FW Lawvere, M Menni (contd.)

0:00 At the same time, he also claims to have no interest at all in philosophy, which is always a sign that people have got a hidden philosophical agenda. So I'm not going to answer any philosophical questions. I mean, Andre asked him a couple of questions, which I think actually would be the best answer to the questions. No, no, no, don't discuss. I do not discuss philosophy. I was just going to say, I thought there were glimmerings there that, I think if I can say it without sounding big-headed, because I have been banging it into his head for the last year, that he's at last beginning to get some, you know, some glimmerings. He still says some daft things, but I... He's been working on it enough that he has made some progress. He was. And, and, exactly. He was. Well, I don't think I have any role, much role. But I have pumped him very hard in the last year. This is what he should be. He's got to understand this before he opens his mouth on these subjects. Yeah I think he is beginning to definitely to move in the right direction. I don't know what the quotes about from Byers were doing there, but let's give credit where it's due, he's definitely moving in the right direction. Yeah, definitely moving in the right direction. But, uh, no, this, this, this... Why didn't you tell me that? Because I thought it was pretty good, but then, of course, you were praising me all the time, so an answer to that could be biased. No, I don't, he wasn't praising you all the time. Anyway, just, I'm considering it was about the... He was saying that I was worthy of consideration, which is already incredible praise.

2:30 Well... You could nonetheless see that objectively he has been making some serious effort to at least orient himself solidly in the subject. He still has a long way to go but he's yes true yes he's certainly much and he is doing some good work in paris in trying to get uh the necessity for studying and learning category across to the people in the history of math community there but uh no as i say although your course replies to this young guy's questions were very very Clear and revealing, good, but I would like to have a discussion about something just than what are the applications of topos theory to finiteism, you know, I mean, on my list of the hundred most, what did he say, just like his, you know, the billion, on the list of a billion questions about topos theory, I would like to have been asked, that would have been somewhere near the bottom. Yes, I particularly like the reply that there are actually only two. No, this is an old thing, because this came up already in 73, I think it was, and early on in France, that, you know, this is the actual meaning of the logic society, the logic, the ASL. So, it was kind of the first case where the logicians were publicly, of course, Dana Scott was already talking, they publicly... ...acknowledged that, well, maybe category theory could be something like a foundation, something like set theory, but they immediately said, well, of course, there could be any number of foundations, there could be any number of foundations. ...politically blurred in those days... Only two. ...precise thing, two. Ah, so that was an old quotation. Hang on, is this where they've gone in? Well, it's an Indian restaurant. We're supposed to be in an Indian restaurant. They've disappeared. I was just wondering how... Just let me look quickly inside. In case...

5:00 No, I'm so sorry. We're looking for our friends. We thought they'd come in here. We were mistaken. Sorry. No, they must have gone down the street. No, no, no. They were ahead of us. They're further down the street. Oh, there they are. Hi. Sorry. Sorry, Richard. I'm doing the same thing now. We knew it was an Indian restaurant, so we thought we might... It turns out there are three Indian restaurants within the space of about 20 yards. Yes. Oh, this one looks very nice. I directed a couple of people to the Indian restaurant at the bottom. Oh, including Andre, because he's going to try and come on after he's finished using the computer. He's not too bad, because he has my mobile number. We think Irina is with him as well, so that would be nice if she could come. But I phoned James. That's the main thing. I'm pleased to see that you're still smiling after all this rearranging, organizing people's movements and people don't want to move. Of course I'm smiling. Completely unruffled by now. As long as people are talking, having fun, enjoying the work going on. Thank you for watching. Yes, so I was going to say, shall we go in? Hi, we think we will be six, but it's just possible that we may have two more joining us, though it may end up being eight. So did he let you know that? Okay. Well, I'm going to wait till people... Yes, we think six, but possibly eight. Or some way to go, or not some way to go, it's called winery, which you can contract and choose as well. Go ahead, Bill. Do you want to take the seat of honour? Go in the middle.

7:30 I'm trying this one now. I see. Well, that is the same, Bill. That is in the same spirit, so to speak. Yes, yes. Yeah, that's okay. I'm sorry, kind of you, sure. I think we probably need to get some more. They're probably going to lay some more seats, I can imagine. I think there are going to be some more. Ah, yes, you're going to explain to me about the horizon. I think there is more than enough. Even if we have two more people to join us. Well, we think we're only going to be six, Anders, but there might be two more, but even so, there should be enough room. Now, this notion of minors... ...more of it is described as a secretionalism. A minor is something you can get by iterating these three steps, these kinds of steps. Sometimes it's hard to see the objectives, but it basically is a kind of subculture. That's to see you again, Joe. Sorry, I didn't mean to. I was just checking the other one across the room. Oh, right. It's the first time. Yes, yes, yes. That's exactly what I was trying to discuss with you. No, no. The notion of graph itself is quite ramified. There can be two edges between two points. Are there loops? Are they graph-oriented? Yeah, yeah, that's right. That's right. What I'm saying is that they sort of move from one of those categories to another. They say we could collapse this thing. Thank you. Sorry, James, go ahead. That's all right. Thank you very much. Oh, thanks, okay. Great, yeah, no problem.

10:00 Oh, yeah, I see, yeah. Thank you very much for your time, and I look forward to seeing you again soon. It suddenly occurred to me that the other thing I should have tried

12:30 ...was van with a gap between the van and the quine, but there may be, well I'll ask Richard when he gets it, maybe that was the mistake, maybe we should have, because I just did it as one word, so possibly that was the mistake. Because it didn't say does not recognise password, it didn't reject the password, it just said, kept saying try usual format.

15:00 Oh, is that the same thing? Okay. Just another way of telling you that you've entered the wrong, oh okay, well. Anyway, we'll ask Richard. If not, as I say, we'll just use the computer in his office. There are quite a few buses. The one I booked was the one at 11.25, which gets into London at 20 past two. No, tomorrow. And then there is... We're in London to Cambridge at 3.30, which gets in at 5.55, but you might want to get an earlier one if you do decide to take the bus, because the registration at the conferences starts at 5, doesn't it? You might want to get there a little bit earlier. It's a pretty regular service, I think there's at least one bus every two hours from Bristol to London, maybe even more frequent than that. That certainly wouldn't be the earliest one in the morning, there would be at least two before that. It's not far at all. It's in the center of Bristol. It's about 10 minutes walk from here. Maybe 15 minutes at the most. Yeah, the main bus station is about 15. Oh, it's where you came in. It's where we met you the other night. Yeah, well, you know, it's not that... I can't come back. No, no, no, I know, but the point... I go back to the triangle, right? You go back to the triangle and then walk down towards... This is an objective way. Now, I once stupidly wrote in an email to the blog that there are these two ways of thinking of the things that can be done.

17:30 You can take all the definable classes, or you can take, given a model, why not take all subsets? That's something like a model. Yeah, but it's a model, it's a model of something much stronger than three, from a logical point of view. Well, all these are going to be, well, yeah, from a logical point of view, GB just has definable classes. Well, that's not what you see. You want to, what, there are two distinct things. What you would join and what you would join in any way of structure and statements about it. If you took the whole power set, you might make only very few statements about it. But there's an error in this. The actual axioms will be true. Because Andreas Voss pointed this out to me and I've been trying to figure out a good way to explain it and publicize it again. Suppose you had a caliber model. Then the class which... It puts up the bijection between the model and the natural numbers in it would be one of these classes that you added. And this would immediately conflict with the simple comprehension. Notoriously, GP does satisfy induction with respect to arbitrary quantum. I think that's related. I think it's very closely related. Particularly thinking of models and countable models. So this has to be, the idea of taking all subsets has to be replaced by taking the natural subsets, that there's the natural process. Well you know, as a general rule of thumb, this already goes back to the island bringing the planes here, but there's a certain correspondence between the definability of the natural. In simple situations, natural tends to be the objective state, but it's exactly the same thing as the final. There's a simple function. In other cases it's not, but it's kind of a parallel notion. And so certainly there should be parallel construction. It might turn out, you see, make both of these besides. And I haven't seen either of them. But probably there's somebody who knows what I'm talking about. But it might turn out that these two constructions do give the same extension of the arbitrary model.

20:00 That would be surprising and interesting. Yeah, it would be a special case if this was a simple-minded interpretation of rationality. Or the one might be bigger, but it can't be too much bigger. Have you ever looked at Zermelo's paper, the 1930 paper on Zermelo? Which is kind of his final, I mean, in a curious way, I mean, for him, no model of set theory is, I mean, it's very anticipating McLean. He talks about methods set theory instead of metacategories. The word methods is a way of escaping, you know, you see an obvious difficulty in the prefix method. Even as early as 1880? Yes, that's the thing. And every knowledgeable logician that I've asked, they say, where did mathematics come from? They all say it's Hilbert, you know, around 1900. But actually, there was somebody who was using that word before, but Cantor doesn't say what it is. But he obviously hates them, but he doesn't say what it is. I'm just wondering if anyone wants to speak to space. But you see, my main interest in this whole discussion is that more precisely, what you would join could be a little bit stronger than Gb, but not proof-theoretically. Gb is not supposed to be proof-theoretically strong. It's just more expressive. In fact, the axiom for that is that it's always supposed to be relevant. And that's proved objectively with that tiger from the start. Well, why not have classes of classes of classes? Because you've got your model, and you've assumed consistency, and so your model exists in a certain world of models, and you can adjoin it.

22:30 Again, naturally, this idea of just that it's really your native vetting of the original model thought of as a small category, of course, you're embedding it in a topos, so you have iterated exponential vetting, too. You know, and that, under the setup, exists. And so you could have joined all the iterated, finitely iterated power sets, or better functions-based power sets, and all of that. It couldn't run. A relatively consistent theory of the category was Cartesian clothes and their ordinary natural sense of lambda conversion. There are presentable functions. It has a single object which incorporates the whole previous. And so that's the sort of background I have of claiming that what I was talking about is relatively consistent with what everybody else does. Yeah, well that seems plausible. I was told that the event theorists have this thing they call DG+. I've never seen them in print, but the BG plus is exactly that, and it's like GB, except you have classes and classes, finite number of steps, you know, they freely use this, and it's only natural, you're going to be talking about molecules and these things, it's totally distorted not to do that, because you want to apply romantic inversion freely in several minor cases, so it's kind of, it's a mobility for you, it's pre-mobility for you. The point is always being able to form functor categories when you need to form them. Yeah, that's what I'm saying. You want it to... That does the work. Freely formation... ...is what allows... ...a slow screen for saying we want functor categories. ...and make sure that you always have functor categories available when you need them. Functor categories with the normal transition, translation... ...because there are many, many things that say, well, there could be these functor categories... ...that are too low for the small categories, then consider the full functor categories. And that's even important constructions. But that doesn't have the simple-minded, you have to check something before you can apply it. You don't want to have to check that sort of thing.

25:00 That's what I mean by free mobility. Exactly. Alright, well, I don't think, I think I'll mull over what you're saying. But the program's for today, wasn't it? Yeah, yeah. We'll come back to this next week. That's the reading. And then, during this discussion, I would like to point out some questions or something for discussion, but I hope we can discuss it next week. So, the essence of what I would like to have an ambitious think, but I think it can be useful to understand. So, towards synthetically known mechanics. We understand what synthetic continuum mechanics is, indeed we should at least have a vague idea of what continuum mechanics is. And of course what theories, various theories of continuum mechanics. So let me start just with a slogan and say that theories of continuum mechanics are something that they need to be studied of the motion of matter, scales...

27:30 At which the spatial extension of matter. Let me point out the various meanings of the key words here. What does it mean, what do we mean by study? Of course, here it is something very related to what Will says about mathematics yesterday. Of course, also from here, mechanics, there is a rather... There has been a sharp distinction up to now. There has been a sharp distinction for people who are interested in the so-called foundation of continuum mechanics and the applications. The domain of application of continuum mechanics is, in one sense, the general discipline of philosophical litigants, but in the other sense, the application of continuum mechanics are suitable for... I think that one of the important features of synthetic continuum mechanics can be, as you said, the foundational applications of continuum mechanics can be regarded in the same way. Let me point out that what could mean engineering parts. I'd like to draw a little diagram, which is a completely naive diagram, but to me it's... It's really suggesting what application of continuum mechanics can be. Of course, there are some problems that are based on reality. On reality, there are some questions in which the determination of the motion of matter is based.

30:00 That's it. We can have problems like design of something. We can have problems like the check if a real body can undergo some... Some processes and so on. So continuum mechanics starts from questions that are, that listen on the reality, real matter, and then of course we answer this question by modeling, of course. So continuum mechanics, no, continuum mechanics in general is with models, and then all models want to, and then the result of computation must be used. Not connected to reality by some kind of interpretation of results, in such a way you can answer the questions between the real and its crucial for a proper understanding of what it can and is and should be. Motion, what do we mean by motion? We mean that we place ourselves in a context where matter has a definite position in space. So continuum mechanics are clearly deterministic to mathematics and have a definite position there. And of course, various things, various theories of continuum mechanics do various kinds of motions. Not all the motions of mathematics can be confused by a single theory of continuum mechanics, but there are various theories. Each one deals with a particular kind of motions. Of course, defining problems in motion.

32:30 Which is characterized by some properties that we will see. And then I think a crucial point is the word scales. And what are the scales with the plural? Because, of course, for the same matter, there are various ways to see. Matters are different, so there are different scales. I teach different scales of course different features of them. For example, in continual mechanics, so the basic feature of continual mechanics is the fact that spatial extension of matter is relevant. So, what is relevant is the fact that the function of matter that we analyze in continual mechanics has a shape, have shapes in space, and during motion they can change. So, the basic point of the name we use. The path of matter with static mechanics can change, can default, can change shape, and of course, finish from a limit case of mechanics, which is what is more traditionally called mechanics. The limit case is the case in which the portion of matter has no shape, let's say, are particles. So, continuum mechanics up to now has been rather separate from the limit case which is... Models as a proportion of matter. Up to now, we used to think that particle mechanics should be kind of a limit of a single shape, but no, no, we are clustered here. We are cleaning ourselves in the word motion as a definite meaning within it.

35:00 It might be interesting also to see a little timeline of this, of the foundation of quantum mechanics, but just to give you an idea, the first work of what I call the foundation of the invention of mechanics, of quantum mechanics dates to 1958 with the thesis teaching the duties of water and oil, and then they consolidated up to the first years of science. So, these years have been the basic setting of the foundation of continuum mechanics, and then there have been some other things for evolution, but let's say, I would say a strong statement that I can say here, but after the first year of 1992, of course, there are a lot of styles of foundation of continuum mechanics, but... I don't think anything is really new from the foundational points that we know happened after 1992. Of course, there are a lot of people working in a different direction, but my opinion on that, I think that after 1990 there has been not much of a break.

37:30 Okay, what the foundation of truth and novel does, the classical continuum mechanics does, of course, first feature is that it deals with the single. So it gives the foundation of the theory regarding the matter at a simple scale. So there is no issue of seeing the same matter at different levels. The second aspect is that it makes use of the basic category. Use is a very, very fine dimensional matter. So most of the developments are based on that. This kind of foundation has been very useful to study the basic, let's say... If you want, if you are interested to study the behavior of simple material, I could go to the simple, simple in many terms, simple materials in simple conditions, in the sense that we were interested in simple conditions, in simple motions, if you want to describe this, of course, it is more than appropriate, of course, if we want to describe simple materials in simple conditions, okay, issues also about scales are known. However, in the last years, in the study of simple materials in extreme conditions, for example, if I want to study the behavior of a piece of steel, if I want to reproduce its behavior under little forces,

40:00 This is the matter. But I want to reproduce this behavior near failure, for example, in extreme conditions, of course, the matter, things are very... Near failure, of course, if you don't think about it in that manner, the bodies expire. They're not eternal. They change a lot. They need another qualitative change. So let me quickly say another little thing. The fact that experience with this pointed out that if some portion of matter is studied at the same scale, let's say, as a basic scale, and this basic scale is a continuum, in a sense, then we can study its motion, which is the evolution of the... I'm trying to say that the same portion of an amount of matter can be seen at each piece of time as different. For example... If I see at extreme scale, of course, there will be, if I zoom, and I can see the same portion of matter, that means it will look like a figure. On the other hand, if I look at the same body, this matter can appear, and we can lose its shape. There is more. The same matter can appear in different ways, even at the continuum. The same matter can be seen very differently.

42:30 What appears, when we look at each time, the effect of the observation is that the same matter can exhibit very different, can describe, but then the problem of changing the scales, one scale over the other, change from one resolution over the other is done by doing, let's say, one scale, one contribution, then one can define the field. You're using resolution for change of scale. Yes, yes. There are also some precise meaning there. For example, there are some approaches that you can use also. Resolution has a parameter that, in a sense, is your time. There are some approaches, there are some differential equations in the resolution variables. This is also... So, resolution is a kind of, you can think of it as a number which expresses the order of magnitude of the accuracies of the instruments by which you can see the matter. And of course, if you push forward, some body can disappear, in a sense.

45:00 Yes, at the same scale, if I look at the heart from outside, a lot of bodies have zero resolution. That's why actually I heard that, you know, cosmonauts, they can see roads, even if there is no way they see them, but they see them as a regular line. Yes, that's a regular line. Mathematical physics is wrong, this kind of implicit suggestion on a world-consistent force, or whatever you call these particles, because there are theories at all levels. But if you have a deterministic theory at the speed level, for example, and you perform this average, it's no longer deterministic. So, for a deterministic theory at the intermediate level or even more extreme level, there's more involved than just aggregating. The aggregating is the definition of what it can be. But also, what about the law of evolution? Because there's no deterministic law that explains the motion of the averages. So, in some sense, there's a real gap of extremes that is absolutely deterministic. The idea is always in the background, you see, that now you've got to lift up the laws.

47:30 I think that each change of scale, if you want a deterministic theory at a higher scale, you must have some additional information. So, the generalization of classical continuum mechanics has been, to take into account this, we can call, is often called the continuum mechanics with microstructure. The common way to deal with this kind of work indirectly has been done, as we call it, continually with microstructures, continuing mechanics with microstructures. It happens that to study some complex materials, the points at a given scale are no longer, the behavior of points is no longer determined only by its position in space, but something else. I can say that everything is always interpreted as a kind of residual on the change of scale, to get the mystic theory at some level points are no longer points in the sense of particle mechanics, in the sense that their configuration is completely determined by its position in space, but you must have something more. There are additional, and of course it requires additional laws of evolution that must be postulated over their basis of work. So the continuum of microstructures are things in which points are no longer mere geometric points, in the sense that I have said, but have an additional piece of freedom that introduces information about what happened at the lowest, at the lower level, at the finest. The general flavor of these applied mathematics talks is 90% of the time is something about this.

50:00 For example, suppose you have a truck full of coal and you want to pour it into your basement. This coal is almost like a fluid, so we could treat it like a fluid, but then again it has granules, you see, so some of the granules merge from, or are sort of, as you say, a residual aspect of the charcoal, the coal, as a fluid. Does it flow like water? Not quite, because it has this granular aspect, so there's sort of two levels, in this case two levels of scale, and that's roughly what bifurcations mean. Two levels of scale which are one sort of subordinate to the other. They're both involved and it's needed to get an approximation to the actual behavior. And there's just an incredible number of work with all its other features in it. Okay, well that's actually, I was beginning to wonder whether Hawaii's class was... So that's what I was saying. Your example shows... The first course that I had from Fusdell was about statistical mechanics. There was classical statistical mechanics. There's also Boltzmann's equation, which is on a coarser level. Now, to derive Boltzmann equations from classical statistical mechanics, the major problem was not solved at all at that time, even though one sort of presumed that it was followed. There was a whole list. And he had several versions of a list of simplifying or approximating assumptions, adding two classical and statistical dimensions. You could derive, because Boltzmann's equation works in twelve dimensions, or six dimensions, whereas classical works in six n-dimensions. So there's a huge change of scale there. But there's, you know, you have to, there are whole books.

52:30 There are a number of meetings, books by Italian mathematicians and others on this question of understanding and justifying it. So it's both a question of sort of proof theoretically justifying and also seeing to what extent that looks like proof. That's already in those two levels. Both of those levels are prudently called statistical mechanics. And in that way, in connection with that, also, I always also think, well really... We can have thermo-mechanics that really should be thermo-mechanics, because somehow, the whole emergence of heat and temperature has to do with the change of scale. There's a residual. A residual, when you idealize the cloud of particles into the actual body, there's no residual. It's just temperature. In other words, kinetic theory is the idea that heat is real, just motion except microscopically, really saying that already. Microscopically or microstatically, there's an additional ingredient. Okay, so this simple observation points out that there are different descriptions of bodies of matter, should be described by objects with a non-linear equation. In the continuum level, often numerical solutions, even problems, are done by using the approximation techniques, mainly what I call the fine element metals, starting from a continuous body and, of course, passes to different bodies modelled with a different kind at each continuous stream. So they are at least a bit disastrous.

55:00 At extreme scales, there are discrete contents. Creation at the continuous scale, of course, continues, but also approximations done by numerical methods also uses different descriptions. Is the change of scales itself continuous or discrete in some sense? Yeah, and I think there are some interesting theories that treat that as continuous. But this can't be a differential equation. No, no. Oh, it is a scalar resolution that satisfies the... So if I may preempt, I mean, you may be understanding what's this. The point is that you make these finite element models, you get a new kind of... The general laws of mechanics should be such as to apply directly to that as well. So a lot of care has to be for general laws. By changing all kinds of things, maybe things like numbers, discretized version is also a valid, discretized not by looking at the particles, but by looking at what's available for computational purposes, pumps and cold. Equally, yeah, pumps and cold in objective version, subjective process of trying to compute both of those, not because of a sort of excluded from pure mechanics. For example, things that satisfy similar laws. Okay, so this simple observation points out that the basic feature that synthetic continuum mechanics should take into account. Of course, there are different levels of them. So, synthetic continuum mechanics, what do we mean by synthetic continuum mechanics? It's some kind of development of us.

57:30 Here is various theories of continuum mechanics based on a suitable category of space, we mean a kind of category of space that has a precise level of good with respect to a lower level, in the sense of axiomatic religion of good. What does it mean? It means that each category of space on which continuum theories must be explained is characterized by... Several funcors enable us to do the most basic relations of, let's say, some watermarked topology. Each category of space is related to a less cohesive category by an inclusion in which Objects of S are total discrete objects that are included in the calculation of discrete by a function. Then, I will point it out several times, there are better joints to this. For example, the equation gives the discrete inclusion at least three joints, in the sense that there is another function which is better joined to the discrete, which is... Components of spaces in a general sense are something that can have a basis, can have a factor, and in some other cases you can have a fuller joint which is called, it's called, it's called, it's called, it's called, it's called, it's called, it's called, it's called, it's called, it's called, it's called,

1:00:00 Of course, the first thing to do is that if we focus on what we call continuous cohesion, this kind of space must satisfy some more structure. We can say that it should boast what is called symmetrical differential. The theory of mechanics should be based by the theory of space and the function of those in spatial geometry. A very important part would be to express this vision by describing it. Right here is a first point of discussion. I start with the most general level. I hope to end with a very concrete example. We hope that we can climb back up to this level. Of basic, of the models of asymptotic differential geometry and triangles, it might be possible the level of, for each rewriting, a level of... So, if... So, of course, we will focus, of course, on the classical computer science. So, the category of space must have. Let me say, what is the advantage of doing this?

1:02:30 One first advantage is that prescribing the classical continuum mechanics in the language of synthetic differential geometry is certainly a great advantage that synthetic differential has. For example, to build a mill within 50 dimensions, the tools must be similar to 5 dimensions, and it can work, and so on with mills. The crisis is really intermittent. Well, yeah, almost. The thesis has been considered as something separate, more difficult, and different in character. Putting things in this way would mean to explore new ways to think, and I don't understand what I would mean by this. There are many, many different ideas about my case. The basic point, the very basic point of the continuum mechanics is, therefore, the construction of a suitable category of theory. So that my mind doesn't try to... I'm sure you're going to come back to this, but...

1:05:00 What was it called? Resolution. Is something that you're planning to do inside E, or is something you're planning to do at different categories? I mean, resolution is a geometric morphism, or is something that happens in E? I think that the change of scale, that remains beyond the continuous collision, I think, and we've done an incredible single category of space. But radically change collision from discrete to continuous, I think it requires different category of space. Could there be a tower of all? Yes, all kind of little spaces, yes. And level of precision, different level of precision. It's like a tower of geometric morphisms. Of course, many particular geometric morphisms are really internal. Yeah, yeah. Any bounded geometric morphism is really... The richer one sheaves over the first one. So that sheaves are described in it. The site lives in it. The site for the richer one... I'm just glad I've been turned into it. Actually, your question is both. I didn't have in mind to suddenly change the category. All right. I was already implicit when you drew that at two levels. The category of space is intrinsically, as it says, is constructed out of an algebraic theory. As intrinsically, it is some notion of the category of space you can... You see, by the way, the qualitative... The sort of interesting ones are obviously a jump, because the truth value object of the upper one is projected down to the one, so that's something like that can never be of the form e slash x, that's something like that is describable internally in a complicated way, but not in a simple way, just by math.

1:07:30 Because you go to B and B slash X and you put I0 and the truth value up there, you get something bigger, you know, it's one dozen blocks to the point. If you're an axiomatic collision, that's called ample collision, you know, it's a fourth axiomatic to the number three. It's qualitative, it's a clear, a clear expression of the idea that quality may be more subtle expressions. But these are the ways of most of the models of scientific research in Germany, which is so important that I think we deserve some... For example, the discussion is this. If you start for any clear theory, a direct theory, then you can construct models of that theory in the battery of a cell. This is a concrete category of a cell, in the sense that... There are three models, but there is a very important class of models, the fine presented models, where fine presented models means that these are models that can be presented in three models, in the category of models.

1:10:00 And then it happens that... The defining meaning of the preservative counter on the defining presentable models, in the sense that the defining presentable models are the completion of the defining beliefs. And then it happens also that, let us write it here, the category of models of the theory tends to be reflected in some category of the defining presentable models. Reflexive means that the very basic construction, which is called the Isbell adjunction, gives an economically defined material adjuncts factor, covariant factors on the final presented models. And then, of course, we can consider shifts on final presented models with respect to suitable topologies.

1:12:30 And, of course, this is another interactive one. So, and then, roughly speaking, models are the algebraic side of the picture, whereas these sheets are candidates for suitable genetic objects that can describe space. Candidates for geometry. And then, depending on what we want our carrier space, we can... Look, we can do two things. On one hand, we can look for suitable, of course, if we do only this, we can say, for example, for the canonical topology for which only the images of these functions are achieved, not necessarily this kind of, whatever the feature we want in our theory. But then we have two possibilities. We can look for suitable sub-categories of science, physics, and mathematics, and then we can impose suitable topologies on this, so that we get different kinds of topologies for this, so that we have something like this. The basic construction starts from an algebraic theory, but this construction by, I think... These two choices, which is one is this, and the other is the topology, geometry, construct a category that deserves to be called the category of space. And of course the basic construction has three ingredients. The basic comes back theory, the sub-category of defining the design of the model, and this very canonical construction gives out a lot of possible variations.

1:15:00 G you're considering it to be full or just any subcategory finitely present models? This is different category of the base and of course depending on base. For example, D in the basic example, the basic theory is range from two. For example, one can, for a basic example is take D, algebras, and make it in the category of C. So that's the operation of the theory of ingestion, which is nominous. And the other is the operation of the theory of analysis, which is both passionate about the real number of things. The second point, the basic point of discussion, the basic construction in a kind of specific model. So these two points are more or less one in the most... Sorry, I'm sorry to... Okay, there is also an analytic theory. There's a paper which improves this way. You start right with the same, put it down to the same theory of complex numbers. However, in my opinion, you're wrong. You should talk about analytic functions mapping.

1:17:30 You need a distinction of some type. The whole problem is always linear. Extremely good thing, it's also extremely powerful barrier. You could talk about the analytic functions from the unit ball to the power n into another unit ball, that's a single-sworded theory, and actually the categories constructed in this style from that single-sworded theory, I think, has never been carried out. It's kind of implicit in the O-minimum. That's one thing. I mean, there's no particular virtue in single-sorted theory. I'm all in favor of multi-sorted theories, but when you can force them into one dimension, it's just, you know, it's easier to understand. The intermediate, the distinctive thing about C-infinity geometry is that it's a polynomial term. C-infinity can prove these different types of other things as well. Just to join two of the polynomials, the muck functions, have the muck functions, just complications of its own, you see, so there are many. The other remark is that this choice of G, if K is a field and we look at K-algebras, among the presented ones are those that are linear, so it's sort of a natural G that pops out. That means the field extensions, well, those are just talking about infinitesimal spaces. So the idea that there's a contrast between macroscopically smooth things and a subcategory of that, which is not all just infinitesimal space.

1:20:00 Something like the so-called myococcal, something there are many small variations of. In some sense, the most simple one is just to take all finite dimensional algorithms, the tiny subcategories that I'm going to present. So you really have to prepare it for those. By getting a pair like E over F, you get three totals, and by just by this I am achieving this subcategorized idea of a scale. And also the idea of scale is somehow here, because how you measure scale by means of functions, the way theory functions in space as you sort of allow it to be used, and sometimes you don't have the idea of what the level of the scale is. The scale means many different things, and that's one aspect, gross aspect that we just did. Yes, so the operation of the algebraic theory, which is in the background, in a sense, embodies the experience of the community of measuring things. Yeah, yeah, yeah, yeah, right, right. And I have space, in a sense, bootstraps. Yeah, yeah, the bootstrap, the Eulerian bootstrap. This is probably well known, but what's the simplest example of adding a bump functions to an algebraic theory? Not to an theory, I mean, I don't know how to even think, imagine it except for starting with polynomials. Right.

1:22:30 You know, e to the minus 1 over x squared, you could construe that as a unary operation. Oh, all right. Added to the polynomial and consider all combinations. This is a syntactic to join that and its and its relatives and all compositions of those with the polynomials. That's an algebraic theory. Yeah. I mean, I say that is, I mean, it can generate an algebraic theory and take as action all the things that are true in the standard model of real numbers. Forty years ago, nobody took it up yet, but... It's sort of in the spirit of classical algebra where you study solutions on the three equations by adjoining them a little bit instead of going all the way to an algebraic solution. Instead of going all the way to C infinity, you could just adjoin that equation and see what qualitative features it has. There is a differential equation, but you could think of it as a fragment of the adjoining solution. There's a certain differential equation that it satisfies. That kind of differential equation is considered to be a little bit singular. So it's like an algebraic extension. Exactly, like an algebraic extension. You see, differential algebra, there's a subject called differential algebra. You can join the two rings with solutions of equations. You have derivations of maps that satisfy Leibniz's rule, but without any underlying cohesion. You can imagine the majority.

1:25:00 The process doesn't sort of accurately express the intuitive idea of adjoining solutions, say it's a ring extension, it should be an algebraic theory extension, because you're interested in compositions of that new function with your old functions, self-composition, with several variables, not just one variable, composition is a natural operation of these, as well as just all forms of products, old functions with your new functions. I'm very glad you brought this up so I can carry on asking very simple questions. I'm sorry, so you mentioned if you take K a field, you take K algebras and then you can see there Field extensions as a G. Have I got it right? No, you consider it. You couldn't do that, but just consider all commutative K algebras. It includes the algebraic extensions in the field, but it also contains... In the theory of wild algebras, we sort of said, well, we've got the reals, we're not going to take any field extensions. So the wild algebras are saying, you're not going to vary the field part. I see. I mean, there's a theorem that says every finite dimensional algebra is a finite product of local. There's a field extension with some nilpotent subjoints.

1:27:30 All the nilpotents form the radical. So there's a structure theorem that sort of breaks down any such thing in terms of fields and nilpotents. You might as well just start with finite dimensions. Take all the finite dimensions. Qualitatively smaller things. Whatever your field was, if you do the complex numbers to start with, then what you're doing is entirely about infinitesimals anyway. On the other end, if it's Q you're starting with, then there's whole information about all possible algebraic number fields in that as well, plus the null focus. But the sort of general virtue of that is, I think I mentioned this before, to say, Shanyuel told me this, but I never quite realized it from any book. If you consider any finite number's presentation... We have certain generating polynomials and several variables, and we have certain generating parameters, and we try to have an algebraic variety. Well, now of course if you're doing calculations, and there's a whole industry in computer science about Bergner bases, polynomial approximations to exponential, to sines and cosines, and to some giant complexes of electrical chain, blah, blah, blah. Systematize the drawing of conclusions as you hypothesize that certain polynomials are zero and that implies that lots of others are. Abstractly speaking, it's just the ideal generated. But this process of generating the ideal, you suppose you want to verify that a particular polynomial is zero. Well, the classical, an extension of the classical Nostradamus such, and or Birkhoff's... Weakness theorem for universal algebra, which talks about sub-directly irreducible algebras. See, there's a limit that sub-directly irreducible algebras, which are also finitely presented, are actually finite-dimensional linear. This is, again, you know, the research paper or a book is even so concerned to get the most general formulation,

1:30:00 that you have various complicated hypotheses, but essentially, that's the essential idea, is that sub-directly irreducible and finitely presentable. It implies actually finite dimensional. And Birkhoff's theorem says there are enough maps to those. So what's the conclusion? The conclusion is, if I have this presentation, and every basic polynomial is going to show a zero, I can assume that I'm working into a finite dimensional algebra. If it weren't true, there would exist at least one finite dimensional algebra, at least one normal work, where it would be false there, too. So if it's a sort of generic finite dimensional algebra, that's the same thing as verifying it there. On the other hand, of course, there are a vast number of important tools that are possible only for finding conventional ones. The whole linear algebra comes into play. The fact that every element has a trace is a fantastically powerful thing to be able to take the trace. You can't take the trace of an element of a finite-dimended algebra. Why complicate the theory at this level? Why not just say, instead of saying, do you need to consider arbitrary fields? Start with complex numbers. No, no, no, that's not the main issue. Traditionally, we've limited ourselves to the viral algebras, the duty. It suffices to consider the infinitesimal case, if you ask. You ignore the real and complex between Q and C, which you're suggesting, and which we, in effect, do in the usual print. Then the whole matter is by assuming that the variables that you're starting with are infinitesimal. You don't know like which degree, but if some degree is infinite, all infinite, all nilpotent, then you won't prove anything more. In other words, you will be able to test whether or not an identity is true just by assuming that you're in some place where all these variables are nilpotent. No, so that's a very powerful computational tool. And of course it's also valid to vary the fields as well.

1:32:30 Especially for our purposes, we're not so interested in varying the field, we're interested in varying the degree of no-potency or nothing. There's a huge difference between arbitrary no-potency and none at all. Even though it's still finite-dimensional, but by Birkhoff's theorem or Nussbaum's theorem, it reflects back to the present moment, which is sort of the case. In universal algebra, that's sort of the second theorem they prove, the Birkhoff's theorem, but nobody ever points out that. The sub-directly irreducible algebras, although you get them in the general context by effective theorem, is probably equivalent as long as Lemma. It's sort of like an active choice. But if you restrict yourself to finally presented algebras, there's a whole different thing that becomes much more concrete or even computational. And in good cases, they're not just sort of, well, they're certainly qualitatively smaller. They may be actually, for that reason, susceptible to a much more powerful... There are many G's that you can think of, but this is one of the iron G's, just take finitely, in general, give it a T. You can take G to be the finitely presented sub-directed irreducible element, or finite products of those, finite products that are finitely presented, a natural part of the site that arises if you're described by... Universal algebra to you in the first place. And you have Birkhoff's theorem.

1:35:00 I don't mind this. Okay, so this is the basic construction. Let us move on to some more concrete. A few more things about features. One of them I already, you must post synthetic differential geometry. What does it mean? This means that it should contain an object. It plays the role of geometric. Here this radical is synthetic, this object is built of. Anyway, geometric line, what does it mean, geometric line? It's something that can be used as basic to do the basic construction of differential geometry. And in this case it means that air satisfies axioms, some family of axioms that are usually called cochlear axioms, that can be stated in very... In various ways. This kind of axons just says that the function space, that's stated in the most basic form, is that the function from objects that are, in some sense, that can be made precisely infinitesimal, map objects from infinitesimal order to R must be, in a sense, by a dimension. For example, if the basic object... Basically, the infinitesimal object is the object of two numbers, tangent and backward. Then, this means that maps from the infinitesimal object to are, in a sense, affected by a finite number of...

1:37:30 And, of course, this kind of co-operations can be done for any kind of infinitesimal object. For example, all variables, for example. The magnetic line, first of all, should contain infinitesimal objects and then should satisfy all the real objects. Well, it's sort of wrong to say it contains infinitesimal objects. They just define d is the set of elements of square zero. Yes, it's a tree, I would say. Well, that follows from the axiom, so you don't have to say it should contain. They're just there whether we want them or not. Whether they're satisfied, yes, but that's another issue. That's an issue. This is three of them, of course. This is three of them. Three of them. Three of them. Three of them. Three of them. Three of them. Three of them. Three of them. Three of them. Three of them. Three of them. Three of them. Three of them. Three of them. Three of them. Three of them. Three of them. Three of them. Of course, I think this is a point that leads to a quick question. Besides to contain the geometric line, I think the category of space, since continuum mechanics is also the measurement of space, should contain an object that should be called the object of pure quantum physics, which is the object that in some sense represents the result of measurements. So, and here is a clear question I would like to pose. So the question is, is the geometric line the same object of the objects of two quantities?

1:40:00 It seems that they are the same, but whenever we deal with the kind of measurement, let's say, metric notion, of course... Usually we add further axioms on air, in the sense that further axioms that can be, for example, Pythagorean axioms and other kinds of axioms, but it should also, if it fits us to represent measurement at some point or another, one also has to use new order. This, and of course, lead to integration, kind of. So I think that the category of the space should be an object that is suitable to do synthetic regression and geometry in this sense, but then there are, you see, these two level axioms. So I was wondering if, in a sense, there should be two objects, in a sense, related. And this reminded me of the distinction between the opener and the ending. Identical reals or something, in any of the pure tokos, you always got this thing, which could serve maybe as in between the two, a rational number good for ordering. The math from R, which is like a test, is really an instance of the native style of representation.

1:42:30 And even the condition to be identical to that is that somehow you take all the rationals greater than a certain thing, that's the same as all the rationals greater than or equal to that. So that itself is a sheath condition. This is what literally sheaths themselves. It's always two-valued sheaths because they're closed sets, they're not more than or equal. So the truth, I should say, the truth value among the sheaths on the tree sheaths on the ordered set of rationals, the non-negative rationals, there is this, you could say, the Dedekind Apology. It maps to that. More precisely, the non-negative part of R maps to that in that itself with the grid. In other words, there's sort of an intrinsic relation between this present-general topos theory and any given r provided r in order on the map. That could be a model of your measurement. So perhaps when we introduce magic, maybe it can also remind us that magic could be valid in mathematics? Maybe there is this map somewhere. And the thing about identity, you see, it's really, you can think of it as the inter-completion. We have a Riemannian manifold, which is evaluated in R by the differential of the energy we use to represent... Which R? What is that one about? The smooth one. It has two points in the manifold, and you can describe the distance between them.

1:45:00 The distance between two points is the infimum length of all the geodesics. It's the set of all the geodesics that connect X and Y. The size of the parameters. This in PIMO is precisely what exists here. So the sort of abstract metric space that you derive from the Riemannian metric, of course, comes out of this. How is this related to just this dividing out by the integer decimals? This is precisely how it's written. That's what it is. No, it does more. This map is neither injective nor subjective. It's not injective because it kills all the decimals. It's not subjective, but it involves the intercompletion of all the other elements, in particular the interdiscipline. It's kind of like embedding a field inside our source. It typically happens in the same community development. And if you want to impose it... If you say you want to invert a certain element, the process is neither ejected nor surjected. It's not surjected because you didn't have one over there to start with. And on the other hand, it's not dejected because inverting F might kill some other things. So the typical thing is neither ejected nor... So here it's important that the boulder is the intrinsic delicate influence because surely the constant will...

1:47:30 All of these are contained in smooth R. What do you mean by the constant reals? The bold R in here is in the topos, but there is a left adjoining. This is the bold R in the topos? From the reals in the base topos. That gives you a real object which is inside the smooth R. The bold R that I'm talking about is the V, whereas... You have the bold R as appropriate to S, and the inverse image is the P of the star. These are constant fields. These are what we call the constant field, but that maps. There's a lot of facts that come out of it. Ordinarily, you don't think that there is a need for pleading. Yeah, yeah, but see, one is more variable than the other one. So, so, you can apply it to the constant. It's got something to do with representing numbers geometrically. This is going on before the cognitive discussions came out. I would suggest it was obvious. Parameterizing motion, differential geometry, is one thing, which is recording the results of measurements.

1:50:00 Like, how do you look at it? You've got a rebounding manifold. You've got, like, two stars in the universe. How do you like your measurements? How many light years do you see? When we measure the distance, we might say, well, let's consider all possible body paths, which are parametrized, and geodesics, the point number of your geodesics, and among all those, there will be this parameter that's smallest, because there won't be among them, really, because it's an end, it's an end. Not among them, but probably the best definition would be different. Both involve this parameter inside these figures. The name to the pure quantities. And we put another name to the pure quantities. Instead of the same name of the geometric lines. This is another pointer. But wouldn't you use another symbol, please? Yeah. Well, all r. Well, r, b. Okay. Then you can have n, q. I don't know how to have that. Another scale of pure is, you know, something else. These are the physical dimensions. Length, volume, mass. Yes, yes, it is in this sense, yes. But first, so the identity requests are not, well, they are pure, yeah, that's right. They are not yet made from actual, made from actual areas. Yeah, also, this is another feature that I wanted to discuss, because also, in the classical quantum mechanics, the dimensional analysis is completely separate.

1:52:30 But I think one other feature that's important in quantum mechanics should be there, is that The theory of dimensions should be incorporated. Yeah, it should be objective. I mean, many physicists actually believe that it's a pure convention. It should have an objective. They mentioned the objects that are not both objects. The objects, some of them make sense on the dimensional model. It's distinguishable. So, length and time are on the dimensional model. They are line bundles. Exactly. One model for this idea is to take line bundles. So, that's kind of a complicated dimension on the whole system. That's kind of simple. Well, a basic objective is so that it becomes a part of well-known homological and community algebra. Well-known in a way. Well, one man's simplification is another man's simplification. No, no, no. All physicists have to know what the mind-bundle is. It's just a matter of the mind-bundle, like objects. And my opinion is that the natural language should be embedded in the very notion of both servers, because both servers are always just thought as a colonization of... Could you say that again, please? This is starting with observer. Could you repeat? I didn't hear you. In my opinion... A complete notional observer should include a choice also of dimensions. This is also a thing that I teach to our students. I think this is a very basic thing because when we deal with numbers, numbers that represent physical properties are not really numbers. And of course there are several choices of dimensions.

1:55:00 That is, there's an actual length. If you choose one length, then you get numbers, but like the distance from here to Cambridge is the length. There is a distance there, but it's not any number of miles or kilometers. Do you think of it as a kind of abstract thing? Well, it's more than a degree of a number. The degree of thought of it is just the line. When you say choice of dimensions, do you mean a choice of unit for each dimension? Yes, we should talk about the basic dimension of the field of mathematics. Go ahead. Yeah, but it's not a choice of dimensions. Or a choice of units. But there is a basic choice of basic dimensions that describes all the physical quantities of the universe. A cross-section of the mind-bomb. I was rereading your paper on the archival mechanics. I thought you were saying something different. The quote, real, that correspond to different dimensions might in fact be mathematically different. Basically, there are categories of modules of some kind. So there could be modules of different ranges. The base range will be the pure quantities. The thing is that the dimensions themselves are sort of one-dimensional, the line-metals, or invertible, if it's possible, but the modules are very, very special. The tensor product of the module with another one is the other one is sort of the inverse quantity, like radius is the time. So, for me, if we talk about Riemannian metrics in a really thoroughly basic sense, they should be valued in a line bundle, in a module over this R, one-dimensional module over this R, because it only gets to be valued in this R with the choice of unit in that unit, which is only possible locally because it's a line bundle in that context. The physical numbers are not different from pure numbers in the sense that the result of measurement is an additional quality of life.

1:57:30 Maybe the Imperial Grand Escort was saying something like this, that there is something that is pure quantity, that there are really number of dimensionless quantities. There is another object of pure dimensions that satisfy an algebra as a vector-based structure and, of course, the choice of units is just... This is an exercise in the choice of units that are speaking of this unit. It all comes down to the possibility of different reels that are measuring different things. Maybe you want a different set of reels, I guess, for engineers, or two different things. The observer. We don't observe these things. We do use, we do use that. Pure quantities could do well. There's a smooth version, a dead version, and so forth, but they aren't essentially the same. We're looking at temperature, we're looking at how high the mercury is. Oh, right. We see that there are some of them. Given a particular physical device, you translate one into the other. You wouldn't be using any kind of historical device. In order to transform time into space, you need a car or something.

2:00:00 You need a physical device to translate one into the other. But somehow the inter-translatability, or the commensurability, they're all modules over one. Does this model give any answer why reals are important? Why people measure things? Is there some insight from this model? The area is different from length, but by choosing one length and looking at rectangles of that width and arbitrary lengths, we can measure all possible areas. This is book two of Euclid. I mean the whole, the whole early part of Euclid is devoted to getting this, this thing set up. It's not trivial. But, but, theory of relation or theory of... I suppose. I suppose. More. And of course. More. I think because in the discussion of... Hold on, hold on, with the pure quantities before you rub it out. Sorry. But don't come out of the reals, really, see? Very simple graphs where there is an R and there is a D. What would an object of pure quantities be? Do they have some, you know, elementary categorical properties that one could hope to find in reflexive graphs, say? That you measure, I don't know. You see my point, huh? Sometimes you might want to measure something not in reals. The general sense logic is that the reals themselves, as we were saying, are sheathed with values and truth values, so you just take omega to the power of q, and using, and not doing that, just taking omega itself for combinatorial purposes, so that you have the hidden component of it.

2:02:30 So is it fair to say that I'm... The algebraic theory of k rigs, where k equals 2, so 1 plus 1 equals 1, but multiplication is not... So there's a perfectly fine dual number object there, 2 bracket consists of... and in fact, the quantum commuter axiom is true. Basic things are true, even though addition is difficult. So, this is somehow close to truth values, but a little bit more than that. That's an idea. If you consider two rigs, in that sense, whose radical is trivial, i.e. no nilpotent elements, what have you got? It's basically distributive lattice theory. So, distributive lattice theory plus nilpotence is a very interesting possibility for a certain kind of element. I have a suggestion. I mean, the fact that you operate with two arms has never been systematically exploited, and I believe this is a way to solve one of the... There are many outstanding problems in SDG, namely we don't have a good theory of integration, and for instance the existence of solutions, all these things where the proof in standard analysis is that we make an approximation, of course we limit the theory of the solution, but the action or the hope of the future...