??/07/89 Discussions, incl. FW Lawvere, G Khatcherian & M Wright on extensive quantity & other crucial topics

0:00 One thing I would like to hear you tell me a little bit more about, Bill, in so little time left, is a really very general question. Again, it comes back to the left here and then straight upwards, to the relation between geometry and logic. This very geometrical way of thinking about logical concepts. You like the spatial origins of our logical concepts. I think if I was trying to provide a brief historical framework for the introduction of the main ideas of the lectures, that's the point where I was talking to the philosophers and it would be in the context of an analysis of how distorted the logical, the way logicians have had of thinking of the variable has been ever since Frege.

2:30 The idea that a quantity lives somewhere is an existential quantifier, not here, as a kind of degeneration of the actual quantity of smoke, an existential quantifier is the optimal product. Well, and the way that sheaves are... Yes, the relationship between sheaves and quantifiers, which I know you've already written about specifically in the 1970 paper. But I think more of the philosophical, even kind of pictorial motivation for those ideas would probably be the starting point for philosophers in understanding the depths of the revolution in the way of thinking of the variable. Yeah, yeah. And also, above all, in saying, you know, we don't have to start from truth, you know, we don't have to start from, we don't have to, above all, we don't have to assume that the sub-object classifier, the truth value, the notion of truth value object is more fundamental than that of a sub-object classifier. I mean, you see, I think the logicians, the... Even people like John Bell still think of it the other way around. They think of the sub-object classifier construction as just another way of thinking of truth values, which happens to give you this, in model theoretic terms, more flexible and comprehensive framework because of all the sheet machinery. But it hasn't actually altered our concept of what truth and truth value are. The idea that they're actually quantity types. You know that there is a naturalistic explanation for the concept of truth, that it doesn't have to have a platonic, that it doesn't have to live in... All this business about the language is in no-quotes, completely obscures the very existence of logical operators, as well as the laws that they satisfy, hiding predicate calculus. These are not a priori, these are things that are derived from the mere existence.

5:00 This implies that intuitionistic logic, including the operators, and, it implies, are constructed out of the purely spatial relationship. Yes, exactly, exactly. But they've missed this point. They've all missed this point. Even people like John, I think, have missed this point. The way of thinking of this is to think of it as validating the intuitionistic conception of truth in terms of provability from the point of view of subjective or objective idealism, rather it's the actual spatial relations in the domains of variation in the world, or the way that spatial relations in fact come out of a deeper understanding of domains of variation. Domains of variation in a very literal ontological sense, not as a metaphor, which is the way that people have tended to think of the relationship between the variable and its domain of variation, really just as a metaphor, something that the undergraduate magician has to be house-trained not to think in terms of these... This is a helpful picture for the stupid students who cannot be trained to think logically, who will never, who have difficulty in learning how to think as true mathematicians. ...and so forth are consistent upon the fact that the variable x which ranges over an object x doesn't denote anything, whereas Anders Koch and I always promoted that x stands for a map, an actual map. Yes, and I think that's the fact that Scott himself, as I know he's a great logician, has always... It's thought of a sheaf as really just as in the first place. The only sheaf he was ever really interested in was the sheaf over the complete Heightening Algebra because it was simply a way of thinking of generalized truth values, but truth value, just generalized truth values as the starting point. You already have the notion of truth and truth value as given in advance, nothing to do with the structure of domains of variation of material reality. Truth is...

7:30 And again that's the Phrygian starting point. The truth is prior to the world. When there were all the speculations, there's a concrete mathematical problem which has not been well solved yet, well explained. If you take prolocalictopos, the truth value object in itself somehow governs the whole thing, you see. But by contrast, an example like m sets where m is a monoid. It does not. In fact, them as a group, the truth values are only two, which tells you nothing about the group and how it's acting. That's sort of an opposite extreme. So this gives rise to a property that you can attach to a topos, a linear topos over a base topos. Namely, is it or is it not the case that the sub-objects of the truth value object generate the topos? Generate, meaning that there are enough, that the family of objects is said to generate it, if there are enough maps from these into any other objects to distinguish maps and distinguish sub-objects, maps from maps, not maps into. In the simplest case, the sub-objects of one condensed into, sub-objects of one already generated. But the next step would be, what about sub-objects of omega? When do they generate? Or sub-objects of omega to the omega. So that's a kind of an expanding sequence of classes of topos, namely those that are, to some degree, in some higher order sense, determined by their truth values. But the only trouble is that they would think of that entirely in terms of the type theory, wouldn't they? Just in terms of... Well, no, I'm saying let's take the concrete problem. What about the first step now? For which topos is it the case that sub-objects of omega generate?

10:00 Definite mathematical problem to characterize, let's say, in terms of a site, what kind of a site on the base topos S gives rise to the topos that is generated by sub-objects of omega. As a very special example, just take any monoid. For which monoid is it the case that the sub-objects of omega generate? Take a new sheet. Yeah, come on, sir. For directed graphs, here you have the object, omega sub m, the right ideals of m, and a times x is an a. All objects generate, i.e., if you have two maps that are different, this is the next level of multiplication beyond locality. This metallic is generated by T's contained in one.

12:30 I say, for example, M is a group, and yet you need G itself to generate. If you take these directed graphs, knowing that the generic arrow does generate, you have two graphs and two morphisms of graphs. These are different. They must differ on some arrow. An arrow and E is a map for I. But what's omega? Omega is just truth values. And it's clear that I is contained in this, for example, here, or here. There's a sub-object of omega which is generated. So, but to give a useful description in terms of what properties does M have, it turns out it must have, it must consist largely of idempotence, and there's one idempotent that's kind of playing a central role and so on. It's the property that a moment like may or may not have. In other words, you see the strategy to to fly in the face of this philosophical speculation by posing a mathematical problem. Well, yes, I'm well-reputed, I know, but remember I'm not yet able, and please God, one day I will be able to pose a concrete mathematical problem in this form, but unfortunately I have to do it in terms of kind of I'm trying to get a handhold through the philosophical ex-speculation, but yes, I certainly agree with your point about Scott. This would shed concrete light on the philosophical... So what does Scott say when you put this problem to him? Well, his method is basically to pretend that he doesn't understand, that there's nothing to it, and then ten years later... Announce it as his own discovery. I mean, all this stuff of Foreman's and his about... He kind of accepts things piecemeal, you know, a little piece at a time.

15:00 All of this stuff about identity and existence in the Scott-Foreman work, that that's the way to do many-sorted logic, just in terms of restrictions. Yeah, sure, you... I mean, indirectly, they are thinking of the domains as... I guess we'd say spatial topos, it's taking the spatial aspect of the topos, but they're not, yeah, but they're not, but they refuse to accept the materialist implication of that, that there really is, we really are actually dealing with the structure of domains of variation in the sense that you and, okay, Jerry, when you say that logic really has to be understood from the point of view of this, this point of departure in geometry, you know, for them, That's just simply a picture about how, I guess, how, you know, to restrict certain operations and still have well-defined, other operations well-defined. I mean, I'm putting that very badly, but you know Foreman's paper on identity and existence in the Durham volume. Yeah, well, that's what I'm saying. And he actually opens that with a short polemic against you. Now, Lorvier has said that we should, that the whole... The question, the whole topos point of view opens the question of, with regard to many sorted logics, of the analysis of the notion of the variable and sport to light as something which the way of thinking of the value of the variable in orthodox logic is obscured. He's saying, no, that's... The homology which they give there, while they do talk about domains of variation and so forth, it's only applicable to localic topos. Yes, yes, yes. At that time, they had not come to grips with the simple examples of directed graphs. No, no. Their analysis doesn't apply to that application. They got enamored with the suspected topos of Hyland, which it certainly is not localic. So they had to expand defensive actions of conservatism.

17:30 There's a story about that. Eventually, events forced him to accept a little bit more, but the principle that you should accept all, you see, since 1962, when the first seminar, he found the possibility that he was categorized. In that sense, he was more progressive than anybody else in that group. But his own, you know, acceptance, it was just, until recently, with his transitions. So admitting that there are toposes which are non-locality, basically. Yes. But I think that his big difficulty in that would have been that he'd always thought of topos as... He'd always thought of sheaves over complete hiding algebra as the guiding example of the topos, because he'd always thought of it simply as a space of generalized truth values. Well, we're still thinking of truth and truth values as something which are... As something not as in any sense a material quantity, or not in any sense a quantity type. And for him I think it would just be quite unintelligible. The whole Phrygian tradition doesn't allow you to think of truth as a quantity. The basic principle is the mere fact that the characteristic mass and the sub-objects are in this one-to-one correspondence. You have a category, you have this one-to-one correspondence. There is a profound restriction on the category, because it implies, on the one hand, that this object has operations like and and implies, and on the other hand, that the sub-objects of any object do form a hiding object, so they hold to break operations on these quantities, as well as the identities that they satisfy already follows from the mere, you know, identity of opposites.

20:00 So propositional functions and suboptimal functions. It's just a mere statement that those are in bijection. Natural bijection implies the existence of the operation of that. So the program, you see, is that the same fact should hold about space and quantity generally. Not just two-valued quantities, so to speak, but real quantities. Sort of bifurcation into analytic geometry and synthetic geometry, which Steiner was promoting synthetic geometry in opposition to the analytic and so forth. Oh, Steiner, who said of Cantor that this, yes, that, what was that remark that Colin quoted in his paper about? Well, no, it's not particularly, but it's just a very, it's just a passing remark, but again, I think it reflects the, actually it reflects the impact which Fred had already had on these people when he says that... The fact that you can add and multiply real quantities could itself be a consequence... Oh, no, I'm sorry, I'm... ...of the actual isomorphism, the direction between quantities in space. As well as the properties of space itself could follow from that, so that the precise, and in a certain sense, the bare statement that you guys are very much fond of.

22:30 Yes, it is. I'm afraid I was wrong, actually. It was Schoen, please, quoting Steiner, which is that even though Cantor, as he himself incidentally stated, borrowed the notational concept of power from Steiner, still the corresponding geometric formulation has nothing to do with the kind of thought which lies at the base of set theory. Mm-hmm. Yeah. But there you are saying that in fact... I want to read this, but... Oh, yeah, why don't you read it? Yeah, yeah, yeah, yeah, yeah. Geometric formulations do have everything to do with the kind of thought that lies at the base of set theory. They provide the correct framework for seeing it and it's true. Yes, for seeing it. Yes. Yes. This is such a tremendously powerful... Oh, sorry. Please go on. It means it has terminal objects, artesian products, equalizers, These are just some of the minimal conditions for a category to be exposed to a numerical general or particular to give a kind of concrete kind of quantity you want to say let's let's take let's take C Any subcategory of X which is closed with respect to coproducts only.

25:00 So an example of this would be C equals all, say, three-dimensional objects. The sum of two three-dimensional objects is still three-dimensional, or in general, n-dimensional. I want to consider several Cs, but this is a... Sum of two compacts is compact. So there's quite a number of C's like this. So now, we want to define first extensive quantity varying from object X. And this is going to be a category, first of all, because we... We pass to quantity in the more usual sense by the Galileo, Schreiner, Cantor, Bern, Cyclone, Deacon, by abstracting the object. I think we have to find a shortened acronym for that one. So this will just be defined to be the category C slash X. In other words, an object or a quantity is a map into X. But whose domain is in C. And amorphism in this category is in any commutative triangle. So this is a certain category. Now why is this an extensive quantity? Because extensive quantity type and extensive quantity of type C is in the object here. This is a category of extensive. Do I make clear? I'm not saying this is a quantity. You're not saying that X is... Its objects are quantities of type C. So again, you have the idea that X is the car, right, and C is the smoke, and the smoke is put into the car in a certain place at a given moment, right?

27:30 Right. So notice that this category—well, first of all, C over 1 is just C itself, so these are the constant quantities of type C, or just the objects of C. That's it. So that's remark zero. Remark one, c over x has co-products. You can add these quantities. And this is the usual kind of triviality. I mean, if you have two things that are over x and another thing that's over x in two given maps, well, you can put those two given maps into a single one. Just because C1 plus C2 is the coproduct in the category X, I'm not even worrying about the structural mass, the placement of the bodies in space. But then you note that all the diagrams still communicate. In other words, the same coproduct that you have in X, in essence, is the coproduct in C over X as well. There's an induced functor, let's call it F or a shriek. How does it work? Again, in a completely trivial way, you have a given arrow from C to X, close it with F, that's over Y. Domain is still in C, because it's still the same C, in fact. But it's a different variable quantity over X than it was over Y.

30:00 And difference in nature of variation over different portions of the domain of variation is something which that sort of Reagan way of thinking of the domain of variation is just a picture of the relationships of structures already there in some universal truth value object and just can't grapple with at all. So if you consider in particular the unique map from X to the terminal object, which in Johnstone's very nice abusive notation is just called X itself again, instead of exclamation point, it's stupid. One, if I apply the covariant punctuality of the epic sense of form to the type C, two, that is the covariant punctuality to this particular F, the unique map to the terminal object, You get to see the total. So the smoke in this room is an extensive quantity in the room, but there's a total amount, which is a constant quantity. An object of C is a constant quantity, so it's very important that we apply it to the pairing between intensive and extensive, which I'm coming to now.

32:30 Totalization. Yeah, the total mass. Mass distribution of the Earth insofar as it affects distant planets, it's largely through the total mass. The way that it affects a satellite circling near the Earth is quite variable in this. It has to detect oil and iron and positive variations in the gravitational field due to the variation in the mass. Intensity, yes. So the mass of the Earth placed in space is an extensive quantity which is quite variable. But the total mass is a constant quantity placed over space one. So this is the basic properties of extensive quantity as you see. Now what are intensive quantities? Ratios of extensive quantities. In general, we have to have ratios. Mass might be one type and volume another type. The ratio is density. Two types are if you want volume and area. That's a quantity type. The ratio between them would be the dimension of length. It would be intensive and the dimension of length, so... Bill, I'm sorry to interrupt. This is so important. I've got to finish. I'm going to get those tapes before you go. Oh, yeah. Can you go on talking to Jerry? Okay. Um, Jerry, if you can... He can explain it to you later. Yeah, you can get as much of this as possible. Yeah, yeah. Yeah. Well, yes.

37:30 But this is just as regards to types, should be variable over x applied to x, f upper star, an extensive quantity of type x, you take the pullback called f upper star, and then you take the composition, so this is something over x, but you require that this object should be in Z2, that's a condition, but then this arrow has a property. Whenever you map C1 into it and take the pullback, you will get an object of kind C2. So, for example, if C1 was anything three-dimensional, then C2 is anything... Let's say C1 is anything two-dimensional. One, let's call it one. One-dimensional pullback to two-dimensional.

40:00 So that means that F itself has a kind of one-dimensional. Well, let's say one-dimensional should pull back to two-dimensional. I mean, this should have like a two-dimensional, because the product of something one-dimensional is some kind of product. So this is a strong restriction on this thing as well, but it's of an intensive nature, because if you have a map f from x to y, then this will give rise to a functor from Ic1. The two of X into Y, same types to X. It's almost a tautology. You have something over Y which has the property that whenever you map a C1 into it, the pullback is C2. Let's first take the pullback along F. This is a possible capital F. Now let's test it. F of X is an example of the Y1. Pullback from pullback is pullback, so therefore this would be in C2. The g of this map is in I of y, since the C1 and C2 are fixed. For all y1 of y1, hence f upper star of g satisfies for all x1, f upper star of g applied to x1, which is g upper star of x1.

42:30 This bi-associativity is in C2 since fx1, an example, or y1. So, in other words, it's just the principle that is pulled back, but then with these conditions, you see, anything in C1 is in C2. It's just that principle, two such subcategories of putting these conditions. I call it f of this little f of this term, which goes back from y to x, so it's contravariant. The x-intensive quantities of this given type is a contravariant function of x. This is one of the main things that... Intensive quantities is contravariant. Extensive is covariant. But intensive has a stronger property. It not only preserves sums, but preserves products. So F over star is like the ring of continuous functions, you see, is a contravariant functor of space, meaning that the pulling back preserves multiplication as well as addition. Pushing forward the distributions only preserves addition, so this preserves products as well as plus. Well, it preserves plus because of distributivity.

45:00 So, distributivity precisely said that pulling back preserves sums. This is not to be considered as trivial either because it's the condition on the category that we started with. The fact that f upper star preserves products because i c1 c2 of y is contained in script x over y. Now, both of these categories have half-products. Product over y is this pullback. Product over x is pullback over x. So, if you have two things over y and you take their product, this is one kind of pullback. You could take the pullback of this product and we would have that upper star of 1 crossed with a y of 2 and there's a canonical map there. But it's certainly an isomorphism. Again because of this, this is one pullback, probably another pullback, so these two kinds of pullbacks commute with each other. It's just the trivial properties of pullbacks really that make this that upper star involved.

47:30 And f-upper-star of 1 sub y is equal to 1 sub x, because the terminal object of the category of objects over x is just the identity map on x, pulling back the regulators. So f-upper-star is contravariant and multiplicative. These are the two basic features of intensive economy. Contravariant and multiplicative. Extensive quantity is covariant. Both are additive. So the additivity of extensive quantity is just this triviality that the coproducts over x are computed in a way that doesn't even depend on x. It's just the absolute coproducts. On the other hand, the fact that f upper star is additive is due to the distributivity of the universe in which we work. So both kinds of quantities in their punctuality preserve addition, but the intensive one, which is counter-variant, also preserves multiplication. So now we come to integration. Namely the fact that you can multiply intensive times extensive. You can multiply density times volume to get mass. Volume is extensive, mass is extensive, density is extensive. There I was giving a different kind of example than the representable example. Intensive quantities are the actual maps into a certain object R, or like a negative. That's a representable type of example. It's concrete in a different sense. You have this abstract object R, and then the actual maps into it are the intensive quantities. And what I'm saying now is not necessarily representable, but the extensive quantities are actual objects.

50:00 Yes, okay, less abstract in that sense, more in tune, more fundamental. The idea is eventually to identify these two, you see, that there should be a category X in which there is an object or some objects which represent this objective quantity. X is any distributive category and C and C1 and C2 are just any subcategories with respect to sums. There are two aspects of integration. You take the integral of f, fd, dm. You have to have two things. Well, f times dm, and then integration, which is total. So what is f times dm? So if f is... dm is... Well, d is just to remind you that it's extensive, really. Dx is an element. And f is the intensity. Yeah, yeah. So I've been using small x for this. Okay, dx. How about that? Good notation. That could be misleading. I wanted to denote it by some kind of fat dot. So this is again just pullback. You have a f, you take the pullback.

52:30 And the composite, which is again over x. This is f dot x. It's another x. So this operation is independent of the choice of c in a certain sense. It's just pullback. But note that if x is in fact in c1 over capital X, and if f is in... The definition of intensive of type C1 and C2 over the same x, then F dot x as in extensive of type C2 over the same x, always over the same base, because the definition of this was that the pullback of a C1 is a C2, the definition of intensive is of this type, this type. So the integral over x of fdx is just the total of f dot x. In other words, f dot x is still an extensive quantity varying over x, but if you compose it with the map to one, you get an object over one. So this is an element of, in this case, just of c2 operation here.

55:00 Space x in which things are going on. Now if you change the x of my map, you have two, you have three things going on. You have the push forward, the covariance functoriality of extensive quantities, you have the contravariance functoriality of the intensive quantities, and you have the fact that you can integrate or more exactly multiply intensive times extensive. So how do these three things interact? That's what this projection formula says. So let me take, okay, so I can start with little x, which is extensive over x of appropriate type, and f, which is intensive over y. Now what can I do? I can take full back of big F, so that's intensive over x. I can therefore multiply it by little x. No, no, let's call it g, sorry. F upper star of G is an example of the big F. Therefore, it can be multiplied by X in this sense. Now, that's another extensive on X, so I can apply forward to that. I have extensive on Y. On the other hand, I could simply have pushed X forward, giving extensive on Y, which could be multiplied by G in this sense.

57:30 This is the formula. Now, why are these equal? Well, it's again because pullbacks and pullbacks and pullbacks. The F-lower shriek merely meant to sort of consider the thing over Y. The something was given over X, but you'd use F to consider it over Y. On the other hand, both the multiplication of intensive times extensive, as well as the counter-variant functoriality of intensive. Just two different cases of pullback. So since pullback of pullback is pullback, that's why this formula is true. Again, like this definition of f dot x, this formula itself doesn't really depend on which types, which c1, c2 you take. But it's easy to see, of course, that if you do specify these types, then everything is preserved. It's just saying that... The objects C belong to certain categories. There's no further remark required about the types. It's automatically preserving the types. So you see that in effect one can construct a sort of reformulation associated.

1:00:00 These objects are one of the types in tensor quantities. Extensive quantity types. These morphisms are the intensive quantities. In other words, the types, in my concrete example, the types were these subcategories closed under addition. More abstractly, you could think these are just indices, names for the types of quantities and actual variable quantities. In other words, to go from area to volume, you could multiply it by a length field.

1:02:30 A field on X is a variable length. You can make this multiplication. Then you could multiply it by another intensive quantity, which was a frequency field, and you would get a ratio between areas and volume rates, volume divided by time, but varying over X, you see. The type is independent of x. It's a subcategory of the whole category of a specific object is singled out. If you do single out an object, then you could have these variable fields capped to that. In other words, multiplication is related to composition as well. Objects, morphisms, this whole category should be called the category of variable quantities varying over x. These objects are the intensive quantity types. This is independent of x, so the objects are the same. No matter what x is, this category has the same objects. It's like an algebraic theory, you always have the same objects, even for different theories. But then, so if you have a map, you change the space, then you get a puncture.

1:05:00 So this is just like the homomorphism of rings, the homomorphism of continuous, the rings of continuous functions is a special, is a fragment of this. You know, in other words, we have not only scalar quantities, we have tensor quantities, so they form a category, not a ring. You know, different types of tensors are different objects. It's not a ring, but it's... The whole idea of saying that quantities form a C star algebra is an obfuscation. Because to say it's an algebra already means you have pure quantities, so you've chosen C and H to turn velocity and action into pure quantities. If you don't choose a unit, then even for scalar quantities, so to speak, you have really a category. Those quantities that would multiply action into velocity or multiply velocity into mass and so on. Multiplication of scalars is really composition of arrows in a category of types, but then there's also the tensorial aspect or the vectorial aspect. You want vectors of dimension action and tensors of dimension action, so there's another direction in which the quantities really form a category, not a ring, but it's an additive category, so you can add things, you can multiply things. You can only multiply them if it makes sense to do so. The punctuality of quantity is contravariant.

1:07:30 The normal publishes when the time is right. I mean it's just a, it's a series of tautology definitions. It's a guideline. Yeah, yeah. Is there an expression for the general abstract of the universe? Decisive abstract general relations from Marx, 1857. This is it, huh? This is the... Yeah. Yeah, that's right. The distributive category, extensive and intensive. Yeah, the essay of Marx. So this is the... And then you kind of look forward. Yeah. Yeah. Well, I mean, the kind of place you're at, this is another question. This point of view is most often used as in case theory, where these extensive quantities are, if C is the n-dimensional spaces, then my extensive quantities of type X, the isomorphism classes of these things, is a kind of homology of X, whereas the intensive, C1, C2, is the cohomology of dimension n minus n, or...

1:10:00 This is a kind of example to represent these abstract things called homology and cohomology classes as actual spaces over a scene. A homology class could be like the smoke placed into the room. This object is a homology class. It's called representing homology classes by cycles. The abstract is just directly from those objects over x, rather than some more homological mysticism. A side comment. I read the review of Karubi's cave. The man was saying the whole idea of cave. Is that a good idea? Yeah, there's something to that. In other words, that's a way of coarsening the thing, right? That's right. I mean, I should have said that even. You see, this is just an additive monoid taking the actual coproducts of objects. You can add them, and they're zero, but there's no negative objects. But once you make the abstraction, you just consider isomorphism classes of objects and call addition this coproduct. Then you can then you can tensor it with z and introduce negatives. Well, yeah, I have vague memories.

1:12:30 They call this the Grotendieck construction. I mean, it's just one detail. The real Grotendieck idea is to precisely to use the actual objects as the ingredients. But then there's this, it's a kind of coarsening process. It makes the thing smaller, easier to compute because You know, two things which may not be isomorphic as actual objects, they might, they might, when you introduce negatives, they might become equivalent, you see, so there are fewer classes, it makes it a more, more qualitative thing, you see, and this smoke here is the same as that smoke there, even though those are not isomorphic as, well, why is it the same? Because there's some negative, some formal difference of... There's a lot of fun in these, which might be zero. The formal difference might be zero, might be forced by the mere fact that you force it to be an abelian group, instead of just a commuter monoid. And the same thing here, which you can do that to both homology and philology, to make it a group, additive groups, not merely additive homonoids. Hence it's coarser, easier, pragmatic. More qualitative, partial classification of objects. That's the idea of tensoring with Z. So that's certainly one of the important ingredients on K-theory, but I think it's passing from the Burnside rig to the Burnside ring. The ring applies to the contrarian aspect, of course, because that's the one that preserves products. I mean, it's that passage from the rig to the ring, which introduces, as I say, a coarsening, a more easily calculated, a more qualitative classification of the objects. That's only one ingredient. I mean, the basic idea is to actually to consider these objects at all. I mean, as opposed, you know, as opposed to the other kind of calculation of homology and .

1:15:00 Which is a very formal algebraic way of calculating what in many cases turns out to be the same thing as this coarsening of the actual objects. See, but I mean, the fact that cohomology acts on homology, I think it's called the CAP product, which you would name sort. It's... Cup product. Yeah, it's just the notations they've picked. Yeah. Cups. Cup is a multiplication of... In the absence of conceptual notes. What is the comma category, for Christ's sake? I hate it! What should we call it? Slice? Same. Are you going to come up with a name? I'm going to have a name better than comma. So cup product is the multiplication of two cohomology classes to get a cohomology class, which in this kind of concrete representation is just this pullback of things over a given x or a given y. But whereas the cup product, no, the cap product, is the one which multiplies intensive times extensive to give extensive. In other words, in this case, cohomology times homology to give homology, and it satisfies. This formula should be true in any case, but it's obvious in this example, this concrete representation, because cup is also pulled back.

1:17:30 Both cup and cap are pullback, and pullback is pullback, so the formula is easy to establish in this example. And that's why you should want it to be true if you choose instead to calculate in some more abstract way, you know, using cycles and co-cycles and all that stuff. You still need that formula because it's a reflection of This one. This is the basic example. You're striving. Why should you want to compute homology and cohomology? It's what these French guys never ask themselves. It's just something you do automatically. What you want to reflect is exactly this. This example is what you're trying to compute. No matter what the circuitous route to arrive at that computation might be. Which has a representability, not with omega, but something like KC, KC1, C2. So it should be, it should be before, so that the, it should be a, so that the actual morphism that acts on a certain object, some bijective correspondence, like characteristic functions of subsets, in the case of omega. It's an analog of omega for actual quantities, instead of just truth quantities. Is it good example? Well, I call it K because of Eilenberg-McLean space. Eilenberg-McLean space in the case of cohomology, which has exactly this property.

1:20:00 Well, I'll tell you what it is right now. So that the maps from some space X into that fixed thing, which is independent of X, should come out to be the cohomology, in other words, should come out to be the intensive quantity. In other words, what was this? This was an f over x, which had that property that whenever you map c1 and pull back, you get c2, okay? This should correspond to a characteristic map into k, and there would be some total space, but just like true, you see, to omega. Now it's no longer going to be 1, but something that's fixed, I mean, independent of that, so that the pullback is the f. So if you classify these intensive quantities of this type by maps into a fixed, this would have to be a preferred element of intensive quantities of that type varying over that particular x. So there should be one space x, namely k, and one intensive quantity of that type over k, such that every intensive quantity of that type is a pullback. This should be the basic axiom of mathematics, in the sense that omega is the basic axiom of logic.

1:22:30 This is the next one. It's the next one. But you've had these ideas in some form for a long, long time. So topos was just only one particular... Following the line is you do logic first, mathematics later. It works in detail. Yeah. In this one, right? This cube tracker, yes and no. Not the actual value of the quantity. This is more physics in here. Those of you in maths, physics, and so on, I've learned more than you'll find in this. I'm just trying to see how I can actually achieve the omega theory as a special case of this. True, the logic is a special case of quantity, right? Namely, right, I mean, we always have the option in this of fixing on one C, so that C1, C2 is always C, right? So then this category of intensive quantities just reduces to the monoid of endomaps of C. Generally, the intensive quantities went from C1 to C2, C2 to C3, but as a special case, these could all be the same. You just have one C, and so they are working on this one C back to itself, of course, depending on X.

1:25:00 So if you have a bigger X, you have more endomaps of C. A smaller X, you have fewer endomaps, but make it a monoid instead of a category, an additive. Additive monoid, i.e. a ring, a rig. Well, maybe a non-commutative. To make it commutative is a more delicate thing. Why should the endomaps of something be commutative? If it's one, okay, you can understand the endomaps of one are commutative, but if it's a bigger thing, it's a real condition, some kind of extra information in the fact that the endomaps of the sea are So there's something about the foundation of quantum mechanics in this too, which C's are the endomaps actually commutative, surprised they're actually commutative, in principle they should be non-commutative, in other words you could just say I, we could fix, and then I, I of C, C of X, just called I of X for short, is a non-commutative, we can add things because again we add over X, not just adding them. All of these can be multiplied as we pull back on F. Now what I want is to get the F's to be... So I take a fixed state. So, for example, if there are no sub-objects of 1,

1:27:30 then this will just be sort of a category of finite sets. I have 1, 1 plus 1, 1 plus 1 plus 1. All of these are expressed as a finite co-object of one. More generally, perhaps we should take sub-objects of one. If there are no sub-objects of one other than zero, then it's essentially just this. Now, what is now an intensive quantity? Well, an extensive quantity is just a finite set placed in space in a certain way. But particles are understood to be a finite discrete set, but they must be placed, they must be distributed in space, and that's an extensive quantity. Yeah, yeah, yeah, okay. Yeah. More generally, we have continuous bodies, but let's just take discrete body, which is a finite set of particles. This is co-variant, because if I map one space into another, for example, if I map the space into itself, well, this will move the position of the particles. It might also cause some of them to curl less, if you want. We don't think it's a lot more of the area of Archimano, right?

1:30:00 You don't see me. His eyes light up. Your eyes light up. Ah, yes, particles in space, yes. No, it's just connection, you see. You sense that. You sense that. You went into Katie and so on. You went beyond my... The next tensor problem is the distribution of finite set of discrete particles in space. Yes, that's one of the things I wanted to ask. You see, you noticed my lights. No, because when you went to 2k, it became a bit, I mean, the tensor was Z, right? So then you have formal differences of negative particles. I thought about that. Well, actually, one of the things I wanted to ask about taking some of the amplitudes in the quantum field theory of extensive quantities, particularly things like, well, occupation state number, which would be an example of just how the notion of extensive quantity is more fundamental than that of collection and extension, because you don't have that. No, you know, you don't have the members of a flexion and extension in that case. It actually becomes itself a variable, depending on what base you turn it to. You touch it with z, or you touch it with 2, or you touch it with v2. Yeah. So I was right. This Mabry insistence on extensionality is all found out with atomism, and I'm quite certain that that is. Well now wait a minute, I think he's actually moved beyond that, you see. Yeah, when I say his hang-up book, he used to position us, he sent out his papers, notes, yeah. In this letter he says, these arithmoids that are attached to the object depend on the unit. What is a unit, you see? Choice of unit, right? I mean, the abstract set of maps from n to x, that's another abstract set, but you've chosen the unit to be n.

1:32:30 You know, the hydrogen atom could be a unit, or the pair of shims could be a unit. So the actual abstract set, the arithmoid that's attached to it, you know, depends on that. So, I mean, I think he's right. It's just that the units are objects in the category, the generic figures. If you interpret, if you read his letter favorably and say, unit now means... Generic figure type, I mean, it's a chosen object in the category that you now use to measure other objects. You see how you get different abstract sets out of the same object, and you use different units to map in. Yeah, I absolutely agree. I'm just saying that that is something which I absolutely agree. He has now definitely had this understanding as a result of listening to you. Yeah, I mean, this is the clarification that I've been stuck with. What I'm trying to say is, I think I'm basically concepting this as a program to rehabilitate Pythagoras. Because at least, even if it turns out in the end that Pythagoras was a raving idealist, as everyone says, such an attitude toward investigation will bring out many things. Sure. It wouldn't have been brought out before. So as an example of extensive quantity type, we could just take finite sets or finite coproducts of x to the 1, and we have these distributions of finite numbers of particles. Now what are the corresponding intensive quantities? This is an interesting thing. So an intensive quantity of type, finite sets to finite sets, Subject is the finite number of parts that are placed in X. Take the pullback, it's still finite. So, in other words, the bold I of X is all the finite fibers, which is a small generalization of sub-object, you see.

1:35:00 Well, not very small, but interesting. Because sub-object means fibers are singleton or zero. That's great. Yeah? Uh, well, well, yes. But I mean, a monomorphism is one whose fibers are either singleton or zero, roughly speaking, and there also might be sub-objects of one, which might be complicated, but I'm studying the locale equal to zero. The crucial point is that a monomorphism is something whose fiber is either singleton or zero. Singleton or zero is an example of finite, so a slight generalization of a sub-object. All of these would be a map whose fibers are finite. Right, rather than adjusting zero to one. But these form a rig, you see. I mean, in other words, if you take two maps whose fibers are finite into x, take their pullback, their product over x, still has finite fibers, because it's a sub-object of the product of finite sets. Each fiber is a sub-object of the product of respective fibers, which are... No, no, sorry, the actual fibers of a pullback. That's why it's called fibered product, because the fibers of a pullback are the products of the fibers, the actual products. So here we use them like this. Well, we never do anyway, sorry. This is an idea which I've been struggling to try to understand for about three years about to see whether there might be some way of generalizing the discrete vibration in the case of, I mean, okay, discrete vibration, in the case of metric spaces, discrete vibration is just equivalent to a set of abstractions. I mean, you mentioned something that you say quite concretely, I mean, your generalized...

1:37:30 Think of set abstraction in the case of logic, logic in the case of set theory. It's just a special instance of discrete vibrations. Yes, right. Yes, right. Categorically discrete. I mean that would be topologically discrete. No, no. Categorically discrete vibrations. I'm just wondering actually what the, for things where the unit, what does that translate as? You see, for menger, which were the units, you know, that in comparison with which each of the entities, the kind of, so to speak, is called one, is not, is, where's that red line, I forgot something important here, the only good teacher here is the, the product, multiplication is graded multiplication. In terms of these types thought of as subcategories closed under product, there's an operation on them which corresponds to adding dimensions or degrees.

1:40:00 You have these two subcategories. Now you sort of consider the convolution in all objects which could be represented as finite sums of binary products, one from this and one from that. So when you take the internal product, it'll change its type in that way. If F1 is of type C1 and F2 is of type C2, then their product will not be, it'll be of a new type. It's like degrees of polynomial for dimensions. I forgot to say that before. There's a tensor product on this category. The types actually form a category, because these C's, one can be included in another, but I put it very generally, so one example would be all n-dimensional spaces, and so between n-dimensional spaces, between three-dimensional spaces and five-dimensional spaces, there's not much of an inclusion, but in principle, you could take the union of those two, that would be it, the union in the sense of all possible finite co-products. That would again be a C. So there's a vast number of types of quantities here. But when you take the product of two quantities, it will change its type in sort of an obvious way. I mean, like, length times area is volume, it's not quantity. In particular, length times length is not length. So for a fixed C, in general the thing is not closed under multiplication. But my example of finite sets is, though, you take a finite sum of products where the first one is a finite set and the second one is a finite set, it's still a finite set, because it's not like... I don't like length-area giving volumes. It's already closed under this multiplication. The only thing I assumed in general was that C was closed under sums,

1:42:30 but this one happens to be closed under products as well, and therefore you don't get outside of that type and you perform multiplication. Right, and that is just one of the things with singles-ounce sets. No, not really. I mean, there are lots of other examples of that. It's one of the things which is similar to setters of time. Yeah, I'm sorry, singles out is the wrong expression. What I mean to say is it's one of the ways, it's how you should think of sets as a further quantity type. Yeah. Well, the mere fact that it's closed under sums means it's an eligible quantity type. Starring operation, which is like an addition of dimensions. So by fixing on one type, or by restricting the theory to one type, we get something that is multiplicative as well as additive. So, you see, we get a second type of grad infinity, meaning with natural numbers instead of truth values. If we impose this idea of having a kind of Eilenberg-McClain space. This K is not the key of K theory, it's Heideberg and Feynman's K. Why don't you write that down for me? Yeah, sure, I'm going to make sure I keep copies of all of this. Well, it's your copies. Well, I'll get them, I'll copy them. In fact, we can copy them at Heathrow and I'll get them. So you can turn them back to me. What's the take of time? Did you say a takeoff time was 6, Bill? I thought it was 6.45 on the air-to-natural. Yes, maybe. Well, I don't want to stop, Bill, but let's just finish what you're saying. No, no, I'm just thinking about it. Which you're probably going to be thinking about in the next talk. Okay, some C exists. It is semi-continuous natural numbers, which contains the omega as a special case.

1:45:00 There are two ways to do it. One could consider a sub-object, omega, to be like the usual omega, or one could take... Once you take just this thing as it is and consider tensoring that with two, I mean, this is the rig in which one plus one equals one, okay, so this is a way of getting it important and intensive, this idea of particle distribution, by simply, you see, saying true or false means are there any particles there. Isn't that exactly what people in the field theory do with the amplitude operators, particularly the occupational state number? I mean, that's how they think of it. Without knowing this, obviously without knowing this machinery, it can't be true. So there's sort of two ways of imagining, getting back to logic, you see, there are two ways of imagining weight and potent quantities, which have this, roughly this idea of, is there anything there or not, or where is it? What part of space is... One would be to actually consider, as is usually done in totals theory, consider only monomorphisms from the start, and have an object that classifies those, and if you want to get the support of something, you first take its end, you have a map, which could be a more general quantity because the fibers have various quantitative sizes, take the image of that, which is now a monomorphism, and you take the negative value of that. This is one idea. The other idea is...

1:47:30 We consider these finite quantities outright, these particle distributions, which might indeed be representable as well by some kind of semi-continuous natural numbers out there, and then we abstract from that or we squash that by tensoring it with the infogriff, Chan-Weltz, and so on, and compare these. They probably aren't quite the same without further assumptions. Annihilation and creation. Well, yes, exactly. I was going to ask you, isn't there, is there a big connection to that as a creation and annihilation operators? That's the thing I've been striving for for the last few years, to try and see how I could do. Basically, how well can we use spin statistics in the class of topos? The method of finite fibers are very general, continuous things, infinitely continuous and spatial. They just happen to have finite fibers. But as such, they're not at all finite. Well, this is the classifying topos of the... No, no, the classifying object is not in the category. Yeah, the classifying object is in the category for finite vibrations. But the classifying object for... They're not required to be vibrations in any sense. They're only required to pull back finite to finite. Yeah. So, you know, this is just to illustrate the big contrast between extensive and intense. For the extent that it is, from this point of view of the same doctrine, the same example, the extensive things are the really finite things that are placed in the distribution of finite number of speed particles.

1:50:00 Whereas by contrast, the intensive quantities are everywhere, really everywhere. But their values are sort of one that only point-wise are what the finite sets. A double covering, you see, would be something which was sort of an intensive quantity which was a variable two. It really varies over the whole space. It's not at all finite in the way it's placed in space. So the intensive and extensive are quite different. The integration process there was just a question of... You have the distribution of particles in space. You have one of these finite vibrations that's called, and you take the pullback, and the pullback will be possibly with more multiplicities, another finite particle distribution, except that, in other words, any place where this intensive quantity is alive, you multiply its value by the particle. You can duplicate the particle. If the intensive quantity has fiber of cardinality three at the same point where there is a particle, you put three particles instead, and then you get a new particle distribution. Its total value is just the total number of particles that are thereby created. That's the integral of the intensive, the discrete, and the extensive. Of course, that is just how creation operators do work. I guess so, yes, sir. I mean, this is a bunch of people are thinking about it. Except that the operators are abstract quantities already. Everything is a quantity. Well, they call them operators. Well, they're operators on quantities. They're really operators. Yes, but you see, that's the point. They ask quantities themselves in some sense. You were dealing directly with the geometrical thing. Exactly. And really, this is the first deep materialist explanation. This is the first deep materialist explanation of the subject matter, particularly from your way.

1:52:30 I must find out more about this, but it seems to me that this guy is up. With problems which have plagued me ever since I first heard about fermions. I've been trying to reconcile, I mean, that's why I got involved with English Channel that time, because I couldn't understand how you reconcile fermionic creation and elevation operators, fermionic amplitudes in quantum field theory, which I was told was the laboratory of metaphysics in the modern era. The thing where I think there's a lot of identity and a lot of quantity and space must be formed with the understanding of discrete quantity and the check. Yesterday the world. Yeah. Yesterday quantum mechanics dominated the entire world. Well, as of today, do they not dominate this room? No, I don't think so. Yesterday, the world. They're already doing that tomorrow. The whole universe. And I heard this. The Nazis rule the world. Well, I think it's going in a bit to describe one of these theories as a product of that Nazi ideology, but certainly I understand what you're saying. So, at least one of them. Well, it certainly ruled my way of thinking for a long time, in the sense that I actually thought the idea and the process became mathematical as well. Well, I agree with you. I think it's one of the best spike claims to the contrary afterwards. Oh, you, you know about that. You know about Heisenberg's, uh, language. Oh, what absolute pathetic language.

1:55:00 You really want to bomb and all that. Oh, dear me, oh, God, yes. And not only that, have you ever read, have you ever read about these pathetic possibilities? Have you ever? I mean, that's so... I tell people about them to get away with it. I mean, uh... You know, I first became aware that there was this thing called National Renewal in the summer of 1933 when a student who had been coming to my lectures, sort of wearing a brown shirt, came into my room and asked me, Professor Heisenberg, why are you, as such a gifted and intelligent man, not more active in our movement for national renewal? Oh, tell me about this thing I said. I'm not exaggerating. He had a long talk with this sympathetic young man, and he explained to him, this is Heisenberg in the 50s, you know, he's always been the good German, you know, why, how much happier he, this is Heisenberg in like 1954, I mean, how much happier he would be if only the Germans would reconcile themselves. You know, it's the category, confiscation, it's the nationalist category. If only the Germans would reconcile themselves to being a member of, surprisingly you said, a common European house, good members of the common European family, and why can we not just be just like the Danes or the Dutch ones? And then afterwards I told them, well even if you're Iraq, I think we're all Indian. This is Heisenberg writing his memoirs in 1950, in the 1950s... This thing called America, you've forgotten this thing called America. You've forgotten, you'll have forgotten it always. This is Heisenberg writing his memoirs, ingratiating himself with the Americans in the 1950s and reconstructing a conversation, a mythical reconstruction, a conversation he was supposed to have with a young Nazi in the early days of the Nazis, actually in fact already after the regime was coming apart, which time he tries to give the impression he was living a life so absolutely devoted to.

1:57:30 The contemplation of the platonic verities of the pure science that he was totally unaware of the political situation of Germany to the point that he had never spoken to an artist before himself in 1930. Part of the very brags in another place. Part of this proto-Nazi movement. Yes, yes, in 1919, exactly. And he then pretends, as I say, speaking obviously to an American audience, an audience of American admirers of the great, great Sniders in the 1950s, that with great foresight I told him, ah, but your hero has forgotten his dreams of the world. There is this thing called American great citadel and fortress of democracy. I'm not making it up. I'll get you the passage. It's not legal. We believe you. And it's not beyond the need. We'll hear from his followers. What a way of putting it. It's as though there's this group of people who are mesmerized by one man, you see. Not that there's some scientific content, but they're all trying to understand. Venom is Korea, crony. Oh, you're a tax mania, isn't it? He says drivel is a major attack. This is already a major attack, because it's suggesting that category theorists are these kind of people who follow a certain spirit.