Daseinization and the Redemption of Quantum Theory

0:00 As you all know, he's a professor of physics at Black and Black at Imperial. Chris has spent much of his professional life at Imperial. He did his undergraduate degree there in physics after having a brief career as an electronic engineer working for British Telecom. He also did his PhD there, and apart from, I think, a three-year gap at King's College, is that correct? Yes, correct. His professional life degree in Imperial. Of course, he's extremely well-known for his work in theoretical physics, and he's made important contributions to quantum gravity, quantum gen, retro-dynamics, blue quantum gravity, and the other things. He's one of the pioneers in the application of category theory of physics. He's also, of course, a friend of philosophy of physics, in fact, is a philosopher of physics, in fact, is a philosopher. And his work, for example, on time, some of this in collaboration with Jeremy Butterfield, these papers are compulsory reading for anyone interested in the foundations of quantum gravity, the foundations of quantum mechanics. And again, I refer the papers by Chris, and in some cases, joining with Jeremy on the application of topos theory, the foundations of, the foundational issues of quantum mechanics. So it's a great pleasure for us to have Chris with us today. I'm not even going to begin to repeat the title. There's a word in the title that I didn't believe existed, and I pointed this out to Chris, and he demurred it doesn't exist so I stand correct over to you Chris thank you my phone doesn't exist I even know how to conjugate it I dasanize, you dasanize, he dasanizes we change, etc well thank you very much of course thank you for the invite and thank you for coming are you aware that England are playing football at this precise minute so I'm doubly honoured that you've chosen me rather than the other possible entertainment so the actual title of this talk is dasanization and the redemption of quantum theory. I did actually realize that Tim emailed me in the middle of the week and said, what's your talk about? And I realized it's not actually totally obvious in the title. I like the title, but I feel it's completely clear.

2:30 So, in fact, what I'm going to talk about is work with my young colleague, Alvaro Storin, who is here from Frankfurt, a mathematician. And it's about, in a way, you can see it as a very big extension of the work I was doing with Jeremy. It is about Topos theory and its application in quantum physics. And I hope towards the end of the talk you'll see why the verb to dasomise is not actually so silly as it might sound at first. I think that's quite a good word. How should we start? Well, let me begin by saying that classical physics is extremely boring. If there was nothing in the world about classical physics, I would have stayed an electronic engineer, that's for sure. In fact, it's so boring that we do something to it. I do like the way in physics we have verbs. I've always been struck by the fact that we talk about quantising systems. Something you do to the theory, you quantise it. And the clear implication is when you quantise it, it's better for being quantised by and large. It's an improvement. So you could say that quantising liberates the theory from boredom to quite the opposite. In fact, when he overdoes it, because it then becomes incomprehensible, unless you're an instrumentalist, which is great for making money if you're intel, but if you feel that you wonder about the meaning of what you're actually doing as a theoretical physicist, you want to go beyond that. If someone like myself, who spent most of their career working in quantum cosmology and things to do with quantum gravity, it clearly doesn't work to have a theory which fundamentally, You would say it has to be interpreted only in instrumentalist terms. So, we have this curious situation then. You start off with classical physics which is boring, then you quantize it, it liberates it, but makes it too incomprehensible. So the theory now needs redeeming. We're not going to get it back to the classical level, where it would be boring again, you see, but somewhere in between. And hence the, um, an attack for the verb deducsonism, which is what it means in this context. Now, from a technical point of view, one could say that all early attempts to make quantum mechanics intelligible, if you'd like to bring it back towards the more sort of realist view of classical physics, founded technically on the infamous Crockett Specker theorem. Physicists do differ a great deal how they view this theorem. Personally, I think all first-year physics students should be told about the Crockett Specker theorem. And in fact, the Chapman Imperial College who gives the first-year physics course on quantum mechanics

5:00 very introductory, gets me to give a guest lecture each year. What it's about is the Coughlin Sprecher term. I think it's so important they should know about it. And I'll explain a little bit later why that is so important. So this was the situation really, which Jeremy and I tackled six or seven years ago now, was to try to find a way of understanding quantum mechanics which would be, well we introduced the verb neorealism because it sounds better than realism or beyond realism. So non-instrumentalist and not just a naive realist. But we got hung up on various things. We didn't actually confess this in print, but in fact, there were two gaps in what he did, and we sort of both led on to different things. Jeremy moved here, and I did something else. Anyway, we've come back to this now, as you'll see, and say, working with Andreas, we've got a whole new take on this now. So let me start by reminding you of how logic comes into classical physics. I'm going to make what, for me, is an unusual statement, is that, in fact, what Andreas and I are doing, we think, is not just constructing a new way of looking at quantum mechanics, but a new way of thinking about theories of physics in general. Now, those are my friends in the audience who know I don't normally make that sort of enough, unless I mean it. So we do think, actually, there's something really quite interesting here, about the way you formulate theories in general, the mathematics that you use, that this is all based on how logic comes in. I want to explain, and I apologise if it's a bit rudimentary, this, but remind you of exactly how it is that logic arises in classical physics, first of all, and also how this is related to what you might call the philosophy of naive realism. So naive realism is basically the view of the world my mother would have had. My mother was the most quintessentially normal person you could imagine. How I appear is a mystery, but my father also was the same, in any event. Both my parents were naive realists. Everybody is. You are, because you're professionals, but outside this room, everybody is. I think the world exists. You say something is true, and if it's true, you've got that it's true. It's obvious, really. Why do you need philosophers? So how does this work in physics terms, if you are a naive realist? It works in the following way. We start by talking about the notion of the state of a system. When I was an undergraduate, I remember actually being very confused as to what the word state actually meant. My experience was very early on in my undergraduate career doing physics, the lecture would talk about states of the system.

7:30 There were classical states, there were quantum mechanical states, there were thermodynamical states that I found really confusing. people talked about. Well, the idea, of course, of a state is a state is meant to be something if you know what the state is. Somehow you know all that is possibly known about the system. That's very simply the idea of a state. And so you always have in classical physics this notion of a space of states, which I called calligraphic S here, and then if you know, if it's the mathematical states, if you know what the individual state is, you know all that is known about the system. Now, what does it mean to say orbit is to be known about the system, well, it means in particular that you can say for certain what is the value of all possible physical quantities you could imagine, energy, momentum, and so on. If you know the state, you know the values of all of those. Now, how is this modeled mathematically? It's modeled mathematically by saying that physical quantities are represented by real valued functions on the space of states. So the idea then is to each state S you associate to a real number, which is the value of that physical quantity if that was in that state. So, because I want to be very careful here, I've been careful of notation. The physical quantities I'm just referring to is capital A, and they represented this mathematically as A tilde. So A tilde is a mathematical function, but what it represents is energy or momentum or whatever. So that's how we do that. How, then, do propositions come into all of this? Well, propositions in physics and classical physics are statements of the fact that a certain physical quantity has a certain value, or, perhaps more generally, a certain physical quantity has values that lie in a certain range. Now, how is that represented mathematically? Well, if, again, I want to be careful here, we think of the proposition, which I shall write like that, as the energy belongs to a certain range of values. How do we represent mathematically? Well, the answer is they're represented always by subsets of the space of states. Why is that? Because we simply say that the proposition is represented by all those states for which it is true in that state. The corresponding physical quantity does actually have a value that lies in delta. In mathematical terms, they're simply saying it's the inverse of this subset delta, the real line. So this yellow area here is all those states which is true that a tilde of s belongs to delta, i.e., which the proposition is true in the obvious sense. And so we see immediately that propositions in classical physics are necessarily represented by subsets in the space of states. Now, in a way, that's absolutely obvious, but it has the most extraordinary implications immediately for the way we talk about the world of necessity, almost like priori, by using this mathematical framework.

10:00 have to talk about things in terms of Boolean logic. You have no option about this. This is what my mother would have taken as obvious. Because what we're saying in general then is that properties are represented by certain subsets. So let me use this notation. P is a property. Yes, subscript P is the corresponding subset. And now we can simply ask, well, okay, suppose we have a pair of propositions, say P and Q. What would represent, say, P and Because if P is represented by SP, you'll see the yellow shaded bit there, Q by SQ, then P and Q is simply all those states which both propositions are true. It's just the intersection of those two subsets. So, of necessity, the logical conjunction of P and Q is represented by necessity, by the intersection of the sets. You do the same thing for the OR operation, it's the union of the sets, and in the same way, the negation operation of logic, not P, is simply the complement in the set-theoretical sense. It's all those states which lie outside the set of states which P is true. And so, by this means, what you've actually done is you've mapped, in a sort of essentially one-to-one way, of propositions into subsets of the space of states. And in such a way, you also know what the logical operations are. Now, this has one enormous immediate implication, because the logical structure of the subsets are a set, as well known as the Boolean algebra, Boolean logic. So, of necessity, therefore, you can say that in classical physics, of necessity, the way that all these theories are set up is such that, necessarily, you We would talk about the world using Boolean logic. Remember, at any starting point of it, really, was the idea of what I call naive realism. You knew the state of a system, you knew everything about the system. It led to the notion of functions, etc., and so we built up this mathematical model. Which is why, I suppose, it seems so difficult for people to think about physics in not these terms, because all classical physics is like this, and it just, of course, fits in the way we think about the world in general. But that's how it works if you're a physicist. In particular, of course, you get this distributive law. The E and S or B is E and S or E and B. If you're wondering what the letters E, S and B stand for, the answer is eggs, sausages and bacon. When I talk about this to you, I'm the graduates. I'd like to imagine I'm... No, I'll wait for it. You're too dignified to say this to you, anyway. I imagine I'm going to a hotel and I'm offered a breakfast of eggs and sausage or bacon.

12:30 You see, the question is, what are you going to get? And if you're a classical physicist, it's this. If you're a quantum physicist, it's this. You can't eat nothing at all. But anyway, I should pass off the night. So this is how classical physics works, and you can see very clearly how this has come out of the mathematics. Now, this is very simple. This is how you must represent propositions. Now, let me come up to some high-level discussion, which is how we talk about truth values of propositions. The reason we're doing all this in this detail is I want to argue that in general, we should think about replacing sets with objects in the topos, which is a huge generalisation of set theory, but we still maintain the same underline logic this is why I'm doing all this in sub-kill so let me come up now to talk about the truth values of propositions now, first of all, let's talk about this in the point of view of physics if you're a physicist, classical physicist and you're a realist heart, then the basic propositions as I say, you would interpret as saying well, the value of A lies in some subset delta real numbers, and then would say, generally speaking, for any particular state, S, that proposition is either true or false. Either it is true in that state of the system, and the energy has that range of values, or it's not true, it's false, simply that. And let me be very careful and denote this truth value by this square brackets with little s at the top. So this means the truth value of the physical proposition, A belongs to delta, given that the state is S. So this is asking about, if you like, philosophical questions about the world, not mathematics Now, if you're a mathematician, on the other hand, where do truth values come from? If you're a mathematician, modern mathematics is essentially, part of my about, say, later on, grounded in set theory. And in set theory, the basic relation is one quantity is a subset, or belongs to another set, basically. One set belongs to another set. So we have this classic notation here, is that if we imagine having a set X and K is some subset of X, then if little x is a point in capital X, x epsilon k is the mathematical proposition now that the point x belongs to the subset k. Now again, of course, in ordinary cities, it's either true or false. Either an element belongs to a subset or it doesn't. So in that sense, it models very well the classical world, which is hard to supply them. So the two things are very, very close at this point.

15:00 But because I want to be so careful, I'm going to use a slightly different notation from this. I'm going to leave off the quotes. The quotation marks here means the physics statement, and here it means it's a mathematics statement. Also, I've used the word in here to remind you this is about physics, and here it reminds us about mathematics. There's a reason we've been so pedantic, believe me. Now, in fact, from a mathematical point of view, these truth values are given by the so-called characteristic function of the subset. This is a function from x into the point, say, 0-1, which you can read as being false and true. and then the truth value x belongs to k is simply this function acted on x this is based on the value 1 if x belongs to k nor otherwise, it's totally tautological since it's very elementary alright, so on the other hand we've got the physics idea or if you like the conceptual idea of physical quantities having certain values where that proposition is true or false then on the mathematical point of view we've got the notion of the fundamentals of set theory in particular this fundamental x epsilon k statement which is either true or false How are these related in physics? How do we actually lock these two things together? Well, the answer, in a way, is sort of obvious, because I've just said it just now, in fact, on the previous transparency. At least, one of the answers is obvious, but there's another one which is much less obvious, which turns out to be the thing you need to focus on. Because, clearly, this proposition that the statement that the physical quantity A has a value lying in delta, given the state is X, is the same thing as the mathematical statement, but little s belongs to this shaded area. It's almost tautological, but this is none the less it's mathematics. If you like, this is physics, if you wish. Um, so A's the mantle of deltas, all those states s to which A children of s does belong to delta, and so in a sense it's obvious, but none the less not totally true. So if you like, we just call it that. So we can say for each subset, as I said, each proposition has associated subsets. So this is really the critical thing. Here, you have worse, A is an energy, so A stands for a word in the English alphabet, or German alphabet, if you wish, whatever. And here, these are just mathematical symbols. It's how they're related. I hope that's absolutely clear, because... Now, this is what doesn't work in quantum mechanics. And it turns out the Koffer-Specker theorem can be interpreted. In fact, there's a breakdown of this relation. That's really what I'm going to develop today, this concept. But, more to make this clear, let me say there's another way of talking about things being true. Now, Now, a physicist would never think of this, that's for sure. He or she wouldn't see the point of doing it.

17:30 A mathematician would think about it as a matter to you, because mathematicians always think of things that have no point. That's the whole point of being a mathematician, of course. What it is for philosophers, I wouldn't, I tremble to say, but... I have many friends who are philosophers, so... All right, here's the alternative way of doing it. The first thing you think, this is really crazy, but it's not. Let me associate with each point little s something called capital T, which I call the truth object. And the truth object is all those subsets of s, which do contain little s. So if you like, it's all those subsets of s, which it is true, s belongs to them. Now, if you ask, what is this thing? Well, it's a collection of subsets of s. So what they actually belong to, or it's a subset of, p of s. Now, p of s is simply, p stands for power. It just means the collection of all subsets of S. This is really important at this point, although it seems totally tautological at this stage. So I'll simply define T of S as being the subset of P of S, all those collections of sets of S, which is true, that belongs to that. Now, under those circumstances, you can equally well say that, of course, that is true. Also, you could say this proposition is truth value. That is the same thing as this statement. Now, again, this is a mathematical statement. Now, rather saying that little s belongs to something, It says that this subset belongs to that collection of subsets. Now, at this level of mathematics, these two statements are completely equivalent. But the thing to notice about that is that this thing basically refers a little less directly, in the sense, this is what fails in the Cotton-Specker film. There's no equivalent of these in quantum mechanics. However, this thing doesn't. It has this thing instead. This is a proposition. Now, propositions exist even in quantum mechanics, even if, you know, sort of deterministic states better. It turns out this thing generalises to quantum mechanics, whereas this one doesn't. That's in the one way I understand the Crump and Specker film, right? I'm sorry this seems a bit abstract, but it's actually fundamental I'm going to talk about, is that that and that are equivalent in classical mathematics, if you like, in classical physics, and they're going to be different in the top-level world. Okay, so that's the basic starting point, right? So we talked about how logic arises and how truth values are associated with states of the system. right now then what in fact goes wrong with quantum theory so here I want to briefly review the Copernes-Becker theorem

20:00 as I say since our first year undergraduates get taught this these days I thought I would have a thousand about it so the Copernes-Becker theorem now the Copernes-Becker theorem is a mathematical theorem that's what it is so what it means is something quite different mathematics. I mean, I often thought I should have been a mathematician, actually, because no one asks you, what does it mean? I mean, you don't think it's not a problem. You just say it's like a true thought, broadly speaking. With your physicist, you suddenly say, what does it mean? Now, what the Cops-Decken means is actually quite delicate. But here's what it says. It's a piece of mathematics. It says this. It arrives out of the attempts of people to attempt to construct hidden variable theories in quantum mechanics. So what they were trying they went to say, well, suppose that we have more or less the familiar mathematical apparatus that we know of quantum mechanics, who operates on human spaces and stakes, that sort of thing. And although, on the face of it, the formalism has no room for saying things have values, I mean, what it gives you is probabilities and nothing else. Nonetheless, they said, maybe there's bits missing, given variables. If we knew what they were, in fact, we would know what the value of things were. And so they came back to say, well, do such things exist at all? In other words, can I try to have an operator A in, say, the quantum mechanical structure of some material, some operating in some building space? Can I associate with it a real number, which will be interpreted as the value of that physical quantity in some way? Whatever these hidden variables are, whatever these unknown micro-states are, nonetheless, can we still do this? Now, yes, of course you can. There's millions of ways you can associate operates with numbers. So you've got to be before precise in that. and you add two certain dishes. The first thing is you require the value of the operator A belongs to the spectrum of A Now there are very good reasons for doing this in quantum mechanics because of course in the usual sort of simple interpretation you're told that the results and measurements can only be eigenvalues and so on and so on and so on. So this is, I'll take this as ruins, we'll be going to say it's true so B belongs to that. And then the second thing, and the critical thing goes by the delightful name of Funk I was much cheered up to see the word funk written on the blackboards in our research group two days ago. I thought there were only two people in this group who knew what the word funk is. Not me, it must be Andreas, indeed. He's giving lectures to our students. So what does funk? Well, funk says that if you have some operator being which is a function of another one, the value of what is associated with that is that function of the value of i. So, for example, if I'm going to say I'm going to associate a value to the energy squared,

22:30 that value will be the square of the energy now you might think that's obvious, but you might think isn't that true? Of course, no, of course it's not true because you're defining what you're doing behind these steps but it seems highly plausible so this is what I'm going to assume too this is what everybody assumes, this is funk that the value of energy squared is the square of the value of energy there are some immediate implications of this which are quite intriguing firstly you can show quite elementary algebra you see if you have a pair of operators that commute equals 0, then the value of A plus B is the value of A plus the value of B, and the value of A times B is the product of the two values. That follows immediately from just these two conditions, a little bit elementary algebra. So that's kind of nice. And what it means in a technical sense is this, is if in fact, this is very important, if you take the algebra of operators, which in physicists' terms will be things you can measure simultaneously, remember one of these sort of if you like a lot of the characters, if things could do, you could measure them with simple terms, if ever that may mean. By measuring simple terms, you could do it. And so in particular, if you take off an algebra, what this tells you is this thing, in fact, is a homomorphism from this algebra into the real number. So that, in a way, is an elegant way of stating what funk is all about. It says that you will have homomorphisms from community adapters into the real number. That's very nice, of course, because that's a very clean mathematical statement. So that is what any valuation would have to satisfy if it existed. and this most wonderful Cochrane-Specker theorem which said it was also implies the P's projector, the BP, it was 0 or 1 so we should tend to attribute it as false or true, so this is where true false values come into this sort of stuff anyway, you have this wonderful Cochrane-Specker theorem that says actually there are no such things if the dimension of H is greater than 2, so that's the great blow to realism if that theorem were false, quantum mechanics would be completely uninteresting, just like And you'd do something else to make it interesting. A new verb to splot or something, I don't know, and you'd have to do something else to make it interesting. Fortunately, the theorem is true, and therefore, from the character, it becomes interesting. Now, what are the reactions to Coddance-Becker? I'm very keen on Jung. You may groan, some of you, but I have to confess my sins. And Jung had his own intriguing classification of the human race personality types. But something Jung didn't realise, there was another quite refined classification

25:00 of human race into three personality types, which are according to your reaction to the clock and speck of things. You're either a realist, an instrumentalist, a nostalgic realist, or a neo-realist. Human race divides into three sorts, you see. Now, the instrumentalists say it doesn't matter. Why should it matter? You see how much money Intel makes each year? Do they worry about the copper spectre theorem? Not much. So it doesn't matter. But in fact, famously, about 20 years ago, when I was thinking of 15 years ago, switching more time to working foundations of quantum mechanics away from hard core quantum gravity, I met the head of that then-hazard department outside the lift shaft. He said, oh, I shouldn't. What are you doing these days? I'm thinking of working on the basic problems of quantum theory. What do you mean? Oh, no problems in quantum theory. nuclear physicists, they distinguished them. But for him, there weren't any problems because he was at what he was doing. They built bubble trainers and stuff, and it all worked fine. So, instrumentals, it doesn't matter. Now, the nostalgic realists, they are still somewhere around. There are those who think, actually, the world is simply boring, basically. There's various interesting psychological reasons as well. You might be a nostalgic realist which one can go into. No doubt you're a father and things like that, or Freudian. In any event, our nostalgic I called Calligraphic Dudley the collection of all sort of commutative algebers of operators, so they're the things that look at a classical, you can assign patterns in the same thing. And they will say, well, you just pick one of them. It's pretty desperate, I think, as a thing to do. Nonetheless, some people do it, and I'll come back to this later. They'll start to realise, if they say you should just pick one of these commutative algebers, then the nearer realists, like Jeremy and myself and Andreas, they actually, in spite of the cotton speck of theory, we still use all of these smoothies afterwards, but you put them together into a topos, and so you have everything working for you at once. Now, you may think this tripartite division of the human soul is a little bit extreme, but I invite you to consider which one you think you belong to. So, nostalgic realists, then, typically, if you like, in terms of the cotton speck of theory, they would choose only partial valuations with restricted remains. A good example is the Brougham interpretation of quantum theory, actually. That's a very classic one. In the Brougham interpretation, you essentially say configuration verbs have a fundamental ontological status. It's denied everything else. They do exist in some real sense.

27:30 Well, I mean, you know, maybe that's right. I don't know. I just find it uninteresting, personally, but I'm less presumably right. So that's one of the nostalgic readers to do, and we, as they want to get well away from that. Okay. Well, then, what I need to do nowadays to, um, this is now jumping considerably, um, you should talk about the top of standardization for theories of physics. This is, um, in a sense what Jeremy and I were trying to do when I didn't quite knew it at the time. It now becomes quite clear. Since this is the legal answer I'm trying to do, is to find a new way of talking about theories of physics in general. See, at the moment it's slightly odd. If you use the aspect of classical physics, from a structural point of view, all classical physics theories look the same. They all have the same basics structure, the space of states, physical quantity represented by functions and propositions of subsets and so on. Of course, different systems have different spaces of states, but at a structural level, they're all the same. And then quantum mechanics is completely different, and quantum mechanics is not like that. You haven't heard the space of states, and the meaning of the states is quite different. I mean, physical variables are not represented by functions on the space of states, rather than about operators and so on and so on and so on. So on the face of that, you would say that quantum physics and classical physics are two quite different things. Now, I won't want to claim that, in fact, there's a different way of looking at physics from topos theory, where it actually, you will see, the classical physics and quantum mechanics, both living exactly the same, surprisingly. The only difference is they work with a different topos. The classical physics is working in the topos of set, which is the ordinary mathematics, and the quantum physics is working with a different topos, but apart from that, the actions look identical to those in classical physics. This is very, very surprising. I mean, this seems highly unlikely at first sight. Nonetheless, this is what we think is happening. So, this is the first time I've spoken about this in public, and if you knock it down, okay, you knock it down. So, here's how it works. The idea is this, is to start off thinking about objects without points. You see, the Cognos-Becker theorem basically tells that there are no microstates in the classical sense, or things that would tell you that that is a more physical thing. So you might think, well, in classical physics, the microstate was the point in the set. So it's a bit like saying you've got a set with no points. Now, is there such a thing as a set with no points? Well, of course there's not. A set, apart from the empty set, has points. However, one of the great sort of adventures in mathematics in the first half of the last century was category theory, which made this jump immediately. It's talking about things, if you like, but things like sets without points. I mean, the trouble with sort of classical category theory goes much too far, actually, and topos theory is kind of in between, as I'll explain. In any event, here's, if you like, the vague starting point to say, well, maybe the Kotlin-Speck

30:00 of theorem is telling us. We still want to talk about things like subsets. We still have propositions in quantum mechanics. We still want subsets but we don't have any points. Now, of course, kind of a subset of that many points other than the empty set. But as we'll see, in top-off theory, it's all too possible. In fact, it works well and well. So here's the idea, then, is can we find a set without points to replace the set S of classical physics? So this then suggests this use of category theory. Now, Jeremy assures me, he says, your audience will all know what category theory is. And Jeremy always tells me the truth, so I would take you for God and you know, run away from category theory. I guess in case you don't, I'll just remind you very roughly is very clever, really. When you first see Category Theory, you think, oh, it's just acting like nonsense. But when you get into it more deeply, you know, it's actually not about that at all. It's really quite profound. What Category Theory does is it takes the sort of structural relations that you use in Set Theory and it keeps the relations without the sets. So the most fundamental thing in Set Theory, really, is you have the notion of sets. You have the notion of functions between sets. Now, in Category Theory, you rather than having sex just thought of things for an optic, it would not be thought of necessarily as made up of a point. But you still have arrows between them, or morphisms of arrows. So these are the things that play the arrow of a function. I mean, roughly speaking, as what Caterpillar theory does, if you have simple rules like you have an arrow from here to here, and one from here to here, you can compose it with another arrow from here to here. Just as if they were sex, you could compose the functions in some ways. It's a very useful part of mathematics for many reasons. I mean, apart from anything else, it's many different branches in maths in a way, it's different examples of different categories. For example, a common use of categories that most people meet is a category that appears as objects are strictly sets with structure. So, for example, the category of groups, the objects are groups, which of course is that same, and these arrows are simply structure-preserved in maps, like homomorphism between the groups. So this is just a way of talking about structure-preserving things, and in this sense, category theory is made useful just as a tool, just dealing with groups and rings and everything else However, what I'm interested in is not that, but situations where it's not that sort of category because there are other sorts of categories that are quite different. A good example of this is a particle order set. A particle order set is just a collection of points where arrows go from one to the other. In fact, it's less than that. Now, you can think of those points between objects in the category, and an arrow retrieve them only if there's a... you can go from there to find the order of relations. So that's certainly not an object with a set in structure.

32:30 Now, there's lots of things you can do in category theory which try to sort of emulate things you can do in ordinary set theory but in particular you do have this notion of an element now I'm going to do this sort of straight away because it's so important there are things called global elements of any given object an eclection of all of them is called gamma and how are these things defined? they're defined like this there's a very curious way, talk about the elements of a set you wouldn't think of unless you were a mathematician, I'm sure If I say, take a set with three elements, and of course you know what I mean. If I say, well, what are the elements? You say, well, you know, if x could be done with that, that's what you mean. Well, that's another way of thinking about elements in terms of function, which is kind of trivial when you first see it. You say, let's take a set with a single point, just start. Let's take all those fundamentals from that set into the other one. So imagine you've got this set here with, say, 27 elements. And here you've got this set with a single element. They're all functions from there into there. What can they be? well it's only a single point here just 27 possible functions it goes that point, that point, that point, that point, and that point i.e. it's just a set of elements so you could think of elements in a set as being functions from this one special set of just a single point into the set now if you ask why on earth would you want to do that of course, I mean, yes why on earth would you want to do that and the answer is because you do it in category theory otherwise you wouldn't think of it, I'm sure but Miss Howard does have a very important point because in any category you can ask for the analogue of a set of a single point now I went to such a thing that's called a terminal object I went to the definition here, it's not important, I'll just give the critical example later on but there is a carrying analogue of just a single set, an object just one point if you like, where you define elements to be arrows from there to there and the claim is, this is what Jeremy and I proved, in fact, the common specative is equivalent to the same statement there are no arrows at all in the case of quantum theory anyway, it's just the definition, but I'll come to that reference there is actually to sort of directly set like things. So here's what we have in mind, is that we're going to try to replicate the basic structural things, like I said, were the foundations of all classical mechanics. And the difference is, rather than having sets of functions, we're going to have objects, values and a category. So I'm going to say, well here's a possible generalization of physical theories in general. Rather than having a space of states S, can we imagine that we have an object in some category, and can we imagine that propositions

35:00 correspond to sub-objects, which we suppose are defined in some way, which I would like this, and what's more, physical quantities correspond to arrows from this object, the state object, to something else, which we might call the spectral object of each physical quantity. So basically the values are right, if you like. Well, it's not necessarily going to be set, it's just objects in the kettle. But could we do this? If we could do this, very much everyone, this is the Chris Eichmann and Arnereff doing axioms of physics, you see. Now, what's really fun, if someone stands up and pretentiously proclaims a series of axioms, you immediately prove a contradiction in them. That deflates the speaker, rather than speak. Of course, you can't do that to me today, because classical physics satisfies these axioms, of course. If this really is a our functions and everything fits. So classical physics certainly belongs to us. What's the interesting question is, does anything else belong to this? Or is it just an extreme extravagance that has no interest? Of course, there are lots of questions to be asked. So the first question is, well, as I said, there's already, can, in fact, the Coffman-Specker theorem be discussed in this way? That is to say, can you show a Coffman-Specker theorem is equivalent to the statement that there's just some category of an object that has no points? And that's what the chaos theorem says. And the answer is, yes, you can. That's what Jeremy and I and I'll come back to that for a little time. However, that yourself is encouraging. It doesn't get you that far because the obvious question is where does logic come from? One can't say this too strongly that the fact that the subsets of a fact from a Boolean algebra is of fundamental importance for their use in mathematics and physics. Absolutely fundamental. Now, given the general category, even though they may have things called sub-objects, they will not have any particular logical structure. Maybe we're mighty sure. So that's not interesting. I mean, it's just too general. So that's a very non-trivial question to ask, how does a logical structure arise? And the next question is, well, how are sub-objects, even if there were such sub-objects, how are they related to the values of physical quantities? Because remember that the propositions were things like, well, the value of A belongs to delta. Well, what on earth could that mean over here? Even if I managed to say, well, delta is some sort of sub-object of the possible range of values, whatever it might mean, maybe I could say that, but how on earth could I relate that back to a sub-object of sigma? I mean, a classical set-through, you just pull back the set function. But in general, you can't do that. So it's a very non-trivial question to ask, too. And of course, thirdly, it's the obvious question, is if sigma has no relevance, what could you mean by the truth values, anyway,

37:30 associated with a sub-object? Now, you remember I said there's two ways of defining truth values in classical physics. One is that little s belongs to something. Now, if I know little s's, then he can't belong for anything, obviously, because that doesn't work. So any possibility, if you're saying you have no points, is whether or not the second way would work, where you use these so-called truth objects, and rather than having little s belong to that, you have a truth object that belongs to something else. And in fact, that's what it does work. This is what's interesting and curious, is that there's a split at this point. Even though you have no points in these sets, you're still going to have truth sub-optics, as Jeremy and I have discussed. And you'll see how this fits together in a moment. Anyway, these are the sorts of questions you might ask. Now, in a general category, you get nowhere at all with these things. But the thing about top-ups theory is it's precisely set up to do this. In fact, if you'd imagine going right back to the beginning before top-ups were invented, and I'd got to this point and said, I'm awfully sorry, but that's the end of the seminar, because I don't know what to do. You might have gone away and invented top-ups. It's been precisely those categories which all these things work. I mean, there's something about this subject. When I first started reading it, I thought it really fits. Sometimes when you're a theoretical physicist, if you're interested in exotic mathematics, I tend to be in my career, sometimes you feel your force in it. Of course you write papers and you get published and so on, but even so, in your heart of your heart, you know it's a little bit of a force. It is something that's quite different. It's like you put your hand in a glove and it just fits. You feel the something about this mathematics is absolutely begging to be used. Of course, it may be useless, despite all that, but nonetheless, it does fit. So basically, a topos is a category which, in a sense, all those previous requirements are satisfied. And the reason they're satisfied is you have this wonderful thing called the suboptic classifying, suboptic classifying objects, omega. The way that works is this. Remember I said that subsets of a set are in one-to-one correspondence with these characteristic functions. These are functions defined on the whole space, taking the values all from one. They simply pick out the subset. Now in a topos, exactly the same thing is through, although these are no longer functions, just arrows in the category. The suboptics are in one-to-one correspondence with these arrows. sigma is going to be the analog of the space of states in quantum theory, and of course it has no points, because they're 100. And capital omega is going to be the quantum analog of what was 0,1 in classical physics, true or false. So we're going to get logic in here, and it won't be ordinary logic, it's going to be whatever this logic is in this topos. So that's the first general statement. And the second general statement, which is very, very extraordinary, is that if you take the collection of all sub-objects,

40:00 the products in the top of us, they do actually form a logic. Now, this is completely amazing. I think this works. You have the analog of AND and OR. Even if they're not subsets, even if there's no points anywhere inside, they still have a logic, which is almost like Boolean logic. It's not quite. It is distributive. This is what's called a Haitian algebra. The only difference between Boolean logic and Haitian algebra is, broadly speaking, you don't get excluded middle. You can have alpha or not alpha. In Boolean logic, it's 4 is equal to 1, thought logically. That can imply 1, but not equal to 1. Another way of saying the same thing is that alpha implies a couple of alpha, not the other way round. Apart from that, it's just like a blue logic, particularly it's distributive. So in a haitian algebra world, if you go to a hotel and you order breakfast, eggs and sausage or bacon, what you want is what you expect, which is eggs and sausage, orex and bacon, it's not nothing, which is what happens in a typical quantum hotel. Now, this is extraordinary. In a sense, this is like saying out there in mathematics, there's this huge generalisation of set theory, it's almost like classical set theory, it's almost the real, but not quite. But the thing which it differs from is the very thing you can live with. The only difference that actually makes in practice, if you do mathematics like this, which is, this is the mathematics that actually underpins the intuitionistic program the long, long day, is you can't prove, you can't use proofs by contradiction. To use You can't use proofs by contradiction. You say what the proof says. You say, suppose it's not true, and prove it's not the case, and that's not the case. That doesn't imply the original thing, because that is less than equal to that, not equal to it. So you can't use proofs by contradiction. If you want to prove something like this, in Haitian algebra, you've got to actually prove it by hard work. You can't say, well, suppose it's not true, and I'm sure that doesn't. You've got to actually prove it by constructive means. Apart from that, it's as good as ordinary mathematics. In fact, these days, I can't think it's better in ordinary mathematics. because you can dasanize into this but you can't dasanize into Boolean algebras that's what it boils down to, as you'll see right it's also true that P sigma exists now P sigma, P s is the power set, it's a collection of all subsets now atop us you can show that an object, which is called P sigma, which behaves in many ways like that thing does, of course again it's not a subset in the sense of points, but nonetheless it works, that would be very important to talk about truth Well, if you have a global element... See, we know some global elements don't make sense. I mean, some objects do have elements. They may not have enough elements. You see, in ordinary sets, you have this comprehensive principle of basis,

42:30 as a set is given uniquely by its elements. In a topos, that's not true. You may have no elements at all, which is what happens in the Capra-Speccanthor. Or it may have some elements, but they're not necessarily enough to specify the thing. But nonetheless, we have some. And if that's true, we have things like this. set theory. The difference is that this thing has truth values which don't take the values of nought and one, but instead, the intellectual or global elements of this thing. Now, this thing always does have global elements. So you do have this a Haitian algebra. So the net effect of all of this, as you find in the topos, this isn't too like a critical start to think of the cropos thing. You still have things like little x beyond the capital X. The difference is that now these are global elements and these are objects where that thing still has a logical value. But you see in this general Haitian algebra, which is bigger than 0-1. So you have this notion of sort of intermediate truth values. Something that quantum physicists have toyed with many times over the years. And so now therefore, we can genuinely say new way of thinking about physics is to extend classical physics to flip-floss physics. So we repeat just what we did in classical world. I mean, now the action is saying, rather than saying that the states of the system are represented by a set, You say instead they're represented by an object in the topos. Propositions are represented by sub-objects, which means you have the H in algebra. The values of things, and I can call them that, take their values in another object in the topos. Physical points are represented as arrows between them. These things are sub-objects of these, but in analog of the deltas. And in the topos, this does pull back. So just as A to the minus one delta makes sense, it makes sense in the topos. And so you begin to realise you can build up a framework which looks exactly the same as classical physics. And you can imagine it back in physical propositions, like the energy belongs to some range, into the structure, and ask what is its truth values. You can imagine doing that, and of course there's a big generalisation of classical physics. Now, the obvious question to ask is, well, are there any interesting examples of this other than classical physics? So I'll say, the action is not empty, but classical physics satisfies, but as anything else, and of course, obviously there is, or I wouldn't have given it at all. And the answer is, yes, rather surprisingly, quantum theory can be written in exactly this way. By this wonderful act of decimalisation, because we're getting up to the definitions now, quantum theory can be hurled into this world of existential realism, or neorealism. What I'm going to show is that in quantum physics, there exists a thing like this, as an object in a certain type of fossil, this stuff exists.

45:00 And each projection operator, which corresponds to a quantum observable, can be Dacenized. So Dacenization is something you do first of all to projectors. When you Dacenize it, you turn it into a sub-object that this topples. And then it satisfies a distributed algebra. That's what we're going to do. So the claim is that, what's more, the claim is, which is only, I have to admit, not totally proved yet, but I think it's true, the genuine thing is true, is that you can need to recover quantum mechanics from this. In any event, you'll see. I'll explain that exactly. So can quantum mechanics be written in this way, or at least represented in this way? so in that sense quantum physics and classical physics are two examples at the same time, actually you can think of both as being you've got a topos which you're working with you have objects in the topos which is the space of states, the difference is that classical mechanics works in set theory, whereas quantum mechanics works in a different topos apart from that they look identical, which is rather striking as we didn't realise ok, so topos quantum theory so now what I have to do The actual, of course, which we use. Now, this is the one which Jeremy and I worked on some while ago. It looks a bit technical at first, and I don't know any way of really making things look non-technical. When you've done this stuff for a long time, of course, it seems absolutely obvious. To such an extent, you think, well, surely the Milton must know what a spectral pre-chief is, you know, someone goes, he's ringing my door to read the gaspies, he says, what are you doing? I said, I'm working on the spectral pre-chief. He says, oh, yeah, of course, you know, I watched that last night after the football. not like that, I know. So here's how it's going to work. Let W be a commuting algebra of self-adroned operators. So these are the things that, in ordinary quantum physics, you would say are simultaneously meant. And let sigma W note its spectrum. So what that actually means is simply all valuations defined on W. The Coffman's Becker film doesn't stop you defining valuations on commuting of subalgebra of the operators. You can do that. So let's because that's the best that you can get, really. And this is the best that you can get. Now, if you like, if you want to, because these W's, I call them windows on reality, they're like Booleans, if you like, snapshots of the world, things that you can do, and still use ordinary Boolean logic, even in ordinary quantum mechanics, by cutting down, as I say, if you're a nostalgic realist, what you do. And so you can call these things, well, context, world views, or even better, Is that correct? Say it properly.

47:30 First you should be... You should look up. You should look up. You are so wrong. I looked up on the internet to check. That's terrible. I checked it on two different sites. I do apologise for insulting the German nation. I hope you preach some footballs. I think that's all I can say. Anyway, you know what I mean? Now, what is interesting, all these words begin with W. You see, earth and sound, weakly on reality, worldview, context doesn't, that's experimental error. All physicists allow one. So, that's a very mystical fact, see, everything begins with W. I think this is very profound. This is absolutely true. Since a paper to referee by someone who wrote it, it was about field theory. paper was, physicists used the word field in two different ways. There's fields electric field, magnetic field, and there's fields of rings and fields and things, you see. So these things must be connected together, because we use the same word to describe them. But this is very deep. So I like that. So I take this as an indication from above that there's something deep there. I can't see about getting that wrong. I looked it up. I really did. Alright. So let me call calligraphic W properly for me. It's... Belt and Schoen. Belt and Schoen. That's what he said. So take the collection of all of them. So this is the collection of all commutative subalternals of Opway. Each one of them is a world view for, like, context, a classical snapshot of reality. Right. Now, the actual partial order set, that's the first simple, but mainly important, mark. Just all of them, I say, with that, to his less root, they'll be run if it's a subset of that. I mean, that's obvious, and that's easy to do, they can all pass you all to set. Now, what we're going to do is, the thing that Jeremy and I did, is construct what's called the spectral pre-sheet, or the spectral bearing in set. And what consists of this, is that you attach to each one of these W's, let's have a fibre bundle, it's not quite, you attach a copy of the spectral. So these evaluations defined on that subset, it's like what classical physics would look like, if you could just start with this Boolean sub-algebra, two-thread algebra, what classical physics would look like. Quantum physics becomes classical, we'd just stick to that. And then we can associate one sigma w1 with sigma w2, because if that is less than that, it's easy to see a homomorphism, well, that restricts the homomorphism, it's just more

50:00 true that you can do this. So you end up with this rather like the picture, which I'm very fond of. This is a varying set. This thing will be the object in the topos, which is the space of the analog, the state space of classical physics, in terms of all. So here dots. These are the Nelson-chong, spelled properly, and this. And over each one of them we have the valuations, so simple naive realism applied to each point, if you like. And these naive realist views of the world are related in a very simple way. And you just get this, mathematicians always say appreciate, or sometimes they use the word variance set. You can see why you call variance set, because these set changes from point to point, so it's like Now, one of the really beautiful results in mathematics is that if you fix this collection W, the collection of all these pre-shoes is actually a topos. That's what's beautiful. The collection of all possible varying sets over that given base set of category, context, is a topos. So the objects, so what you have to think about, this whole lot, this is what That has been used to this whole collection of all these maps and so on. So it's a single thing in the topos. It's simply an object. You just call it a name. But from an external point of this, what it looks like is a great array of things. So all that's packed into a single letter, which you call sigma. And that's the thing which represents the state space of quantum physics. And it has no point. It comes out in a moment. So that's what we're going to do. This is the topos we're going to show. So this topos is based fundamentally on the idea. that you could take a quantum physics. They're classical snapshots. They all fit together in a certain way. And the whole art taken at once actually has a very non-trivial structure. Each individual one looks like a classical world. Take the whole art together and think of it in the topos language and it makes a whole new dimension of sort of meaningfulness and so on. Now, this is a statement that this always makes my students grow in my lecture. The definition of something is obvious, because it's never obvious. Of course it's not. the analogue of the object 1 in this case is in fact where to each one of those base points the W, you simply attach a single point just one point, just matching up like that, and that's as it is obvious and then global elements are defined to be matched in that thing into that whole lot this is enough to check out, if you do that what a global element is, it's an association

52:30 basically to each point well, you say something such as this as it were, putting this thing over here. Each one of these sets here, you state at a single point. That's why these things match up. So it's a matching point. And that's what the oval element is. So that's what you actually do. And then the thing which Jerry and I based all our work on is the statement that the Coffman-Specker theorem is actually equivalent to the statement that this thing has no global elements at all. So that object, thought of as a single object in this has no elements whatsoever. And that is the Kroppenspecker theorem. It has no points. It's pointless mathematics, as the people working on it, say, with tongue-in-cheek. So that's where Jeremy and I based our thing. I quite like to use this sort of diagram, thinking about this. Just tell me, what is the dimensioning of the grade of two? Well, it's not dimensioning of the grade of two. So there are global elements. Right, so the special quotient spectrum is restricted to much greater than two. Is there any way of the cotonology theory? Ah, I'd love to know that. The whole thing screams out the cotonology theory. Well, let me actually say this. It's a very good point. I'll come straight back to that, because this is a bit like here. This is a statement's arm of a fiber bundle with a double twist in it. And if you ask, can you construct a global section of this fiber bundle in a continuous way, you can't do it. So you start down here. go up there, you go around like this, you don't know what, you go up here like that, all the way around, but you come around, you come back up to there, it's a discontinuity. And that's because this type of boundary is not a trivial boundary. In a sense, the same thing that's going on here, it's up to the same thing that's going on here, is that it's like, it's pretty much about it here, actually, because Roger has this picture of the Coffin-Specker theorem using the, you know, these Escher diagrams, yeah, it's the same, that twist, you see. Now, what this is really exciting is that the Coffin-Specker theorem, in this point of view, is a twist, but it's not a twist in the theory of bundles, it's a twist in the theory of pre-sheaves. Now, the reason I mention this is exactly the answer to your question, because in the case of bundles, the way you've shown, the unique mathematical way of classifying the fact you can't do this, is in terms of cohomology theory. And then, ultimately, there must be a cohomology theory of pre-sheaves, which says a certain cohomology group, or whatever, is non-trivial, and that's why. And then the statement would be in dimension two, the group is trivial, and it's not true I don't know what it is. It must be there. I know it's one of my PhD students. There was a PhD problem some years ago.

55:00 He quickly jessiscent it. It must be doable. Yeah, 10 minutes. Is that right? 10 minutes, yes. It's fine. Well, I will skip because my experience is that no one ever understands what I'm saying when I say this at all. That doesn't determine electors, have you noticed? I mean, they know it's not, no one's going to understand, but they still say it. It's always fun to be curious. It's like yesterday, I was at a Senate meeting, of all things, at the Imperial College, I have to attend from time to time. I had to give a report, a big moment, and some gastric wish I was on. And the previous person was speaking on the Equivalent Committee of Engineering Faculty, I was doing Science Faculty. He would run, run, run, run, run, you see, about, well, this could be pathway, this degree, course could be changed, that. No one's really interested in this at all. No one in the room. Why is he doing it? He did it. They always do it. But it came to my term, I thought, well, this is very interesting, I should pass this by. Something like that. So I'm going to pass it by because I know you're very interested. What I want to do instead, as to what you're all itching to know, is where does redemption come in? In other words, given that this statement is true, how do I go from there to developing a whole theory of quantum mechanics based on objects in the top? I'll complete this sub-object satisfying painting algebra. Because at the moment, all I've said is what we discovered. isn't it so nice, isn't it elegant that there's no global elements, but where do you go from there? So the answer is you redeem projectors. So projectors operators are in urgent need of redemption, that's perfectly clear. As Cleve Classically, they were very classically, they were just subsets, the space is set, so they boolean, they were just boring. Then somebody had the idea of quantizing, like Dirac, actually, and he turned them into projectors, and then they became incomprehensible, so we've got to get back to Earth in the way. So what we want to do is this, we want to identify sub-objects of sigma with propositions. somewhere. But we are, for the moment, just doing ordinary quantum mechanics in this way. So how we can do it? Well, what we're going to do is to exploit the fact that we do do this. The sub-objects of sigma are hating algebra. That's what we're going to exploit, which of course is very different from the quantum logic we're familiar with, if none of us want to do it. So, or if you like, the global system is hating algebra, then we can look for truth values in this. Now, if you look at this carefully, if you ask, well, look, we've got the spectral come in anywhere. What you can actually see with elementary maths, really, if you look at the subset, imagine that picture over each point, you've got the spectrum. Any subset of that spectrum does actually correspond to a projection operator. It's easy enough

57:30 to check in the algebra, in the commutative algebra. So you've got zillions and zillions of projection operators there. But the problem is, how do you associate a single projector with a sub-optic of sigma? So you've got a single projector. You want to associate it with a projector in each one of those algebers. Right, so you've got always commutative algebers world you respond to mechanics. You give me a single projector, I've got to tell you in each one what that thing looked like. So how do you do this? And the answer is you dasanise. Now, that's where the word has come from. It is, I explained, a very old English word actually. It is very hard to find, but if you go back to a very ancient English text, about 10th century or so, you'll find dasanise explained there, along with parasolific, another word which I'm very fond, this comes from the same general stable. In fact, I could say Daseinization is a contribution to the personalistic debate about the nature of quantum mechanics. I'm sure there's a subtitle, actually. I forgot to mention that already. Why Daseinize? I hope you can see why Daseinize, because we're going to take a projection of it, and we're going to hurl it out into the world of classical snapshots. Now, Dasein, as you know, I say as you know, that's a silly thing to say. Heidegger, if you like Heidegger, Dasein was man, I can't being in the world. Now, the important thing there is the hyphens. It's being hyphen, in hyphen, the world hyphen. The hyphens are a very important sign. Without the hyphens it doesn't mean anything. So, datinization is taking the projection operator and making it partake in being this in the world hyphenated. So, that's very important. So, you do that because you associate with each projector, projector, this is serious stuff, aren't you? Don't smile. You associate with each projector. Projector in the local world view, which is classical. So, it's being there in the world. So how do you actually do it? Now, he asked as well. When Andres, when I started working six months ago, he was, like as it were, a bride coming to join the groom with a dowry. The dowry he brought came from him and his supervisor, Professor de Grote, who's a distinguished German mathematician. And in fact, the dowry was precisely dasanisation, which is what we should have thought of doing, and we didn't. So I should completely confess our errors here. In a sense, it's almost obvious when you don't know what you're doing. Think about it. You've got an algebra of rockweights that can use, including lots and lots of projectors. Here's a projector. you want to represent, not represent but somehow, that's what I say the word flows so freely from one's tongue

1:00:00 you want somehow to kick the projector out into the world but you want to be as close as you can to where you started I suppose, as good a fit as you can, so the obvious way of doing it is well, let's take all those projectors in W, which are bigger than P, like proposition you say P implies Q, we take the smallest of them this is, see it's like I've got straight disambigured So what we do then, is in each world you represent the given projector by the strongest proposition you can, which is implied by the given proposition, but lies in that particular world. I mean, you know, it's a natural thing to do, but we can think of it, and the rest, and the supervisor did. So, okay, you can do that. What is, of course, not object, has to be checked. This is a much like I call Gravy. It's in fact, this is a sub-object. That's what's so beautiful. If you check, it all matters. And so by this means, you associate with a projector, a sub-object, this object is in the top of us. And therefore, it partakes in the internal logic, which is there in the top of us. This could not have worked, right? I mean, you would think of millions of ways you might think of. Or try to, as it were, And most of it wouldn't fit together. You see, if you think of your subject, where's the match-up together? Everything has to match. All these arrows, you see. It's a simple thing. But it works. And therefore, you can immediately use this to talk about quantum mechanics in this language. And in fact, what you can prove is this. As you might expect, it doesn't completely represent quantum mechanics. It couldn't do. Because quantum logic is non-distributive, which new logic is. But you can prove that your operation goes lost exactly. This one is weakened. So, obviously, fundamental and satellite effect, would be called it, that could really do it. Also, you can focus this one-to-one. This is important, actually. It means that nothing is lost, in a sense, by doing this. A project is mapped uniquely into a sub-object of the topos. You're not losing any information. It's like taking a sort of a picture of the world, which is curved, and sort of projecting it onto a plane and skewing it around, but don't actually lose any information. It's not in there, but codes in a different way. So, that works. It's one-to-one. So, what we in fact have is this chain. We start off, this is the idea then, this is the use of quantumism. We start off with the physical proposition, the entity belongs to something, whatever. Then you do standard quantum mechanics, you take the operators that represent that physical quantity, you take the spectral projector, this is the projectors of those eigenvalues lying in that range.

1:02:30 that's all you rather can to decilise that and that's the thing some of the top you now say represents the proposition in this new way but of course now the logic of all of these things is a hating algebra it's no longer non-distributive so you can work like an ordinary logic that's what's striking now just a couple more minutes let me just say very briefly about the notion of truth I know you won't follow if I say it now let me just say that just very briefly how this works I said at the beginning there is two ways in classical physics to talk about something being true let less belong to something and the something belongs to something else now in this topos what you can see because there are no global elements you can't say let less belong to something but the other thing used to that capital T that truth topic works perfectly and that's in fact just the thing Jeremy and I used what we were doing. So that fits into this thing very well. So you can still talk about things being true, but the way it works, you still say proposition is true, and you still represent something that that I'd like to belong to something, but the whole thing now is interpreted on top of us. Apart from that, you can manipulate it just like you can in classical physics, so it's interesting. there's a final thing which Professor de Grota has done which is to once you start dasonising it gets quite compelling actually you think what else can I dasonise can we dasonise the English football team for example make it a bit more effective what we've said so far here's this object each projector there's a sub-object now What about physical quantities? Now, what the voters showed is that decimisation can be extended to fit not just to projectile parameters, but also to train operators. I won't go into that, because it's not enough time, but the net effect of that is you do essentially complete the size. There's still some slight debate about this thing, but it's almost there. Andres is a perfectionist, that's the trouble. He likes theorems to be true, because he's a mathematician. Theoretical physicists like theorems to be saleable. It's different. Anyway. We're almost there. So this looks like it's going to work perfectly. So this is de Grota's work, I think. All right, where do we go from here? So what we've done, then, or claim we've done, is that we've shown that quantum mechanics can be embedded in a literal sense, in a one-to-one way,

1:05:00 into a stop-off scheme which looks exactly the same as classical physics. Exactly the same, except that you're using pointless sets. But apart from that, it looks exactly the same. In this sense, classical physics and quantum physics are just different examples of the same underlying structure, none of this does appear to be true. Now, having said that, this is really the research program underway, and there's many things we're trying to do. One is that not every sub-object of sigma is of a form delta of something. There are like generalised propositions. There are propositions which are essentially contextual in nature. There's no way of looking at all these stringed sub-algebers, is that you have a very strong contextual component of physics. And there are interesting propositions you wouldn't normally talk about in ordinary physics, because you could represent and contextual physics. What do they mean? The uncirculating relations, now, there simply has to be a nice way to look at the uncirculating relations. What we would like to do is so that the uncirculating relations now become logical statements between the values of I and B, purely logical. I mean, it's not a silly thing. You see, it's very frustrating to say, well, frustrating in society, really. Take this object signal. You see, it's a spread out collection of sex. Now, there are no global elements, so there are no lines you can draw, But there are sub-objects. Now, as a sub-object gets smaller and smaller and smaller, you think, well, can't you make it smaller and smaller and smaller until you get just a thin line? No, you can't. It's like a sausage. So you make it smaller there, it sort of bulges up there, sort of thing. So are there minimal-sized sausages? Well, like in physics, we talk about minimal uncertainty things, the coherent states, you see. So is there an analogue here? And if so, can we take two of these sausages and say, well, this sausage intersection with that sausage is bigger than another sausage. I'm absolutely convinced that that's something like this. And what I just conjecture is more than that. In any top-off model of physics, there'll be the analog to the uncirculations. And it'll come from the logical form. It's just that in set theory, it all trivialises it right. That's what I'm pretty sure is going to be true. That's a conjecture, I have to admit. That's not proven, it's just a conjecture. But I mean, the whole thing, you know, I'll say it fits. So that's an interesting thing we're working on. but measure theory I don't know if there's any topos theory I don't know if anyone's done measure theory in a topos but even a strong desire for doing this because you see if you start to get carried away by this you think maybe the whole of quantum physics everything can be answered in this way for example quantum information theory is very fashionable these days you want to get some money to do physics in quantum physics these days you pretend you're doing

1:07:30 quantum information theory now so we have at Imperial College we have a huge string theory group Any string theorists in the room, before I insult them in my... OK, good. String theorists love to get bigger. I mean, more of them. And we recently have a post in our group which might be in cosmology. So all the string theorists, in nomination as candidates, all these string theorists claim that they're interested in cosmology, string cosmology. So you don't have to talk to a real cosmologist. They pretend that it's true. In the same way, if you want to get money these days in quantum energy, you pretend you're getting quantum information theory. You might be under the chance. However, maybe this is not so trivial because, suppose you did classical computing, this is one for you, too. Suppose you did classical computing theory in the topos. In this particular topos, this didn't look like what people called in quantum computations. There's simply got to be a link, because this link between the quantum and this topos is very tight. It's not just, it's very, very tight. There's also intriguing things come out of this. The thing that helped me to see most of all this is find this thing. See, my real motivation for doing this personally is not to explain ordinary quantum mechanics. quantum mechanics is wrong. Like Roger Penner, I think very deeply, actually, one of the problems in quantum gravity is that quantum mechanics itself doesn't work down at the point of scale. And you're just committing category of it all the time, and you try to apply it. But what do you generalise? And I like enormously the idea that this is what you generalise, that you work with a different topos. No longer the topos are included in algebras of something, but topos is in general. They're all going to give you back work for pain. You see, if you're a physicist, at some point, you've got to make statements about the world, every topos allows you to do that. And I really am quite excited about the prospect non-operator-based purposes, if you like, at this time, which we can still do as some others, because it's just my own personal motivation, and that's a pain to do quantum gravity. Right, I'll stop there. Thank you for your attention. Thank you very much. Thank you. Yes, no. I've answered the question. So you're expecting a yes or no answer, aren't you? Yeah, yeah, yeah. And the answer, of course, is not that. The answer is something different. the answer is a I mean, there's an answer to the question but it's not yes or no this logic is not yes-no logic it's multi-valued logic can I just ask

1:10:00 what is a way of putting it you're giving a framework thinking about what you would narrowly call open space quantum mechanics How important is Hilvers space? Does it work for a Van Asch space, for example? Well, it certainly works for, I mean, one of Andreas' passionate interests is obscure upon Neumann-Alger's. And it certainly works for that. The whole point in the sense of what I was saying today is that you don't have to use Hilvers space as the tool. No, quite, but you do claim a fit. Oh yes, for only quantum mechanics it fits. Right, and the interest is partly, how close is that fit? And that's why I say ask, what about Van Asch space? What about Hilvers space, say, the wheels are supposed to come up? So it's general with respect to... doesn't it include that much of this? We have a spectral theory of Benox, so you could do commutative algebra. Yes, I don't see why not. Where it wouldn't work would be for truth objects. That's hardly surprising, because the truth objects is where the states come in. What about C-Steralgebras? Well, as I said, it works for albative von Neumann algebras. I mean, if you take any von Neumann algebra, take all the exclusive von Neumann subalgebras, it fits exactly with this. Hang on, but then let me just ask about the circuit as universality claims for C-Store Algebra. Is what you've done better? Because the logical structure is distributed at the end. You see, the thing that's always fascinated me is the logic. You see, if you ask a C-story algebra theorist how do you use quantum mechanics, they'll say use an instrumentalist term. People like Harb, for example, are very strong instrumentalists. My whole desire to get away from this, and in a sense, what stops you getting away from life is the non-distributed nature of ordinary quantum logic and the Chaos theorem. The whole point about this is it maps you into a distributive logic. That's the point. It's very, very different. I mean, C-story algebers are a useful tool, but they don't really... It's a very minor shift. to go from Hilbert's base to C-Store altitudes in a structural sense is really quite a minor difference it looked very similar to predictive analysis paradarity sorry I can't hear you very well oh yes yes yes is there any symbolism in your approach and present approach and does there have a more specific testimonial input that it wasn't

1:12:30 Yes. Oh, yes, yes, yes. We're very hot on existence. Given existence, the question is, is it very similar to what it looks like to do with that? Well, I mean, you could... The quantum logic people became very fond of general orthomodular lattices. I mean, all sort of working quantum logic has been done, not using Hilbert's basis at all, just taking logical structures that look like quantum logic, which are orthomodular. Now, any one of those you could apply this to. I mean, it's quite easy. given the sort of thing I can't pronounce his name, anyway was talking about yeah, no, I know what he did yeah, no, I know yeah, but take any of that school of people working in the 70s, for example, who were working in quantum logic in the 80s, who worked with, say, general orthomodular lattices, take the collection of all Boolean sub-algebras of those take those as the base elements in the category, they have a spectral theorem, the whole thing goes straight through immediately, so the answer is yes, it does it would certainly include that Yes, that's what we're claiming. how does that enlighten the realist problem what is the content of neo-realism yes it's very it's one of these it's hard to state this the statement is that you are making propositions using a logical structure which is distributive which means in particular you can make a deductive system out of it so you can talk about proving things in a logic like this you have the notion of proofs and so on and also there's a proper notion of representation so when you have these truth objects there are in effect representations of this logic in another logic of the same type so the situation is that you can talk about the world in terms of propositions these propositions have a logical structure and you can talk about truth values of these propositions and these truth values take their values not in 0-1 but in the 18 algebra now because the propositions themselves form a distributive logic, as I say you can make deductions, you can sort of deduce things like that, so in many ways you can repeat the sorts of things at least in principle you do as a physicist, you don't think about it quite these ways, no one sits and writes down a proof

1:15:00 step by step, but in principle you can so you can do all that where I suppose it gets strange is when you ask what does it mean, if I can use that word what does it mean to say a proposition is true with a certain value in one of these altitudes and the answer is well I'm not sure I knew what it meant anyway to say something is true I'm not just skipping the question here I do worry about this even in all D0-1 logic I can manipulate these things so the superposition principle could be assigned a principle well a superposition principle is one of the things we have to look at down the line I mean superposition principle is asking what is the relation between the truth objects for one state and the supposition of two states and that's a mathematical question we've still got to address but it would have a role in all of this it ought to because of this the fact we represent these projectors injectively is very important, nothing is lost when we map this thing into the world we don't lose anything in doing it but I don't know the answer to all these questions I mean I often think about this because I always say well in quantum cosmology we would use these sorts of things what would you actually do to thinking in terms of true or false. You'd have to sort of be like a computer. You'd have to follow... You could write down the rules for deduction and reasoning. And there are complete rules. I mean, usual completeness, there's some work, but there's sort of logic. But if you ask, what does it mean? Well, of course, you can't map it into the normal world. It's difficult. It's not like that. So I don't know. Well, Chris, I know almost nothing about top losses, but I do regularly recall reading an article once Oh, yes. Hello, John. Yes, yes. And it was actually in the context of fractal geometry. And my recollection was that he included that the notion of infinitesimals in, you know, in the real mind or whatever, which is obviously relevant in the fractal geometry, could be described by one of these... That's correct. And this actually had this property of these nonfaith and truth values. Yes, that's correct. So, I guess my question is, have you thought about sort of linkages between the sort of top losses you're discussing and these other types of geometric? Yes, I have. Only at bed at night, but I can't sleep. I did have a student once a few years ago, actually, who did a PhD on synthetic differential geometry, is what the subject's called.

1:17:30 It is, I suppose, if I get a bit wild at my old age, it's quite fun to imagine. Well, suppose, for example, you took one of these topos and tried to do a general relativity in it. So, what would you do? And, of course, I think the answer is you would do what you suggest, is you would try to make the topos one where you have infinitesimals and work like that. So, I mean, it's not impossible. I mean, it is, I have to say, a wild suggestion to the way of quantizing gravity, but you could actually do it. You could imagine. Sorry? There might not be any singularities, though. Oh, indeed, yes. Well, that's always been one of the possibilities. No, that's one of the attractions, of course. Yes, that's right. Yeah, some questions, well, lots of questions, very fascinating, but I mean, coming back to the delta operation of delzionization construction, do you have the slide? Yes, I do. Yes, it's quite simple, really. Right, so that's a particular study faultful. One minor point is it doesn't point to the fact that conjunction isn't preserved exactly. Does that somehow relate in the end to the non-distributivity of the generation? Yes, well, you could say it has to be like that. Absolutely, yeah. It's where it comes in. It's where you lose the non-distribution difference. Exactly, yes. Now, I mean, so you've given a particular definition here, but is this in some sense a general construction? And does it have a characterization as to why it's this construction? Generally, in what class? I don't know. I mean, at least that this is the sort of, this is the greatest sort of substance that you're associating with. It's the greatest idea we've had so far. There's another one, I could tell you a stretch away, which the Grota again thought of and which didn't, which is the other way round, you could take all those Q's from less than P and get the bigger one. Sure, of course, and so then you would say this is the greatest and the least of something. They in fact turn out to be isomorphic Q's. Yes, so we have those two, they in fact map into each other, the isomorphic objects it turns out. Can you just remind us what the Q's are? Yes, just projectors. So, W is a world view. It's a really intuitive relative of operates, which has many projects, of course. And you just take, you know, all the cues which are bigger things, take the smallest off them. Why isn't that P? Sorry? Why isn't that? Oh, because he doesn't talk to W doing this. Otherwise, it wouldn't be decimizing.

1:20:00 You're localizing. Yes, you're localizing. Of course, I mean, the language... In fact, what John Bell talks about his local set theory, and he would like... I suppose a related question is, I mean, is, I mean, so there's quantization, and there's dozenization, I mean, can you make a general construction, because if you started off with quantization as kind of making physics interesting, so can you see, can you see a reflection of the idea of quantization as a construction you're doing? I mean, are these, are these adwork functors or something? Well, that would be... about that, actually. It is, of course, tempting for me to try and quantise this again, see what happens. I think that would be a bit OTT, don't you? I was just thinking, you know, is there a way of going from, I mean, this is, I mean, okay, you're plucking this out because you know what quantum mechanics looks like, but what one would really like is something that explains how it arises in some general considerations. Yes, I mean, in a sense, you could say what's really good is contextual logic, I mean, that's one way of thinking about this, is that you're trying to build a contextual theory of logic, so that things have their, things have ordinary truth values but in context, and putting the whole lot together, that becomes like these, that's one way of talking about it, it's still a bit too vague, I don't know a definitive, I know what you're asking me now, but that's a vague question, I understand what you're saying. Well, the relative thing in this cluster would be, and you're talking about the uncertainty relations, so, I mean, are you going to make H or something pop out of somewhere, or...? H doesn't have to be in there, actually. I mean, H is all of units, in a sense. But some idea of the sort of current onto a nice on-scale thing... Yes, there will be. I mean, of those sigma of H, these are the spectra. It's really where they... See, things have units in physics, right? So somewhere you have to put that in, and that's true. But, I mean, since I haven't done that anymore. I'm going to vote for Nick, then Jeremy, and then David. A proper decent hard note of quantum realism would demand that a decent real expression of quantum scale that it says nothing about measurements. If it's thousands. Yeah. in the basic process of the theory of thought and it wasn't clear from

1:22:30 all this to me whether you accomplished that or how. Did I use the word measurement anymore? No. No, absolutely. But nevertheless Ah, well that's where, sorry, I should have explained that from the beginning that's one of the whole points about this, is never to use the word measurement what replaces measurement are these generalised truth values mechanics is contextual and when you make a measurement you choose which thing to measure that picks out a context you measure a physical quantity a you pick out the commutative algebra of a and all its functions there and what this is doing is taking all of those things together so you could say it's all potential measurements if you like it certainly isn't about measurement no but the context when i picked up on context that seemed to me to be standing in yes but you see the whole point about topops thing there's all those different contexts that I can pack together into that single object, sigma. That's the whole point. Every possible contact is in there. If you unpack it again, of course, you can still want to be tactile structure. But how does that satisfy... Well, I didn't use the word measurement, that's for sure. But how does it satisfy a realist to have it packed all these contacts back into... Well, because this is, in a sense, the question I was asked just now, about what does it actually mean to use this language for real? you don't talk about there's no counterfactual statement if you make a measure instead someone gives you say a quantum state of the system you work out the corresponding truth objects I didn't go through and then I can give you a truth value for the propositions the proposition P is true in that state so that truth of value if you like contains within it all possible you can extract out from that all possible results of ordinary quantum physics is lost in this structure. It's all in there, hidden in there. But it's presented in a way which gives you a logical structure. That's not the point, I think. I guess I'd need to show you in detail. It's a lot to take on at once. So once you've got used to these things, they seem obvious. But it's not at first, I don't know. I just wanted to make a comment about the logical form of the uncertainty relation, which was to suggest that maybe a series of papers by Kutowski do not recall in your language. about Cochin-Specker ways of thinking about uncertainty. I think he calls it the indeterminacy principle. Certainly in general, that happens for about 99, or something like a logical indeterminacy principle.

1:25:00 There will be a series of things which might be just... Yeah, I wasn't aware of that, but I'm not. They're not translated, yet to be translated. Thank you. Okay, check it out. The answer is yes. I mean, in a sense, what do you mean? The thing about topos is it takes from getting used to it as an internal logic and with an external logic. When I throw these pictures of all this stuff spread out like that, I'm using the language of ordinary set theory. They're sets. And that's like the external language, like a meta-language you talk about. And that's ordinary mathematics. Then when you work internally, you find the logic is intuitionistic. now the way I actually construct these things I'm using the external logic of ordinary mathematics to construct objects like these pre-sheaves and I'm saying that they are these things internally which then you deal with intuitionistically so in a sense, one way of answering your question insofar as this is meant to be a different way of looking at ordinary quantum mechanics the question is whether or not the mathematics of ordinary quantum mechanics we've been about intuitionistically there are some things you I ask my mathematicians friends this from time to time they always get very vague answers the mathematicians have developed the theory of Banach spaces intuitionistically very well for some reason they never seem to look at Hilbert spaces it always amazes me I don't know why that is there's a whole raft of stuff on Banach spaces and their theorems and the intuitionistic analogues Whether they find a Hilbert space, I don't know, and what's more, the people who did the Bannock ones didn't know either, because I asked them. So they answered the question, I don't actually know whether or not the external mathematics can be developed intuitionistically or not. Do you know? There is a very constructive curve, please, and you can interact with respect for the Hilbert space. Oh, I know that, I understood that a lot on that, but the thing is that it's not that, it's whether the constructs of the Hilbert spaces themselves. Yeah, whether that can be done constructively, that's what I'm not sure about. See, a Hanbanach theorem doesn't hold, for example, in general. That's not a constructive proof. They're constructive analogs, but they're not as strong. I just don't know. It's a spectral theory, yes. As I understand, non-classical logic is easy to premise on the assumption that we're just actually wrong about classical logic.

1:27:30 And it feels that way as if there will be a sense of consistency in the long run. You're ultimately saying, we're making a mistake with this external mass we're doing in this classical logic. We probably shouldn't be sure we've been all done by external and internal. It won't be discovered as the actual biological logic. have a student who did his PhD on synthetic differential geometry and quantum mechanics and there he had to work constructively throughout. It's quite interesting because you find the spectral theorem, there are different forms of the spectral theorem, somewhat bizarrely. Whereas in ordinary math is just one, there's different types. I didn't, did you know that? You'll notice if you knew that. Yeah, I didn't know that anyway. They did, they bifurcate. He still did it. I mean, he managed to build up, certainly for finite dimensional human spaces, a constructive proof of things. But how far it goes I don't know. Not completely. I'll put that slide to the end with you. Which one? I missed the guidance from Jeremy. In 3, the delta P there, it's not going to be delta subscript up. Right. I'm trying to get a quick one. Yes, that's right, essentially. If that's the case, one of those w's ought to include p. Yes, that's right. Yes, that's right. Some of the best terms are. I didn't say it was a deep result, I said it was important. Yeah, that's right, exactly. Yes, yes, I've been thinking about that in my spare time. I haven't revealed this to Andreas. I think that's in my spare time. I don't sleep very well at night, so I think of all these things. It's a clear analogue of this for history theory. You can almost write it down immediately, actually. You just change the base category a bit. And in fact, it's the philosophers originally talking about things called sets in time. I mean, the topos theory came, one branch of it came from logicians, philosophers, people thinking about things changing in time. It's one of the origins, actually, of this type of logic.

1:30:00 Okay, in fact, exactly, yes, that whole, that's right, exactly, yeah. So the answer is yes, you can certainly write down immediately. So in this sense, history theory looks exactly the same as, okay, you've just changed the set. That's the reason why I asked, because I think I may get some grip on your neorealism in history space terms, whereby perhaps different values are attached to different history spaces. So, can you perhaps try to say again what your realism is or near realism is in history-space terms? Well, be careful there, because one of the problem is that it's the truth object. You see, in ordinary quantum mechanics, you have the notion of this truth object, which is associated with probability one-statements, basically. Now, probably in history theory, you have this notion of consistent sets, and you can only really apply truth considerations to consistent sets. So you've got to say something about that, first of all. But apart from that, if rather than having all possible sub-adultures doubles, you had consistent sets of propositions, for example, then you would get something, I think, similar. So you would be saying that propositions were true, but in the context of different consistent sets. You've already addressed the question briefly, but nevertheless, as far as I understand, your claim so far is that classical mechanics and quantum mechanics are different representations of the same structure. Apparently, yes. And in how far is it possible to also establish a similar relationship between classical and general relativity and already existing quantum gravity approaches on the other hand? Oh, it's trivial. I've seen no idea No, it would be a good question to answer I've got absolutely no idea Dynamics Oh, that's relatively easy at this level It's just an automorphism of the algebras The way this is set up It's like non-relativistic physics, if you like, by talking about space and the states in the first place. Dynamics is relatively simple to implement.

1:32:30 It's just, if you like, an automaton and logic. But isn't where, basically, in the crux of the problem, one interpretation is that the experience is that the states change in dynamics. The truth optic changes. The truth optic moves around. So the question, when somebody says we'll confuse her group, what it means to have a true value, it isn't 0 or 1, it's like saying, well, whether 30% chance on Friday would be different. When you get to Friday, now you say, well, what is it? I've seen the weather, like this has that relation to 30% chance that I predicted three days ago. What is your description here stated? Is that what you're asking for? What? Are you asking? That may be another way of rephrasing it, but it's more... I just know that, intuitively, that your statements about... Basically, you've set up a propositional capitalist, You said that I can tell you what the true value is for any proposition.