Topos theory

0:00 Welcome to the first workshop on Catergiz-Mortinian foundations of physics. First of all I want to say that it's really great that you have all come here, thanks for the interest which is really exceeding our expectations by far. We have an international workshop and it's really great to have you all here. Maybe it's also a good time right now to say that I'm standing in front as the organizer but there's so many more people involved today I didn't know it would ever work out such an event without the help of many people and I would like to mention Katsira Dinodai who organized and helped me a lot with this stuff and Jamie who's on the camera today and Jake who set up the website for me. And here's the completely unreadable schedule, but I've made enough printouts outside, so you may have seen it on the web page, we will have six talks a day, in three sessions, so in the first session it's Isaac, it's Herman, and then at half past one we'll have lunch, and this is one thing I'd like to ask you for, just to cover our courses as we're running this workshop on virtually no budget. Please contribute £5. This will cover your lunch and coffee and juices and whatever. And we will go up to the 8th floor in the same building. So just use the lift or the young person to stay up there. Unfortunately, there is only one lift running currently because they are very close. The 8th floor is a common room so you can come out of there.

2:30 In this workshop, we don't have to be there. You just put a red base there and put your five pounds in there if you want. Okay? Then we'll have 4% for Wednesday and I'll start them. And after that, we'll have a discussion session. And I just want to hope this works out. It would be great if people would just come and maybe start to think about it. Questions there, considerations, but then again, this discussion session starts. Then there's a coffee break from 4.30 to 5.00, and we will finish with two more talks with Jenny Vickrey and Louis Crane. And after that, tables are reserved. You are very welcome to join us to the Prince Regent, which is a pub close by. I do have the map, but of course it's on the printer, I want you to look at it. It's actually quite close to the A.I. people request through Queensgate Terrace. It's really easy. Okay, that's all I have to say. Chris is what he's like. I'm going to go to Chris. I have a spot question. I have a dead one. I like it so much. That's okay. Yes sir, first I want to thank Andreas for doing all the local work and setting this up. And the first speaker today, he doesn't really need any introduction, it's Chris Ashton, who started the Topes Approach Foundation for Physics somewhere in the 90s. He's a leading figure in quantum gravity, and it's an honor to have you here. Thank you very much. Actually, the more correct the slide I was, I began in 1970, a long time ago. I worked in it for about 30 odd years, but during that time I became more and more concerned about something very complicated, way beyond any particular research programme at the time. I was really wanting to found out what is a complex matter, what is a mathematical foundation? And for that reason, in the 1990s, I did start thinking about topology, and then I stopped doing cubics.

5:00 It's one of these wonderful things, you know, you wander into the back of the library. I was most interested in the models of space and time. You see, for a long, long time I've wondered why is it you model, as physicists, that we model space and time in sets, because sets are quaint elements, but I mean, is there really such a thing as a point in space, a point in time? I mean, you ask any physicist that, they'll say, well, it's an approximation. No, it's an approximation. What is an approximation to? What's the reality? So I started reading the idea of thinking about mathematical space, but then my brain was still appreciating it, and I'm very excited about it. And that's what I've been doing ever since. So, again, I've worked on it. So, Jeremy and I, Jeremy Buckwell and I, worked on this in 2000. But then he got stuck at a certain point, and you can't see this in your face. You'll still be with me. So what I'm going to do is to give you a, well, a fairly general point to start off with about why, how, is, anyway. There's a lot to see or concede about the use of topos theory in physics. Now, it's not the only one in case you've read it. There's not only a great deal of talk on it about topos theory in general in the field of physics. I think one might try to do that, but there's nothing quite different. It's actually quite fun to say that already, there's so many people working on it, it's not that easy. So, at the end of this talk, I'll get a bit more technical. So, let's get started. I'm not used to doing it. I have to rush overhead.

7:30 The Imperial College of Wisdom, to celebrate the 100th anniversary of our founding last year, gave its trust again to everyone. Morse and Reyes seduced me by introducing me to some of their heroes. So you can now deduce what they'll like to count my presentations to. I'm not very good at this, it's not like that. Let me start with quantum gravity. Now, everybody likes to talk about quantum gravity. Anything vaguely dubious, what kind of is it? Anything interesting, anyone, any mathematician trying to find something of interest against the niceties of physics always has the heart of quantum gravity. Now, why do they do that? Why do we do it too? Well, there are good and bad reasons for that. The good reason is that actually, the problem of quantum gravitation is still a major issue in quantum physics. There are certainly basic equations, but at the basic level, there are still certain fundamental questions, which most people agree are there actually. But to worry, they cross across the different probes that are quite uninteresting and transcendent in particular. It does raise deep problems as opposed to the mathematical and the philosophical kind. And we start off with general relativity. We know that in general relativity there's something called space-time, for whatever reason, which is represented in mathematics by the word for manifold. And that's the mathematics of it. So you've got good old set theory, you've got manifolds, you've got differentiable functions, the whole whack. And from a philosophical point of view, in a sense, general relativity is the ultimate classical theory. It was the ultimate sort of apotheosis really of classical thinking, general relativity. And in that sense, it's the ultimate realist theory. Well, having said that, the reality of space-time points to the fact that we're in Jupiter is meant to be real, isn't it? Why does it go to the capital of the Earth? The Earth is actually very formative, very intangible, coherent, and it's good for space-time, if you want. Now, just let's walk these points around. It's saying that points have no physical meaning whatsoever. And there's a certain sense that's actually true. The motion space kind of point is the point of the manifold. There's no physical meaning at all. And that's correct.

10:00 So that's one sort of, it's a cautious part of that point. Another one is something, again, it appears as if it's been said for a long, long time, Custom space, phi, general relativity, and a whole sort of thing. Use real numbers to place them and say where you are. Now, the real numbers are very abstract. Think about it. What do you capture from these little bubbles? Mathematics, reports, and so on. The real numbers are very abstract. If you go through the technical secrets, they're abstract. But if you've done a series of courses, you know where you're going. It's not very obvious that you've got to model things in physics. It's not obvious. Is there some sense that we can say it's an approximation? If you ask any of our 500 graduates, there are a number of places here that might be able to do a much better work than an approximation. Can you say that the question of the root of pi is essentially the same as here? Well, it's a possible thing. It's an approximation. It's what? It's a possible thing. So it's a long time. This has been somewhat dubious. We've gone all the way down. Let's do a version I've taught you. Okay, that's general relativity, and then we have quantum theory. Now, one of the problems of quantum reality is that quantum physics is normally formulated, works within a given fixed space-time factor, and there's always the ordinary possibilities, you know, we've got psi x at the x belongs to r3, that is the fixed space in which we're working. Then we have the T on Trolley's equation, that's the fixed T, so there's an external factor. You have quantum computers doing quite a second thing in class, externally in class, without things done in space time. And the reason for that, one of the main reasons is that power is partly a perception-international interpretation of it. It works very well. The problem is that it does not in itself, in its normal way of thought about it anyway, say how things are. It's in terms of what would happen if you were to make measurements. And measurements are things that take place in space and time. You technically need to do it somewhere else. So, place and time increase and pose. They're given and fixed. Now, so where do they begin? You can see from a very, very general perspective. If you try and bring together quantum theory and geometry and relativity, this is very well developed. And what's exciting, comes to me on this slide, is how can we apply these things to space and time and so on, you know what I'm saying.

12:30 The best point, anyone giving a talk on these things, always looks at the audience face, meaningfully says, Plank length. Plank length. Now the Plank length is the length which you can just get if you take this combination of things that arise in physics. g is Newton's constant, that's gravity, h bar is Planck's constant, so that's quantum theory, and c is the speed of light, well... You take the square root, and lo and behold, you get the sum of the units that you make. Now this is mystified by people for decades, but it's true. And if you turn it into seconds, it's none, because it takes a very tiny distance. It's very tiny. The wisdom is that you say, well, something dramatic happened. This is a sort of conventional analysis of physics at the very heart. We don't really know what that is, and therefore you can motivate anything. Is there any research where any graph would apply to all kinds of data? You know, a chapter of promotion, an interview, and so on. Someone says, why are you doing this? You say, blank blank. Something mysterious happens. And if that seems to be true, the question of what that is and not so is, what it is. Now, if you look at the main... I mean, at this current stage, there are more than one... There are many research programs on gravity. The biggest one, like quantum reality, is string theory. The next biggest one, in terms of the kind of people working on it, is loop quantum gravity. And there are other things, too, which are a good one. Now, in their own separate ways, they both negate the idea, roughly speaking, of the game of points in space and time.

15:00 They both suggest that motion space by points is not really that fundamental. I mean, string theory obviously is even loops and stuff. So, in some way, these are both suggesting that... Points don't make much sense, and the idea is that if you were to get down to this point, then it would always suggest that what we need is a non-manifold, well, it's in the point of wisdom, of course, that's a bad habit. You can use... I don't know what you'll have to say, I don't know, everyone knows something. We move on, I can't with that slide. Now, what are we going to do with theoretical physics? Well, remember I've already told you primitive, which is the collection of three things which physicists supposedly knew as a construction building. What is the real world? Then we have the data. Then there's the mathematics we choose to use in constructing the physics. And then there's the conceptual framework, if you like, the philosophical framework, within which we interpret and formulate the theory. And this triangle underpins all theoretical physics. Now the problem in quantum gravity is The first one is very largely missing, although there have been books. This one, real world data, although there have been some interesting discussions about this in recent times, it's very largely the same. So all we have is mathematics conceptual framework. Now, that's a very strange way to think about it, to develop a theory, because all you've got to do is think about what has happened over the years in quantum mechanics. People have said that their general relativity, their quantum theory, they don't fit. What would we do with this? We may just invent something. And they're so strange because it means that no one really knows what they're looking for. I've got to remind myself, if you just woke up one day and you'd gone through a pile of papers under the desk and somehow you got this mystical message that this was a constant field of gravity, would you know whether it's correct or not? And if someone wrote a theory that's unified all the elementary part of this explanation, when you understand it well, it picks all those answers up to your constants. You'd know at once what it was they've done. But the point is that it's not so sure what you're looking for.

17:30 Which is interesting actually, I think that's what I've heard from the students. For a lot of people, they always think that's a good one. No, that's the thing, that's the question. These days, I'm one of the college deans, and it's great fun actually. I get more research time since I was a student. It's good fun being a dean. And deans aren't friendly girls. They hang with deans whenever they interview people. If you get someone in front of you who reports a work you've done, they say they've done this and that, how do you judge whether it's good or bad? Because you can't say Wisset fits the data. Well he didn't pretend he does, but suddenly Wisset agrees to it. Now I often wonder the same thing about key mathematics. Some years ago I was sitting next to Mike McTeer at a dinner. And I said to him, how is it that what profession decides what is good mathematics? He said, what the good mathematicians choose to do. I thought he was going to say, slightly circular. It's kind of like music, you see. It's kind of like a composer writes a nice tune. Why is it that people like it? All I've always seen is if you get a scientist and you invite them, they go their way. Now, I've been working in quantum physics for so long now, I actually didn't see cyclism as happening. For a while, people turned to cyclism and all the cyclists shouted at them and they'd come to think of it. They'd die their way. Now, if you're lucky, if you're a young person, if you're lucky, if you're working on something which at the moment you've got to be quite a bit, it's thought to be some sort of built upon from people. Because it is curious, and I was once to go to the Potsdam Institute in Germany to do a lot of generativity stuff, and there's a lot of feedback about the Institute, and when I was thinking about setting it up, I was invited to go there and give a lecture to the committee within a quarter of a month or so, whether it was worth it or not. And I was asked to give a talk on quantum gravity. So I've been honest with the chair, and I stood up. I more or less started off by saying this, I said, really speaking, there's no quantum, there's no quantum science at all. The way that Popov would recognize it, you can't falsify it.

20:00 Some have gone to science. My close-up absolutely followed it. Anyway, I gave it to mine anyway, so... What people do? Well, a lot of work is on my knowledge of other theory. Quantum theory, quantum dynamics, anything you happen to know. I mean, you take that and you start from there. Now the other thing is people simply indulge in personal mathematics and personal philosophical criticism. And if you only, you see, if all you've got is those two... What else do you know? Paul Bagus once said to Q3, he said, each year I walked into the maths library, closed my eyes, pulled at the bottom shelf. That became maths I'd use for the next year. Anyway, I could say a lot about this, but I won't. But just to clear this fact about research in this area, is that it's not like ordinary science. And sometimes it's worth re-resonating, because we ask most people in physics, we tend to forget that. Obviously, human physics is bad. So then we come to the, well I think that's one of the main questions. What is the strength of ordinary quantum theory? When applied in common gravity, and this is the thing that's worrying me for a long time, is that most of the, say, one of the big sets of people won't stand up for it, do please suppose, all of them absolutely won't. Now, in the one sense it's understandable, because what else can they do? It's what you've got. But nonetheless, it's an extremely suspect. There's two critical issues that have been at least partly motivated by what I did here in Los Angeles. What is the a priori use of real numbers? If you think about it, one of the standard formulations of quantum physics is the Hilbert space, incisor, algebras, deprivation, quantization, geometric quantization, quantum logic, path integral, whatever you want. All of them more or less presuppose the standard use of C and R, the properties of C and R. And that's one issue that I find very, very problematic actually. I'm very beginning in trying to formulate a theory of quantum gravity, presuppose the existence of real numbers in the continuum, and that seems to be very, very difficult. And the second thing, which is a very old and actually very important interpretation of quantum theory, is that quantum theory as normally understood is a fundamentalist theory about what would happen if you were to make measurements, but whereas the star space and time are concerned, in particular, one of the ways in which we are, we don't actually need the space and time to actually do anything we need to make a measurement on space or a measurement on time. How long would it take to make a measurement on time? You might measure time with a watch, but that's not the space I'm curious about. I'll reboot this.

22:30 What I said was big, now it's small. Okay, so we have these two problems. Now, to roll with real numbers, please. And how do we actually use real numbers in physics? There's many different ways you can probably look at this and say this, but it seems to me there are at least two different ways. The first is the values of physical quantities. We assume energy, momentum, position, all the things we talk about are real numbers, in particular. If we're units, of course, we take that into account. It's a person. The second thing is that they arise in quantity, the classical definition, is values of probabilities. And the third part of the book is to play off our fundamentally agreed known models of space and time, in the case of geography, in the case of geometry, in the case of classical physics. Now, I will claim that the use of real numbers, complex numbers, is actually a reflection of these two, and indirectly of this as well. Let me come back to this question, first question. Why are physical properties assumed to be of real value? In the case of the silly question you asked, it's actually why the use of curves with the energy of something is so classic as a real number, like root, pi, joules. Why do you suppose that even makes any sense at all?

25:00 Now, I think the reason for this is certainly traditionally a highly-dimensional rule. Well, broadly speaking, you measure things, right? You measure things in space and pi. All of them themselves have a continuum spectrum. Why do you want the extra-negative spectrum? Because that's what you think space is. So you go into quantum mechanics, you want to get something about the structure. If quantum space is not like that, the whole thing is absolutely, it's not that you're using one to make this idea for another. There's no actual logical agreement between them. And the more I think about it, the more I think that this is true. You make a very, very serious error if you start off, you know, always use it really and get that sort of thing all done. But then, of course, the question arises, well, if the quantity of quantum space is not real, What happens to the Hilbert space formula? Because it's very rigid, it's very hard to change that, so I'll miss that one. So the big problem is this, as I see it. The standard quantum theory, in its essence, in its formulation, is grounded in ideas of Newtonian space and sum. I think that's a category error. So the question is what do you do about it? So how can you modify the formula, or generalize it, so as to be, if possible, realist? In inverted commas, which is important that we do. But in some sense, we're able to see, put their space in front and see if they're awake.

27:30 But also, not that we depend on people who are awake, none of the other comments, none of them. And here's the, here's the hint of something, something I'm deeply fond of. I mean, um, now that we have Ferdinand, Ferdinand, Ferdinand, who works on causal sets. Now, causal sets are an interesting thing, because they're discrete, they're discrete models of space and time, which are based in that part of the set. And the... The order of these can't be some sort of causal flow of things, so these things are involved in some sense based on points or units, points are a great place, and these indicate the causal links between them. Now, supposing for some reason, for something new, that there was such thing as the causal set of space, then if that's what we have in place of Newtonian space, what would be the appropriate type of quantum theory for that background? Because I'm starting from scratch. What difficulty is really in complex numbers? I have to go back to the idea of making measurements, values of physical quantities. The values of physical quantities have to be determined in terms of the structure of this thing. Just as in ordinary quantum physics, we determine the values of physical quantities in terms of the given determined space. So I think this is a huge question to ask, and it's a totally fascinating question. At this stage, I've no idea what the answer is, but it's enormously, I think, very intriguing challenge, and certainly one of the motivations for what we're doing. It's very difficult to answer it because the... Hilbert space formulation is very, very rigid. So it just raises the general question then, what actually are the physical properties? I mean, if you say you want to get beyond quantum theory, what is it going to keep? Now, as I've said, um, Ambrose has perverted me from the use of this, this B-metric thing. Now think about B-metrics as it looks like a powerpoint presentation. Now, there's a B. One of the things I get to sit through is endless powerpoint presentations. It's not a powerpoint presentation, it's a quantum presentation. The thing about mathematics is that it's fantastic. They're very presentational. They're the worst part. They're lovely pictures and quotes and very expedient. Now, I cannot possibly reproduce that. Now I have a slightly, slightly, slightly two-set. Anyway, I can't resist having quotes.

30:00 I always have quotes in PowerPoint. Now, here's the point. I think I've thought about this. I mean, I actually rather, I'm interested. I'm very interested in this. Of course, it's captive. Here's a particularly interesting book called What is a Thing? And that is the fundamental question. That's all we're talking about. What we're actually talking about today, is what I'm talking about, is what is a thing? What do you mean by that? Here's how it is, and there's a book called that. It's gone quite early on. It says, from the range of the basic questions of metaphysics, if you're here, ask this one question, what is a thing? The question is quite old, and that's true. It plays a wide amount of a piece of credit. Because whatever example you're about to use, it immediately must be asked again and again. So it's absolutely wrong. The whole history of physics is about asking the question, what does it mean? I'm asking the question today, but I'm going to give you a topos answer, if you want. So I'll ask you, maybe one of these. Now, his idea was answered in his own question, if you look at page 60. This is very, very interesting. A thing is always something that has such and such properties. Always something that is constituted in such and such a way, that this something is the bearer of the properties. There's something, as it were, that underlies the qualities. So his concept of a thing is something, you know, the alphabet is something, you know, a thing is something which somehow bears properties, so for him a thing, a thing, as he says, is a fundamental property, and a property is the way things are. So for physicists, a property would be the energy secret of things. So that's what a thing is, it's nothing but that. Now that's interesting because That's precisely the classical view of quantum physics. That's exactly what goes on in quantum physics. The second, the first, let me just say that as far as construction theory and physics is concerned, in general, what we need to do is get mathematical models for three things really. One is the space-time-space-time of quantum physics. The second is physical properties. And the third is the notion of states in the system. Now, the state of the system, the state is what is something we specify as the way things are. It can be a history of things, but it doesn't have to be anything, does it? What do you mean by the value of physical property? I'll come back to that question again. What do you really mean by it?

32:30 How is this related to the value of the topology? Say that something has a certain value. Of course, how does the state of the system? These are the fundamental questions that physicists always ask for any theory of physics. In fact, there's a long history of physics. Now, let me now introduce a bit more mathematics in terms of classical physics and show how in classical physics Heidegger's perception is totally real. So, in classical physics, what do we do? Well, we have something called a space of states, which is this calligraphic S at the top there. The physical quantity A is represented by a real value, and the interpretation of that is that the state S, little s, determines the way things are, so in both the time, and the value of the physical quantity in that state is a real number. So, any state A, you can determine the value of a physical quantity by working out what A of S is. And that's how it works. Any physical quantity has one of these. Therefore, the very human state you can predict, or say specifically, what's the value of everything. And that's naive realism. In particular, that means that everything has properties, because if you take, say, a subset of delta, for example, then you can... I can't say that the physical property of A belongs to delta. That's saying the value of the physical property of A belongs to that subset of delta. That's a proposition about the system. I mean, delta could be a single. How do you represent that? It's simply A to the minus one of delta. It's all those sets in S. Well, sorry, it's points in S. In other words, this represents a subset. In other words, that's the fundamentally important part. Thus, physicists set up the mathematical retune of necessities, propositions, or property of people, represented by subsets of a cell. And then the mathematics comes in, because, of course, you know very well, the junction of all subsets of a set is the Boolean algebra. And therefore, of necessity, they follow that. So Heidegger could have added, and he would have thought, and by the way, the answer is Boolean. But that's exactly what you have to do, isn't it? You can't get out of it. As long as you think of yourself weirdest in this way, and things have factors, it's very, very important.

35:00 And also, of course, these problems should form the deductive system, so you can argue about the system using these propositions. That's our classic principle. The great entrenchment of the fact that it doesn't work in quantum is as famous, though, as it's not expected. You certainly can't do this if you can't assign a consistent truth to this, right? There's more propositions in quantum. In other words, you cannot talk about properties of things as highly as you want to do, as you want to do, as highly as you want to do, as you want to do, as you want to do, as you want to do, as you want to do, as you want to do, as you want to do, as you want to do, as you want to do, as you want to do. It says that it's impossible to physically simplify it, but the thought isn't what it is. So from Heidegger's perspective, there is no way to do it. Socrates told Heidegger this. He said, well, ask Thoreau. Well, the word is the way it is. I mean, the business of other things, the people of other things. And of course, that's in the sense of what Charles Paul Planck had to say, because instead of talking about things as things actually being there, he had done it in Dementors. He talked about measurements. So what would happen if you did measurements? Of course, you should make measurements in the human space, for example. This is why you get the sort of depression. We have to get rid of this, what we call mathematics. Now, here we get to even some toposciences, because let's step back a little bit and let's take a step back and look at what I've said about how we represent physical, physical concepts in general theory. So, we start off, again, with classical physics, right? So, we're looking at these propositions, A belongs to classical. Classical physics, simple. There are state-spaces, manifolds, and so already A is represented by a function, C is a smooth function, and then these are propositions, you know, which are subject-space, they're called quantum mechanics, so it's very different.

37:30 Quantum mechanics, state-space is a Hilbert space, two-space, you know, physical quantity is represented by a specific operator, and the thing that represents the propositions that you want to develop is in fact the spectral projectors, So it's projected that way, that several projected ways. It's a non-distributive, not something that's historical, it's a logical thing. That's why I wanted to do this. The only thing that strikes you about this is, my goodness, they look totally different. When you first teach politics or learn as an undergraduate, the first thing you're struck by is how completely alien politics is. My first encounter was absolutely polarizing. When I first started lecturing to students, I always found that some students were confused and blitzed. There's nothing wrong with that. And they said, no, no, you must be wrong. I said, because they were highly heroes, you see. They were so convinced the world existed. They didn't believe me. Well, maybe they were right. I mean, you know, but I think it's kind of silly to just establish that. Now, isn't it curious, at least if it was so difficult, wouldn't it be nice if they both did like two different examples of the same thing? Now, how did that possibly happen? Well, because there's a possibility that arises in the structure. Now, you're here because you're just in a category, so I'm going to defend this to you. But I was giving a lecture to a physics department. This is going to be a pretty tough point at this stage if we don't walk out, but you all know anyway what category this is. So I'll simply tell you what I'll suggest about how I mean I'll draw this from what we're trying to do. So let's take a classical picture and let's generalize it just to see what happens. Let us suppose that Blattman doing this, this and this, we instead say well let's say that we have two objects in some category called sigma and r. Sigma is the analog of the classical states, r is the analog of the... And then, physical cognitive scripts by an arrow. Now, that's certainly true and just, but on the face of it, it's totally not true. So let's try and do that. And then, also, let's suppose that for any sub-object of R, there's a proposition, a, b, c, d, which is represented by, hopefully, a sub-object of a signal.

40:00 Is this possible? Does it make any sense? Now, to use this at all, you certainly need some sort of logical structure. Now, in general, it works. But in general, the sub-objects do not form. Mathematical, I mean, this, uh, canonification, as I call it, do, do, canon, can't say, canon, can't, can't, I don't really think I have a theory. I think, this is what I've seen category four. It doesn't work, well, not always, because not in general. Some of them do not, because they're doing the topos. I'm assuming most of you know what, what the other word sort of topos is. Uh, and many of you, many of you may not. I'll simply say there's a category which behaves very much like sex. That's the critical point. Topos behaves like sex. I mean, it's one of the most critical. Ways are important for theoretical physics. That's why it's so accurate. The way in which topos theory looks like a set theory is exactly what you need to use it, which is what you first read about topos theory. So anyway, you know what I mean. What is important really for us is this thing, which is especially important, which we've emphasised, that in the set theory, in the set B, there is a subject B. So it's just finding a whole thing, and then if it is quite correct in B, if it belongs to A, you can say pi of A is 1, if it doesn't, it's 0, and that determines it. Now what's absolutely fundamental to the chop-off theory is the same thing is true, but something called the sum of the classifier, which is, the analog of 0 will be first, the analog of true or false, and we're injecting physically, with the properties, Now, as I've said, the thing about topos is that you do have this logic. Everybody will know that, of course, but you have these two things. First of all, the sub-objects form a H-magnet.

42:30 A H-magnet is almost a blue magnet, quite, well enough for physicists. And the same thing applies to global elements. There are some points where they do form a H-magnet. Now, how do you know if it's a distributive class? That's what's important in physics when you do distributives. It looks like classical logic in that sense, but not quite. The only difference is, as you know very well, you don't have to speak it. But it turns out, well, you know this very well, most of you, but it doesn't matter. All that means is if you have the two theorems constructed, you cannot throw the theorem back on the diction. There are many methods to state that anyway. And so it starts in physics itself. That's why the whole business makes sense, because there are sub-objects of any object that we can relate to. In particular, we have this sort of state object, a significant special object, and the sub-objects of which we relate to. So we come to the mathematics of neorealism. As we be careful here, particularly in maths and physics, what we should be studying about the physical world, what about mathematics, about the way things are out there, what about mathematics, about the way things are in there, or in the heart of the universe. Anyway, it's not actually out there on this table. Well, maybe it is, but that's something. Anyway, you know what I mean. So, if you look at theoretical physics very carefully, what we're going to do is, you start off with things in mathematics. You carefully look at the interpretation of the physical terms.

45:00 It's very important, actually, to keep those two things separate. Let me start off with Gauss's propositions. He's going to accept it. So, again, this won't give a subset K. I said X, so there's one position X belongs to K, once I thought, of course, I don't want to take the part of the philosophy, and the then Bishop of Rome, poor Bishop, a big man, and some of the audience said to him, well, what happens after death? I said, well, you're dead to dead. You're dead to dead. You're dead in the fire, see. None of this heaven stuff. Just you're dead in the fire. I need to do you. There's quite a large moment outside here, you know, in this, you might have an idea of this sort, this valuation, which the value of this proposition, mathematically, is one if it belongs to the K. In other words, if this proposition is true, it's going to have the effect that it belongs to the K. You might think, well, that's because it's orthological, but nonetheless it's true. Right, well, what happens in the topos? Well, here it was fascinating to me that the proposition is going to be partly true. So when I read this, I thought, well, I'm going to miss quantum theory, because in the sense in quantum theory, we have superposition subsets. In fact, if you say the spin is up or the spin is down, neither is true. Neither is absolutely false, you know, if you put the top somewhere in between, you can't say that the spin up is true, you can't say that the spin is false either, because the top is zero. So it's sort of in between, and somehow it stops like this in an absolutely tailored way. Now the way this works is very simple to say, if you have an object in the top box, it certainly does have a global element, there are some global elements. This is just the composition of this. In fact, it's just this. Any points of here, you have this total of one, which doesn't matter much. And the truth value, explained from this point on to that, is just the correlation.

47:30 And it's absolutely topologically important because that's, in a sense, the kind of way truth values come into science. Any use of truth values in physics, everything to its propensity, we're going to have to go back to this. It indulges the global elements, won't it? The global elements of omega are going to be... What physicists, we should say, are the finest physicists, and our claim is that if you do this, you can think this as a certain type of, we call it neo-realism, because it's not exactly realist, in the sense that things aren't just all thoughts, and I don't dare even look at it, in between states, which seems to be physical. Now, here's our main contention. Actually, this is a problem. Our main contention is this. You can profit somewhat, even, if you consider polygenalization as a breach system, each given theory type. My theory is not being classified, it's not present at all. De-physic, de-physic is a theory of physics, doesn't matter. Existence of S, which in theory is applicable, you can formulate it, interpret it, realistically, in a particular topos. So in the Claiborne exam, I think they were quite part of a movement. There was a topos, which was one topos you would use for quantum physics, and a third topos, so you make sense of it. They're different theory types, they're different top-way representations, so set theory, you have a certain representation system. So classical physics, you can always represent things in sets, but then in... So that's the idea, and then within that structure we'd say that we have in any given representation of this type two special objects, these sigma and i, which are the analog of the state objects in classical physics and the real value is thought of as the values of physical problems, these two objects, and any physical problems represented by an arrow, which was always true in classical physics, and then the second plane is the probability that s is represented by sub-objects of sigma, and these form...

50:00 So that's the logic you have to use in propositions of logarithms. Like I said, it's perfectly usable because there are many places in this library that it is anywhere you can use it. And then the third thing is that there's something, the analog of a state is something called a truth. What we call it now, what Jeremy and I have now written down, the truth object, but we're now calling it the pseudo-state because it makes it more clear. But there's an analog of a state in topos, which will give you... Truth values of things. You have to specify what they're just a bit more subtle. That's the blame. And then, politicians themselves are all going to be assigned truth values in. Now, it would be pretty idiotic to say that every theory you're interested in could be real. What we're saying in this space is that what proposes a very general talking is the construction of new types of things, which is certainly a world beyond understanding. I mean, well, I'll come back to that later. There's no question about that. You could invent new types of things. So, the theory expresses the way it looks like it has to. Except classical physics always employs the topic of success, whereas other more subtle theories, including quantum theory, we assume, grab you to the point. And if you want, although I only talk about sigma-9, you of course can also include space-time objects, which you should do as well, which you certainly can, in which case ideas are simply junk. So if you want to do that, include that sort of way of thinking. But that's the idea. Now of course the interesting thing, as the experts will note, is that topos can be used in the foundations of mathematics itself. When I first read this I was absolutely shaken, because I wouldn't have placed myself on it. I always believed there was a universal transcendence, a beautiful world out there, lots of lovely people living in it, lots of good art management, you know, lots of close-ups, no rectors, that sort of thing, lovely place, where mathematics wouldn't exist. And I thought, you know, the perceptor is this sort of universal outlet, and I discovered these things with the topology. These terms were used as a foundation for mathematics itself. I was absolutely shaken from the core of the level world. Now I thought about this problem, can you have a form for each individual thing, if you start to just make it ridiculous. So I still feel a bit psychologically weak about this question.

52:30 No, this is true, it works not very well, but topos can be used as a foundation theory. And in fact any topos has an internal language, which is very similar to the formal language that can be used in set theory. Now our claim is that if you interpret a theory, you would use the internal language of the topos to interpret the theory physically. So it's kind of why it fascinates me, because it always has these two perspectives, the external and the internal. It's absolutely fascinating to see how, just to forget about the three hours of lectures that we've had, because I think it's been a long time, isn't it? But it is, I'm trying to find the next feature. And so, now, let me define, let me say just very briefly what we mean by a truth object. I don't need to be speedy about it, because I'm actually running out of time. Let me come back to what we said earlier. The basic mathematical propositions are solved, belong to K, as I've already said, or equivalently. Math is a subset of K. They're classable, exactly, because they're equivalent. They're not the same in the form in topology, and this turns out to be very important, actually, in the concept of pseudo-state physics, because pseudo-states, it turns out, are associated with these things. In quantum mechanics, the object you construct in quantum theory represents a state, so it has no point at all. Nonetheless, there is an analog to that statement, which is later to be... Now the way that maths and physics interlock is this. If you, this is the mathematical state, x equals k. This is the physical state, the value of a equals delta. Now the true value of this is viewed physically in the state s. It's one, the value of x minus the delta, where a is a thing that represents. Now, this is a mathematical state. This... It's mathematics. This is meant to be physics. So the statement that the physical quantity belongs to the delta in the state s, physics, is one if mathematically that is true. That makes some good pedantic to insist on that. In fact, it's very important to finance that when you come on to a more subtle topic.

55:00 Now, the problem is in the topos. The state logic may have no global elements at all. So there aren't human choices. In fact, there's the case in the topos. What's the outline of the state in general topos? These are things that we called truth objects. Now the base value you hear here is actually quite simple. Maybe rather than go through these phonographs, I'll just... Let me just make this a little bit different. Let me define, given, in classical physics, given the state S, let me define something called Ts, which is this. But Ts, except for the supersets of S, even physically, it's all those propositions which are true if the state is S. Now, it's easy to see that, if you think about it mathematically, if I call, remember, this A to the minus one delta curve, That the composition, say, S denoted K, is the same subject. It's the same thing, it's equivalent to the same thing, but K denoted T, S. Most mathematically, those things are identical, it's thought positive, you're obviously right, S denoted K, K denoted T. At the same time, it's equivalent to saying that the, you know, what happens in a general topic, which I'll discuss in detail soon, is that, yeah, I should stop there, I'm a little bit younger than I am. I was often had to share a session with Professor Penrose, but I was very down with him. He was very serious. He was a nightmare. He'd never stop. He'd be confident, you know, everyone's lying down to speak. It doesn't help to make a difference. I would try and stop. Now, what we do in a general topos is that you generalise it, but in a general topos, you aren't in the lesson. So instead you have to construct this. And that does exist. You see, let me emphasize this again. The only propositions you have available to you about mathematics, really, are from something that belongs to something, a little extra belongs to something, or you might consider a psychosecurity problem. And that's all we've got, and all your physical propositions have to be rewritten in that form. Now, this is not available to you in the general subject, but this is. And it turns out so is this, and in fact what has briefly flashed up in the press, is that in the general subject, you have a truth object.

57:30 It's a collection of sub-optics, you see. It's like you've lost a class. And then this thing has a k with ontology. It's a k with its proposition in the topos. That has a truth value because that's what that thing's all about. The second version, the truth-optic, is associated with the sub-object of sigma. Now that's like this is, you see. Here, this state is nothing but a subset of sigma. Well, I've been acquainted with this thing. I've tripled the difference from one to the other. So you have the second version of the truth-optic, which is in fact a sub-optic. So, the analogue of this, which you couldn't use perfectly well in classical physics, we now, if we want to, can use this. Now, I think, I used to relate it, you know, I said, yes, 1.3. In the case that we started off with, well, the T was where we started off, you know, which we played with. You just take the text of all the sub-logs, take the AND of all of them, and lo and behold, you get this thing. So, these are things we call pseudo-states. And they're, in some sense, the closest you can get to states. Now, how is all this working? This is a very, very brief. What we actually do, we decide what we do, is we represent a physical system with a form of language. That's what we actually do. Each two physical systems with a form of language are the best. And then different theories and translations of this, in terms of language and physics. So you think in terms of translations of language, that's what physics is. Physics is a translation of language. So in particular, it's a translation of what we do, effects, classical physics, different topologies, quantum physics. We've talked a lot recently about the book. I think that's the case with some of them, and it's like we compare it. We think that what we do is equivalent to saying there's always a translation from here to here, so if you go from English into Greek, Greek into German, or sometimes I say if you go from English into Greek, English into German, the claim will be there's a translation from German into Greek. But we think there's a lot of class to say, I mean, it's very interesting to know that it's true. So that's what this is all about, the construction is all about. How does this work in common?

1:00:00 You have this stuff, you see, what are you going to do? You're trying to construct something, you're trying to capture the essence of quantum theory, and you're also leading to ways beyond quantum theory, new types of construction, you've got to pick out something in quantum theory, which is the most important thing, what's it going to be? And different people will choose different things. Some people will choose superlative principle, some will choose this, some will choose that, but we've chosen contextuality. Which is implicit in the fact that things only have value in the context of certain things. And the idea is that in all the quantum mechanics, which is very well known, is that you can, if you want, assign values to things, but only if the operators can use them. Now, this gets confused in the way quantum mechanics normally talk, because you're talking about things in memory. That's because we're so used to teaching the conventions, we put it that way, when in fact, there are these very simple modal interpretations in there, where you pick on a particular thing which is its subset, and say, this has special status, which may be vulnerable to see. Configuration space has special status, and there are many such modal interpretations. So what Jeremy and I did was, we let us take a clutch of all possible modal interpretations, all possible commutative sets of pathways, all possible ways in which we could talk about the system without contradiction, without recurrence restriction. Maybe you can capture everything up to the other end. And we did this by constructing this complex, the semi-topically topos-appreciative. It turns out that the topos-appreciative of the collection of all commuting sub-algebras, or at least all non-sub-algebras, is in fact the semi-constructed. And we constructed successfully these key objects like sigma-halides. Now, I won't say much more about this because Chris Collins took immediately after me, but we'll tell you about it soon. That's a very interesting one they did. Which is quite a big way of thinking about it. So we constructed this object by hard work, and we worked a lot of functional analysis, and they've done something quite different.

1:02:30 What they've done is they've taken the same topos, and they've looked internally at commutative, except from the ordinary view, and lo and behold, it's something similar. This is very gratifying. It also turns out, and some of you know the work that Christopher Holdman did, but especially his, his final quantile watches, you think it's the same thing. So now we've got three different ways of guessing the same thing. It's the way we've done it, it's the way that, and it's the way that the problem's done it. Now it's always very gratifying actually, it's the same thing but three different ways, you see. The key thing that I've said that's missing, I'll just take two more minutes and stop. It's what we call gastronisation, because if this thing is to be, if this thing is to be, if this is to be replaced by a system of space-to-space in quantum mechanics, it's this object in this topos. The propositions about the system have to be represented in some way by sub-objects. How do we do that? We have a jet problem. Now, we construct something called decimization. I think when the world comes to be viewed by an overall contribution of science, I dare think that I'll decide anyhow to introduce the word decimization. It's done partly as a joke. I don't have time to explain it in detail, but the idea is to take the jet problem, and then the context, the signal phase in the spectrum of it, and the signal phase is the spectrum of the inorganism. So that's the projection of things. And you approximate the projection of things as close as you can get in that set. That's the idea. As close as you can get. If you have this projection of things which is up there in this horrible non-distributive object, you're hurling it into the existential world of all the things that can be, that's why. Existential, you have to hurl it into the world. You have to hyphenate it. So you do that. Anyway, it works. In other words, there is a commensurate with the spectral theory. Well, as I said, that's what Moldy looked at once we get here. This is non-believer. What I think is just remarkable is that each state of science, each quantum of science, gives a true thought.

1:05:00 Now, the truth of this, we wrote this thing down originally, and it's not just, I mean, it works, but it's kind of... If you go to the animal, now the pseudostate, which is just the smallest object you can get, so it's just a nationalization of this platform. You take a pure state of science, take the closest you can get... See, you've got this pre-shift here. You're trying to construct a global set, but you can't. Well, you can't construct this. Now, I mean, if you're a student, you'll notice that this letter is some obscure gothic W. And when we first discussed it, we called them verts. See, why do we call them verts? Because the only genre is German sausages. The verts of the sausages are German sausages. If you try and squeeze it here, it's bigger there. You see, if someone tells you you've got this sub-objective sequence, it tells you to get to a point. So you say, why don't you squeeze it a bit? Make it smaller. We can do them at any stage, you can split them, but if you do, it's bigger than the world, it's just like splitting a sausage. For a long time we were called to these things to accept that's not it, verse is a step. If I'm wrong, please call them verse, I don't think I'll be right. So we called them to these things. So that's, roughly speaking then, and I'll stop at this point, is how we've conceived the programme, which is to try to represent theories in general as physical objects, as arrows and categories, as signals and paths. We've shown that quantum mechanics can be written exactly in that way, and nothing's lost when you do that, you can prove that it's always not conjectured, it's always not conjectured, it's not conjectured, the whole of quantum mechanics is there, you don't look at probabilities anymore, they don't necessarily, instead you have these generalized truth lines, and of course it's always going to talk about this, in general, but what's important is how you can start to solve it with sigma, and because sigma is part of the whole thing, except with the analogical space, it stays, so any given use of this, you need to try to construct sigma. And that's why the next organ is so important, because, as Chris is about to tell you, there's an interesting way that he's trying to consider using the internal language of the tolerance, which I think opens up a whole new way of thinking about it. Otherwise, it's a bit, well, I should stop. Thank you. How does dynamics proceed?

1:07:30 Oh, sorry about that, Tom. Oh gosh, I can say so much about that. The synapse involves time. So, Gretchen, do you mean ordinary time? What do you mean time is seen internally? See, one thing that's clear, and this is a quantum hypothesis, is that space is not ordinary space. Look at it internally. Space is not what you expect. There's a role to the physical space internally. It's internal. Now, what about time? If you think of time as external, then the easiest way of doing this is like throwing a picture. If you ask what the Alfred Heisenberg equation is, there are some, but it's a bit more complex. That's the idea. Of course, the interesting question is really, because this is external, but you can do it. That's easy to say. As you mentioned and also emphasized, there's this very elegant thing about Topoi, that they yield this nice formulation of what it means to pair a proposition with a state. The state is just morphing from the terminal object into some object, which we call the state space, All of that, the composition gives us an object of the truth. That's right. So that's extremely elegant and we could say, well, what's a state space? Well, it's a point of topos, it's a topos, and I pick one object, that's my state space, and then what is out of that are my propositions, and into that are my states. Now that's very nice and elegant. But then you discover, or you say, well, my state object is supposed to be sigma. Then you say, oh, it doesn't have any states, any global objects. From that point on, it gets a little less elegant. You have pseudo-states. I thought we were the evident stars, actually. Let me ask a question before you answer. So my question is, why are you so sure? I mean, sigma certainly plays an important role in something. Why are you sure that you have to identify this as the state object? Why don't you look for a state object which does have... Well, see, partly because we saw not a genuine quantum candidate.

1:10:00 I mean, this wasn't meant just out of the blue. It was something that actually appeared out of the blue.