Narrative & rationality

0:00 Do you want to put it to push them? I mean, is it working physically? Well, you have to calibrate it from there, going back, and then it goes. I don't know. No, don't worry. No, I've got nothing to say. If you could make it with the finger. Yeah, the finger. How long do we have? That's what I would like. Good question. I really got you. Yeah. You can choose. So it's sort of an hour and a half for each person? Yeah. Okay. Yeah. Thank you for your attention and see you in the next lecture.

2:30 Well, thank you all very much for coming, and thanks to Ivan for inviting me. It's always a great pleasure to come back to Paris. This is where I fell in love with philosophy of mathematics. I was here studying psychoanalysis at the time. I started reading a lot of philosophy of mathematics, as you do, including a lot of Lakatos and a lot of Lautmann, Albert Lautmann, who I quite like. I thought this was a wonderful club. So, I mean, you notice how extraordinarily culturally variable philosophy is, because when I returned to London to study for a PhD, of course some people had heard about Lakotosh, but just no one had heard about Laughlin. It didn't exist. So I pursued the Lakotosh line, but it'd be good one day to come back to Laughlin. In fact, you can see these very strong cultural differences that occur in mathematics. Where's all the arrows gone? Why are they gone? Here we can see a rather clear case of these strong cultural differences. This is John Burgess writing in, there's a Princeton Companion of Mathematics which will very shortly be published, which is going to be a wonderful book when it does emerge. Slightly strange in the philosophy contributions to the book, which I think are largely made up, one by John Burgess, which is an analytic in philosophy and mathematics, and then somehow one by an American mathematician living in Paris, Michael Harris, who's written a rather idiosyncratic and interesting philosophical book, but there are no other representatives of other schools of mathematics. So here's John Burgess. Philosophy of mathematics, today a smallish speciality within philosophy of science, itself a smallish speciality within epistemology or theory of knowledge, played a much more important role in the past.

5:00 You get this idea of this tiny little, tiny little area, which I find a little disturbing to have this notion of us as working tiny, tiny things. To give you a sense of the activities that I personally engage in, I see myself as engaging in two tasks. One is examining mathematics as a tradition of inquiry. Perhaps you don't want to see it as a monolithic, single tradition. You may want to see it as a cluster of traditions. And secondly, to actually participate. This is something I rather enjoy doing. I mean, to the extent that you're interested in mathematics and the rationality of mathematics in terms of thinking about it as a tradition of inquiry, you're going to at some point have to study a certain section of the history of mathematics. Personally, I find it easier just to engage with contemporary work. I have great admiration for historians amongst you. It's very hard work to learn the language of the past. So personally, I engage with some contemporary mathematicians. And the project that I'm involved in goes by various names, sometimes higher dimensional algebra or in category theory. For various reasons. I mean, one reason you could think, well, this is such a good new conceptual framework for mathematics, possibly it could have the same effect as the foundational period did a hundred years ago. You think what happened to Frager-Russell from the analytic philosopher's point of view, this great transformation takes place due to the development of a new logical apparatus, and philosophy gets transformed. So, I mean, it could have been that I find, you know, I think something, there could be a sort of second revolution, something along those lines. But I'm not so sure. Anyway, I ought to give you just a little idea, though, about what goes on in this program. So there's a nice slogan. And that certainly gives away quite a bit of the motivation of this approach. Every interesting equation is a lie. Whenever you see an equal sign, question it.

7:30 Because behind an equal sign there must be something behind the scenes, some sort of process, some sort of way of going from one side to the other side. If there were no transformation going on, if it were just 3 equals 3, that wouldn't be an interesting equation because there's no movement, there's no change going on. So already in a very simple thing like that, 2 plus 3 equals 3 plus 2, if you start to think about what's involved in In the process of moving, of commutativity, of shifting from A plus B to B plus A, you know, and you could demonstrate it to children very easily, what you do, you put two piles, a pile of two things, a pile of three things, you just swap them around, but then you can, so in a sense you almost could represent it in time as some lines twiddling in space, but then you think to yourself, well how, were there different ways in which two plus three equals three? What if I had switched them the other way around? Does that make a difference? And by starting to think along these lines of how we get from one side to another, here's somebody I work with, he's a mathematician, John Byers, you get these, as he says, these very, very complex structures emerge from the simplest of things, very, very simple things you take as you move them around in different ways, and you start to, I mean, quite quickly you can get up to thinking about not only the study of lines, one-dimensional things in a three-dimensional space, And the idea of n-categories is to try and capture this notion of processes and processes between processes. I've never been satisfied that you've ever hit the top layer. It's always rather reassuring, I think, because Yuri Manin is a very prominent Russian mathematician. Quite reassuring that he's not seen as a great partisan of the n-category movement. It's rather good that he seems to be on our side. The following view of mathematical objects is encoded in this hierarchy. There is no equality of mathematical objects, only equivalences. And since an equivalence is also a mathematical object, there is no equality between them, only the next order equivalence, ad infinitum.

10:00 This vision drew initially to greatly extend the boundaries of classical mathematics, especially algebraic geometry, and exactly in those developments where it interacts with modern theoretical physics. Okay, so it's certainly not just those in a rather small sort of school of thought that think that this is an interesting process worth developing. And any time you find anything like an equation, as I say, you can try and look behind the scenes and find isomorphisms. If you find isomorphisms, you can take that a step further and look for something more subtle, more subtle types of equivalents. And it's always, it's always something, it's always just to curtail, to cut, cut off this ladder when you actually never find an equality in the equation and you've reached your top level. But there's always the possibility to lift the ladder up. Yeah, I mean, as I say, you know... It's got all the signs of being a programme that you really could develop. I think it's a great new language to help you with metaphysics. Here's a recent paper written by Hyatt Center. Physics, topology, logic and computation from the Rosetta Stone. Category theory serves as a lingua franca that lets us translate between certain aspects of these four subjects and perhaps eventually build a general science of systems and processes. So you're finding this great commonality of structure. His approaches to physics are not just his, but many people are conceiving for a series of physics, in the sense the objects are systems, the morphisms are arrows, processes between positions and states of the system. We're finding these great similarities going on just in the mathematical field of topology, manifolds and cohortisms, that's some sort of spaces that mediate between these two endpoints. And they're very closely linked to physics and the topology, and in fact we've seen these physicists, people like Edward Witten, helping the mathematicians by these parallels between the physics and the topology. But then there's a lot of work going on in computer science to try and bring in similar ideas between these different subjects.

12:30 Now this is only two layers of the pie, this is just the objects and the morphisms, the processes. Of course you can take it to another layer. There's now three layers, and this is computer scientists doing this, looking at terms in a language. Then they've got rewrite rules that allow them to rewrite the terms in other forms. And then they're looking for confluences, ways of mediating between these two, the computational paths. Positions that you're, your states that you're... You're performing computations on different paths that computations take, and then you're interested in the relations between the paths. There's always processes between, and again, some approaches to physics, seeing space mediating between matter. Space-time is mediating between space. One slice of space is another slice of space. So as I say, yes, on the face of it, it's a great candidate. One could certainly do a lot of work. I'm believing this to be a new form of, a new metaphysical language, a new language to drive philosophy on. I mean, even if it doesn't work, and I've always been very suspicious, to be honest, of mathematics used in philosophy. Sorry? Ah, yes, well it's lovely stuff, isn't it? Where he likens, he says watching analytic philosophers use formalism is like being in a grocery store and watching somebody pay with monopoly money. So this idea of, you know, you as a mathematician understand what it's like to use symbolism properly, and then you see these philosophers and they're saying, you know, let's take a bit of language, somebody washed their hands, and you say, there exists a time, t, such that t is less than the time now, and at time t, somebody washed their hands, you know, you know the sort of thing that goes on. So, yeah, by and large I've always been pretty sceptical about this, the reduction of natural language using logical tools. I suppose you may wonder, given I just mentioned to you that I was studying psychoanalysis here, you may wonder, am I sceptical about Lacan's use of mathematics?

15:00 Yes, I am. But, as I say, there's another role for doing this. I'm fascinated by mathematics as a subject with such a tremendously long history that wants to rethink its fundamental concepts. What is it about a discipline? There is a slight danger, I think, if a lot of the resources of... History and philosophy of mathematics get devoted to the 19th century and earlier, but we forget that things move on. Just as historians study things when they were new, one must remember that the material today is new and will become the material for the historians of the future. So here are some questions that I've always pondered over. So where is this rationality of mathematics? How do mathematicians choose which direction to take? Where's the freedom? What are the constraints? On the face of it, there's a lot of freedom, but that leaves you vast open spaces to study. What determines why you go in this direction and not this direction? What determines who wins the prizes? Who gets their papers published? Who gets their PhD? There must be some considerations. Which are hopefully rational considerations which come into the assessment of what counts as good mathematics.

17:30 But logic, I feel, is a use that people are spitting in. People could be producing plenty and plenty of logical truths. But that is clearly not the point. It doesn't tell you what are the interesting things. You get this line when you present this to mathematicians. Yes, I mean, they're happy, everybody doing their own thing, if you like, doing their research. But at some point you can tell that something's going well because, you know, some... Famous or some outstanding problem will get solved. These are the benchmarks. These are the times that we know they're onto something because they've solved an important problem. But of course, you know, this raises the issue of why does a problem get to be taken to be an important problem, which pushes the question a step further back. And of course, as we know, there's often a very interesting history to the status of these problems. They say the history of Fermat's Last Theorem. And find that at some points it's not regarded very highly. Gauss famously regarded the problem very highly. So there's interesting questions about why do certain problems get to be seen as... And personally I favor a line of thinking. This is Bill Thurston, American mathematician. So he's considering the problem. His notion is that the aim of mathematics is... So in a paper written in 82, he's talking about his program, classification of three-dimensional manifolds. And he set up a conjecture about the way you can cut up these three-dimensional manifolds to support various geometries, and this is called the geometrization conjecture. So if you look just in terms of what he's actually saying about the conjecture, it's likely not to be resolved quickly. But I hope it will be a more reductive guide to research on three manifolds than Poincare's question has proven to be. So clearly, a problem is seen as a tool, as a means to an end. It's not the end in itself. So here it's seen that the Poincare Projecture was hoped initially to be something that could act as geometry, but for Thurston at this point in the 80s he's saying no, it wasn't a good one, and I thought, well I hope it's a better guy, but the acid test is about whether it does in fact lead to a better theory of three man-at-lops. So the question is not, you know, the geometrization trajectory is not the end now.

20:00 And so, in a sense, it's going to be detecting, you know, if you've understood things properly, if you've understood, he's hoping it's a good sign, a good guide, and will show us that if we understand three-dimensional mathematics properly, we can solve this problem. And this is certainly not a rare point of view at all. There's a very strong focus on the extending of the understanding to be the key, the aim of mathematics. You see this in Atiyah's, Michael Atiyah's work. Michael Atiyah wrote, you know, at the end of the millennium of 1999, one of the prominent mathematicians invited to write articles, he wrote a very nice mathematics in the 20th century, very noticeable there, I mean, how does he go about talking about what happened, what he considered the good things of the 20th century, and what's staggering is the number of times he used the word story, or stories, 16, 17 times in 17 pages he used the word story. Another sign of this, this, this, so this is an idea I'm developing that to anything, anything, so this is a step I'm moving on, I'm trying to go with this idea of Thurston's that the aim of mathematics is to try to make the connection there to the notion of story, storytelling linking to the telling of better stories. Quite a good bit of evidence for this recently, Terence Tao, you know, the very young Australian originally.

22:30 He was also awarded a Fields Medal, very, very young. But he thought, you know, when people get given a Fields Medal, they think they should start to write about mathematics, to do mathematics. And he wrote a paper called, What is Good Mathematics? And he sets up, he writes a long list of types of good mathematics, of ways you could think about good mathematics. And then he sort of, you know, he sort of gives up on that, and then he writes a very nice story, a nice history story, of a certain Lema, with a very difficult Hungarian name. And it was rather rapidly taken up by Alan Conn, York field medalists, who, these people are blogging now, this is the future, all these field medalists writing blogs now. So Alan Conn takes up this article by Terence Tao and says how embarrassing it is, this list, you know, this is like an artist trying to write down what is a good painting. A horrible thing to do, you know, it's just not a good thing to do. But he says, but that was a good story. Okay, so we're thinking about these, you know, what sets, what determines, what dictates the way a field develops. These might be two extreme points, but it seems rather natural to fall into one or the other for some reason. So what one line would want to say, you know, discipline is, it's not that problematic in a way. Discipline is directed by... Rather straightforward and equivocal natural problems, and the answers to these problems are dictated to by the nature of the problems, putting an extra flourish on it, to judge by any rational being, and you can then... Absolutism. Sorry? Absolutism. Right. Rather extreme sort of rationalism of some kind. But I mean, okay, that's extreme, but there are people not too far away from this who certainly... You know, when I... A lot of times when I talk about... The need to think about the history, the tradition, the narrative, and all these sort of aspects, they say, but you know, but aren't, isn't it just really the problems just dictate the answers? We don't need to know all about this history.

25:00 And the other end, I mean, and I suppose, you know, that was quite a bit of the work of, that took place in the history of science at a certain period. Certainly some of the people I, colleagues I had in the HPS department in Cambridge for a time, very much this is their work, is to, is to show you how much effort is expended. It's important to make the problems look natural, to make the solutions look transparent, and they saw their role as historians to reveal that the political nature of the work, you must get behind it, and understand, you know, so behind the scenes, although on the surface what is taken up often by the public is a rather simple story of progress, actually there's a lot of work that's being done in making these little stories, in providing the support for these stories. And actually when you come down to it, I mean, to the extreme versions of this position, they're not going to find any rationality here at all. It could be down to certain interests of various groups. It's extreme form. So the question has always been for me of trying to find this sort of middle way, which I suppose one way one could put it would be like that, of maintaining a position which retains a sense of rationality for historically situated discipline, which modifies its questions and accepted answers by engaging in a socially embodied argument. Leading to improved understanding, including an understanding of the limitations of the earlier stages of the discipline. So I suppose it's something I inherited a bit from Lafatoche. I can't conceive of the rationality of the discipline in any other way except through understanding it as historically situated. And then start to wonder a little bit about the structure of the discipline, the social structure of the discipline that allows it to progress in a certain type of way. So recently I've been rather struck by an American way of thinking. Alasdair MacIntyre, a moral philosopher, wrote books like After Virtue, Whose Rationality, Whose Justice, Three Versions, and Moral Inquiry.

27:30 Who presents rather like what we had here. I mean, in that book I mentioned, three versions of moral inquiry. He's rather saying two of the versions are rather well represented, in fact, by those two extremes. The second one he called genealogical, and the first one encyclopedic. They're linked more to the Scottish encyclopedia Britannica of the late 19th century. Where life is quite simply, you just amass facts and in a straightforward math fashion laws will emerge which will be recognized by anybody rational. Yeah, yeah. And he's tried to develop a middle line again for his moral philosophy, moral inquiry, which is trying to, again, try to tread this middle line, something rather like that. Okay, and it comes along with quite a lot of things which I'm not sure, I'm trying to work out at the moment how far to go along with him. I mean, certainly that's fine. I'm happy enough with that. Inquiry is historically situated. There's a strong realism to him. And I suppose that's something... there's a realism I have as well, to a certain extent. Well, no, no, no, I think that would be on. About mathematics, I do have a notion of the carving theme of Jamie Cameron. Picking out from Frager, Frager was interested in carving concepts, hoping that his logical apparatus would carve concepts correctly. Perhaps it turned out not to work so well. So there's a side of me that holds to a kind of realism as sort of demonstrated by a certain kind of practical term. Typically the analytic philosophers start wondering where are these things and if they exist and how are they and they engage in a rather sort of blanket form of blanket realism or blanket nominalism. Either all mathematical objects exist or none of them exist. But I'm much more interested in the notion. You've been beautifully illustrated by Andre Bay.

30:00 There are two fragments that appear together in this collective work. There's a letter to his sister, and then there's another fragment of a letter. In one part he talks about working on the axioms of metric spaces. He says it's like a sculptor working with snow, making his shape too light. But in another part of my work, it's like carving hard stone. So there you see within the same mathematician, he's working with the distinction of something. There's a lot of natural, something of emotional kind of carving work going on. But sometimes you don't, you're just trying to do something that you can shake any old house. It serves a purpose. But you haven't got something intrinsically right. Right, so other dimensions of mathematics. So there again, I mean, typically there's a divorce of those people that want to focus on the people that think that you have to, that it's important to look at the social structure, the types of communication that are possible within a discipline. And that is having a role in the supporting of rationality. Your analytic philosopher typically has one. This is perhaps the more problematic. This third line is for him. I mean, yes, so that's good in the sense that we're looking at the approved understanding as the aim of discipline. He's got an Aristotelian notion that's coming in. You know, these debates that take place about whether one should talk about just moving away from where you begin, progress from. Must there be a direction or can you just have a pattern? Okay, but there's also, I mean, it's in the long line, isn't it? Going back a very long way, certainly Hopper. That is important. The aspect of rationality is to expose your weaknesses, expose your theories.

32:30 And particularly, almost in a sense, expose your weaknesses in terms of the past as well, require a narrative of the past that's led you to where you are now, but not a self-glorifying kind of narrative, not one of those narratives of, oh yes, of course everything had to lead to where I am now, with the possibility that, in fact, you should learn from the past, you should throw your contemporary perceptions into question, better history of the past. And that leads him, this is... You see, he seems to have been quite heavily influenced by that philosophy of science of the 1970s, the Laptosh and Bayer-Albert. So it's more rational to accept one theory or paradigm than to reject its predecessor when the later theory or paradigm provides a standpoint from which the acceptance, the life story, and the rejection of the previous theory or paradigm can be recounted in a more intelligible historical narrative than previously. An understanding of the concept of the superiority of one physical theory to another requires a prior understanding of the concept of the superiority of one historical narrative to another. The theory of scientific rationality has to be embedded in the philosophy of history. So that would raise a few eyebrows in an Anglo-Saxon department of epistemology. That's just reviving a line of Collingwood, when Collingwood says something incredibly similar. Okay, and clearly there's an awful lot to say about this time. I may come back to this idea of what sorts of history, how are we supposed to be writing here, is there a straightforward notion of writing of history relies on the philosophy of history, what history should be, how it should act, how it should write, do its work. So perhaps we'll have a little time just to have a few thoughts about the philosophy of history. I just want to talk though a little bit, so that theme though of MacIntyre, that one needs to think in terms of aspects of the organization of the social group, which is part of the tradition of inquiry, and one needs to think about communication in the group and between, you know, with rival groups and so on.

35:00 So I think this is something that's really excited me about, so I also run a blog. Part of the wave of mathematics blogs that's just extraordinary. It's really expanded, it's really taken on. I mean, exciting in a way because, you know, you can read Lakatos very commonly. He was complaining about, you know, all you get to see are the finished products. You get to see the journal articles and they're written. We have no idea where the concepts came from. You know, all the interesting and informal dialogue and stuff, we don't get to see them. Things are always rather difficult. You can pick it up in various places, but it's not there very easily accessible. You can pick it up with book reviews. That's one of the best places to go for this kind of material. So one chapter in my book, when I was looking at a debate, should we extend the group concept to group void? A small extension of the concept. It's very, very hard to actually find in print the size of the dialogue, about yes, definitely we should, it's the essence of symmetry, you know, it's a group voice. Another side saying, hmm, it's useful but don't, you know, it's nothing more than useful, groups really are the essence of symmetry. And then another side saying, no, no, useful is a waste of time, unnecessary, abstraction, pointless. And yeah, just very frustrating, you couldn't get hold of it. But now we're swamped with this kind of discussion, which is quite good in a way, and I'm sort of quite pleased to be promoting it. It's like, as I said at one point, it's rather like living in a Lakotoshian dialogue. You're one of the characters, except you end up being politer to each other. We don't have to be rude, we politely discuss ideas. There's also, yeah, sort of slight political aspects to it in a way, about the ownership of mathematics and why... Rather than individuals beavering away on their own projects, we have this idea of doing research in the open. Why should it all be closed off? Why not do it in the open? Or if somebody's written a paper, we can put it out there and invite people to talk about it, discuss it, all in the open. And also, it stops that great divide between the professional mathematician and the non-mathematician.

37:30 It opens those boundaries. Various ways, so that many, John Byers himself has always seemed to have had around some people that haven't been sort of standard professional mathematicians. So, but, you know, interesting when, you know, perhaps after some work, you know, you have to resort back to the journal article. But in terms of acknowledgments, it's becoming harder to do this. You end up acknowledging all the people on the blog, I guess. He claims that we're pushing a new way of thinking, a very N-categorical way of thinking about a large bunch of ideas in maths and physics. I'm very excited about this because I can see how much potential it has. We're also simultaneously pushing a new idea of how to communicate ideas. And the combination is actually really, really interesting. So I suppose what I've always wondered about is whether it was my choice when I chose about 10 years ago to start studying this material N-categorically. Is there a connection, if you like, between that subject matter that I chose to work on and this other interest of mine over the rationality of mathematics? I mean, it certainly was the case that John Beattie has always made it very easy for me to, or made it much easier for me to get engaged with this project just because he was always very keen to explain ideas. But possibly there's something actually in the very subject matter that has something intrinsically, intrinsic interest. In reducing things to their simplest form. So possibly it isn't just by chance that I'm interested in this particular program. So, yes. Right, we have these two lines. I sort of wear two hats if you like. I'm sort of participating in the project. And then I'm keeping an eye out of what it is to participate in the project. So like anthropologists who try and get involved in a certain people they want to study. Again, you try and keep a detached eye and look at what you're doing.

40:00 So let's think, I mean, something that's rather, I mean, it's possible that wherever you went, I mean, I could have latched on to a different one maybe. Maybe Alan Bond's non-commissative geometry. It's got a blog going. Perhaps if I'd chosen a different path, I could have gone down that line. And maybe we could have been doing something similar. But I have a slight feeling that there is something special about this approach. For one thing, it chucks up these rather interesting storylines. These templates that emerge in the writing of various types of history associated with this movement. I mean, the first one, recovery of lost information, that tends to belong to it when you're writing in a rather kind of mythical way. You know, you write about the myth of when we come, when we, when we, when our ancestors discovered numbers. You know, before they counted their sheep or whatever, or written. They've had some stones and they put them into correspondence with the sheep and, you know, through their various stages they come to realize that the number can stand for them. But you can see, I mean, you can see quite a lot of work in contemporary or 20th century combinatorics is actually going back against doing that. That just because you find the same, you're interested in structures of a certain kind and you count them and you can find that they have the same number, but that's not enough because you want to know there's an isomorphism between them. You really need to know that. And that's really important extra information. So we're getting away from just using abstract numbers to count, to think about underlying sets with isomorphisms between them. But that kind of story template, as I say, is rather often used in this sort of mythical guise about when our ancestors took a certain step. Other lines, we get the shadows idea that's quite common, that what we're doing... There's something that's existed in the past and they are just a pale reflection of some sort of projection of something that's larger. This is a sign we're on track. If we develop something and we find that a reflection of it has already been done, it's encouraging. So some of the principal results of Rotter's theory are auto-theoretic shadows of more general categories of theoretic facts.

42:30 Those themselves are very simple types of categories. Arrow going from one place to another. This is a common line we get as well, that you'll only really get to understand something properly if we put things on their right points, on a certain kind of ladder. It's rarely explicitly admitted that the Picard group is really a two group, so is the Brouwer group. The fact that I use one suspension in the definition of the Picard two-group and one taking an isomorphism class in the definition of the Frou two-group all indicates that the two-group nature of Picard-Frou is still not the whole truth, and even better, both fit into a certain three-group. There's a notion of a group, as most of you will know about. We categorify it, as we say, we come up with a definition of a thing called a two-group, and then we can take it a stage further to a three-group. And the idea is that you've seen fragments of these things around before anybody even invented two groups, and it's only by putting them in the proper context of two groups, and then even putting those together into three groups, that everything sits properly and you understand it, and you'll get it to work properly and you'll be able to develop the idea properly. There's a common line, the sort of, you think you're doing this, but really, which you retrospectively say to someone, this is... Now, you should think they're playing around with linear operations between vector spaces, but they're actually playing around with spans and group words. So it's like, you know, in a sense there's a ferment of little stories going around, and I suppose the history of mathematics in some sense is the solidification of certain storylines. Why does some storyline win out over others? So here's somebody like Baez, he's battling to have his storyline win out. And part of those storylines always will involve this sense that we can really understand something that our ancestors did in certain people's lives. This is the better way of thinking about this case against lost information. Yeah, again, very common. You notice things in terms of your ladder. You now see that certain constructions from the past occur on different points of the ladder,

45:00 and then you see that still there's certain similarities between things going on at different stages of your ladder. So then you get very interested in that concept because it's featuring in some form down at one level and also at another level, so we should expect to really push that one up the ladder because it's occurring at another level. So here's something, a very similar, this is an observation due to Bill LeVere. This is a certain condition, something that, some sort of commuting going on in predicate logic with an existential quantifier. But it's really the exact same structure which you brilliantly observed that's going on in group representation theory. And then you try and push it up, as I say, you try and go higher up the ladder. It's also, yeah, it's got a sort of strong heuristic sense. So there are things called bundles, there are things called gerbs. There's a sort of framework in place, and it's not an algorithmic process by any means to bring yourself up the ladder. You're constrained in many ways, but certainly there's certainly some freedom there. And yet there's sort of a feeling that when you get it right, although all the bases are different ways to go, You'll know when you get the right label, but it's certainly not an automatic process. Okay, so, Boas in particular has developed this line. You know, just as you have gauge theory for particles, all about fiber bundles over spaces, you need to do something new with strings going through space, and you need something beyond that for two particles. And it's making rather nice connections back to the stuff mathematicians were doing anyway, in terms of much rather reassuring. This was the project that got the whole thing going, really. I thought of a topic to... See, I tried to adopt this line of... because Biles and colleagues worked on this idea of categorifying arithmetic in sums and seeing how you could understand those as shadows of something that are up. So I imagined somebody, the opponent, the extreme opponent to this program saying, well, okay, perhaps you've done it for arithmetic. Yeah, but what could you possibly do for geometry? Given that you've written quite a lot actually already on Klein geometry, so just a little glimpse of Klein geometry, Klein's geometry from the 19th century, seen in the modern guise, so imagine you have a plane and you've got some Euclidean root acting on the plane, then the stabilizer, it acts transitively, so the Euclidean transformations can take any points of the plane to any other point.

47:30 The stabilizer of a point, the transformations that fix a particular point, are O2, the rotations, the reflection around the line through the point, and then when you factor out that boolean group by these rotations, you get cosets, and they are isomorphic to the space of the point, because isomorphism is R2 in this case. So, yeah, I mean, client geometry from the modern perspective can very much be seen as that sort of thing where you have continuous groups of e-groups acting transitively on some space, and then you're interested in sub-forms of a certain kind. And they're going to fix various figures in the original space, and by fracturing you get the space of those shapes. And then you can look for relations, say, between lines and points in your R2 and understand the ways they, the incidence relations between them. So the natural step was to push it up, up a level. So that's something that we got going when I started a blog. And we were tossing about ideas there, trying to do this research in these two groups, trying to look for sub-two groups. And that's when the idea came to me. We can do it in a better format here. We can support mathematical writing on the blog. We had more intense sort of discussions going on there. But again, you have a construction there already. You've got some pretty strong guidance about how you lift it up with that. But it's not easy. I mean, we still haven't really solved this, actually, after two years. I mean, again, classic stuff. Actually, when I started, I started off going back to that group point. I mentioned that chapter in my book on groupoids. At the time, I was just going to study in terms of rhetoric, it really was going to be a study in mathematical rhetoric, you know, how does one side present their ideas as better than anybody else, with no great thought that any side could be right.

50:00 I mean, not all the reasons a theorist gives for a change in their theory, unless there are good reasons, but I still think there are good reasons for some changes. Now, this is a classic case that people are always doing this kind of thing. You want to suggest that there's somebody, you're doing something sort of significantly new, but there's already a heck of a body of work out there that you're just, you're in some sort of close relationship to. Here is our eight categories. A special, very, very special kind of them are these ones where everything is reversible in a certain sense. These are the famous omega group points. And this is really what Paul Grote indeed got going. He suggested in the 1960s that instead of topological spaces, why don't you think of them as omega group points? Things where there's... So if you have a space, you can take the points as the objects, and all the paths between points as the morphisms, and then all the paths between paths as the two morphisms, and up you go, and that's a capture of that whole space. So he's hoping algebraically to capture the notion of space with his own frequency. Okay, and here's the thought-advice that very commonly you're struggling away, and you find that the very special type of ink category that the homotopy theorist is working on is, you know, we can use them very frequently. Okay, what do we have left? About ten more minutes. I just, I mean, it's all very, you know, so as I say, they could have just, a lot of these things could just become rhetorical flourishes, and even when they're invoking history, you know, many of the little stories they're telling about certain moves that they're making involve telling the story of quite a recent story, but there's the classic case from... Telling the story of, going back to Inerta, you know, the idea of how do you know that there's no homeomorphism between a forest and a sphere.

52:30 And starting off just by associating numbers to these things, that didn't work very well. You put a 2 on here and a 0 on here, that didn't work very well. But if you were allowed to have vector spaces... There's a story commonly told by the category theorists to say that they provide you more information by not just staying down the level of numbers, but by moving up to modules attached to a space. Now you could say, well that's just bogus rewriting history, that's got nothing to do with... Relating to anything like real history. They're certainly concerned, people in this movement, with what they've gone about and actually written some histories you can take a look at. Ross Street is an Australian practitioner who wrote it very much from an Australian point of view. Probably a little bit chronological in some ways, you know, somebody did this and somebody did this and they thought about this and then they did this. Again, I mean it's not a very historically sensitive but I so that we've had some discussions and we need to have more discussions about about the writing of this kind of history and if we take this idea of mathematics seriously then we must write good histories, we must write true histories, we must write true histories because we need to learn from the past because the past should upset us sometimes, often the past should disturb us. So we ought to be reading the work of historians and mathematicians. They precisely should disturb us. This is the gloss on his history, marking out the difference. We are scientists rather than historians, so we're trying to make a specific scientific point rather than accurately describe every twist and turn of a complex sequence of events. We want to show how categories, and even any categories of study, come to be seen as a good way to formalise physical theories, and which processes can be drawn as diagrams.

55:00 Okay, we'll allow them to write their kind of stories, the mathematicians, and then we've got the professional historians and you'll do proper history and mathematics. And there's really not much connection between them. And we never take, we never look at the history that these people have written and take it as history. Obviously we take it as an event in the history of mathematics, the writing of their stories. Say that there really is a gulf between them. There's proper historians of mathematics and histories, and then there's mathematicians' histories, and they shouldn't even be seen as history. Give them a new name. Call them heritage. It's not that it's just a different thing. A friend of many of us here, Leo Corrie, made a rather similar point in a recent paper. He there actually wants to put in these mathematicians' histories and put them into association with mathematical fiction. There are those, and then there's the stuff that professionals do. The thing that has been, which is the singular, the idiosyncratic, is the object of historical research, and historians should strive to understand and convey it in their research. The thing that might be, while a more philosophical and a grave report, is none of the historians' professional business. So there's a very strong, again, dividing, separating of these two activities. Which somewhat troubles me, because I do want... I do want this, to the extent that the rationality of a program requires some historical awareness of the path of that program and a disturbing, the possibility of being disturbed by finding out new things in their history, I'm slightly worried that we're not going to have that opportunity if this divorce, the two-state solution suggested by Gratt and Guinness and Corrie prevails. Just going back to MacIntyre for the moment, I've only got a few minutes. He was going through his, so you remember the encyclopedia, that was his, the hyper-rationalism, positivism, and the kinds of narrative structures that they tell are one dictated by belief in the progress of reason, which often involves the denigration of the past and appeal to principles purportedly timeless, so the narrative reduces the past to a prologue to the rational present, and one knows that kind of history writing, that kind of gloss to the weak history, and it seems to be that kind of writing of history that seems to...

57:30 It's only informing that decision of the Black Bracket Beginners, isn't it? But he does go on, Thomist in a sense of the book. He sees himself. The Thomist's narrative treats the past as that from which we have to learn if we are to identify and move towards our telos more adequately, and that which we have to put to the question if we are to know which questions we ourselves should next formulate and attempt to answer, both theoretically and practically. So this suggests that there certainly shouldn't be two totally horribly divorced fields going on here. At the very least, mathematicians ought to be reading the work of the historians. That would be good. But whether there are differences, it would be interesting to think about differences amongst the historians themselves. Historians are not just one monolithic block. There are variations and differences in your philosophy of history. And how those differences play themselves out in terms of what they could say to the mathematician, I think, would be an interesting thing to think about. I could be going on rather long enough. Yeah, just to end with the kind of questions that I see myself working on, really. So, I mean, I think it's utterly essential that we can't just see contemporary mathematics as this rather featureless landscape that... It seems to be an idea that prevails, but it's really ridden through with these very strong storylines. So what's special about... I mean, there's a question. So what makes them a mathematical story rather than a story of physics? A story of physics, I think, is the question. The question's about good. What is a good story? And why do certain storylines win out over other storylines?

1:00:00 Must we be relativists? Can we not find some historically situated rationality? It's certainly kind of interesting to see how the storylines change. What counts as a good story certainly changes in time. So is that what, from the 70s, you used to talk about learning how to learn? And then these dimensions of the community. I really genuinely hope that this promotion of the blogging is doing something good. It's changing the communication possibilities within the community. That's a good thing. Questions about rivalry. What should you do with people? To what extent should one just... Yes, because often one... I'm sometimes seen by mathematicians as trying to drive them into conflict. I mean, is that just a left-over from 1970s philosophy of science when rival paradigms or rival research programmes are all battling with each other? Is that just a left-over of that? I mean, is it... It's certainly a lot less... It's a lot less likely. We shouldn't really see contemporary mathematics as very simple schools fighting against each other. It's fluid than that. But is it too flimsy? Maybe you could even say they're altered to be a little bit of a school-like structure for the research community. And yeah, five, these relations between mathematicians' stories and historians' histories, there's a lot to think about there. I think that's only the first step to sort of what Graf and Guinness and Leo Corrie have said. I think there needs to be a lot more said about that. We need to come back and think about differences between historians even. And then notions, yes, again, that needs to be worked on. What is the understanding? Does it tend to relate and tend to involve the notion of narrative? And then how do these programmes, how do they give forth these templates for new storytelling? And then how do they get settled? So these are the sorts of questions I work on. Oh, I just ended with a little plug there for my, so I just started working at the University of Kent and we have a Centre for Reasoning. The trouble is, what isn't reasoning in a university? It's kind of everything. But, I mean, it's kind of all, it's only philosophers.

1:02:30 I'm trying to bring this historical side into it. At the moment it's going to this formal direction, formal philosophy of science. Yanking back in a slightly more historical direction. But anyway, we have it. Okay, so thanks.