Reflections on category theory

0:00 What about death, right? Well, it's got to be tempered, right? Well, I mean, nothing, everything has to be tempered. You know, you've got to, you know, because we're just saying this, and this new notion of looking at the world of, you know, actions in a dynamic way is a new way of looking at it, and granted the tools, intellectual tools, that provided, allowed us to look at it in a different way. so you know the notion of function and that sort of thing do have a peculiar genealogy but that doesn't well no there might be that's an intellectual position no i quite agree there might be several ways of moving away from this primitive notion but the primitive idea where it comes from, is what I want to call symbol-generated. It comes from our notation. But, oh, look, art and pair, that's true. No, it's certainly true. Oh, I think it is true, but... What do you think it means that it's true of art and pair, in the sense that what we're trying to capture is some notational and formal, fix the well-defined way. It's just what it is is different between picking out something and distinguishing it from something. It's the notion of something occurring in an artist. Yeah. Arduring is one instance of it, or, you know, relations of heights or distance of power. there's lots of yeah i mean there's lots of questions yeah but but but the actual notation yeah the original idea comes from cartesian geometry and these well there's are in that respect i mean of course there's the first and a second or a beginning and an end yeah or large and a small or whatever but the point the point is that um The original use of these things in mathematics was as labels, and as such, they're perfect, they're quite straightforward. I mean, they are quite straightforward, these syntactical things, except that when they were introduced, there was no sensitivity about syntax.

2:30 people didn't make these distinctions um so i mean for example when you write down a sum you have to put something first and something second so uh the idea that there's a first term and a summation and a second it took a long time for people to realize that and you know i mean i suspect it It's sometime, it was sometime in the 19th century before people noticed that you had to say that A plus B was equal to B plus A, right? Because, I mean, you have to, when you write down in linear form, it comes in, it comes in, it has to come in some order. So the union function, for example, operates on sets. And that's really what salvation is, it operates on sets. So... Because you can imagine, people who propose a new view of something could take a notion from rather a disreputable source, but it doesn't detract from their view that they then create, that they've taken. You know, the category, that McLean says in Category 3, but how do you take a notion of category from whatever it's from, and Functor from another area and things. And I think Functor comes from Carnap or something, isn't it? You could say, oh, well, listen, that's kind of, you know, dodgy. Category comes from Aristotle or Kant, depending on... Yeah. name but if you suppose you said it's the mike carnapp and and font and thought there's such a news team and you know the genealogy of it's kind of dodgy but that doesn't detract from the notion no that's that's that's that's not the genealogy of the concept that's the etymology of the name which is quite different but they claim that there's some sort of you know there's a connection there but that's just an example i mean you can imagine lots different examples That's where you probably get in American politics where somebody says, well, I think we should put more money into Medicare and the extreme right will then come back and say,

5:00 well, this is socialism, this is kind of another communism, the Russians tried this and it failed. Well, that's a ridiculous extension of genealogy. Yes, I quite agree. That's right. it's contemporary but what what what what uh heidegger has to say about it is actually quite interesting there's a long sort of paragraph on this idea he it's he actually uses the german word destruction he's so he wants to talk about the destruction of a concept and of course there's a German, a perfectly good German word for sestorium, which would be the word, German word for destruction. So he's deliberately used this Latin-based word, which doesn't mean destruction, I guess in German. Or maybe not, I don't know. I don't know. But the French happily came up with the alternative deconstruction. I thought deconstruction was the best search. sounds like generally it has to be destructure rather than it could be well that's what destruction that's what destruction means yeah you're taking down something built up um yeah but it doesn't have the sort of um no but i mean destruction of the concept just seems odd it likes like say you take a hammer to it or something that's because we attach concepts like destroying we've made it synonymous with destroyer, which you never should have. Well, it does come from, they're all from the same Latin root, I dare say. Deconstruction tends to have more connotation of being honorable, whereas destruction just means anything that ends up is not that structure in the end. But what he says about this is quite interesting because he's talking about tradition and when you're deconstructing concepts you're deconstructing the traditional notion of this concept and what you're doing is tracing it back and he says if you don't do this these notions that you haven't deconstructed just impose themselves on you as first principles you're convinced that you know they're just as it were

7:30 or just arose out of the nature of things or whatever. And this way, you can sort of disentangle yourself from a tradition that's really blinding you to what's going on. I mean, I think myself that the classic one is number. Yeah. Yeah. I mean, I'm sorry. i think the history of that the history of that concept is actually relevant and it is true that it that the the way it's if you look at the way it's been the tradition has been pat down that has fixed these notions in people's mind and so they can't think around it but then you've got to distinguish that between the aim which is to disentangle yourself from the web of tradition And the means, or one of the means of doing that, which is by looking at the history of the idea and seeing that it's not just part of the furniture of the world, but it was actually constructed. But if there are other means of doing the same thing, of getting to the same result, of realizing that this isn't just a natural and unquestionable assumption, then genealogy isn't the only way. No, it's not the only way. It's just a plausible. It is a way. I mean, because people in the more analytic traditions of philosophy who are inclined to think, well, nothing counts but analytic arguments. But, I mean, you do get down to a situation like we arrived at, where you say, somebody says, well, I might as well take the notion of function as primitive is the notion of sense and well i can't say that the genealogy there is is decisive but it actually does provide information uh it does provide you with a perspective on the thing that's different from the one you get just by just by asserting that sort of thing if somebody says well you know the notion of graph is just as general as the notion of set well I'm sorry I just you know I don't see the analogy you know is some the analogy is

10:00 that you can give an intuitive picture of what a you can draw dots on the board and say this is what a graph is that's our basic idea or you can enjoy the up your five fingers and saying those fingers form a set, which is what you did, right? Of course, there is a rough analogy there, but in order to understand why the concept of graph isn't foundational, where the concept of set is, you've got to think really carefully. And the problem with these guys is that they just grant themselves the whole of logic in in order to describe what they're doing without thinking that that has connections with you know that that in turn has to have foundations I mean maybe I shouldn't say the whole of logic of course we've all got the whole of logic in the sense that we're reasoning logically I mean the whole of formal logic that's the that's the crucial difference Yes. So if Terra Impossibile was a foundation based in graphs or labels, what would it have to achieve to merit the name of a competitor to the set foundation? You'd have to be able to reconstruct, you'd have to be able to reconstruct all the ideas of the foundation in sense in this competitor. Yeah, well, that was Laubierre's idea, that you have this theory of the category of categories, it's a first-order theory, like set theory, and there's some kind of translation procedure. or you translate the essential things or that was the original hello yeah oh hi Andre did you get the message on your blackboard well you haven't come in I take it you haven't come in today Okay. I went into your office and left the message on your blackboard. Never mind.

12:30 I was hoping we could get together this afternoon. Are you around? Well, it's possible. I'm still with Stuart and Richard, but what time are you talking about? Yeah, that's okay. Why don't, why, no, come by my house at 2. Okay, I'll see you then. Okay. I've had time to look at that thing. I've got a lot of suggestions. So I'll see you at 2 o'clock then, at my house. But, yeah, yeah. All right, I'll see you at 2 o'clock. Okay. That's all good. Oh, damn, it's already locked. What were we talking about? What did you have to do to make some... Oh, that's a foundation. Well, that's... See, those original remarks of Walbeers from the prehistory of this stuff, back in the 60s, and McLean. I mean, they just thought that you had to do something, you didn't just produce the appropriate first order theory right right and of course that's just nonsense and i think wallbeer thinks it's nonsense now yeah right this festival theory is already you've got to give some kind of story about what a first order theory is but uh and uh i mean wallbear has done that since the 60s actually i mean he's got these his versions of he's got these this category theoretical version of logic in which it's sort of the logic is just mathematical entities like other categories logical systems but I mean that may be subject to the same kind of objection I mean, what's wrong with the idea that it should be a first-started theorem? First-started is sort of a very loose sense, which is, you know, you can kind of tell when your intuitive quantifiers are just ranging over objects and lots of other... Well, if you want... ...hider or other things, then... If you want... I know, but if you want to say...

15:00 I mean, it's perfectly okay, I think... Well, I don't think it is from a point of view of aesthetics or methodology. But if you're just using first-order language as a kind of method of abbreviating what you're saying and so on. But, I mean, what comes in? So that's just a matter of vocabulary. So you could think of those as just, you could think of your upside-down A's and backward E's as just abbreviations. that's not first order logic that's just that's that's what makes it not formal so what's the problem with this cat's great cat's great maximization of it so if it isn't first that looks like first order formal then we can just create as a non-formal system well you might be able to do that but the point is that it's not what he tried to do that's not what he and mclean tried to do well that's a bad historical error but it doesn't mean the whole idea then so what they produce what works well if i remember mclean was trotting out something called the axioms for a well-pointed tacos don't ask me what that is well that's what it was that was going to be the the ur theory of everything i mean but the idea was that making putting it in a in a first order doing a first order formalization of it made it respectable in the same way as he's because he thinks just as the first order formulation of set theory makes it respectable well of course that's just nonsense yeah but that's irrelevant to whether it can be done oh whether you can come up with a um a set of first principles yeah that just talk about Yeah. So you could, presumably, you could work backwards from the first order theory of sets, you could then say, we know how to understand this in a non-formal way, and we're just going to write some informal definitions of what a set is, which will just happen to mirror these first order definitions. Yeah, but I mean, that's not the way. I mean, that is a reverse of rational art. have a notion. I mean, you can write down axioms, but really, if you're talking about

17:30 axioms in the Euclidean sense, axioms which are first principles, then you come up with the axioms first, and then you can write them down in a formal language if you want to. But it doesn't improve, or you can try to. There may be unformalized, formalizable bits in the axiom. Yeah, I mean, sure. Whenever anyone writes down first-order axioms for ZF, they are thinking of it non-formally. And, of course, if you then go on to prove a consistency in... I'm sorry, and you prove sort of independence results, of course, you've got to treat it, but no one actually ever thinks much about it as a... I mean, actually, everyone just thinks about it in terms of models, so... Even then, it's not. about models than your inside set theory. Yeah, absolutely. But, you know, my point is that nobody ever starts with a first-order theory. Everyone starts with their non-formal theory. And so these guys, even if they wrote down a first-order theory of their Kespier-Kespierre, they were thinking in the non-formal sense. I don't think they did. You don't think in terms of first-order life, because you obviously think in terms of intended interpretations. No, no, I think they did think in terms because the is that you can dismiss all talk about models because you've got the notion of logical validity. So you're only interested in the logical consequences of these assumptions. No, sure. Imagine they're writing them and acting. They're thinking in terms of an intended interpretation. Yeah, but the idea is that when you then formalize it, then you've made it legitimate somehow. Yeah, but that's what I mean. you don't have to have that extra view you could stop short of that and just say right we've got this one from them we've got to assess whether that's any good but then that's the real question yeah but of course they never came up with such things and yeah i mean then if you do it informally like for example if you started to axiomatize the reals by saying well we've got a notion of real number and we've got a notion of multiple adding and multiplying real numbers and we've got this notion of less than or equal and we've got two real numbers zero and one which play pretty important roles in respect to addition and multiplication so you write all this stuff

20:00 down and but then you get to thinking the problem is where does all this come from Now, then you'd go back maybe to the Newtonian idea of Reels' abstracted ratios, and you could sort of make a plausible story out of all this. Yeah, but that's what Frege tried to do in volume two of the Brunegesetz. But that's true of reals, but it's not so clearly true of the category of learning. But the real question is this. What are the fundamental things that mathematics studies? Are these sets bare sets without any structure on them? Or do you come to things, are the primitive things that you study, the structure initially? abstract structure, just a set of objects with a morphology, is that all bound up together? The set theoretical point of view takes bare sets as the primitive thing, and you sort of build up the morphology of sorts of sets of sets, and so on. The category theorist takes the thing structured already as the basic thing, and in the category theoretic sense, the way you discover about these structures is you just use the membership in relation to take bits out of the morphology and take bits out of the set and talk about them. In categories, you don't have a set like that. You've got the structure already in. How do you find that thing? Well, you apply functions to it. You do things to it. So this is a different worldview. And the question is, which of these two things should be our fundamental metaphysics? Except the problem is, of course, again, the genealogies is critical here, because the notion of structure from the original analysis of these theories in terms of structures like groups and rings and things and these were the examples of structures there wasn't any sort of primitive notion of structure but yeah so you couldn't it just did that just isn't intellectual sense I mean the highway I considerable intellectual sense it seems even plausible in fact that we come to things things in the world are already structured we don't come to things just as bare sets this has been an obvious point

22:30 you know, it's too much you know, this analogy to do with vacuums it's quite a lot of energy to suck it it looks like you come to things already ordered I don't come to that set of books it's a bare set of books but I come to them in the order that they are on the shelf well, books are more different from there left to right or right to left well, as I perceive them in a different way I can specify one or the other i don't start the books and then impose an argument no no i agree so i mean this looks like you've got two competing worldviews one is the category theorist worldview and one's the set theorist worldview and how did you know i don't think there is such a thing as the category theorist worldview that's my point and there's no such thing as the set theorist worldview either because the set theory isn't providing world views well it is you've got that It's taking... It's taking... It's taking... It's taking... It's taking... It's taking... It's a different kind of thing. It isn't telling you what mathematics is about. It's a component in a general theory of what mathematics is about. And maybe the... It's the most... least problematic aspect of the... of... of the... of the explanation. And I think part of the general account. I mean, it's got to be integrated into the rest in such a way that we can see these things as part of a general account of how we're doing mathematics. Of course you get a different idea as you make advances. I mean, these things that we were thinking about, like these abstract category or category theoretic definitions like products and co-products and limits and so on. This is a refinement on the older, sort of sloppier idea, that you're only interested in things up to isomorphism. So what makes these constructions legitimate it is that they are uh unique up to isomorphism otherwise you couldn't talk about you couldn't talk about the product right so you had nine isomorphic things satisfying all those arrow

25:00 diagrams then it wouldn't be it wouldn't work right so it's it's a it's a more sophisticated subtler and more useful i mean it's taking this sort of informal idea that well you know sort of mothering, well, these two things are isomorphic, so they're essentially the same, but, so on. And, well, we don't care what the elements of our structures are. We're only interested in our form and so on. It's taking these rather crude formulations that mathematicians used and making them precise by introducing this beautiful new theory which incorporates a kind of generality which hadn't been witnessed before. Because now you see that products are the same everywhere because they're really, the essence of them is in this product diagram, right? Okay. So that seems to me, that seems to me straightforward enough. The problem is that the problem is precisely a set theoretical one. What is the status of these large categories? And wouldn't it be nice if you didn't have to make a distinction between large and small? But as long as people have been working on this stuff, nobody's seen a way around it. And I think all the kind of polemic against set theory has just obscured what the real issues are. I mean, it's not that... I mean, I think, well, Veer's idea is that, his central idea on this, for example, on the reels, is that what's wrong Kantor and Bajostas' theory of reals is that it's discrete somehow. I mean, it's the idea... it's an arithmetical count of a geometrical notion. That's what's wrong with it. So, the really deep question here is can you get some notion different from that of arithmos?

27:30 or sat, on which to ground things, or, you know. And it's not clear to me, A, that anything would, any benefit would accrue if you did. Maybe, maybe it would. I mean, maybe you could solve this problem of large and small categories. But that may just be an inherent feature of the world. There's a difficulty about that. and there may not be any we may just be up against an intractable problem I don't know but I mean to pretend that the notion of category could take the place of the notion of set is just crazy but why have I thought we'd just given a reason why it's so good it may well be possible no because you said you could start with the notion of a I mean this is and functions, other than sets and membership. Well, what do you, um, what, well, I think there's a really serious problem about what you mean by function there. What exactly, this is exactly the question. Is function affirmative? No, I don't think so. Well, you may not think so, but then, you know, we're talking about what Peter and Wiles was here. Yeah, the argument. You're talking about, you know, and these are ancient problems. Well, actually, the problem about functions isn't an ancient problem at all. This is the point, that the function, because you've got this obsessed with the genealogical account, you're not seeing that the notion of function is simply a part of the question of the problem of change. Yeah. And if, you know, if you can see it as that instead of its technical usage, I mean, I don't know, basically, the function is just the common category theory, precise and mathematical usage of something that's... So, in canon... No, I think what you've got... It's like a fact, as in, frankly, the precise notion of... So, the categorical idea is about objects and transformations of objects. Well, actually, why not use the word that the category theorists use, morphisms? Morphisms, yeah. So, the question of what a morphism... Sometimes morphisms are functions, sometimes they're not. Yeah, exactly. So, but amorphism is an idea of a correspondence, somehow. Yes, an action on a... From here to here. From here to here, right?

30:00 Okay. Now, that can be evaluating a function and an argument and ending up with a value in here. Yeah. Or it can be other kinds of things. Okay. Absolutely. So, okay, I'm open to that, but the question is, is that precise enough to... I mean, is there a precise enough notion to produce axioms? I don't think so. I think the axioms for categories come from the original source of them, namely these concrete categories. It's, as Volver says in that quotation we just read, it turns out that if you start thinking, instead of thinking in groups, instead of thinking about groups and their members, about morphisms between groups okay see the essential importance of morphisms i mean this the the central importance of morphisms is much a very complicated business and it was only only you know dedican sort of began to see it and nurse it was probably the first person to really understand it and to you know to see what was going on why you know why for example when you studying groups the study of group formomorphisms is absolutely central because there's a lot of stuff in group theory that doesn't seem to have anything to do with that you know i mean um studying finite um fermentation groups you don't seem at first it does it's not clear why these things come up i mean i when i was teaching stuff i used to use the metaphor saying that it's that uh that these morphisms are like it's kind of like the employment of x-ray it's kind of like x-ray crystallography you're shining uh when you take a morphic image you're sort of shining along on an axiom of symmetry so that a lot of things get turned up on the photograph you get one dot on a photograph. There's a whole line of things in a crystal. Right, yeah. And so the idea is that if you take enough shots from enough different angles, you can reconstruct the structure of the crystal. I mean, all these sort of simplified... Because a homomorphic, sorry, an epimorphic image of a group is a kind of simplified picture of that group. You know, you're taking the things in the cosets of the kernel as being

32:30 identified for the you know you're sort of saying think of these things are all similar in some respect the things in the kernel are all similar and that they're behaving like zero right you know so so there's this whole sort of very rich heritage of why these things are important and it just seems naft to say well we've got this i don't think we have a primitive notion of of morphism of that sort but maybe there's that is an instance of a more general notion of morphism and that's kind of what lord here is hinging at when he talks about when he talks about menha having coherence in the way that sets don't And all that is, I mean, I don't react unfavorably to that. I mean, he's saying all the right sorts of things, but unfortunately the details are not there. And there's, I mean, we go back to Aristotle. Aristotle contrasted geometry and arithmetic in the following way. Geometry is primary for us, but arithmetic is primary for its simplicity. And the reason is there's more, more of the features of the things in question have been ignored in arriving at the notion of arithmos. And that's perfectly true. If I take, if I want to talk about the set of books on that thing, I forget about everything about them. I forget about their colors, their sizes, their locations, everything. I mean, so I've got just something really sort of thin. I mean, correctly, if you think psychologically, you're thinking away certain things. There's certain things that are just the primitive experience of looking at that bookshelf, and the arrival of the concept of the number of books on that shelf, the arithmos of books, does involve thinking away

35:00 connections between them that are irrelevant from the arithmetical point of view. So all Well, it's perfectly true, and Laubierre calls attention to this, he's got, you know, he talks about the coherence of this stuff. So maybe there is something there, and maybe if we, if we, if we could think about it in the correct way, we'd see that there's a more general notion of which arithmos is the sort of one extreme end. Presumably a category theorist is going to say the more general notion is a top class. But we're just, at which science is one example. Yeah, except that, what's a top class? And you, if you ask what a top class is, then you've got to give a conviction. Well, the straightforward definition of it is, yeah, it's a category that has these particular features, but that doesn't mean that that's the only way you can do the definition of it. But if it turns out that there's some definition of it that comes via primitive and intuitive ideas, then... Well, I mean, John Bell and I, that's what John Bell and I was spent all afternoon talking about. Because John was wondering whether, for example, some kind of looser version of set theory, or less, a more liberal version of intuitionistic set theory, as we were talking about. And there's a lot of work that's been done on this volume that doesn't come from category theory, but comes by Martin Loeff's stuff on type theory. Peter Axel has done a lot of work with, so there's interesting stuff. But, again, the question is, you want stuff that's brutally simple, and so simple, in fact, that you can't think of any way of saying it. You can't think of any way of describing what you're talking about in simpler terms. Now, that's manifestly not the case with morphisms and categories because you can describe them in a conventional axiomatic way. It's just that you want to include these large ones as examples, where it would never occur to you

37:30 to include large groups as examples of groups. But strictly speaking, if you're worried about these things, perhaps you ought to. No, you weren't. you remember that's kind of you see it's great right there's anything with science what's that it's another one science oh no well I mean the point is that if you if you are familiar with questions concerning foundations and the role of set and so on in those foundations in some sense you've known sin or you've eaten the apple of the tree of knowledge of the distinction between good and evil and so on so you're not in a sort of you know pre-lapsarian state that's required to be to know what's next you know i don't know what i don't know how to say what natural is but i know it when i see it right so so um But nevertheless, it's implicit in, you know, in elementary group theory, I say, a group is a set with an operation. Now, as I say, in this prelapsarian condition, you aren't aware that there's a difficulty with the notion of set. And there are notions that seem to be close to it from it. on a kind of conventional view of set theoretical notions, then it would be natural to consider, say, monstrosities like the direct product of all groups or something like that. And, you know, would you say to a group theorist who is proving something, you haven't you've forgotten to take into account this particular you know this monstro group the direct product of all groups well i think it i mean it's the the idea of it just wouldn't arise well because you know proof is the general results of that yeah but of course it's problematic because of you know it's all big you know sort of keeps on generate things like what a small group is a group itself and so on and so forth but

40:00 if you got a large one that wasn't in some way um self-referential or didn't include itself in the you know certain over which um it was making a group and there's no problem about it will be included if you've got a notion if you've got a notion i mean if you've got a notion of concrete concrete category. It was the sort of thing Bourbaki was looking for in the definition of what a structure was on it. Okay. If you've got a, if you had a notion of concrete category from the bottom up, a complete notion, like if you bought Bourbaki's analysis or whatever. If you had that, then the problem of large objects would continue to come up, Because an abstract definition of structure is going to produce, as part of it, a completely general definition of structure type. Where, for example, by structure type I mean there are structures of group type, which are sets with binary operations. There are structures with ring type, which are sets with two operations. And there are structures of field type, there are structures of topological space type, and so on. So you've got this most general notion of structure, then you have a completely general notion of what a structure of that type was, and then the temptation would present itself to include, amongst things that were structures of that type, things for which the equivalent of the underlying set of a group would be large. And the point is, what I want to say is that these questions about the logic underlying set theory are absolutely crucial here, because I want to say that there just are no large structures. There's just no such thing. what causes the problem is the is the idea that what i call species and what are generally called proper classes are regarded as things of the same sort of a sort that would be suitable for

42:30 underlying for interpretations of these general right okay so groups springs there would be large who could impose a topology on the... Okay, the trouble is, that's interesting, because even maybe you could make sense of the first over things, like a large group, like this monstrosity of the direct product of all groups or whatever. But what you couldn't... So the underlying... the underlying set of that would be what would it be the underlying set would be all maps all maps defined on the class of all groups such that the value of a given group and a given group lies inside that group and that's it so I mean this is a monstrosity well it's presumably impossible Why? Well, because it's self-referential. Well, this is what I'm getting at. My view on these questions of large-scale questions about sense is that there are no such things as proper classes, and the consequence of that is that you really, if you take that in conjunction with the idea that whenever you've got a structure you can always put a topology on it or you can always talk about the substructures of that structure and so on if you take those two together and i mean one way of taking that second restriction is saying where one consequence is that wherever you've got a an interpretation of a first order logic you've got the interpretation first order language you've got, you've also got an interpretation of the second order language associated with it, or the language of order only associated with it, or whatever. Okay. So there's a kind of uniformity principle for logic. So wherever first order logic applies, second order logic. And there's this other thing that these, a uniformity principle for these set theoretical

45:00 constructions. So whenever you've got a set, you've got a power set, which you need, okay. So what happens then is that the logic of, the logic that applies to the totality of sets cannot be classical first-order logic. Cannot be, isn't it? You know, from those two seemingly reasonable principles i mean if somebody if somebody were doing uh if you've got a large group for example then you haven't got it unless unless there's a unless the class of all unless there's a the class of all classes proper classes is at another level in the hierarchy right, then you can't talk about the subgroup structure of the thing, because you've got the quantifier over all classes of elements of this group. Okay? So you don't, group theorists don't say, well, of course I'm talking about groups whose underlying sets have power sets. now let's consider the ones that don't i mean nobody ever talks like that right i mean the whole point about set these set theoretical principles is that they're ubiquitous you can apply you can apply them anywhere and you don't have to be self-conscious about it and you don't even have to think about it actually the whole point of the group axioms is that they're also applying universally and it doesn't matter whether it's a set or a class that's your underline If something satisfies the group axioms, then it will satisfy you. Well, then, but then you're up against this problem of these uniformity principles, which are violated by that. Because a large group doesn't even have a subgroup structure. Because you've got to quantify over all subclasses of the class of this. Right. So, okay, maybe that's okay. stop where does where does this business of trying to make a set out of something that isn't a set stop well i mean you can do it by getting increasingly large universes yeah but then

47:30 the totality of those was what about that yeah okay so you always have that set theory runs up against this problem category runs up against this problem we know that there's a problem here that trying to crush things down. Yeah, well, what I'm saying, what causes the problem is supposing that you can apply conventional logic to this domain. You can no more apply conventional logic than you can apply conventional set theory. No, right, but... And so that's the problem. And so these large things are something quite... They're not things, even. But I don't particularly see how this is an objection against category theory. No, no, it's not an objection against category theory. What it's saying is that the difficulty, the particular difficulty that category theory comes up with, namely its desire to use these large concrete categories, is what causes all the difficulty. Yeah, I mean, Deceptor has to solve this problem, so category theory will have to solve it. I don't see a particular problem. I mean, you could say, group theory has to solve this problem. Yeah, there is a sort of urge in group theory, even though set theoretically just sort of say, well, set class doesn't much matter. What other way is it? I mean, this is already given away to... There's no way to do group theory other than set theoretically. There's no way to do category theory other than set theoretically. That just hasn't been shown yet. No, but I mean, we can do it. can do it that way but then if something i mean it's been they've been thinking about this stuff for 60 years and nobody's come up with anything that hadn't been thought of in the first four or five years well they've not been thinking about foundations very much and they've been just okay so it's only four years because it's since the 60s but even then i'm not sure that many people have been devoting their time to doing foundations they've been understandably getting on doing that Cantor and Zermelo was, what, 70 years? No, no. Well Cantor was doing his work in the 1870s, and Zermelo was doing his, and 30 years later, it's pretty much the same. Yeah, 34 years, so it's not some, you know. In fact, the problem was just identified 30 years on from Cantor, so. No, it's already

50:00 present the difficulties are already present in his work in the 1880s right so it doesn't get he he well he solved the same problem he played he solved it himself and i think he did and also for a long time presumably a lot of the issues that foundations were thought not to were not even realized to exist because people thought well people have a perhaps an idiosyncratic idea of what foundations consistent and thought well we've already sorted that out so there's not really a problem with that to be well if you think about it the the profession of mathematics was dragged into questions about foundations as it were willy-nilly or against its will it's not as if this is something at all had always preoccupied mathematicians it didn't preoccupied. Greek speculations on foundations were not, it wasn't Greek mathematicians doing this kind of stuff, it was Greek philosophers. And of course there was always these speculations, but it somehow, until the 20th century, or until the late 19th century, it didn't depend on practice. And the revolution in in the concepts of mathematics that took place in the 19th century was much more radical even than the one that took place in the 17th century. I mean, it really tore loose from geometry completely. Geometry has a kind of heuristic role to play now, but not a logical one. Or what people call geometry is a kind of curious analgum of old-fashioned, you know, what you're talking about. It's really, it's a kind of, what is called geometry is a kind of species of algebra. So, and I think, to summarize it, what I'm trying to say is, It seems to me that there are no problems in category theory, per se, except the desire to use large categories, as structures.

52:30 and and and that if you translate that large and small only makes sense in the context of a set theoretical it only makes sense in a set theoretical context which is the context in which all mathematics sits actually except here's a possible solution to this could to what extent So we're talking about the desire to use large categories, because people talk about the category of groups, and the category of sets, and that kind of thing. Could you not reinterpret this as saying, well most of the results that people are actually interested in don't depend on is actually being a category of all groups, a category of all sets, but some appropriate restriction of it, and the people who are coming up with these theories will be just as happy. So if we can find the appropriate restriction, such that things do make a little more sense without having to worry about the species of proper classes, then they can carry and forgetting about where the ceiling of their talk is, because they don't care about things that are anywhere close to the ceiling. They're just looking to talk in as much generality about the things that aren't really interesting. Well, I'm kind of going to a little groups thing. I mean, it's fine. That's one of the... That's an okay... I mean, I know it's going to be large, but that's fine, because the results you get are just the universal results that we get by quantifying overall groups, and no one has any problem with that. It's treating it as an object itself, isn't it? about the category of groups doesn't have any problems because it's an object it doesn't have any problems that don't already rise just talking about groups in general yeah exactly but once you start talking about the category of opposite functions or we know with you know i'm going with um well it's in that quotation we just have to ask it is we have that quotation where walbert says you pretty start by thinking let's look at the let's look at the mortisms and then you end by noticing that even though you started with groups or rings or whatever you always end up with the same general kind of thing in the category which is a kind just like all the other things

55:00 okay but the question is i mean the question is when do they talk about large categories what do they use them for and is there anything problematic in the use of actual use of large that seems to run up against problems if you um but presumably you won't because i mean um i mean presumably you can always interpret your yourself small means below the occurring in the hierarchy below the first inaccessible strongly inaccessible and large means above yeah so and so one one way out of it is what category theorists called growth and deep universes which are just that which is just the version of uh set theory that describes yes 1931 so where you have every that yeah and he says what about the what about the totality in which well he says cavalierly well that's that belongs to medicine theory and i'm not doing medicine which is a happy way of no it's one of those happy verbal solutions to a real difficulty that's a problem you're working on there's always a verbal way out of it you always figure out what domain you're working in yeah and the problems you can't solve big meta in the function of the domain although it's a hypothetical community of people yeah very soon yeah so i mean i know i think hope that you're not working in i i think what we i would i think what you have to do is face this difficulty about the global about the radically different logical status of these large domains and what the global logic of large domains is and whether you can do these set theoretical constructions and at the same time take into account these necessary logical distinctions.

57:30 I mean, this is the point Bell was making when he said, you have to consider that whatever, you know, however loose an idea of object and so on you're going to have, you have to think of morphisms themselves, the morphisms of a category as discrete things to which the notion of identity applies. And it's the traditional notion of identity of being one and the same thing. So, you know, when I draw a commuting diagram, what I'm saying is that this succession of arrows is one and the same thing as this composition of arrows is one and the same thing as this arrow. Well, I think I knew a new one, as well, I guess Bell doesn't look at that small ounce of stuff, but identity is very rare, I think. No, no, that's a big deal now. It's not in, it doesn't, it isn't in Thomas theory, it's not taken as a, because the identity propositions have varying truth, are maps to the to the truth value object just like them yeah but you were saying about the sort of global function even sat there you know even your uh you know your question of how you should do the logic global saturated by density almost decided well i did well there's no decidable identity relation amongst global functions that's the problem so it's not clear how you would do i mean maybe you need to just relax us maybe that's not an essential part of the thing i don't know i mean this is all what the problem about foundations for set theory is but it doesn't it strikes me as being self-deluding to suppose i mean like saying well we've just got a notion of got to, you take the notion of set as basic, and I take the notion of morphism as basic. What's the difference? Well, it's an enormous difference. It's an enormous difference. What is it? It's, well, for a start, it's, the differences are revealed by a genealogical

1:00:00 analysis of what's going on here. The notion of set is the oldest mathematical notion we have the notion of morphism is a re is a is a is a interloper a recent newcomer a guy that only appeared on the scene after all this other stuff has been done it actually comes from the idea of concrete categories but but if you say you've got this notion of set and we've got this notion of transformation. No, no, that's different. That's different. That's different. But then, but let's, let's, okay, let's spell that out, though. I mean, it isn't the axioms for a category, but, well, it depends. And if you, it's not clear that nearly having this notion of a collection of things necessarily leads you to, leads you on a royal road to Exactly that one unique set of one unique formalization of it. Well, the one is a formalization for a category. The four is not a category. It seems for a category, then. You know, so what's the axioms for a category? A bunch of objects, you have a bunch of transformations. Yeah, but here's the... They've got to be associative and they've got to be composable. Transformations are that, I assume. Here's a category. They aren't just the axioms for category. Okay, here's a category. There are three objects. There's Stuart, you, and me. And there are five morphisms, and the identity morphisms are the chairs each of us is sitting on. And that wastebasket is a morphism from you to me, and that one is a morphism from me to Stuart. well that's the category yeah not on this from not on this foundational view it's not well why not well just in the same way it sounds like the airsons for a set okay here's a set its numbers are uh no the the sets the membership relation is that chair yes and then the set is stewart stewart yes so you can do it formally if you can do it formally you're disappointed you can take category theory to be a good tilbert style formal and you can take the notion of satsby

1:02:30 hilbert stuff yeah but there's no what we're trying to do is take an intuitive notion the notion of transformation and give some action yeah and the category there are the captains are that yeah axioms oh well let's read them and then they just you can sort of see where they are you wanted to cash out the notion of what are the if something is going to qualify as a as a a concept of transformation, what primitive rules should it obey, then, well, something that does nothing at all, we're going to admit it as a transformation. A transformation that carries A into B, if we have a transformation that carries A into B, and we have another transformation that carries B into C, then we're going to say that just one following the other also counts as a transformation that carries A into C. and we have three of them two first then we're going to regard the order in which we consider them to be composed to actually be significant so we're going to say that well i mean the order is determined by the numerous coordinates but i mean but no the order the order of composition so if we regard if we regard no it's it's crucial that f circle g circle h is the same as f circle g certainly yeah yeah not something about transformation this is yeah you're just cashing out and shoot your intuitive mission transformation in terms of vaccines and I know what it goes what is this intuitive notion of transformation so I think most intuitive notion of science I mean you know that's quite simple it's a collection of things determined in science that's not simple whole what's the tournament and size but you're really like i will give you a example but it's a terribly so i want you to see how it is okay okay this is a transformation okay okay now describe that in terms of morbid categories it's a it's a try i'm giving you examples of transformations like this is an example of a set as well as a chair no no this isn't it you're talking about it because the fingers on my hand considered separately compose the unit or the units of the set now whether you can say what the finger is it's not so clear because i think they go down i don't know but if you're talking about genealogy the notion of determinants and science is about as bad as you can get and completely imprecise

1:05:00 finally you know as you point out in the book it's made you know it originates in boundary stones and fences i mean this is what our foundation for maths is supposed to rest on You can't appeal to genealogy. When we appeal to genealogy in that case, it just shows up for concept to be totally inter-science. It does show it up. And it shows why the question of a parasite exists to be unanswerable. You've got a clear example of sets and a clear example of totalities that aren't sets, that aren't bounded. boundary between them. You can't even answer completely unequivocally whether Parasat is a universal operation. Uh, no. And it's so unclear, it is very unfair to accuse the Sat and Category Theory's foundations of being unclear when the Sat-theoretic one is woeful. No, it isn't. That's the point. It isn't. It's got it's it's it's got a longer history it's got a it's got connections with intuitive everybody's got the idea of the set of people in this room or the set of cars on the university wall but the notion of transformation is much more vain now there might be a notion is it's age shows it's up to be even worse, because if it's been going for two and a half thousand years and we still can't decide whether it's the Paraset's a universal operation, it suggests that there's something you can't really take at the bottom. So one of the criteria here is how old is the root concept? And if we're comparing something to that, that's not really... But, you know, this is a bit of striking masochism. If one of the advantages of basing things at set is it's a very old notion that goes back to Harris Ross well then the same advantages appear to apply to the transformation perspective because the idea of a transformation goes back even further it's pretty no it's not that it goes it's not that it goes back it's not the fact that it goes back that makes it that makes it that gives it credibility the fact that it goes back seems to

1:07:30 suggest that it's natural and the notion of erythmos is much more precise and well it's much more particular than the notion of a transformation i haven't got what you're talking about i haven't got a a good hold on what you're talking about when you talk about transformations i mean there's the transformation of Daffy into a laurel bush, you know. There's the transformation of Theseus' boat into a boat of the same size and shape made of completely different timber. There's all kinds of transformations. There's the transformation of a tree into a table. you know meet my heart these are really good ones i mean this is you know it's the plurality but you know the myriad of um transformations and telegons and some of the other sets that go with them i know but the sets are all similar whereas these things are all disparate you just brought i think you've thought about such so long you start thinking and it sounds very very different a set of books on that shelf yeah but what you see See, what I'm trying to get away from is this idea is that once the axiomatic method was developed, people thought, you don't have to worry about whether things exist. You describe them carefully enough, and you can use the notion. Now, that is ultimately, ultimately, indefensible. That's not what we're trying to do. Oh, yes, it is, because you're saying if we can set down conditions that these things and you picked arbitrarily the axioms for a category. But the point is, those axioms, we know where they came from. They came from the idea of a concrete category. No, they come from the notion of transformation. No, no, they came from the ocean. I mean, historically. Well, they happen to coincide, but that's where they came from. Well, that's like saying the axioms for set theory came from looking at how people do mathematics. You know, they needed to have the notion of... that that was the point of stimulus enterprise was to find out the set philosophical principles used by mathematicians but but you're not telling me that things like power set would have cropped up on intuitive oh yeah they did they popped up in character but these people who are mathematicians

1:10:00 were looking to reconstruct mathematics they weren't reconstructing intuitive mission i just don't see what you think is different between us taking the notion of transformation you think ammunition set and trying to create. Of course, like we said earlier, that, you know, you're going to have some wooly concepts that are going to be things that maybe fall out, or you don't consider them, maybe things that fall in that you exclude. But that's just because we're taking wooly concepts and making them precise. But there's nothing different between our two aspects. This is the fallacy that is rampant in philosophy departments of laughing up into the stratosphere and looking down and seeing the two people doing something that the good Lord, as the French say, is in the details. Of course, there might be an analogy like this, but just if you look at the details, there just isn't. So I think just two objections are, one, we appear to be gerrymandering our definition in order to meet the end result, whereas if you start simply by meditating under a tree for six days and contemplating the notion of set, then the death of the... Well, it took more than six days. I mean, there's the whole of classical arithmetic, and then there's character. You shouldn't have to look at any of that sketchy, because your initial set doesn't arise from that set. What you should do is you should sit under the tree, you should contemplate things that meet your intuitive definition of set, and then you should trace out from these examples all of the properties that now are what it is to qualify as a set. Well, of course, if we, on the other hand, already have our vision of the axioms of category theory as an end result, then the fact that the primitive, the intuitive notion of transformation meets these is no evidence that that's a good foundation because we're just gerrymandering it. Well, I don't, I mean, if the thing is so, if I say that the reason you can say that it meets these axioms is because you haven't got any clear idea of what the notion of a transformation is. It's too vague. There are too many things that it could possibly mean. Why is it more vague than the notion of set? Well, you could say, this is a question like, why is Madeline Carroll more beautiful than Hattie Jakes?

1:12:30 And you say, well, Well, I mean, you're claiming it's not that sort of question. You're claiming it's sort of a question. No, it's a question of seeing. Here's why John thinks it's more vague. Because you can point to an example. When you point to an example of a transformation. You say, this is a transformation. You say, this is a transformation, or when I put my coat on, it's a transformation, or whatever. And somebody has a scene in front of them. and they say, okay, which aspects of this scene are the transformation and which aspects of it are just extraneous details. So the colour of your coat, is that part of the transformation? Yes, absolutely. This is important. Whereas, when you say the set of chairs in this room, you might even say, well, the upholstery of the different chairs, is that part of the set? And you say, no, no, no, it's just the fact that there are... No, sorry, what is it? It's just the object. The upholstery is involved in the set, because it's involved in what this chair is. Yeah, so the power of my coat is involved in the transformation in the sense that my coat is involved in it. And, yeah, so just like there are all these extraneous details about the set that are not essential to its set-ness, in my part of the strength of the but what do you want to use it it's translation with just like you have to do a lot of stripping away and i'm thinking on a little bit detail that you don't want you're just that's that you would like to have to do a lot of thinking away in any given example transformation get to the bare notion of transformation but there's also many of us that we need to draw a bunch of dots and say look there are no more details It's very easy to get. The question is, how do you specify each? What good way of specifying sex is to specify its numbers. What good way of specifying a transformation is to specify a thing that begins with and a thing that ends with? Yeah. Those can be specified precisely. So the ship of Theseus, moving from its original state to its state later on, is just, you can specify the transformation by specifying the ship of Theseus as the start and final step. Well, you can't do that in which it... But it does the ordered pair. But you can't do that. It might only be one transformation. You can't do that in categories because you've got different, the morphism isn't fixed by where it comes from or where it goes from. You've got to specify the process between, that goes between the two.

1:15:00 Talking about process here. In this world view, there are things called processes, and those are the transformations. And this is a different metaphysical picture, that's the point. Well, give me an example of a process. I mean, a process by which you take off one bit of timber and then puts on another one, you know, that's a process. It's not a sequence of events. It's a process. Yeah, it's a process. Okay, here's what makes this difficult. The business of stripping away the irrelevant features, yeah the things that don't of course if you're talking about the set of chairs in this room these are each of these chairs a particular thing so you can't strip away its features and it remain the same thing okay so you can you can say certain features of these chairs are irrelevant yeah okay so you know all I need to know is that those two chairs are chairs and are different in order to at least chairs and two chairs in the room yeah okay so but now so the process of ignoring or stripping away features uh in the case of the set that's all you have to do there's nothing else left right in the case of the transformation you're still left with a puzzle about what the process is No, no, no, because the target is, the target of what you're stripping away to get down to is the process itself. So if you don't get to know what the process is, then you're a no... Okay, well give me an example, one single precise, clear example. I mean, I'll allow you to do the stripping away so that, for example, we can agree that the chairs in this room form a clear example of a plurality, at least. Whether it's a finite one or not, is it maybe still a disagreement, a point for disagreement? So what's wrong with the ship of Theseus one? that's why there's an object at the start should the thesis and stuff objects the animals they've replaced all the timber and the process is the replacement of each of our timber

1:17:30 okay so so so here's the original ship and here's the so presumably in this case there'd be lots of intermediate ones of course you've got to be absolutely precise about the process for instance if John had come and taken out the bit of plank because it's a different process from the other one. But this, of course, is the same problem that comes up with objects. So the question of when you're presented with a scene, which process you're seeing, is a difficult and imprecise and vague question. However, that's the same with sets. But once you've specified precisely, as you always said about arithmoids, there's no objection to set theory specify, we don't specify members of our sets precisely. Mathematics starts once. It has been specified precisely, and likewise, Mathematica theory starts once the processes have been put. So the question is no longer, there can be an objection to it that we can't specify precisely what the process is. We can give an example of a precisely state of process. Okay, John took out this bit of timber, then some and such took out this bit of timber at each of these times and that's absolutely precise so there's no no it isn't actually absolutely precise because because for example was i just tentatively somebody's come along and put the plank in and i i'm just smoothing around i'm over and i lift it out and i don't know turn it over to see what it looks like and put it back yeah is that part of the process oh absolutely okay so what if i just spin over the side of the boat is that part of the process if andrea had come in earlier and sat in that chair would it still be the same chair you know so with the set containing all the chairs in this room be the same set again in this other possible world the potential part of the chair is the question of whether it's the same chair and this reduces the question of whether it's the same process but once you specify the process precisely then you're off the way and mathematics starts after that just you're clever at pushing analogy but i don't think in com in the concrete the notion of the the notion of a process like this is anywhere near as clear as just the notion of a collection i don't know i think we probably use the notion of objects more often than we use the notion of

1:20:00 We have a solution of set more often. No, what these rely on, you have to specify the object precisely, and we have to specify the process precisely. And I think we talk about objects as the same again more often, and so we have that sort of talking about it. But there are all sorts of problems. The processes involve objects. I think you'd have to take the notion of objects as given. Yeah. And practices involve them. You know, everything has to take objects, isn't it? No, I mean, I think this is much more, it's obviously, clearly, much more general. Lots more things will fall under the notion of process than fall under the notion of aeropause. Well, I mean, I kind of presume that you could do an interchange. You could do an arithmetic process rather than an aeropause of stages or something like that. But an aeropause can't be regarded as a problem. How are you going to regard an aeropause as a process? way around okay and the and how do you get from this come this vast conglomerate now look i'm not saying well i'm not saying the way it's impossible what i'm saying i'm not saying an object and stripping away the process yeah well what i'm not i'm not saying that this is impossible what i'm saying is that you shouldn't be satisfied with half-assed verbal justifications of it you know but i mean that's what we're trying to do is make it very precise we're trying to make it in some way okay so so because i know i should have wedded so i would like to know whether there's anything wrong well yeah i was trying to defend the idea that there wasn't so the other the thing that's just occurred to me just before i've got to go is you you've been told that if you get five rings five rings you have to leave no it's uh the uh snooze part of the alarm i'm gonna go down to the office before you go and open your email and give me the paper when I get home. Oh, right, okay, that's perfect. That's fine. So, yeah, what... You must hear Stuart's part of the show. It just occurs to me that if your primitive notion is process and from that you can get some of the category theoretic axioms,

1:22:30 then once you get to there, your freees start running. Whereas if your basic notion is arafthloss, To get up to set theory and be able to start running, you do have to climb your way over the phony problem of singleton sets and the empty set, which you acknowledge are things that have to be jigged around in order to get the set. Well, or else we have to go through a lot of, we have to do a lot of prolix circumlocution. yes say if the sets not if something pulled into this project did such and such and blah blah blah if more than two things do two or more things do so you you could get around it by all that all that um circumlocution but it wouldn't be desirable exactly no but okay but here or transformation or whatever you call it um made precise and obvious satisfaction then there are very few criteria that you need to place on that notion in order to make it a category. Yeah, there is the problem of constructive proofs, though. I mean, I'm... You know, because this is the point that your friend made us into, John, that there's problems, because the proofs we've seen so far just use the notion of category, I think, and the axioms of category. they don't have any other existence assumptions but there are going to be proofs in category theory where you need existing not unqualified existence assumptions but conditional ones that say if you have these sort of like if you have a set then you have a set of a subset I think it's possible that some category some category theoretic proofs will need these sorts of things This can, in fact, be one of the problems of the project, that they don't have. Well, my feeling is that it's not the axioms are categories. It's probably the axioms that are the topists that we should be looking at. No, I don't think that. But the point is, what I'm saying is, too many people think, or are satisfied with a glib explanation. Well, okay, we've got this vague idea here. okay now these are the axioms we're going to use and yeah yeah and you say well

1:25:00 i know you've got to argue about someone's got to propose something and other people have got to back it and down i think when it happens with um that we just happen to get a genius like so mailer to you know basically put down everything but even he had to have one added to his and you You know, the idea that all of Jermelo's ideas are in Cantor. Yeah, but no, it's okay. Well, then Cantor, the genius, whatever. But, you know, we're very lucky in that situation. We may have been just as lucky in the Anvil to just accept the category of theoretic axioms as the only ones we need. But, you know, that's not a debate. For example, replacement is just, is kind of like, well, union is the axiom of union, It's like multiplication, so some Greek genius figured that one out, because their operations weren't on just pairs of quantities, they were on whole collections of quantities. Absolutely. But if the earliest you can trace the proper setting out of all the formal rules is Cantor, and yet the idea of Arathmos goes back to the Greeks, then obviously it's a very subtle business. but cantor says i'm using i'm taking the traditional aristotelian notion i'm an aristotelian realist he says i'm taking this notion yeah but i'm extending it but i he had to do something he didn't do something there was some work he wasn't just transcribing things that people people in ancient greeks had already written back oh god no no never gave accent no he was changing it by he was changing it by allowing you know by relaxing the condition whole must be greater than the part i mean it's strange that you know euclid did give axioms for geometry but never for awesome well he he gives definitions in this yeah i yeah i i agree he gives something in book seven he does give a bunch of definitions but on your right he doesn't give he doesn't give axioms so he certainly wouldn't want to deny that there's a great deal of effort to going from some primitive, intuitive notion to a properly worked-out set of axioms that came true? Look, I'm saying that there are two things that have to be taken into account,

1:27:30 and neither of them is obvious, okay? And the first thing is that what is called the set theoretical foundations of mathematics, that is much more thorough, much more plausible, much more intuitive than anybody is willing to, than anybody believes, because they've got this distorted view that somehow it's connected with a certain first-order axiomatic theory. And they've got the confusion that you can use axiomatics without having the notion of set available. And all that's just hideously confused. And the second important point is that, of course there may be a way of doing category theory do. But simply looking at these axioms and saying, oh, well, they captured my notion of transformation or whatever, is just wrong. I don't see how it's wrong. It's a starting point. You say, well, they do. As a matter of fact, they actually do capture parts, at least in my notion of transformation. There may be other things other than associativity and existence of identities that I want to put on. How do you know that if you didn't get start getting more of your notion of transformation it would eliminate certain things as categories yes sir it would it would if you know that's very possible in which case it's going through theories off from the wrong so well no these things will still be category everything that satisfies these axioms will be categories because it satisfies the beginning thing but if you put the additional conditions on things that are now considered as perfectly legitimate categories would cease to be. Would cease to be examples of a theory that grows out of your intuition. Yeah, but that's what you've done by, you know, we no longer have complete ordered fields in your set theories. So there's no objection to say that, you know, you may, by making your primitive notion more precise, exclude things which you once thought fell under it. People used to think, or people still do think, the real numbers form a set, but you don't. So that's just a progression of considering more precisely your intuitive notion of arithmetic. No, it's not. It's a rejection. I mean, it's a self-conscious rejection of a point of view that allows you to arithmetize geometry in that sort of straightforward way that Cantor did. Yeah, so this is... So if I... But that's a theory, and then later on,

1:30:00 my intuitive notion of transformation made me, forced me to accept another axiom, category and it's things i used to count as categories were no longer castries then that's just the same as you know yeah so you're no longer counting natural numbers so just as you have none they're not just as you have you're going to do the cantorian set theory and not just on a whim for these reasons we would end up with yeah except the difference is that yeah but the difference is we're talking about things that we can't see yet but the point is that this this business of of what the consequences of being finite are as you as richard's been arguing is really deep and there may be there are points of view that that are even more prosimonious than euclidean secretary for example where you deny that that you think that you can that kind of all those transitive closure or or uh or the uh power set axiom but there may be finite collections such that there are infinitely many sub collections yeah i mean i don't i really don't think that i i really don't feel that it's very plausible but It might happen, right, so much to some other large parts of math. One obstacle was mind-open to any possibility, even repugnant ones, right? Because if the power set goes, then things get really hard. But then maybe things are hard. That's the problem, you know, we've studied nature. There's no guarantee. Well, we're not studying nature, but something, we're super-nature-lating. There's no guarantee that things are going to be easy. well we'll see but I just think that I don't think I think the problem is not to think of analogies so that you can as it were you know you can as it were do a two-cock kind of arguing and somebody would I mean the problem is you should face up to the intrinsic difficulties of what you're trying to do and saying well I'm at least I'm no worse off than Seth here is and then and then you bring up No, but the point is, my desire is to just do it. Just go ahead and do it. But you were putting objections to even trying to do it.

1:32:30 So I was putting, you know, I was saying, well, listen, if you think this is an objection to trying to do the Category Theory 1, then set theory should have been stopped by it. No, no, I don't think it's an objection. I think it's, my objection is not to trying to figure out how to get at Category Theory via some notion of transformation or morphism, or how to make this primitive. My objection is to then just assume that these axioms, that the category axioms are axioms for this. Okay, okay, yeah, sure. So you've got to argue about that sort of thing, sure, absolutely. And in order to argue about it, you've got to have a clearer idea of what a transformation is than we've got. I mean, the notion of a set, it may be dubious, it may be all kinds of things. It's got things in it that are not clear, but we've got clear examples, and we've got and we've got a clear idea that all the examples are essentially of the same kind whereas we haven't got that with the business of processes because we don't know clearly enough what a process is supposed to be so I'm not saying that that can't be done I'm saying maybe the way to do it is to approach that have a different kind of idea where you've got a looser notion of object which hasn't got a classical identity conditions I mean that's that's other way of thinking. Well, this one will run and run.