Non-Commutative Discretizations — Finitary Algebraic Sheaf & Topos-Theoretic Perspective on Quantum Gravity

0:00 the other's joke curse I have an answer the question he said well ask nature an improper question and she will answer back to you with another question and well here I mean take some water here the improper when I was writing this I remembered a very nice saying of Aristotle in the Comanian Ethics it says He is supposed to define and divide is to be considered God. Well, okay, the mathematician, perhaps, kind of, for a little bit. The joke of Samson is... So, let's roll the dice and proceed. I'll give you a nice... Bob's favorite diagram, I call it Bob's favorite diagram because he did a very nice talk in Imperial College a year ago, and you must act more. Okay, well, we have the two pillars here in the 20th century, which is relativity theory on the right-hand side and quantum mechanics on the left-hand side. And the current challenge in theoretical physics is to bring them together in a meaningful way. Actually, let us go through the flow of this diagram, okay, starting from the left-hand side, from quantum mechanics, we have an order that this is one of the tanks that still treats time as an external absence of time. We have the two-page space of a high-exertial manner that is a part of the satisfaction of knowing, and of course, an output of that is what we call quantum logic, which is the logic of closed space and so on. Then we have a phenomenon of non-locality, and in non-relativistic quantum mechanics, and we have the debate whether, you know, we have the information part and the matter part, and the information part is associated with bell inequalities, entanglements, and

2:30 what is called holistic picture of physics in the quantum deep. and of course relativity theory splits into special relativity and general relativity special relativity is a field theory you can think of field theory without gravitational interactions and the unison of quantum mechanics with special relativity gives you quantum field theory as the essential structures involved quantum structures involved in quantum field theory are kinds of products of filter spaces okay let us leave it there tensor brother comes in cognitive theory in these guys if you want to formulate the notion of many particle states you have single particle state which is a subspace and a Hilbert space and then you have tensor wall fly of the other part and then you have subspace that's not important special relativity the quantum field theory we have Einstein the notion of Einstein locality quantum field theory again is formulated in Minkowski space which is the flat spacetime of special relativity and Einstein locality tells you essentially that if you have operator algebras representing your local operations that you do in spacetime and if these two operator algebras are space-like separated, that is to say they are then all the operators in one in one algebra commute with the others. So space like separated actions, operations, commute. This is the content of what is called Einstein locality and of course Pond-Munfield theory is fundamentally based on the Minkowski continuum and because the Minkowski continuum is a continuous space actually what that means is that with an infinitesimal a volume you can pack an infinity of events that results in a very hand-wavy way to say that this is the culprit for the non-renormalizable infinities in quantum field theory and of course these also are due to the fact that particles are treated as point sources of the fields and a point, at least from the point of view of quantum field theory

5:00 I mean, there are very ingenious ways of trying to avoid points in quantum theory. For instance, by considering these fields to be operator value distribution, so in some sense we smear the offensive points to something larger, which later on I will try to connect with the notion of locales, which are essentially pointless structures. but later on okay, then on the right hand side we have, and this is my way of approaching physics on the right hand side we have relativity theory that gives us general relativity which is the theory of the gravitational field Einstein's theory of the gravitational field and which essentially rests on the assumption that space time is a smooth manifold, okay and then we have the assumption of space time as a smooth manifold essentially encodes encodes the notion of locality, of classical space-time locality. Namely, what is locality in classical physics and in general relativity? Well, it tells you essentially that the equations of motion are differential equations. So, differential equation, differential locality means that all your observable quantities propagate infinitesimal infinitesimal, that is to say the causal efficacy is infinitesimal the nearest infinitesimal neighboring events actions connect infinitesimal neighboring events but the assumption of the smooth continuum is arguably the culprit of the singularities, for instance in terms of black holes and with the canonical example of a singularity being the Schwarz solution to Einstein's equation for you, but you are not familiar perhaps you are not familiar with what this is a solution to Einstein's equation that has two singular points at say at radius 0 radius from the source which is the matter of the gravitational field and at the radius equals to 2n Okay, the R equals 2M part of the solution, which is called external singularity, actually it was shown by David Finkelstein not to be actually a singularity, a true singularity,

7:30 it's a coordinate singularity. What that means is we have laid down inappropriate coordinates to coordinate the manifold, but if we use a preferred system of coordinates, then we can resolve the singularity, so it's not an essential singularity. at r equals zero, right on the source mass, the gravitational field. It is a true similarity, and again, it is due to the fact that we do not know currently in physics what it means to measure the gravitational field right at its source, at its point mass source. Okay, but... It does, that was a very nice question. As singularities correspond to black holes, black holes are now considered to be very important. this should be seen as a feature and not a part of this that's a very good point in some sense in some sense you see the black hole the black hole it is an object where the smooth constructions break down from the point of view of the smooth spacetime as a black box, we do not know what's going on in there it is singular because essentially because the smooth coefficients of the differential equation of the partial differential equation of Einstein have that similarity it is again, black hole similarities are characteristics of the continuum manifold, assuming that the structure sheaf, actually I speak in those terms which I understand relate more to my work, the structure sheaf of coefficient functions for the differential equation, for the partial differential equation, are the same functions. It would be interesting to see, for instance and that is one feature of my work currently, it would be interesting to see, to see, we now get this statement, this statement, that the smooth calculus, the calculus of maniples, extended down to very small distances to the Planck scale, it breaks down. That is the model of current physicists, it breaks down.

10:00 My argument, rather, as we spoke of in a victim's timing sense, it's really, to do actually differential geometry to lay down differential equations to apply the tools of differential geometry one need not have the smooth line I mean you can the scheme that my colleague Professor Miles and I have developed the mechanism survives, all you have to do differential geometry, do you have to have the notion of connection, that is to say an external differential, and some functions which are marked by this connection, and you can form equations. The mechanism in a nutshell, the mechanism of differential geometry survives is independent of any underlying space. Now, it is convenient, the notion of the manifold, it has served us well but the work notion of the manifold with the smooth functions on it which are singular must give way to a deeper structure where the basic mechanism of differential geometry still holds I suppose my point was simply that if you replace the smooth setting with singularities without singularity. Not without singularity. Actually, the independence means that actually the base space on which can be ultra singular. You can have a base space on which to localize all the things with everywhere dense similarities. And still, the mechanism survives. So the question is that the ball is in the physicist's chord to tell us what does this mean if I can set up Einstein's equations over very singular spaces what does this mean for physics, what does it mean to have infinite singularities and the structural sheaf, for instance one of the structural sheaf of functions that we can use in setting up this abstract differential geometry quite popular and it's called distributions and they are not it is a nonlinear distribution theory of a

12:30 nonlinear version of the distribution theory of Lorentz Schwartz and where where distributions form an algebra and these distributions are ultra-singular but yet you have a differential you can set up your cohomology theory it's meaningful to define a curvature form and equate it say with some matter contribution or even like Einstein's equation you do not have to count matter yet we encode the we see through the simulation the mechanism survives so what does it mean differential geometry breaks down at the infinitely small when the smooth space time why is it Why do we assume up front the smooth continuum and we have faith in it? For instance, I was reading a famous, he's actually an Oxford professor of mathematical physics paper. It was Roger Penrose, and he says, in a paper of his, he says, we assume that space time is a four-dimensional differential map, right? Smooth map. He does some calculations, and at the end, he says, he finds that there are similarities. and he says therefore nature has similarities no, here is the basic confusion I think of the physicist that he confuses the model with nature, the model, the classical differential calculus of course, is assailed by similarities but that's not, that doesn't mean that nature has similarities, what does it mean nature has similarities Your model is of limited applicability and validity, of course, it breaks down. Smooth functions, half-singulates, okay, certain magnitudes which are smooth, blow up. Okay, revise your calculus. Revise your space. Presumably, if you assume another algebra and another structure sheet of what's called differentiable functions, which gives you a cogent differential so that you can set up a differential equation and even if that algebra

15:00 is everywhere dense in singularities and you can still write equations then you evade the singularity by encoding it in your own calculus but anyway, what I'm trying to say here is it's a confusion of the physicists of the model and with reality I think the manifold model exactly, the smooth manifold model and the differential equations that you can consider, of course it is a similar, but that does not mean that differential geometry cannot be applied in reticular spaces or discrete spaces or other spaces that may seem to be very singular from the smooth viewpoint I mean, we see certain tendencies now with the so-called synthetic differential geometry as well, which is different from the abstract differential geometry of minus, but at a more basic level, they revise the calculus. For instance, they start by assuming, you know, an intuitionistic type of logic underlying, okay? There are some... The way it's almost opposite, because they assume that everything is... Yes, exactly. I can see there, I can see there, low here, all the way through. Okay, but let us continue, because I want to give you the more concrete things. in trying to unite quantum mechanical relativity theory the basic no-goal theory is that there are no finite dimensional unitary representations of the Lorentz group if we want to unite the Lorentz group which is the symmetry group of relativity theory and with essential features of quantum mechanics for instance certain evolution of a rate must be represented by unitary operators in the human space there is a conflict between Unitarity, finiteness, and Lorentz invariance. So the question, there are no final dimensional unitary representation of the Lorentz group, since the Lorentz group is not compact, OK? So the question that now confronts the thesis is what to sacrifice? Do I sacrifice finiteness? Do I sacrifice unitarity? Or do I sacrifice Lorentz invariance? Perhaps at a very small scale, Lorentz invariance

17:30 to be a symmetry I would if I was to decide on this I would certainly sacrifice unitarity and I would keep finiteness because I believe in finite Okay, so many, very, very, very, I don't know the initial pessimism, but when there is a, when there is a new math version of Murphy's Law, what I call it, if there is 50-50 chance that a new theory would turn out to be wrong, then 9 times out of 10, it turns out to be wrong. so this is Murphy's the new math version of Murphy's law and of course I can tell you that already I have managed to quantize I have the quantile theory of gravity but unfortunately the margin of this slide is very small so I cannot do it but we must take it easy because there has been really a long detail the continuous versus the discrete we have many we have of course I guess, I mean, this is from the ancient Greek point of view, which I'm familiar with. And Plato versus Thales, of course. Well, here I can talk for ages because I think that the West took the wrong turning in going with Plato. I mean, thinking that the true philosopher was Plato. I think the true philosopher, the natural philosopher was Thales. and Thelis was you know Plato, I called him responsible for introducing mysticism into philosophy a lot, Plato while Thelis was a hands-on physicist he was conducting experiments he was doing theories, actually testable theories it's amazing so the rule of thumb, in ancient Greece the metaphysically minded philosophers are pro-continued while the physically minded ones are pro-discreet okay, something to think about But of course, I'm going to talk about a bit. He says, what is space? Is it discrete or continuous? The forms of the dilemma were the following. He said, well, he said that discrete space, unlike a continuous space, carries its own metric, namely, just count the number of points.

20:00 While a continuous manifold has a continuous symmetry. for that he was influenced by I think, but in any case he says, therefore nature must be continuous in the first, a discrete matter is its own metric, the metric cannot be prescribed from the outside so nature must be discrete, but he says also I see that there are continuous symmetries in nature therefore nature must be continuous so he couldn't make up his mind but also in an anomalous have a quantum diagram, is it a field that has a continuous propagation, or is it a particle log? Yes, what do you mean by discrete? OK, you know, this is very, very interesting. I always chase the physicists. What do you mean by discrete? Do I mean, for instance, mathematically, you would think that discrete and also discrete comes with topologies, is the completely disconnected set. Sorkin thinks about finiteness, then. Finiteness, exactly. You know, I think Sorkin is a bit his own enemy. Because I think that Sorkin, well, he calls local finiteness. He's interested in locally finiteness. Locally finite, I have told him. Locally finite does not mean that your models are finite. They are locally finite. We'll see what we mean by Locke. Certainly, you are right. What do you mean by discrete? Oh, everything that says a discrete is a discrete. Is it discrete? Is the natural number? Are the natural numbers discrete? Discreteness, is that a design? Is it something that in the context of topology? We do have something discrete. I mean, if it is. OK, the counting numbers, I don't know how it is. Express it in terms of conditions on the components, I've spunked up your space. I could just press the screen. Okay, yeah. All right. I'll try to say that without getting an N. True. But, okay, this... Okay. Let me go to the middle. Okay, no computer, no. That's really tight. Okay, we started with the agnostic algorithm of... The space-time continuum, who gave it to us? I mean, we assumed it. That's a very important question.

22:30 so and if it gives us problems when we say the space has singularity what does that mean? if you assume the space if you assume something which is different and also equally versatile differentially speaking maybe you can evade the singularity but we'll see but nature has singularity I'm the first to doubt it what does that mean? Well, manifold reasons against the space-time manifold, we can concentrate on seven graphs. Right, first is the constants. Well, given a manifold, a differential manifold has a constant, I mean, topology is locally Euclidean, fixed by the theorist, and it is not subject to dynamics. Okay. There have been some ideas in physics, most notably by John Wheeler, who said that it It's a bit short-sighted to think that only the metric structure of space-time is subject to dynamics like in general relativity, and fix the topology up front to the local Euclidean one. Even the topology should be subject to dynamics, and that model originally proposed by Wheeler was called space-time performance. Okay, so we are questioning whether the deeper structure than the metric structure, the topological structure of space time should be relativized, should be subject to dynamics somehow I mean, even because at the level of topos theory, topology and logic is a unifying language I mean, it would be very interesting to address why the logical of the world is not subject to dynamics and this, I'm sure Bob would give very nice arguments for that The pointedness, of course, as I said earlier, essentially the singularities, general latinthian quantum field theory, come from assuming a pointed space, and any point, and that space can be opposed to a singularity of some important field. And the other, of course, is the continuous infinity of the events. Well it is quite doubtful, we have no actual experience of a continuous infinity of the of events, and in our experiments, which are of finite spatial temporal extent, that

25:00 is to say, they are laboratories of finite spatial extent, and the duration of the experiment is final, and we record a finite number of events, so it's a generalization. I mean, it's going to be very, very interesting, I'm telling you, to see what Movere has to see, to say, to these arguments. What is an infinite cardinal, what does it mean? I mean, we say that the continuum, that the spacetime is a a continuum, a continuum. What does it mean? We have no... Well, can somebody say that we have no operation on it? I mean, something like... Okay, the The manifold topology, of course, is based on two-way Euclidean connection between points. And this, I would like to say to compare it with, I call that conception of the topology as coming from Euler, because essentially Euler could cross the ridges of Kettysburg in either way. Well, non-superposing connections, of course, the classical notion of a topology does not allow the connection to define the topology to be subject to superposition, so you have some notion of crystalline rigidity of a topology, and of course, the events are labeled, are are ordinatized by commutative algorithms of coordinates, again this is a classical conception. What we now call non-commutative geometry, essentially all the attempts are to, the commutative algorithms of C-infinity coordinates, assumed coordinates, to be replaced by a fundamental non-commutative ring structure, whatever. And of course the globalness, I don't want to mention too much about the globalness of topology. classical topology is the study of the properties of space such as handles, holes, and when relativity theory told us that all the important physical variables are local. Suggesting instead, okay we have to suggest something instead of them, instead of constancy

27:30 of the topology some dynamic variability of space and topology, of course the pointiness we must in some sense blow up or smear the authentic points by something larger like say neighbourhoods where the continuous infinity of the infinite factors where local finiteness I will explain what that means with some combinatorial structure which which does not rely on the continuum. Of course, the space like this of the connection with something that the topological spaces that we are going to favor should be something like the directed graphs. So there must be something built in, an asymmetry built in in our structures, in our topological structures, which later on, this asymmetry I would like that there is a definite distinction between the past and the future and the connection of the different events and of course if we have an algebraic representation of this of this topological, this discrete finitary topological spaces then we must allow for superposition between these arrows defining the topology and this superposition perhaps can be interpreted in a quantum mechanical way as interferences in the topology And, of course, if this algebraic representation of the topological space turned out to be non-computant, non-abelian, so much better for us because this would give us hints about an underlying quantum-mechanical structure. And, of course, when I spoke about globalness, I mean, we should stick to global considerations. so I will start with finitary substitutes so a true inspiration for me as I think it was I found that it was for Prakash as well was my reading of a 1991 paper by Sorkin it's called finitary substitutes of continuous topology So, we start with a topological manifold, a C0 manifold, and consider a bounded region

30:00 in the manifold. Bounded meaning it has a technical, meaning a region whose closure is compact. It's also called relative compact in the mathematical region. Okay, and we let Ui be a locally finite open cover for X for that region. But locally finite, I mean every point, little X in big X, has a neighborhood that meets a finite number of recovering open sets. And we define the smallest open neighborhood of X in the topology generated by the covering sets, which is, of course, the intersection of all the open sets. I wanted to correct it up there. This is the smallest open set, and the subtle hole was generated by the covering. to which x belongs. And then we write that relation, x, r, y, when the smallest open set containing x is containing the smallest open set containing y. Immediately one can verify that that relation is a pre-order, namely reflexive of transit, related. One can actually, based on this relation, define a topology on basic open sets of the following four. These are basic open sets. And actually, this is actually a theory. More finite topological spaces are pre-orders. I read that in Burbanki. One question that Rafael asks is when is such an apology T0? A topological If you remember, if every pair of points has an open, for every pair of points there are open neighborhoods above them, that they are, that do not include the other points. So, you see, you know.

32:30 You could just say it's the specialization order, that topology generated by this company. Yes, specialization is exactly what I'll be using in the next one. Okay, that's one way of doing it. Well, this is actually hard. This is the tool, but I mean... That's an example of a T0 in modern Hausdorff space is the Sierpinski space. The Sierpinski space. The question that he asked is T0. Well, in other words, such a pre-order will cease to be T0 if they are closed loops. stated in a positive way well, when that relation is a partial order, when it's also anti-symmetric so what Raphael does is define the pose whose elements are the equivalent classes of the points of x with respect to the equivalent relation please note please note that actually the arrow notation is quite literal the arrow notation here x to this post, and then we say x, y, means that the constant sequence x converges to y in the topology of the post, right. so for a finite of course

35:00 we know that for finite topologies for finite topological spaces convergence continuity can be equivalently formulated in terms of convergence of sequence but for infinite spaces you need some more structure than sequence which is our filters but for finite what can You can define converters in terms of... This open cover, where does it come from? Or just kind of... Just a piece of cover. Cover it. Cover it. Yeah, but it's not . Where? Where? It's in... Well, you know, in physics, these open neighborhoods which can be called, are called usually gauges and these are usually laboratories so locally finite over cover means that a point x which belongs which has a neighborhood that intersects a finite number of the encoding points means essentially that the point, the event is recorded is recorded by the final number of times but I will tell you when I will introduce the notion of refinement of then you will see what it means actually this is a process of localization because as I will refine my coverings in a way that So I should envision them as being epistemic things, epistemic or non-autological. Yeah, so you're thinking of the epistemic as observable. Yeah, okay. That's fine. Okay. Epistemic. All right. Then we consider the refinement. We have to not be observed. Both of them observed. These are good questions, but I don't know if it works. I'm not out of time to discuss it. We can discuss. I think it's very important.

37:30 Then we can see that the following net, or we find the net. So, in some sense, what this means is the following. As I employ more numerous and smaller open sets to cover x, I introduce a partial order, a refinement relation between the local refinement of the covers, right, and correspondingly of the posex with sort and solibur that we use in the next one. And such a refining net or inverse system can be shown that it has an inverse limit or categorical. Tell me something. Tell me something that I wanted to ask you. And the inverse limit, which is categorically dual on the inductive limit the categorical limit or the colimit the inverse limit is a limit the inverse limit is a limit the inductive limit, the direct limit is the colimit so in some sense so it is epistemic indeed, because in some sense the more I refine my observations if I was able to give a property such as energy, microscopic energy the more I resolve, the more I refine, the higher energy of resolution for a microscope that I employ to say, to localize x, to determine the location of the point x in x the more I isolate, the more I... And the link, this is actually, this is the dual course of how you define Storch's chief degree. I mean, please make a comment. This is very reminiscent of the work of Mike Smith at the Imperial College in the Green Science Department. You told me something the other day. Right, right. I mean, of course, the general said I was quite close to that of the main theory, where between partial structures and logical spaces through the specialization of them. But he was interested in which spaces can you express as inverse limits of finite parts.

40:00 And if you have a sufficiently flexible notion of refinement, then you can get a large class of practically compact spaces. That's right. That's right. and so that's some fine problem-making work there that's been going on well he's still around I think he's of course these partial orders the locally finite partial orders extracted if I'm not going to find an open problem of C0 molecules from Sorkin's algorithm that they have an equivalent description as simulatial complexes actually, in the previous when I defined the smallest open set in the sub-topology generated by the covering when I defined the smallest open set actually, one could immediately recognize that as the definition of a nerve according to Cech so we let u as above to be a local defined open cover when the nerve k of u is the simplistic complex whose vertices are the element that is to say the open sets in the covering and whose sequences are according to the following a set of vertices forms a k-simplex if they have non-empty interception yes, no sorry, actually the smallest open set is the nerve of x relative to the over-covering view. Now, let me show you. Now, as beings in the visual context, four, and the other one. And any nerve can be viewed as a positive as follows. The points of the positive are the synthesis of the nerve. And the arrows are known as follows. We say P arrow. Do we know if P is a place for you? Okay. to the cushion structure, which are the incidence algebras associated with this locally finite process. So let P be a locally finite process, as before. We can write a lot of the rack.

42:30 The Kettbrough to be, then we can see that the formal linear span of all such formal symbols and we can use well from representation if you would like a field of characteristics not too but yeah I won't dwell more on the complex numbers actually to make contact also one at a time if we chose complex numbers and so the formal linear span of these symbols and then of course the associative product which immediately you can you will recognize as the product here in it in the post that regard it as opposed to Capricorn this is the other product of ours what does it mean which one it's just a product of arrows it means the sequence you may think of this as an operation of first say annihilating S and then creating R and then this operation is following rather this one, from left to right first apply this and then apply this ok, so of course immediately that product, that arrow product is associative because it's transitive, but it is not commutative, right, and it is trivial to see that this algebra and can be analyzing two spaces. One is the A0, the R. Well, the A0 is the span of all self-incidences, the identity arrows, and the R is the informal span of all the others. We'll give a physical interpretation to this, but the transparent physical interpretation will become transparent when I give you the limit theory, the inductive limit theory of this algebra. So A will never have the product defined... A will never have the product defined...

45:00 It is trivial abelian. Trivial abelian. Trivial abelian, that's right. That's right, that's right. This is, I mean, A is, if you like, really a discrete set. Completely disconnected. We can define a homological sort of grade on the X elements as follows. Now the grade, when the element is the number of vertices mediating between p and q, in the path of small distance in the path of small distance in the path of the line of the post. This is actually a homological degree, since as we saw there are situational problems. Is this still so instant? No, this is. Then omega, I mean the answer is to direct some of the following subspaces of dot grade. We have weights of zero grade or degree, the other self-incident system, no vertex mediates between. Then omega 1 and omega 9. Now, A, we call A the algebra coordinates. We will justify short. And R is the module of differentials. That's why also the notation we use omega because they are reminiscent of forms, differential forms. but this will become the important theory in this is by Richard Stanley at MIT 1986 for good for good locally finite substitutes in the sense of sorry the category view of posets of locally finite posets and poset morphics, poset morphics are just monotone maps is there a characterization of what the good posets are just in terms the order, just in terms of the order-theoretic properties. Do you need to start with a question? You know, all of this is going to work just for abstract process. I mean, but I suspect that it's rather found out with where they're coming from as a particular story. And as with all of the work on the structure of these kinds of process, we end up in theoretical computer science with things like event structures and so on. So I'm just wondering how to know.

47:30 I don't know, I don't know. Continuous maps, continuous maps is dual to the category consisting of the incidence algebra and algebra. I should really, this dual here really is really, I mean, actually this correspondence between the two categories is functoria. As a matter of fact, dual pertains to a contravariant these two categories and this sometime ago prompted me to think of pre-sheeps defining pre-sheeps on these low-refinite process of sorting and later I called them finite-facetime sheeps but waited until later so Stanley's theorem was in a combinatorial yes, in a combinatorial these algebras, as a matter of fact were first introduced by Giancarlo Rota in 1968 in his book on the integrative hominophorics and the theory of Mobius' forms. 1968. Then, but I've been told that... Was the assumption there that the concepts were finite or locally finite? Locally finite. Still only locally finite? Locally finite, yes. So what would that have meant there? Would it have meant something like principal ideals are finite? the principal ideas are finite yes I'm going to talk in the context of specialization as you we call it Gelfand specialization but I mean it's exactly we'll go to the representation with the maximum the principal okay so that was essentially that observation by Stanley to think of perhaps some sheath theory, to do some sheath theory in these locally-finding substitutes. Locally-finding of those in certain continuum. And the question is, of course, yes, can one define a topology on omega graph in such a way that the two topologies are equivalent, that that is the topology of the offset and the retains of the subset, and that is to say, this is the

50:00 specialization procedure so first we must identify points in these algebras we borrow from standard theory in algebraic geometry and we define points to be the kernels of equivalence classes to be reduced to representations of the algebras these are primitive ideas in the algebras and you may think of this these ideas are of the following form they are excluding the self-incidence on P then we can define a row of probabilities as well generated by the following relations on a primitive spectrum of the algebra we say that the idea of IP is broader related to the idea of Q if the product idea sorry, you're calling spec omega the set of these of this primitive idea you're defining we have omega P there and what's the spec of omega? I lost the definition of Speck of Omicron. Speck of Omicron is the state of all IPs. So there's a parameterization on P there. Are we just taking for all P then? For all P. Right. Well, the P here, the P here, of course, is a representation. And I call IP. If P is a representation of what I have done, do something. Do something to represent P. Well, IP goes minus 1 or 0. But it's true to say this is the kernel of the representation P. take a representation of the algebra, right? I thought P was an element of the post-sert. Yes, but it turns out that exactly the primitive ideals in the algebra, in the incidence algebra associated with the post-sert, are of this form.

52:30 But they are kernels for some representation, which I call P. all these ideas are, all these ideas in the algebra are the inverse images of zero in a representation what is a representation in the sense of this kind of map, what structure it's a finite dimension of Hilbert's space representation I will refer you to a paper by Zapaterin to see the representation theory of incidence algebra take it from me now, it's not the point is how to define a topology on this primitive spectrum of the algebra and the rototopology is defined as exactly in a way akin how do you define for instance in the prime spectrum of other algebras of other rings it is strictly speaking the product idea is strictly contained in the intersection idea Then you say that I, P, Rho, I, Q. Well, what does generation mean? What means the generation of the Rota topology, the generation in relation of the Rota topology, is exactly, exactly, and here's the crux of the argument. Here means that when I, P, Rho, I, Q, I'll write here. That's why we have it here. I, P, Rho, I, Q, which happens when this holds. For the pole set, the pole is P, star. Where arrow star is the covering relation in the Hassel diagram. The immediate arrows between the points. The immediate arrows. sorting topology on the pole set that's the dual to the transitive closure exactly, that's it the sorting topology is the transitive closure sorting topology is transitive closure transitive closure on that or to get

55:00 R, you just start topology and you take the transitive reduction. Exactly. So the sorting topology is obtained as the transitive closure of that problem. By star. These are the co-ordinations. So it's a kind of stone neurality between the algebra and the post-sense. Oh, is there something like transitive reduction? Well, it has to be a special kind of order. An order which is generated from its that's typically the kind of property that event structures have but not general process of course yes, event structures I'm not familiar, you keep saying that you see, if we have two more weeks for the proposal it's very much of the same flavor as causal sets of course, but I can give you some references after I'm going to speak of the, actually again it was Sorkin who did it for me because I was having these finitary topological spaces, and then, of course, I got exposed later to the causal set. I said, wait, wait a second. One must change just the physical meaning. Now, partial order should not be regarded as, you know, having a topological meaning, but having a causal meaning. what do I get from that? well, you get causal sets and essentially these discrete and quantum algebraic topological spaces the incidence algebra is associated with a causal set now I call them quantum causal sets and then I I don't speak about that let me give it another I know, explaining things which is always a dangerous thing to do it's good for us Yes, but it's okay if we do not manage our discrete differentials. I wanted to just mention to you that our discrete differential multiples in the sense that there exists a nil-potent Keller-Kartan differential operator which takes you from this linear subspace to I plus 1, and of course it is nil-potent. I cannot overemphasize the importance of this. I mean, the observation that we have such an operator given by the border and the co-border operators in homological terms.

57:30 It is the start, really, of a finitary, you can do actually finitary cohomology of these structures. And I have done some. And if you define finitary space-time sheaves, actually you can do finitary sheave cohomology of this. And they have very rich, these actually are forms, these are spaces of this, discrete spaces of discrete differential forms they behave well because at the limit, again, as I refine my observations, as I refine their localization about the original spaces point events, at the limit, again, of infinite localization or refinement of a direct system now because the omegas form a direct system, dual to the inverse system or projective system of the of the poses, not only the topological, the c0 structure, but also the differential structure, the c infinity of x, should emerge. I say should here because I'm speaking here tongue-in-cheek. Look, I have not proved that at the inductive limit you get the forms, the differential forms. that is to say I get at the length A gives the abelian algebra of the abelian ring of C infinity function of X and R, the linear sub A gives the bimodule of differential forms over the structure algebra of smooth function I have not proved that but this is my main hunch I mean I challenge this an open problem I don't know how to prove it This is my hunch coming from having the analogous theorem proved for the topological money by Sorkin. I mean, we have the proof. Actually, this is not Sorkin's proof. It's a check-in-an example. First, they proved that. But that's my hunch. Because we have the differential structure built in, this algebra, at the limit the C0 continuous topology but also the differential

1:00:00 of course I will mention there is a comment one could a sort of pardon the expression philosophical comment one could make about this I mean you started off by saying well you know smooth things do they really exist and so on but here it seems that we should take all these finite and discrete things as approximations to what occurs in the limit which is back to the familiar smooth picture isn't that going rather against than what you were saying. Yes, of course. What you're saying is really that I'm cheating because you're saying that the smooth continuum is lurking at the background and it is obtained at the limit of infinite refinement, at the best approximation. It would be better if one could sort of give the finite approximations independently of any information about the limit, for just having, axiomitizing the properties of the post-sets and how you got refinements directly. But still, even then, I'm kind of saying that, OK, maybe this is what we can empirically observe and finite grains of resolution within the limit. We come back to the familiar picture. I didn't know that it is obtained in the limit. You are right. But that it is obtained in the limit, and I think your criticism stands. It is important. It is not, as I claim, to be inherently finite or inherently discreet, because, exactly, these substitutes are obtained by the result of the background, there is the continuum. Perhaps it also strikes a good balance, because I'm not totally opposed to what Louvier would say. I was going to say, if I can just interject a very quick comment, this itself seems to me to be a very Louvierian viewpoint, his ideas about the so-called Aristotelian intervention. modulo my objection that exactly the smooth structures and the continuum is obtained at the limit the limit itself is non-physical is non-pragmatic what do you mean infinite localization what do you mean infinitely small it's a kind of platonic idea where we're seeing some shadows along the way yes these approximations are mere coarse images of that of that which is not it seems to be to have more the flavour of the

1:02:30 Aristotelian potential infinite than the actual infinite in this construction that you're discussing, but anyway, we don't want to go off too far on the philosophical tangent well, this exists though I suppose if one brought quantum considerations in at some point and you said that you had to sort of leave off at the Planck scale or something you might then land up I don't know if that's the way the story goes. It is, from Raphael's point of view, it is. For instance, for him, the causal set is the fundamental structure, and the manifold is only, of course, approximation of the causal set, not the other way around. Not that the causal set, that these are approximations. So he has a paper saying, does a manifold approximate a causal set? Some may have, so on. yes, these are good big questions I don't know how to answer them still, I'm trying to comprehend what's going on here the discrete I want to all right since we started with complex algebras at the limit, we inevitably will get complex space and coordinates and differential forms at the limit, so reality conditions we know that the classical theory is based on real manifolds and not on Kether or whatever have you so why start from omegas over the field of complex numbers why not say the reals up front but we do not know we cannot make any connection with quantum theory per se real quantum mechanics does not behave very well but what about prime fields I don't know I don't know, I have not investigated perhaps we can define incidence algebra over the but what does a quantum theory log like that look like, I mean what are the amplitudes I don't know well, first of all Pablo was just saying something about negative probabilities do you know something do you know a beautiful paper by Feynman it's called Negative Probability I'll give you the references in the quantum implications actually it is edited by

1:05:00 exactly, that's why I know it and he says you see in quantum mechanics the probabilistic interpretation I mean something which is a probability is a positive matter it is He's intimately associated with a positive definite Hermitian inner product in the Hilbert space. And Feynman had some idea to do quantum mechanics in a Hilbert space with an indefinite metric. So that for him would be also more natural to relate to relativity theory, relativity theory, the metric is pseudo-Euclidean. I mean, it is indefinite, again, you know, the Minkowski metric. He has some wonderful ideas about this. I mean, he says, yes, also he associates a negative probability with a positive probability for a particle. A negative probability for an antiparticle and a positive probability for a particle. And since the laws of motion for the particle are the time reversal of the laws of motion for the antiparticle, he links time asymmetry in some sense the domination of positive probabilities over negative probabilities the time asymmetry because if laws of nature were time symmetric we could not describe the forward propagation of a particle from the background propagation of an anticorps the universe exists with probability 0 which would be canonical but not very visible Yes, yes. The counterintuitive. And, of course, people had also looked at negative probabilities in the context of the Bohm group approach to the Lugol theorem, hadn't they? I'm sure Basel can tell, but I don't know. Quite a bit about negative probabilities in his... Details. Since, well, of course, here we are given... the question for instance in Raphael's work I don't want a universe of console sets one must be able to dynamically vary these poses what does it mean to dynamically vary a pose set

1:07:30 it was I guess let me tell you let me tell you I'll put it back a little story about this perhaps I should maybe this slide his idea was to switch emphasis as I said earlier from topology to causality first, the arrow does not mean topological connection it really means causal event P is before event Q or Q after P and then that was I think his maybe subconsciously I don't know if he was familiar with the word of Christopher Zeeman of course, which is very important to me and Zeeman showed if you model the causal relation between events after a partial order then you can determine 9 out of 10 components of the Minkowski metric the 10th being the cold four-bound part of the Minkowski metric space-time volume. 9 out of 10 components. And also he showed that indeed, if Mikovsky's space is represented by a partial order, then the group of photomorphism of the partial order is isomorphic to the conformant Morenz group. So, I think the original idea of sorting to model the structure of space-time in a point of view really can be grounded to Zeeman, on Zeeman's work, working with partial orders as representing causal spaces. And I'm letting you have locally finite finitary post sets with other causal sets of Sorkin. And my idea was, well, my idea was first to algebraicize the causal sets, so with the locally finite post sets representing causal sets to associate their incidence algebras in the way I showed you before. And then, if you want to vary these what I call quantum causal sets, is to define sheaves of such incidence algebras over a

1:10:00 causal sets, now treated as a background base space, all right? And sheaves. Let me tell it immediately, my hunch. My hunch was that you can vary these structures, sheave theoretically. That was my, my, okay. We know, of course, then I had to define what I mean by finite space-time sheaves finite space-time sheaves then of these ring structures of quantum causal sets of the incidence algebras and of course I was over, after taking Lambeck's courses I was very much in resonance with category theory and I felt that since all classical physics plus quantum field theory can be formulated with enough care in the topos of sheaves over the manifold, sheaves of sets over the manifold, where I can build the rings, the fields, whatever have you, the higher structures, from that the topos of sheaves of sets over a continuous time, which can be interpreted as a universe of varying sets, continuously varying over the continuous parameters, parameter space X. Then similarly, a topos space-time sheaves of quantum causal cells, perhaps it could qualify as a universe appropriate to quantum gravity. What does that mean? Again, it would be quantum causal cells that vary due to gravity, because to a locally finite causal and sort of gravity, because, again, what is gravity? I want to bring to your attention that gravity is a fancy way, at least geometrically, gravity is a fancy way of thinking of a variation of causality. What is general relativity? General relativity essentially rests on the equivalent space. Look. Equivalence space. Which says, the curved or gravitational

1:12:30 the curved manifold space-time manifold locally Minkowski and gravity locally Minkowski so locally flat In a bundle picture, it would be something like that. You have an event X in a manifold. You have a lot. This is the light cone, an event X. And this would be, say, a curve in a manifold. F, which is the curvature of the gravitational field, I'll write that G, is a measure of tilting of the tilt, or of the tilting of the ruby of the Minkowski life cycle. Look. So, the equivalent of the Minkowski life cycle at every x. This is the equivalent of g. At every x. It tells you how the gravitational how it tilts the local causal structure at every x. Another way to say it. What does it mean, is locally we cost? It means that space-time is locally flat. Space-time locally, by Zeeman, by Zeeman locally, is a partial order. Local is a partial order. And this variation is a variation of the partial order. It's how much we tilt the partial order, whatever that means. Whatever that means. But it is a change of the... what is it varying over it is varying you mean what is the base space these are Binkowski fibers these are Binkowski space what is it varying over it varies over

1:15:00 the manifold that's what it varies over the sheaves of sets over the manifold is a universe of varying sets over the manifold the finitary space time sheaves of this I substitute here the partial orders, the partial order the local defining partial orders of Sorokin which by Zeeman I encode all the information about the metric of the of the Minkowski state, local Minkowski space by the equivalence principle I substitute it by the incidence algebra so I have a bundle or a sheave I have no base in the southern each stalk is a non-metall and at the base of course I cease to have a continuum at the base I have just a locally finite topological space and how do you vary this? well to have a non-trivial variation means to have curvature means you must have a non-flat connection so the it is then in the question here I mean you must first define in this in this in that sheaf of incidence algebra so over over locally fine of course you are first define a connection as a sheaf morphism I don't want to go into this forever then this is a non-flat connection then you define you can define because a A connection is simply a differential, the Kellen-Kartan differential that each stock has, plus a gauge potential. and if I have such an object that I have such an object I can construct this then I can consider also the curvature of the connection which is usually defined as b squared you see you see how the equivalence principle works here locally fiber wise my connection, locally fiber-wise, the equivalence principle tells you that locally the curved manifold is locally in cost, is locally flat. Locally flat, what does that mean? Locally, I can find a coordinate system that covers this point, where this vanishes.

1:17:30 Where this vanishes. So the curvature is d squared. But d squared is in your fault. So, indeed, locally it is flat but of course locally locally it is a partial order or the incidence algebra of the partial order which by Zeman a partial order determines Minkowski space because his face is flat thank you very much but if you want to compare entities living in this world I mean in this final in this talk with this talk you must define some notion of covariant derivative you cannot you as in flattening of his face, you must adjoin this. This thesis is called this process of substituting a flat connection by a curved connection. It's called gauging. It's a fancy way mathematicians work with gauging all the time, namely localization. These incidence algebras are inhabiting this top here and this top are completely different worlds. That you cannot compare apples with oranges here. I mean, stock-wise, of course, it's meaningful to add, subtract, multiply. But if you want to compare distant objects, distant parallelism, what is called, then you must introduce some recipe, which is the connection. The connection gives you a method to compare objects living in these two disjoint worlds. And the connection comes with a curvature. But of course, you can test for the root of your connection. If you identify these two stocks, if you bring in this unidentified, then this becomes zero. Of course, I'm not sure. Your connection reduces to the new photon. Here, which means that not only your curvature is zero. Okay, okay. These are geometric pictures. Okay. But the mathematicians work with localizations. Constantly localization means essentially what, for instance, what I was looking at the other day, even Grottenig speaks. I mean, I think Grottenig was doing gravity there. Believe it. The abstract idea of gravity is the following thing that you have.

1:20:00 Well, you're thinking of doing things in sliced categories relative to sort of relative so you're doing things in a sort of fiber setting with something over and over a given in the algebraic setting over a ring or something, pulling things back pulling things back, that's right in some sense it is what the space over which it is a ringed space that's what it is from this point of view scheme theory would be something a sheaf of rings of non-commutative rings so therefore a scheme over the primitive spectrum of a ring that's what it would be and this localization you have independent local rings correspond to because these rings derived from the partial order actually they encode the same information these are locally, they are independent though so I've got a means of comparing, of gluing them together in a geometric way, it is meaningful to say if we take a section here, such as A another section, it's meaningful to say A minus B Stop-wise, of course, if it's A minus B, A minus B, to be able to do that, you must introduce the notion of connection, which is very, okay, it's a very geometrical way of seeing it, but I tried in vain, I think it's probably my problem because he's a very brilliant man. I try in vain to convene Sorkin during the summer that, you know, Schiff theory and Topos theory gives a natural setting for studying the dynamics of causal sets of these locally defined and partial orders and he says but I don't understand this you know, this

1:22:30 algebraic geometric ideas, and what does it mean I say, do you know that substituting a point by an open neighborhood about it, it's an age-old theme, starting from Czech, I think, and then Alexandro from Pontriagina, all these people trying to resolve somehow, and we see in algebraic geometry the various blow-ups. In algebraic geometry, we blow up an offensive point by, say, a space of direction, tangent to it. What does it mean? I mean, essentially, this is what it's been doing, in algebraic geometry since its inception this is not it's not a for a mathematician this is not this is not a surprising thing to say to say yes okay if i consider sheaves and then a topos then the tops are also sheaves of these rings over whatever have you i mean when you're using fiber bundles and so on. That's quite common in science physics, isn't it? Why is that acceptable? I don't know. You know, the only different thing that... There are a funny lot, I think. There are funny lot, in a sense. It is very difficult to convince people that, for instance, for instance, with the continuum. People, mathematicians, have managed for millennia without it. Really, really. Fiber bundle, what's the difference between a fiber bundle and a shift? Nothing. Fiber bundle, the only difference that I would say is of importance is that while a fiber bundle is locally trivial, local is a direct product of the base and the stock, a shift may be locally twisted. it may be also important difference between sheaves and bundles but i mean it's the same thing you have a base base you have the strokes the other thing you call the base space and fibers and you have compatibility conditions perhaps perhaps lazar's definition of a sheaf would be very odd i mean to to define a sheaf as a local homeomorphism that which i can understand though

1:25:00 So I think this is a very nice definition of a sheet. I'll just ask one more thing. You're thinking in terms of taking sheets of these algebras over a particular base space of what you call causal sets. So is that a category of causal sets or a particular causal set? That would be, look, very good. Initially start with a causal set. I recently had some idea to have the category of causal sets and these sheaves over the category to be something like to consider it as a a site, a growth-endic type of topology so you would have, you consider as a site, so you have sheaves over a site and then growth-endic topos that seems more natural I'm getting to that, I have not done any work yet, but to think of it exactly as a site and then a growth-endic because look what would be an actual topology to define all this, the ordered topology for a causal set would be again, as here relative to the open pass the basic open set would be the set of y such that y equals the Alexandrov topology yes, the Alexandrov topology I'm taking you a lower set I'm taking you a lower set Also, I would like to apply some ideas of taking lower set. I have some in there because I want to define sieves on it. Or how do you think they call sieves? I know that mathematicians are not very familiar with sieves, but they are familiar with co-cribbles. Creebles. Creebles. It was the term in the Virgin Big School at some time. Yes, I think primal is just trench for a lower set. Lower set or an upper set? Well, in the case of a partial order it would be a lower set. Lower set, yes. But then people sometimes turn the order around. That's why I said co-criples. Because upper set maybe is the co- They're taking the contra-variant. The contra-variant. Yeah. Okay, so a natural topology would be, the other would be the Alexandros topology

1:27:30 So, you know, the gradient would be A, X, Y, Y would be all set, such as X, Y, Y, right? This you mean the Alexandra topology? No, the Alexandra topology I would take to be the opposites, in fact. You would take to be, as I said, this would be lambda X, would be the causal path. This is, yeah, this is... You would say this, B, right? That would be more like the patch topology. in the physics literature this is referred to as the Alexandros for some reason can I ask you something here maybe look at this which Alexandros the AD I think so the Russian topologist there are two There is an AD, which is the famous one, I think, with nerves, and it's the PS. No, the other way around. The PS, and the AD is the causal space. I see. All right, well, I thought it was the lower one who was the original. This is topology of Minkowski space. I see. No, it's the other one. The other one. Nerves. That's the one I'm sure they were. Right. Yes? Okay. By the way, in domains, I mean, this is the kind of, also the kind of thing we do, except that we have limit points on the post-sets. Then you have things called domains, and then you take the Scott topology, where you have to respect the structure of the limit points as well. The Scott topology is, I don't know. Well, so the point is where you have not just the partial or structure, you also have limits of directed sets, least upper bounds of directed sets. in the realm of sober space, it's not just team order, but sober. So that the, so it's not just that the specialization is a partial order, but that, for example, it means that every directed set of specialization ordering has a least upper bound. Yes. So, they are maximal points. and there are maximal points

1:30:00 yes, yes so you're including the limit points the ideal things as well and then the open sets should respect that structure in a sense that not only should they be upwards closed in the specialisation of the ring, but if a limit of a directed set so the limit of some infinite process is in the open set, then some member of the directed set also has to be in the open set so it's giving us Is that the dinoscot? No, is it? Sorry I apologize. I thought there was a different Scott involved. It was the Scott who collaborated with... It's Dana Scott. Okay, right. There's another Scott called Phil Scott. Yeah, who collaborated with... That's right. Yes, so there's a symmetry also with the confusion over the Alexandros. And there are a lot of Scots in the north of England. For God's sake, don't say that. let me give you the same thing so I guess we should let me close with this I think it's wonderful you have correctly grasped the drawback that the continuum brings if the molecular view of matter is the correct appropriate one, that is to say if a part of the universe is to be represented by a finite number of points, then the continuum of the present theory contains too great and meaningful possibilities. I also believe that this too great is responsible for the fact that our present means of description miscarry with quantum theory. I mean, here, the problem seems to me how one can formulate statements about the discontinue without calling upon a continuum space and as a name. The latter should be banned from the theory as a supplementary construction not justified by the essence of the problem, a construction which corresponds to nothing real. But we still lack the mathematical structure, unfortunately. How much have I played myself in the way of the manifold? 1916. Imagine, 1916, Amelia. One year after General Relativity. I mean, this man was great. This man was great. The manifold served him so well. I mean, imagine his surprise when he was taught Riemannian geometry by Grossman, Marcelo Grossman.

1:32:30 Imagine that all his ideas would be put in the way of the smooth man, he would form differential equations. Yet, as soon as he was ever ready to abandon the theory, that served him so well, ever ready, wonderful. I mean, this is very inspirational to me. But also, let me finish. This is one year after G.R. Let me give you one year before his death, which is amazing. Einstein's prophecy, okay. One particular good reason why reality cannot at all be represented by a continuous field. From the quantum phenomenon it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers, quantum numbers. This does not seem to be in accordance with a continuous space-time theory and must lead to an attempt to find a purely algebraic theory for the description of reality. Who Einstein? time. So much he resisted quantum mechanics, and so much he abided to geometrical principles. I mean, there are the background space, it's there, you can have lines, points, wonderful. However, he says, nobody knows how to obtain the basis of such a theory. Einstein's meaning of relativity, appendix deep, 1954. It was beautiful. It was beautiful. I mean, you could add to that that the numbers themselves should be a finite precision, if I want to take that to a little bit. So that's, with that addition it would be as finitistic as you could go, which is very interesting. Okay. It is important to bring such ideas to the table when Bill comes. Sure. To see. see. What are their reactions? Also I'm planning to bring also Tassos to see, because I think then too they are going to have a good discussion, because Tassos also is one, he's a very mild man as well, he's not like an asshole as I am. I do not push people to be... I mean... Let us see what he sees. Let us see what he sees. They are real things. For him, natural numbers are the constructs, really, of human mind.

1:35:00 So has Malios sort of looked at synthetic differential geometry on standard analysis and so on? He has, yes. Did he sort of consciously look at them and decide he wanted something else, or was it that he just had a direct path to doing the work that he did on this? No, he was not motivated by this. It was ever since he was, I think, at the, I mean, this abstract differential geometry really, he originally started by considering shapes of topological, he comes from topological algebra's background. And originally, he conceived of that back in the 60s, when he was at the Institute of Limbo and Karniwitz. These people. I mean, his work is really old. I mean, now it was given in a book. It was crystallized in the book. But no, no, it hasn't got almost anything to do with synthetic differential geometry. Nothing, nothing. Essentially, his motivation was to write down calculus without calling the smooth. to show that you can do almost everything in calculus without assuming any notion of calculus at all. Just by using the algebra of functions. Yes, just by using the algebra of functions. And if you have, the motto is if the algebra is right enough, that is to say right enough gives you a nice differential. It's more in the spirit of things like differential algebra that seems to take it Yes, yes. Differential algebra, yes. It's not wholly disconnected from the synthetic differential geometry approach so much as one's looking at the structure in the ring as determining the space in the properties of the inverses. These are ideas really, at least for the commutative cases, Keller all the way. I mean, Keller said that every commutative ring has a differential. Yeah. In the basic point of view, synthetic differential geometry is smooth, I mean that's the basis Actually, synthetic is all actually synthetic is the differential it's all transformed You have these linear approximations

1:37:30 everywhere Just as people thought back in the 18th century You mean like Well I was thinking more of people like Tourette some of the precursors of the calculus this principle that in the infinitesimal infinitesimally small every function is perfectly smooth and perfectly linear I forget who it was who propounded that but was it Oyn? I'm not sure Certainly though Rothendick had insisted on including Niel Potens for sure I mean these what's called Lovier-Cock yeah the Lovier-Cock axiom right the basic axiom says that you always have this linear transformation to an arbitrary final of course that's in the setting that you have a set theory because you have 0% ring with 0% square and 0 and of course that has all sorts of knock-on effects on the calculator for instance it means that you lose the immediate value theorem well yes I mean you lose excluded middle tends to have an effect yes but it has many effects but at least one of the effects orthodox calculus is that can't prove there's immediate value theorem here we have a example think course with the structure, with the incidence algebra, that they have the differential, and at the limit, okay, you recover, there is a strong feeling, I have a strong feeling that you recover, I have not proved it, as I said, that you recover the smoothness. It's interesting because it started with pose, it's not simply pose, but if you associate a ring, and if you take with some pinch of salt keller's inside, that every commutative ring here is non-commutative, but every commutative ring has a differential, so you unveil also differential properties of the atom. so at the limit you get a differentiable space

1:40:00 and it is interesting to see that in the limit you would have that very basic motto that I was taught at least in high school that you know differentiability implies continuity well not the other way around differentiability implies continuity it's interesting, very very interesting I find of course I had a deal at the end I had to just ah, this I shouldn't forget, because it is such a big audience of course I would like to thank very much Yes, but it is the, it is the defining, I was defining, okay, this is a great time, she used it, and the differential structure, small ones, and I feel like, I wanted to add to Milner, because there are some, there are some results by Milner, that he said, that if you start from discrete differential algebras, you start from discrete differential algebras and you take some limits then you may end up with inequivalent differential structures things do not take what I said I may be wrong I may turn out to be wrong but you get at the infinite refined injustice move I mean I'll try to prove it I'll try that you get the infinity at the end well we are I think we've been the longest talk we've had this seminar but from my point of view certainly one of the most stimulated so thank you very much thank you that's great it's very fascinating to me many of the things using tools that aren't familiar from a particular platform but even

1:42:30 Yanni, if there's any consolation the last time I was at a seminar with an audience as small as this, and in fact it was exactly the same size exactly three people, was in fact when Bill Lourvia spoke in Florence on Midsummer's Day 1998 about extensive the first hour of four of us oh I'm sorry yes well actually come to think about it I believe there was a totally silent member of that audience as well who was somebody who just wandered in and sat at the back for about 15 minutes but otherwise there was just myself John Bell and Alberto Pruzzi you're in excellent company because you know the story that when Newton gave his great lectures at Cambridge which became the his lecture is on arithmetic he started I think with three people in the audience he finished with one this is what I say the lecture is arithmetic so you're in very very fine company Thank you. He only circulates selected The ones that Bob vouchers for I'm not able to get to your talk Bob, that would look extremely interesting If I know your email I can give it to Bob It's very easy to remember It's mpbw.currentbund.com I'll give it to Bob, if that's okay. Okay, that sounds good. Yeah, I'll give it to Bob, yeah. Should we go and have a look at the...

1:45:00 Yeah, should we like to have a cup of tea or something? We'll have a couple of... Look, I'm really apologising... Oh, no, not at all. It's easier when there is a small group to go into things. Yes, location of Bingo. Yes, absolutely true. I'm telling you, I'm very sorry. It didn't, I'm possibly going to know. And that's Joy Christian, isn't it? Yes, yes, that, that, that, ah, we're going to have a cup of tea then probably We should all go to a couple of advanced stairs. if you want me to scribble down my email now I can do it absolutely yes that's where I can eat ah, immediately my pencil's just marked thanks I get very excited when I see from other disciplines people that speak almost the same language

1:47:30 in the sense that they use structures I must show you some of the stuff about event structures also maybe we can dig out I don't know I know absolutely nothing about that nothing about the main theory either which sounds like it's something that I certainly ought to try and learn a bit about yes I mean can we certainly like to know are you in the classroom? no I'm not, I'm not anywhere now I was many years ago at Cambridge and then later at London but I'm completely I'm a small travel business specialising in conference organising but I try and stay in touch with foundations of physics particularly with capital which is my first principal interest and the foundational significance there on so it's out of a philosophy and math background of a mathematician when I realised I was never going to do anything like it enough to do a television search. I did a retread as a philosopher of mathematics. Who did you do that with? It wasn't very long time ago now. I was in the incentive. That was, well, my supervisor was actually Hans Kahn, but they were quite a magician. Yes, my, well, there was Wilfred Hodges. Yes, I saw a lot of Wilfred Hodges. Yeah, yeah. I was thinking, who was the supervisor Grassman? he was also a theologian he was a theologian or definitely a teacher and also of course he always created the modern theory of theology really? he was the most important grammar in history yes he was I mean, Grosven was the man who declared Sanskrit but we never had to read Sanskrit The man was arguably the most powerful man

1:50:00 of the 19th century. It's quite incredible. Not only was he clearly in a very great algebra working, unfortunately in such isolation, in such a level ahead of the internal mathematical culture that he was quite, for a very long time, not understood. Indeed, he was dismissed even by the editor of his own collected works as a family of fellowship. Why don't we all go downstairs? But he also found the time to be Is it possible that I can call my life? Absolutely, why don't you make one all night? I just don't mind when I start my life. Yeah, I think almost all the fundamental rules of phonetics and phonology. Really? Phenology. Okay. And as I say, great, great. Thank you there, Ian. And to the column. Paul, do you foresee more than an hour that we can stay? I think... No, I think I'm glad he's saying that he translated the... He was the first very translator of the Sanskrit. Right. Not there anything about it, but he discovered several of the less fundamental laws and yes well it's I think, in fact, he was far more appreciating his lifetime for his work as a grammarian and as an orientalist than he was as an altruist. What would he write about? Oh, maybe I'm planning on tea or coffee. It's related to tea. I actually would rather prefer tea. Right. And it's astonishing how recently... In fact, it's popular to be the work of Ratham in this school, isn't it? Some of Ratham's insights. A very interesting paper by Bill Wilhelm Ratham, do you know it?

1:52:30 I'm on the service, Mr... I'm getting all these papers off you. Is there any... They didn't have a paper. I don't know. I don't know. I don't know. I don't know. I don't know. Well, I've got to go quickly on Sunday, so I'll be awake for about a week. Have we had a little bit more of it because I'm going off to France to see where it works here. Are you having a little bit of berries? No, no, I'm not. That's going very well. What is Tuesday? Oh, well, Tuesday, I don't know. Well, it might be that way, then. When is that? Thank you very much, yes. It's something I don't know. Ah, well, there was a game that was a straight time. I don't know. What day is that? Can you remember? What day is that? He said two weeks ago. 26th of March I'll be available again but I'm going to be back around from about the to the 27th I must and I'm sure I'll turn over there but there's a it's actually called grassland dialectics and category theory and it's in the proceedings there was a conference in 1997 for the 150th anniversary publication of the in 1994 but the publication of the proceedings wasn't from 1997 Can I ask you something? I don't know what you're going to do. Do you know the answer? Yeah. What is it? I'm published. It's made all of those kids. Well, it's published. It's published a long way. It's a good word. It's a good word. That's the first time I heard it. Yeah, it's a good Greek word, too, of course. It's a good Greek restaurant. Yeah, exactly. It's a good word. It was a good palatable. I break the symmetry, I choose any? No, it was definitely Hugh Musliet. Hugh Musliet? Oh, well, Grassman did. Grassman did. Grassman's crossbar in 1877. It was my best, because I...

1:55:00 Grassman was a very marginal. No, it was... In fact, it was Grassman's first published mathematical work, and he published the in 1862 he published a second revised edition and then he published the Foreman Era which I think was about and he died in 1877 I think he was actually I think he just revised the third edition of L'Australian Square before he died but it was certainly published in his life, in fact more than 30 years with more is there. As a matter of fact you know the papers that they are coming Finkelstein's. One paper is based, I mean, the Austen, the other time that I was here, Samson told me that some of that quantum set to you, Finkelstein, which is a lot of it is based on Grasso's ideas, but how far it is, yes. He said, tell me, you told me it's a linear version. Well, it's kind of a linear version of set theory, yes. I mean there's an inductive definition which is I mean it's the tensor algebra symmetric algebra and exterior algebra the linear algebraic versions of lists, multi-sets and sets and he's closing up under those inductively so you get the hereditary structure of sets and he then wants to have a kind of thing, which would then be like solving a domain equation, as we say in domain theory, which is also a place where inverse limits come in. Yes, yeah. And clearly he does have a way of thinking of inverse limits and indeed thinking of all the logical structure very much in terms of the algebraic setting, well in the first place, and then the geometrical meaning of those constructions. the logical meaning of construction seems to fall into place within that framework in some ways it's a remarkable anticipation Nick Clifford knew of Grassman's work yes he certainly knew of his work I don't know very much about what the extent of the influence was I mean because Clifford's product is nothing

1:57:30 but Grassman's product I think that Quentin may have discovered that independently. Grassman's work was pretty much neglected, as I understand, in his lifetime. And then later, of course, after his death... Well, of course, some work is of the nature that you need to rediscover the ideas yourself before you can understand what the other guy was doing. Yes, yes. And I think Clifford Pizzitti would have been in that position. I don't understand, unless I can work on it. So you can sort of forgive his contemporaries, maybe, for... Oh, yes! I think he was extremely, extraordinarily difficult to build a mind of half his groundsman's at the time. And also, probably, I mean, I can't say I've read his work, but I mean, I could... Unfortunately, his philosophical style was clearly mainly influenced by Hegel and Firmarker, so it was barbarously impenetrable, and even by the standards of the 19th century German idealist in their philosophical writing. And also, of course, he had no patience whatever with the axiomatic method, which has already become the dominant position of work by the time he was writing. And he was physically intellectually very isolated all of which contributed to making it easy for him to be marginalised. How did he make his living work? As a school teacher I think he inherited some money from his father He was actually trained for the ministry. He was trained as, he was expected to enter the Lutheran ministry, but decided he didn't want to. Intellectual crisis of, well, not exactly a crisis of faith. I think he remained a believer, a theist. had problems with some of it. Who first defined the group? Galois. Galois, surely. Did he define... Well, I mean, there's always the matter, what does one mean? I mean, in terms of who...

2:00:00 If you mean he wrote a definition not thinking of it as a group of permutations, that may be a little rather later date. But not much later. Really? I guess by the 1860s. So who would have written down that definition, then? Oh, OK. I'm just guessing. I'm not a history or a historian of mathematics, but I would have thought probably. I can imagine it would have been a few decades later in terms of really giving the abstract definition. An associative product with inverses an identity. I think it's Galois. I think so. I think he just did it like that and he didn't talk anything about permutations. I think so. Are you sure he didn't say there was a permutation of the roots were closed under composition and inverses. I mean, that's what I would strongly suspect, but I might be wrong. I can't say I've read this. There was a phase of people writing down abstract, algebraic, axiomatisations of things just in the last decade or two of the 19th century, the first decade or two of the 20th century. Quine used to like to quote some of Huntington Huntington who was one of the leading and he was writing down a lot of these definitions then of course there was piano I think it was very important for axiomatisation and that was towards the end of the 19th century Piano, did you say the piano? The famous Greek mathematician told you the joke We are at McGill now, and I'm taking a complex analysis course. And the complex analysis is actually a famous guy. His name is Drury. And he first starts with the first course. And he says, Peano, he's also English, Drury. actually he's from Oxford or from Reading and then he says Peano the famous Greek mathematician and I stand up and I say excuse me sir but Giuseppe Peano is obviously Italian, he's not Greek

2:02:30 and he sees that I'm a bit and he says it's alright are you Italian? because he thought I was offended he was Italian I said no I carried the nation there I said no I'm Greek I upset Yanni terribly the other day by saying but of course certainly all Sicilians and Neapolitans are honorary Greeks anyway of course there's a Thales University in Turkey because he lived in Asia Minor from Miletus right the birthplace of western philosophy yes actually even Mania as you said Sicily in Agrigento I know they were big who was from Brazil? Aristotle didn't he work in in Sicily Aristotle was well Aristotle was actually he was in Sicily of course he was in throughout and then of course Pythagoras spent most of his life in Sicily as well incredible Swiss people I saw a paper by Max Chalmer, I have a very many... Sorry, just because we have a very good time. I'm so sorry. Yeah, I agree. In different ways, people are coming over into life and they are visiting the imperial. He said, yes, because they did not have all the time. We have had all the time. I agree. He made a mistake. He said, how can we understand this? Aristotle said that the natural state of motion of a material body is addressed. And he didn't go for so many years for coming here, but he didn't go for so many years for coming here. He didn't go for so many years for coming here, but not only addressed, but to do that, that can explain to me through some kind of fiction. I said also in... We're not quite certain days yet because of the fact that we have a national congress, Yes, which takes place, and we're not quite sure what the dates are, which is important.

2:05:00 To the extent that he could and he was bothered to do it. He didn't explain it at Einstein, but probably he didn't. Anyway, he definitely didn't the first time. And he stopped and said, yeah, probably. And he'd be in Italy. The other thing is a bit unnatural. I mean, I think it's a bit of a caricature of, for example, I mean, well, Archimedes was quite around thinking of instruments of all, quite practical. And of mensuration. Was it Eric Tostadius who measured the earth? Yes, it's a concept. we'll focus very much on the foundation so these are quite impressive points that we mainly interested but of course it's true that there wasn't the you probably talked about social so Colin McClarty is giving a talk about general history of the whole After this year, we can say that in a racist way, I think the emphasis of the theory and how it emerged from what we're playing through a growth and deepening I think they found that to actually explain, to get your friends dirty is a bit... And what was it about a year or so ago, you were telling me Everything can be captured by the human mind, but pure thought, that was the idea of lightning. It's still so beguiling nowadays. We're having some inclusion of... ...vibrations, shushies... That was what I mentioned, the kind of enriched perspective. Yes, yes, yes. Yes, they are in the... Exactly, yeah. Well, I would be fascinated. I mean, would you be interested in talking about that? I'm thinking, yeah, on physics. Yes, exactly, which is right. You can't be playing with a spin, you know. That's right, that's right. And you get a nice trip out of it.

2:07:30 I mean, as I say, I can't promise that we've been able to... You see, the thing of things is a bit polemic. I mean, at the moment, the only people are never going to have enough money for a private college. that's why he didn't understand because this is more dirty stuff it's in some sense I'm going to think you might be very much giving up a deterministic theory but the quotations that you gave were fascinating to show that he was willing to explore things that went And then, he's a philosopher in Florence. All these polemics of Einstein that say, well, he was again popular with his... No, just read, a big read. He was ever ready to abandon the class. But he has done some quite interesting work on application. Relation pedigree. Called pedigree. I'm really... He'll be giving a talk. There'll be a couple others. We don't have to keep it small, but maybe for financial and yet... The generation after really felt that he hadn't stayed up with... Give me a wee moment which I should have brought and as soon as we've got to have a look at some of these things I was mentioning I mean it's certainly going to be within the first 14 days of August 2002 you see with the domain theory there's a lot of stuff about 11th is looking the less likely dates we just have to check a few we know that those tend to be available at that time and so is Colin, so am I, so is Alberto, so is Yanni which is to do with sort of taking an inductive process of sort of building up layers. Well, that's not a few things about the thing. Then close in the past. If they trim it, if they trim it, then it will tell it. There was this paper I was working on since then.

2:10:00 You've now got maybe 18 months if you want to finish it in that time and then give it to this meeting when you're on. So this is my cooking. actually if you want to say just replace okay that's great is this department I'm sorry this is I'm sorry I'm forgetting that that's great thank you very much and do you want to just come around the dates yourself you can make functions and apply them but everything is in the same it's not time so you can apply them to yourself definitely in the first half of it was 2003 the moment would probably have served to the 11th on the slightly dates but in the end so what this needs if you think in terms of building a kind of space or set that would model this is a set which is isomorphic or a set or a space of some kind which is isomorphic to its own function space because then you can use to make sense of self-application because everything is a function everything is an argument and so on now Cantor's theorem tells us that we can't solve that equation in classical sets actually it turns out one can in various topos what he did was to find a topological space where because of the condition you actually have a topological space that is homeomorphic to its own function space so a space x which is homeomorphic to x to the x yes and that cannot well i mean it's an interesting question whether that can be done in a in a space with classical separation properties so certainly that it's already a non-trivial result to show that it that there's no compact house dog space which non-trivially has that property and it probably is true of any house So, therefore, one had to go to more general kinds of T-naught spaces. So, really, in a paper by Scott from 1970, which appeared in one of the, actually, in the, well, it was one of the volumes edited by Lorvier. I think it was the sort of sympathy conference when he got fired by Halifax.

2:12:30 Yes, it is. It's 1972. It's spring of... Yeah, Thomas was able to say geometry and logic, I think. So, then you find one of the first papers by Scott. right right very good in there there's one of Scott's early papers on this he had actually talked about it at the big logic meeting big logic colloquium right it really goes back to 69 as I was saying that's what I mean in fact he'd written a page given some lectures in Oxford saying that the type free lambda calculus doesn't make any sense here is Here are models for it, including fixed points, and so on. And then he sort of, as he likes to say, had some kind of nightmarish vision of this where he realized that you could make sense of this type-free calculus. But then this is exactly to do with starting, you know, once you're in a suitable category of spaces, you start at a suitable place and you iterate the function space functor, and so you get a sequence of things, and you take the inverse limit, called D-infinity models so things satisfying the isomorphism isomorphic to X to the X there is actually a nice description of the construction in the lab at Scott and it allows for self-application allows for self-application but you have to keep very careful track of retracks don't you well one of the points in order to make function space into a covariant functor I mean, it wants to be contravariant in its first argument. So the point about using retracts is that that makes it into a co-variant function. You just twist the things around. Right, or actually use things which are both retracts and junctions, embedding projections. But it's exactly the idea. And this then led to a sort of whole, I mean, you could then use the same tools to construct the wealth of such things, which are, and the idea is you're really closing under constructions and finding something that's sort of satisfying this functorial isomorphism. It's a fixed point of the functor. And so what I was saying about the Finkelstein thing is you can sort of see, you can do the same kind of thing, closing up under forming.

2:15:00 You can also think of the way you construct the set theoretic, universe, maybe with some bound on the cardinality, well I mean even if you don't take a bound, we just sort of work in a suitable instead of thinking of topological spaces and continuous, you think of classes and sets where smallness gives you a sort of way of thinking about continuity, a way of thinking about continuity so you can then think of just of the power set and if you look the least fixed point gives you the usual universe of well-founded sets and actually another thing that out in recent work is that you can also give good meaning to non-well-founded sets which is the greatest fixed point which is kind of like taking a projective limit rather than the inductive limit the limit points which gives you these sort of things where the membership goes all the way down so that's the kind of thing he's not so far from his considerations at all so in fact so listening also does this kind of Finkelstein like he doesn't do that anymore he doesn't do it it's absolutely fascinating you brought that up you probably know Lord Veer and Steve Shannon have got a book on sect theory just coming up from Cambridge University Press it's actually impressive than ever, which is revealing the whole set theory of course, as you might expect from their point of view, precisely revealing the whole thing around the fixed point of construction. Oh, that's interesting. I mean, Lauvire and Joyal are kind of have antithetical views on many things. This is Lauvire and Steve Shannon. No, I know. I mean, I know that they're sort of I mean, they wrote the other book together, but I mean, meant to say was that Schwayal has what he calls an algebraic approach to set theory which Lorvier hates and Schwayal wrote a book with Iker Murdyke called Algebraic Set Theory which came out a few years ago which is Schwayal's view which is so I mean it would be interesting to compare why does Lorvier decide I'm not familiar with the

2:17:30 Murdoch Joao book, but I would have said that his approach, from what I are listening to, that his own approach to the Secretary was very on the break. Apparently, I wish you were. He sees the whole thing in terms of it. It may ultimately be a personality. I think it may have less to do with the content than with the... Because Joao has been interested in years and so I think apparently but he was giving a talk about them and he was very angry he involved capitalist concepts and so later I think at the same conference where he stormed out he made a point of having a paper with money games in your face kind of thing so I think it may be I think he could be trying to sound a personal side but I mean as you say I mean, after all, the basic idea is that you take sets as a free... The idea is you have an algebraic signature, this is a trial Murdoch theory, where you have sub-lattices with one unary operation. You think of the unary operation as singleton. And the whole point is singleton as a free object, in the case of the practice of sets. Right. So, anyway, so I mean, it may be that they're not that different. So this is coming out soon? at the moment, because I got a long email from him about it. He was actually asking me if I could track down a photograph of Sammy Eilenberg for the cover. The photograph, the cover, they wanted Cantor and Dedekind, Eilenberg and Maclean on the cover. But they couldn't find a photograph of Eilenberg as a young man. They wanted them as young men. The earliest photograph they could find for Sammy Eilenberg was from about 1948. I think he's going to have to settle for an early middle-aged something like that. He's going to be allowed to actually hold up to the appearance of the book for a long time. It's actually it's all set up. This is an advanced book, I take it. It's nothing like the conceptual mathematics book that he wrote. The reason he wrote that book was I think just to dispose of the challenge which was regularly thrown down to him by the virulently anti-capitalist theory people, like Steve Simpson, that

2:20:00 whatever else could be said with category theory was clearly pedagogically useless. There was no way at all that you could teach elementary mathematics using category theory. And I think that was the whole reason you were able to put it. Whether that book disproved the claim is a very different question. There's a lovely old boy. I see what's his motive for writing it. I'm going to say he succeeded in disproving this paper that is why we know that it will never be used as the basis for the pedagogy those challenges are always dangerous this reminds me of Wittgesheim dropped a philosophy and he became first a teacher at a mountain village in Austria I don't remember he was teaching he was teaching them sons which were not about money and one girl in his memoir he said one of my greatest sins but this is one the girl could not you know form i don't know the junction or he she calculated wrong and she he yes he did yes he was expelled yes he was he was dismissed and of course these days he would have ended in jail of course i mean yes i think so um yes actually people i understand that bernstein scholars have kind of gone back and looked at incident and especially in today's atmosphere right of course of course i really don't think that sometimes somebody has intellectual attention of this at all suited being a school teacher so i guess I know what you're saying. I'm still going to try.

2:22:30 I'm sorry to say, I might probably be a babysat in my office, because my mother sent him to my office, actually. I don't know. It's half a version of that. And I didn't go to my office today. The stuff I really want to get my hands on, of course, is those things of Chris's. Ah, Chris's for very little bit. Well, isn't there much? Let's see what I love. I'll be back around. I'll be back around. I'll be giving you a service. You have a coat, right? A coat? Yeah, yeah, yeah. I think then you ought to go down. All right. I want to sit around upstairs. I'm sorry. What I'm going to do is go home. Can you come back for me, honey? Yeah, I'll just give you a reason why. You can either give me a report over here, but I can always leave my stuff if you want to pass. I'm going to pay you. No, no, I'll just come back. I'll never talk. Oh, yes, thanks. Nice to see you again. Thanks for the fact. Yes, for the point. Let me do it. Plus, there it is. I don't know. Thank you.