're physically natural. And yet these played a big role, a significant role, in the history of 20th century mathematics. And the mathematicians wouldn't have got there without the physicists. So that's going to be an easy way of dedicating the idea of this is mathematically unnatural. Okay, well what I want to do next is to suggest that the partition function isn't as mathematically unnatural as Ruelman has made out. a little bit of maths, not too much. Some of you have been to any of my talks before. You may have heard some of this stuff before. But there's a big movement going on in mathematics to shift from set theory to category theory. There's a nice translation of two columns that you want to translate. Where elements of sets go to objects and categories. over from one to the other. This is very much, this is the quintessential place of where that to how. Way, way down the list comes there's a set way over the set side, a free commutative a ring isn't a ring without negatives. A free, one of the free commutative ones of these with no generators are the natural numbers. There's a nice characterization of the natural numbers. There's just some initial, unique, free type of voltage. If you translate that over to the categories, you get the category of sets. That's why sets are important

32:30 because it's a very simple idea. We just need, without a huge amount of detail, we need a little bit more. We need If we stick in one generation, these things are now going to be series, series in one variable, x, where the coefficient of x to the n over n factorial is going to be an integer. And the translation over there of these things are going to be called structure types, species. Let's give us some examples of that. So there's a nice basic sort of one example of one of these things is the exponential function. and its correlate is set which is a very simple structure if ever a collection presented it to me I'd say are you a set yes you're a set so it's a very very basic baby set baby sort of species or structure type it's a rather trivial one it just says are you a set and that leads to the exponential function seriously this series to, so I take ULock and I stick you in order as many times as I can, all the quotations of ULock. That corresponds to this. You can then play games of adding these things together and multiplying them together, but what I want you to focus on here is that the derivative, so I've got my, you're going to come in and I'm going to put a structure on you in as many ways as I can. Well, the equivalent, everyone, on this side of you taking the derivative is bring somebody else in from outside, and then I see you, and then I try and put the structure on this new group, and then somebody else come in to the room. That's why I'm taking the derivative. It's something pretty simple. I'm just tossing an extra thing in and putting a structure on it. So, beautifully out of this, you may have thought that the fact that the derivative of the exponential function being itself It took you till 16 maybe, till you knew that. But on this slide, what it's saying is, if you are a set and somebody joins, you're still a set. That's quite a nice little, simple thought. Okay, now there are all these, these are the various ways that you can kind of combine these.

35:00 Combine these species. And it occurred to me, this is great. a fan of John Byers, the mathematical physicist, who he likes to dabble both with, I mean he's almost one of the favorite loop of quantum gravity approach to quantum gravity, so that's him as a physicist, and then as a mathematician he loves this stuff. So having written about this, I thought, I was wondering, this is quite a famous function, where you get x over actually the other way around. So it occurred to me, well you can just build it up quite straightforwardly, out of these complicated, combining these two functions. So then I'm trying to make a bit of sense of it. So then we entered into a bit of a dialogue. I thought I'd be able to do a dialogue in CAD. So here's what I was talking. Yeah, I've been meaning to think about this, because that function I just showed you. I think there's a link to things called the Pannui numbers. the coefficients that come out of those. So he said he'd be mean to think about the series since he read A. Conn, the hero of mathematics, comments on Baloui numbers in this book. This is quite a fun book if you've read that. Conn points out there that if there's some H, this is all known as Berlin I do, H is the Hamiltonian for some sort of particle in a box and is the inverse temperature, then this is the operating, you take the trace of to get the partition function. And he claims to pondering this, explains all appearances of this function and Benouin numbers in topology. This is a bit of the mathematics that Atiyah would have said, ah, yes, character of the class is excellent, but very much developed in a purely mathematical way. Okay, but this book where they wrote about these things happened before quantum theory invaded topology, and everyone's talking about quantum topology. So we have to sort of, how are we supposed to make sense of what quantum is just saying? So then I come back and try and put a construction on this composition. And the essential bit is, I mean, if you, it's actually down this bit here. So it's some sort of composite Of what I said was sticking things in order

37:30 Applied to Just being a set So when this thing encounters anything It's got to say Wherever you are, I'm going to split you all up Into sets And then I'm going to put those sets in order So if it's only given one thing The CTDS can start saying There's the one thing say there's the one thing, or it can say there's the empty set in that one thing, or there's the one thing in the empty set, or it can go empty for one thing, and so on. So it's very much like sticking a ball in a certain number of boxes, in one box, or there's two ways of putting it in two boxes, or there's going to be three ways of putting it in three boxes, and so on. Okay, so you carry on doing that. You're obviously going to get in for the number of things, aren't you? From 1 plus 2 plus 3 plus 4. Is that worrying? Well, it's not worrying. It's getting with translation over. You don't have to worry about divergence. It's like going back to the early days of Euler when you just played around with the diversion series. You've got amazing results out of it, but you can do it properly. So this is lit to the zeta function. Which, yeah, which if you continue over, I mean, usually, I mean, you know, in this original form, you weren't going to think the z-tomption is a negative number, but if you continue it over that way, you come out of this thing. Okay, and everything works hunky-dory, and everything works out fine, and the right kind of efficiency is now. So then he comes back, so I've got another interpretation of the Bernoulli numbers. Again, more and more confused and tantalized by the relationship between Bernoulli numbers, species, physical mechanics, and quantum theory. quantum arc is that he was emphasizing the relationship between the new numbers of partition functions. Partition functions in statistical mechanics and quantum theory are often just another way of looking at generating functions in species, which, again, is something of absolutely beautiful simplicity. If you haven't thought about this yet, I can perhaps shed a little light on what's going on, though not fully penetrating all the milk yet. So then you think of a system where you've got great energy levels. So we used to do all this, blindfold, won't you? and then you form a partition function by summing these things you know sorry what's a species that is talking about generating functions yeah he's actually changed the word to structures it's rather like

40:00 that thing that comes along and wants a different sets of, it's presented with you in the room and I start putting different structures on you maybe I put a trivial thing on and just call you on a set or if I'm a different species I want to put you in order at different times as I can. Or if I'm a different species, I want to put a group structure on you in many ways like that. Okay. So, right, and this is all known to you, fortunately. This is forming, so you can do lots of great stuff with the partition function. Maybe by differentiating it with respect to to that constant, that reverse temperature. Then if we take out the zero point energy on the harmonic oscillation of quantum mechanics, the allowed energy levels now equally spaced from zero. So when we form that sum there, because these are just going to be multiple of themselves, you can play that trick. So you can see emerging, don't you, this thing that I was talking about before, one over one minus e to the end. There's a problem of b equals zero, so you tend to multiply by b, and then outcome there's Bernouinums. But what does all this have to do with species? I'm not sure. It's tantalizing for two reasons. In my work with Jim Dolan, the harmonic oscillator itself is categorified. It's Hilbert space becomes the category of species. So this is taking the Hilbert space of harmonic oscillator, and its categorification is a category of species. the creation operator translates over as an operator between vectors in the middle of space, translates over to some operation on these species there's going to have to be something funked up and if it turns out to be just that derivative which is just putting an element in it's going to take you to from the space and then it goes to n plus 1 Annihilation is the same as it turns out to be translated over to taking elements out. And the number operator of doing them in order translates to just multiplying by a number. It's called the number operator.

42:30 So they each try to make sense for this commutation relation, for the annihilation and creation operator. And what it pans out over on the right is saying, if we have a box with some balls in it, there's one way, one more way to put an extra ball in and then take it all out, then there is to take the ball out and then put one in. So if I'm dealing with you here, there's one more way of the porter coming in and then me choosing one of these people in the room. Land there is, me choosing somebody in the room and then the porter comes in. And that's all there is. There's a computation which has been wrong with you for so long. There's a little bit so slightly reminded here of but he's talking about how well really I was getting their aesthetical sense and why some of them don't see eye to eye on this and he's saying it's really because people have bottom line mathematicians have a bottom line my bottom line is placing balls in boxes these are called the troilers so is it we can see how he was taught by him because Bayer was actually trained in MIT but Roto I think yeah because Roto was a joint professor of mathematics and philosophy and Bayer says he only went to hear the lectures on Heidegger he didn't go to the mathematics lecture does he mean counting by there placing balls in boxes is that the point there's a lot of rich work Does he say that notion of bottom line in connection with his idea of the question of the triviality, is that what he means by bottom line? It is where that particular intellectual temperament finds the satisfaction of grounding everything in the trivial, or does he just mean that is where that particular intellectual temperament likes to stop, but it's not through the triviality? his particular brand of aesthetics leads him when he's making everything to be trivial leads him so it is related to your previous quote this use of the phrase bottom line is linked to your previous quote about triviality does trivial mean tautology well this is the kind of that's what we've got that's just tautology isn't it so this is going to be a curious thought

45:00 in fact I mean stop short of reducing to triviality because people will just say, well, that's just trivial. So the common thing to do is stop a way short of that and then just spin out basically because you know really what's going on and you spin out stuff and you think how deep you are. But you need to, you know, get it down to sticking balls in boxes and that's what it's all about. It's about clarity, isn't it? The trivial... Sorry. Yes, it's not triviality No, it's just I mean, it's crystal clear Yes, it's presumably to go through the route from the start through the complexity and come out of the other side and see that it's a triviality, so-called. Yeah. If you like. Yeah, crystal. In other words, it's obvious. It doesn't mean it's dismissed. You don't have to dismiss it, that way. You've got to go follow the path before you can get... Follow the Buddhist, almost. Yeah, that's right. The way. Yeah. Anyway, carrying back, so on he goes again. In your post, so there are two ways. He's been doing it one way to try and interpret this VN, and I've been doing it another way. That's the contonality of a set of structures on the N elements. You get a diversion series, basically the Riemann zeta function, evaluated at a negative integer, and you resum this to get the right answer. This trick is common in quantum field theory under the name of zeta function regularization. So I never knew I was doing that. speaking in prose so there are two possible connections they don't fit together neither one is fully worked out though yours comes a lot which I put in there not just for the ego boost but just the thought so just taking his notion that one can you can end up saying something in a field that's generally understandable