Discussions incl FW Lawvere, G Khatcherian & M Wright (contd.)

0:00 There was always such explanation about destroying your... Back to your reality. So in general, an intensive problem, this is just a, this should be variable over x.

2:30 Variable. For star, quantity of type x, you take the pullback, and then you take the composition. So this is something over x, but you require that this object should be in C2. That's a condition, you see. This arrow has a property, whenever you map... If we put C1 into it and take the pullback, we'll get an object of kind C2. So, for example, if C1 was anything three-dimensional, and C2 is anything, let's say C1 is anything two-dimensional, oh, one, let's call it one, one-dimensional pullback to two-dimensional, so that means that F itself has a kind of one-dimensional nature. Or let's say one-dimensional should pull back to two-dimensional. I mean, this should have like a two-dimensional case because the product of something one-dimensional is some kind of product.

5:00 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, C2 of x. Now let's test Y into the 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. Well, f of x is an example of y1. Pull back and pull back and pull back. So therefore, this will be in c2. To say that g with this math is in I of y, the c1 and c2 are fixed. This means for all y1, g upper star of y1 is in c2. Hence, f upper star of g satisfies all x1. F upper star of G applied to X1 which is F G upper star of F X1 by associativity is instant 2 since F X1 is an example of Y1.

7:30 In other words, it's just the principle that pullback, pullback is pullback. But then, with these conditions, you see, anything in C1 to C2, it's just that principle, two such sub-categories of conditions, so you get this functor, we call it f of this little f of this star, which goes back from y to x, which is contra-varying, the x, the intensive quantities of this given type, is a contra-varying functor of x. This is one of the main things. Intensive quantity is contravariant. Extensive is co-variant. But intensive has a stronger property. It not only preserves sum, it preserves products. So, f of the star is like the ring of continuous functions. It's a contravariant function 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 the distributivity. In other words, the operation of f over star, you see, is just pulled back. So, distributivity precisely said, over base x, distributivity precisely says that pulling back preserves sums. It's not considered as trivial either because it's the condition on the category we started with, but the fact that f-upper star preserves products because i c1 c2 of y is contained in f over y, so if you have g in here, f-upper star of g is in f over x.

10:00 So, if you have two things over y, and you take their product, this is one kind of pullback, you apply f upper star, now you can either take a pullback of these crossed over x, or you can take the pullback of this product, f upper star of 1 crossed over y, 2, and there's a canonical map there. Certainly an isomorphism. Again, because this is one pullback followed by another pullback, so these two kinds of pullbacks commute with each other. There's just a trivial property of pullbacks here that make this f upper star. 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. So, F-upper star is contravariant and multiplicative. These are the two basic features of intensive quantity. Contravariant and multiplicative. Extensive quantity means covariant. Both are additive.

12:30 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 co-products. 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 contravariant, also preserves multiplication. So now we come to integration. 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 what the actual maps of a certain object are. That's a representable type of example, but it's concrete in a different sense. I mean, 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. It's less abstract in that sense than it is more intuitive. The idea is eventually to identify these two, that there should be a category X in which there is an object or some objects which represent this objective quantity. But at the moment I just work with X as any distributed category and C and C1 and C2 are just any subcategories that close with respect to sums. So now integration, total is one, there are two aspects of integration. To 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?

15:00 Well, d is, d is just to remind you that it's extensive, really. I've been using small x for this. Okay, dx. How about that? Good notation. Now that could be misleading. I want to denote it by some kind of fat dot. So this is again just pullback. You have f, but you take the pullback. 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. Just pull back. But note that if X is in fact in C1 over alpha X, and if F intensive of type C1, C2 over the same X, then F dot X extensive of type C2 over the same X.

17:30 Always over the same base. Because the definition of this was the pullback of C1 is a C2. Pullback over C1 is a C2. It's tight. It's tight. Now, so the integral over x of f of x is just the total of f dot x. The map to one, you get an object over one. So this is an element of, in this case, just of C2. Now this fundamental formula, which says that if you have a map, this is all with a fixed space X in which things are going on. Now if you change the X on my map, you have two, you have three things going on. Push forward the covariance functoriality of the 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.

20:00 Speakers include all kinds of mathematics, including mathematics, geometry, algebra, algebra, mathematics, physics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, So, I can apply, push 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, g times f. And these are equal. This is the formula. Now, why are these equal? Well, it's again because pullbacks and pullbacks and pullbacks. I mean, the f-wheeler shriek... In the early 20th century, the term intensive was merely meant to sort of consider the thing over y, that 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 contrarian functoriality of intensive, are just two different cases of pullback. So since pullback is pullback is pullback, that's why this formula is true. So I won't start out with the statement cool-da-kee. So again, like this definition of f dot x, this formula in itself doesn't really depend on which types, which c1 and c2 you take.

22:30 It's easy to see, of course, that if you do specify these types, then everything is preserved. It's just saying that the object c belonged to a certain category. Remark required about the types is automatically preserving the types. So you see that, in effect, one can construct a sort of reformulation to each x associated with the category. These objects, extensive quantities, types, but its morphisms are in the intensive quantities. Extensive quantity types, morphisms are the intensive quantities. In other words, so, the types, in my concrete example, the types were the subcategories posed under addition.

25:00 But more abstractly, you could think these are just indices. These are the types of quantity. The actual variable quantity, so, in other words, to go from area to volume. You can multiply by a length field, a field on x, which is a variable length, then you could multiply 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. In other words, the type, the type is independent of x. It's a subcategory of the whole category. No specific object is singled out. But if you do single out an object, then you can have these variable fields, capital F, and compose them. So, in other words, multiplication is really composition as well. So, object's morphism is composition, multiplication.

27:30 The category of variable quantities varying over x. These objects are the intensive quantity types, which is independent of x. So the objects are the same. No matter what x is, this category has the same object. It's like an algebraic theory. You always have the same object, even for different theories. But then, so if you have a map to change the space, then you get a decumptor. So this is just like the homomorphism of the brains. The homomorphism of the brains of continuous functions is a special, is a fragment of this, in a way, in general. In other words, we have not only scalar quantities, we have tensor quantities, so they form a category, not a brain. The whole idea of saying that quantities form a C-star algebra is an obfuscation, because to say that 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 don't really have a category. Those quantities will multiply action into velocity or multiply velocity into mass and so on. Multiplication of scalars is really a 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, tensors of dimension action, so there's another direction in which 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 when it makes sense to do so.

30:00 But the punctuality of quantity is a contravariant. You're going on a partnership when you're trying to cut it aside. I mean it's just a, it's a series of tautology definitions. It's a guideline you can proceed on. Yeah, yeah. Any expression of what you mean by tautology? Do you use general maths? Like, do you use metaphors? Do you use a phrase? The Decisive Abstract General Relations of Marx, 1857. This is it, right? Yeah. Yeah, that's right. The distributive category, extensive and intensive. I have that essay of Marx, so this is it. And then you can look for the various embodiments of it. Yeah. Yeah, well, I mean, the kind of place where this is most often, this point of view is most often used is in case theory, where these extensive quantities are, where c is the n-dimensional basis, and my extensive quantities of type x, the isomorphism classes of these things, is the kind of homology of x, whereas the intensive...

32:30 The type C1, C2 is the cohomology dimension M minus N, or N2 minus N1, correct. This is a kind of example. In other words, to represent these abstract things called homology and cohomology classes as actual spaces over C. A homology class could be like the smoke placed into the room, so this object is a homology class. It's called representing homology classes by cycles, to realize homology as actual objects over x's with the facts, so it's something more abstract, or rather the abstraction is just directly from those objects over x, rather than some more homological mysticism. A side comment. I read the review of Karubis in the book. The man was saying that the whole idea of K would be the notion of man. Is that a good idea, or was the man just saying that? Well, 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. This is just an additive monoid, taking the actual co-products of objects. You can add them, and there's zero, but there's no negative objects. But once you make the abstraction, you just consider isomorphism classes of objects and call addition this coton, then you can tensor it with z, you see, and introduce negatives.

35:00 This is what they use for Grotendieck, too, don't they? I have vague memories. They call this the Grotendieck construction. I mean, it's just one detail. The real Grotendieck idea is precisely to use the actual objects as the ingredients. But then there is this... it's a kind of coarsening process. It makes the thing smaller, easier to compute, because two things which may not be isomorphic as actual objects... They might, when you introduce negatives, they might become equivalent, you see, so there are fewer classes. It makes it a more qualitative thing, you see, and this smoke here is the same as that smoke there, even though those are not isomorphic. Why is it the same? Because there's some negative, some formal difference of quantities which might be zero. The formal difference might be zero, might be forced by the mere fact that you force it to be in a billion group. Instead of just a community of homo, so, and then the same thing here, that you can do that to both the homology and the cohomology to make it a group, additive groups, not merely additive homoids, and so the coarser, easier, pragmatic, more qualitative partial classification, partial classification of arguments. 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 not really a reference. In other words, it's passing from the Burnside rig to the Burnside ring. The ring applies to the contrarian aspect, of course, and that's one of the other products. But I mean, it's that passage from the rig to the ring which introduces, as I say, a course in a more easily calculated and more qualitative classification of the objects. That's only one ingredient. I mean, the basic idea is actually to consider these objects at all, as opposed to the other kind of calculation of homology and cohomology, which is you have these cycles and co-cycles and t squared equals zero and all that stuff, 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.

37:30 See, but, I mean, the fact that cohomology acts on homology, I think it's called the CAP product, if you have any mystical names for it. It's a cup product. Yeah, it's just the notations they've picked. Yeah. Cups. Cup is a multiplication of... In the absence of conceptual rules, you choose the notation. Yeah. Like the comma category, for Christ's sake. Yeah. I hate it. What should you call it? Slice? 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. This is the one which multiplies intensive times extensive to give extensive. In other words, in this particular case, cohomology times cohomology to give homology, and it satisfies the so-called projection formula. I mean, in other words, this formula should be true in any case, but it's obvious in this example, this constant representation, because cup is also pulled back. Both cup and cap are pullback and pullback and pullback and pullback, so the formula is easy to establish in this example.

40:00 And that's why you should want it to be true if you choose instead to calculate it 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. But 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. So one might speculate, you see, that there could be an X. And so on and on and on and on and on and on and on and on and on and on and on and on and Since the analog of omega for actual product is considered just truth quantity. Well, I call it k because of Isenberg and McLean space. Isenberg and McLean space in the case of cohomology, which has exactly this property. 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,

42:30 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. So this should correspond to a characteristic map of decay. There will be some total space, just like true. See, true is a map from 1 to omega. Now it's no longer going to be 1, but something that's fixed on the independent of that. Yeah, so if you classify these intensive quantities of this type by maps into the text of that type, this would have to be a preferred element of intensive quantities of that type varying over this particular space, that particular x. So there should be one space x, namely k. And one intensive quantity of that type overtakes such that every intensive quantity of that type is pulled back. So this should be the basic axiom of mathematics in the sense that omega is the basic axiom of logic. This is the next one. This is the next one. This is what's moving, yeah? But you've had these ideas in some form for a long, long time.

45:00 So topos was just only one, I think, the power in the line of the logic first, mathematics later. Plus it's the one that works in detail. In this one, just keep track of yes and no, not the actual value of the topic. This is more physics-y. In fact, philosophy, maths, physics, and so on, I've learned more than you'll find in this sort of boundary as well. 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... 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 to C3. But as a special case, these could all be the same. They just have one C. And so they are working on this one C back to itself. Of course, depending on X. So if you have a bigger X, you have more endomaps of C. Smaller X, you have fewer endomaps. Make it a monoid instead of a category. An 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 that endomaps of one are commutative, but if it's a bigger thing, it's a real condition from some kind of extra information and the fact that the...

47:30 Endomaps of this C are really commutative. So there's something about the foundation of quantum mechanics in this too, which, for which C's are the endomaps actually commutative? Surprised, they're actually commutative. In principle, they should be non-commutative. So, so in other words, you could just say I with the fix, and then I, I of C, C of X is called, and X for short is a, is a non-commutative grid. We can add things, because again we add to x by just adding things to x, but also they can be multiplied as we pull back. Now what I want is to get the, I want to get the f's to be, so I take a fixed, okay, so for example, if there are no sub-objects we want, then this will just be sort of a category of finite sets.

50:00 I have one, one plus one, one plus one plus one. I have all the objects that can be expressed as a finite co-variant 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. Just a finite set placed in space in a certain way. The distribution of particles. The 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, okay. Yeah. Marginal has continuous bodies, but let's just take discrete in the body which is a finite set of particles. This is covariant 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 coalesce into one. We don't, we don't say it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, it's a, You see, you went into chaos and so on. You went beyond my competence. When you came down to the next center, you don't know about any distribution of finite set of discrete particles in space in a certain place. Yes, that's one of the things I wanted to ask, actually. You see, you noticed my lines. No, because when you went to 2k3, I became a bit... Tenser was z, actually. So then you have formal differences of negative particles.

52:30 Yeah, yeah, yeah, I have thought about that, yeah. What, um... Well, actually... Taking... The quantum build theory is extensive, particularly things like, well, computation state number, which would be a good example of just how the notion of extensive quantity is a more fundamental matter for collection and extension because you don't have the members of a collection and extension in that case. It actually becomes itself a variable, depending on what base you choose. And that's, you see, so I was right, Mabry's insistence on extensionality is, he's all bound up with alchemism. Well, now wait a minute, I think he's actually moved beyond that, you see. Yeah, no, when I say his hang-up, what, he used the position as he set it up in his papers, notes, yeah. In this letter he says, these arythmoid that are attached to the object depend on the unit. What is a unit, you see? Choice of units, right? The abstract set of maps from N to X, that's another abstract set, but you've chosen a unit to be in. You know, the hydrogen atom could be a unit, or the pair of shoes 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 now objects in the category, the generic figures. If you interpret, if you read his letter favorably and say, unit now means... I think that's a generic figure type. 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 in these different units that you 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 years. He didn't have it before. Yeah, I mean, this is the clarification that I've been struck with.

55:00 What I'm trying to say is, I think I'm basically, in sympathy with this, for a man to rehabilitate the tactics. 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 that wouldn't have been brought out before. I'm kind of in sympathy with this. So as an example of extensive funnery types, if you just take a finite set of a finite coproduct of an object we want, and we have these distributions of finite numbers of particles. Now what are the corresponding intensive quantities? This is the interesting thing. So an intensive quantity of type, finite sets to finite sets, a variable over x, an object over x, with the property that whenever you take a finite map, In other words, bold i of x is all the finite fibers, which is a small generalization of sub-objects, you see. Well, not very small, but interesting, because sub-object means fibers are singular and are zero. That's great. Is that? Well, yes. I mean, monomorphism is one whose fibers are either singleton or zero, roughly speaking, and there also might be sub-objects of one which slightly complicate that. I'm setting the locale equal to zero. The crucial point is that monomorphism is something whose fiber is either singleton or zero. Well, 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, but these form a rig, you see.

57:30 I mean, in other words, if you take two maps whose fibers are finite into x, take their pullback, their product over x, it 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 the respective fibers which are finite. 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 product. So here we use the... Don't say anything too important. Well, we never do anyway, sorry. Not true. See, 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 notion of discrete vibration in the case of, I mean, okay, discrete vibration, in the case of metric spaces, discrete vibration is equivalent to a set of abstractions, I mean, you mentioned something that you say quite concretely, I mean, you've generalized a lot of the metric spaces better. Think of set abstraction in the case of logic, not in the case of main set theory. It's just a special instance of discrete vibrations. Yes, right. That's right. Categorically discrete. They may not be topologically discrete. No, no. They're categorically discrete. Well, I'm just wondering to see what the... for things where the unit... Sorry, that's not the internet.

1:00:00 You see, for Menger, for Menger where the units, you know, that in comparison with the teacher, the entities, the terms, they should be called what is not, is. Where is that red one? I forgot something important here. Only the teacher uses it. And all the students. The product, the product is multiplication, is graded multiplication. Length times area, you get volume. You get a third type of thing. So in terms of these types, thought of as subcategories to those hundred products, there's an operation on them which corresponds to adding dimensions or degrees. You have these two subcategories. I sort of consider them the convolution. They include all objects that can be represented as finite sums of binary products, ones in this and ones in that. So when you take the internal product, it'll change its type in that way. If F1 is the type C1 and F2 is the type C2, then their product will not be, it'll be of a new type, like degrees of polynomial dimensions and things. I forgot to say that before. There's a tensor product on this category. Types actually form a category. These C's, one can be included in another. If 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, if you could take the union of those two, that would be a, 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.

1:02:30 But when you take the product of two quantities, it will change its type in sort of an obvious way, like length times area is volume, 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. So in other words, if I call that operation star, this is S again. If 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. It's not like length-area giving values. It's already closed under this multiplication as well as this sum. The only thing I assumed in general was that C was closed under sums, 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's one of the ways, that is just one of the things that singles out sets. No, not really. Well, there are lots of other examples of that. It's one of the things that singles out sets. 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, if the mere fact that it's closed under sums means it's an eligible quantity type, the observation that it's moreover closed under products means it's one of these quantity types which is stable under this starring operation, which is like the condition of the measures. 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 the second type of bad infinity, meaning the natural numbers instead of the truth values. This idea of having a kind of island, bird, and plane space. This K is not the K of K theory, it's the island, bird, and plane of K. Why don't I write that down for you? Yeah, sure, and I'm going to make sure I keep copies of all of this. Well, it's huge copies. Well, I'll get them, and I'll copy them. Tell me if you can copy them, and I'll keep them. Well, what's the table top?

1:05:00 Did you say a takeoff time was 6, Bill? I thought it was 6.5 on the actual connection. Yes, maybe. Well, I don't want to stop now. Let's just finish this. We should probably be thinking of going within the next couple of years. But you can add true to true and get two. Too true to be true. Too true. In other words, it's a classifying object for a mass of finite fibers. Oh, okay. So now one could, one could, one, there are two things one could do. One could consider a sub-object, omega, could be like the usual omega. Or one could take, one could take just this. All of these things, as it is, and consider tensoring that with 2. This is the rig in which 1 plus 1 equals 1. So this is a way of getting impotent and tensors, this idea of particle distribution, by simply, you see, saying true or false means are there any particles there. But isn't that exactly what people in field theory do with the amplitude operators, particularly the optimization state number?

1:07:30 I mean, that's how they think of it. Without knowing this, obviously without knowing this machinery, can't we think? So there's sort of two ways of imagining, getting back to the logic, there are two ways of imagining Maven-Boden quantities, which have this, roughly this idea of, is there anything there or not, or where is it? One would be to actually consider, as I usually do in topos, to 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 would be more general-founded, where the fibers have various quantitative sizes. Take the image of that, which is now a monomorphism, and take the omega value of that. That's the one idea. 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. Then we abstract from that or we squash that by tensoring it with the Shandwell construction of other dimensions and compare these. They probably aren't quite the same without further assumptions. And so on, and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, They just happen to have finite fibers. But as such, they're not at all finite. So this is the classifying topos of the... No, no, the classifying object in the category. Yeah, the classifying object in the category for finite vibrations.

1:10:00 And 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. To have finite fibers and such. So, I mean, this is just to illustrate the big contrast between extents and intents. The extent is, from the point of view of the same doctrine, the same example, the extents of things are the really finite things that are placed in the finite number of discrete particles. Whereas by contrast, the intents of quantities are everywhere, really everywhere, but their values are sort of one and only point-wise, but are between finite sets. So you can do something which was sort of an intensive quantity which was a variable two. But 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 a distribution of particles in space, you have one of these finite vibrations, let's call it, and you take the pullback. 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. If there's a particle there, you multiply by that, you 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. It's total value is that the total number of particles that are thereby created at the integral of the intensive in respect to k. So then it's a kind of creation operation to do what? I guess so. Only this is a bunch of people that are thinking about it. They come out and imagine it. Except that the operators are abstract quantities already. Everything is a quantity. Well, they call them operators, but really they're... Well, they're operators on quantities. They're really operators, but you see, that's the point. They enhance quantities themselves in some sense. Here we're getting directly into the geometrical thing. Exactly. And really, this is the first deep materialist explanation.

1:12:30 I didn't realize this. You realize this is, this is, this is. I mean I'm not exaggerating. This is, this is, this really is the first materialistic theory. No you don't, no you don't, no you don't. The materialistic explanation of stopping matter. Take me further away. Do you note yourself? Carry on, I just will. I got the reference. Do you note the reference? No, I didn't actually. It's very bad, though, insofar as bibliographic references don't tell you where he found out about that. Anyway, you know a little bit more about this. I must find out more about this, but it seems to me that this guy's up with problems that have plagued me ever since I first heard about thermionics. I've been trying to reconcile it. I mean, that's why I got involved with Finkelstrang all that time, because I couldn't understand how you reconcile thermionic creation and annihilation of thermionic amplitudes. Which I was told was the laboratory of metaphysics in the modern era. You'll think where our ideas are like that, I don't think. The space must be formed with the understanding of discrete quantity. Yesterday the world. Yeah. Yesterday quantum mechanics dominated the entire world. Well, then, how about today? How about today? Did I not dominate this room? Tomorrow. No, I don't think so. Yesterday, tomorrow. It is. It is. Tomorrow, tomorrow. That poor universe. Can I have this? ...not be true to the world. Well, I'm... I think he's going a bit to the square of quantum theory as a product of that Nazi ideology, but certainly I understand what you're saying. Well, at least one of them. One Nazi, one proto-Nazi, who decided to study physics in order to... Well, it certainly ruled my way of thinking for a long time, in the sense that I actually thought the idea that... And the process was the American idea as well.

1:15:00 Well, I agree with you. As a matter of fact, I think one of the most spright claims to the contrary afterwards. Oh, you do know about that? You know about Heisenberg's claim that he... Oh, absolutely. I really want to borrow from all that. Oh, God, yes. And not only that. Have you ever read any of these pathetic companies in the world? I mean, it's unbelievable the amount of work that we get away with it. I mean, you know, I first became aware that there was this thing called national politics in the summer of 1933 when a student said they'd be coming to my lectures... So he's wearing a brown shirt. He came into my room and asked me, Professor Heisenberg, why are you such a gifted and intelligent man, what's 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 to explain to him, this is Heidelberg in the 50s, you know, in his country, having always been a little German, you know, why, how much happier he, this is Heidelberg in about 1954, I think, how much happier he would be if only the Germans would reconcile themselves. You know, this is the category, the obfuscation, the nationalist category, the national type, national script. If only the Germans would reconcile themselves for being a member of the common European house, good members of the common European family, and why can we not just be just like the Danes or the Batraves, and then afterwards say, well even if you are right, I think we're all aging, particularly in tears, because we all know that Führer has forgotten about this thing called America. This is Heisenberg writing his memoirs in 1950, in the 1950s. This thing called America, you've forgotten this thing called America. You have forgotten, you have forgotten always that there is America. The bigger part of it. This is Heisenberg writing his memoirs, gratiating himself with the Americans in the 1950s, and reconstructing a conversation, a mythical reconstruction conversation he's supposed to have with a young Nazi in the early... There is the fact that he was living a life so absolutely devoted to the contemplation of the atomic narratives of underlying nuclear science that he was totally unaware of the literal situation in Germany to the point that he had never spoken to an artist before.

1:17:30 Yes, yes, in 1919, exactly. And he then pretends, as I say, speaking obviously to an American audience, an audience that America admires, in the 1950s, that with great foresight I told them, ah, but your Fuhrer has forgotten those dreams. Well, there is this staple American great citadel and fortress of democracy, which, uh, I'm not making it up, but can we take a look at the passage? It's unbelievable. We believe you. It's not beyond belief. It's as though there is this group of people who are mesmerized by one man, you see, after some scientific content that they're all trying to understand. Oh, he attacks me, doesn't he? Right? Not that there is a mathematical science which we're all trying to understand. No, I mean, this is really... Well, I don't understand it because Moishe is not... You see, Moishe is a... He was a very good commentator. I don't know if he was trying to trust me, actually. I mean, he's never trusted you. He was distributing Trotskyite literature. Was he? Well, I know, they always argue with him a lot about Marxism, um, in that I thought he was distributing, he was a typical Trotskyite, and in the ten minutes after distributing openly Trotskyite literature, he maintained he was not really a Trotskyite. Oh, he's always, yeah, I didn't know that.

1:20:00 That's what they always do. Because they imagine, they imagine they have some contradiction with some leaders, I don't know what they imagine, but they always say that. I mean, Smith is the first one I ever met who actually said, I'm Trotskyist, and I was proud of it and so on. Almost all of them say that they're not. Even in the very act of promoting Trotsky as a great orator, great whatever, or even in his case, distributing Trotskyite literature. That's a serious thing. Yeah, yeah, I take your point. I didn't know that he did that. I mean, whenever I've argued, whenever I've fought politics with Mark Snyder, it's always been, you know, him and me with anti-communists. Or it's been, you know, all we've been arguing about is the Israeli elites. But even if I didn't know that, I would have to read this immediately. So what's the polite term for it? Tendentious. This very formulation is extremely interesting. Yes, it is. I must say, so I said so. I was very disappointed in my math. I don't know when I got this. When I see that, I know that what comes after is really an attack on category theory, whatever it is. Well, it is. You can recognize that from the first written idea. That was the opponent, the scolium. This is an object that he's instructed me to study. You see, this is extraordinary. I have a cushion because you are the key. Your program, with Bill Orbea, is motivated by economy. That's extraordinary. Apart from this common ideology, the two projects, Bill Orbea and Wil Barton, you are two wings. Two spirits, yeah. You are two wings. What happened to science? What happened to science? Absolutely. The two projects also share something which is in fact common to not only to them but to all programs on the foundations of mathematics, and in fatigue goes as far back as as far back goes in this area, the desire for economy, the wish to reduce all the notions of mathematics to the smallest possible number of primitive notions. I said, you know, I don't think, well, this is what I wrote back to him. I don't think the lawyer has an overwhelming interest in primitive notions, in economizing primitive notions just as an exercise in seeing how to achieve a common goal. In fact, sometimes Shannon will... Rather, he is interested in understanding why these notions are primitive. And that's what he does.

1:22:30 Right. Decisive abstract general relations. Yeah. There might be a lot of them. Yeah, there might be many more of them. It's not a priori that there's... It's not a priori that there are one field. This is the old medieval, you know, occultism. It must be so, because God, you know, must have created the universe as elegant things. Marks 1857. Can I just get the... We might have to move. Yes, we might. Well, now... Yeah. I put your stuff in the... This is my most immediate priority Bill at the moment that we've taken you to get your plane.

1:27:30 It is to find the record of the lectures and to get them to you. So can you give me the address? And it'll make, I'll do it via one of the courier services that are very reliable. And I'll also make a document. Send it.

1:30:00 I don't have the postal as well. Yes, okay. Yes, I will. Bill, there you go. 085, you've written it down, 3164. May I take that? Yeah. Okay. I don't mean to be unpleasant about it. Bill, you're not being unpleasant. I can't tell you how upset, but can I just look at you? See, normally... Listen, I'm not going to let you down, okay? Normally it would be okay because I wouldn't be able to write anything like that. Yes, I know. Anyway... I know she has. I know. And this whole completion of this whole project is so important. I tried to turn over a new leaf. I will not let you down. In the past I was always late in finishing the papers for the volumes of proceedings and so on, but now here we have this opportunity to actually make a major progress on that in a week. And major progress is something which is This is, to me, the most important thing I've done in my life. Bill, I'm not going to let you down, OK? I can understand why you're annoyed. I know you'd be too. OK, I'll see you in a moment. And you're going to be there, well, certainly by Tuesday. What is your U.S. address?

1:32:30 Oh yes, 106, Dieffendorf. 14214. 14214. Dieffendorf, I suppose. Dieffendorf, down there in the middle. Dieffendorf. There's a common number of these things. Years and years ago I rang them. You've probably got them. I don't know who. I don't know who. It's kind enough. It's just a... Yeah. They also identified a lot of people in some psychology classes. The first time I tried ringing you, they didn't let me in. Actually, you made me shut in on your language. Okay, so... Yeah. All right, okay. Deeds, not words. I'll give you the reference for the Heisenberg conversation with the Nazi another time. But it's quite disgusting. It's such a threadbare piece of special pleading in that anybody with the slightest historical sense must know it couldn't possibly have happened that way.

1:35:00 Bill, can I say something? I hope you don't think I've gone into this whole thing at a dilettante. No, I know I haven't got the knowledge or the qualifications to be able to put the questions to you that I really want to put, and to formulate them clearly. No, to anyone who used your dilettantism would be completely off. Okay, but if I was in your position, I would be beginning to think, though, this man is a dilettante who doesn't take things seriously, because I should have. I should have made sure I had those. I'm well familiar with the, shall I say, the foibles of being an intellectual. I too have rooms full of stacks of books that I'll never read, and I too have more commitments than I can hope for. No, but not on this one, Bill, not on this. This is the biggest, for me the most important thing I've ever been involved in. No, I really am fully convinced of that. So please, know that I will get them to you. Fully convinced of that. Okay. Right, these are the ones you want to copy, are they, Joe? Yeah, when the time comes. Well, we'll take them with us. That's Markova's letter, and Mabry's reply to it, which is here. And the list of questions I wanted to ask you, which are now never going to... Well, they will get asked, but there will be another time. Yes, as a matter of fact, that's one of the things I'm going to do. Once I've got the profit from this next big tour in, that's one of my priorities. I'm going to buy a fax, a telex, and a photocopier, instead of having to use them, you know, have it all here. I'm going to have it all in that room there, what was my dad's old room, clear all the junk out of there, have the photocopier, the tex, and the felix, have the library in the bedroom, what was the bedroom. Which is why all the books in the garage are there, because the bookcases have just come, so I haven't had a chance to put them in there. Get the library organized. Get all the tapes organized. Get the reading lists organized. And probably, if there's enough money, have a lot of conversion done, so I can have a bedroom upstairs.

1:37:30 You can get a conference in there. Well, then you see there's some capital there that if this opportunity ever presented itself that I could spend three or four years going to seriously learn mathematics, or learn the mathematics I need to know to be able to tackle these really serious issues, then I could do that. But it's all problems of somebody trying to do things for themselves, isolated as an individual. Organizations, certainly not in this country any longer, left that I can take seriously politically and at the same time trust, I can either trust or take seriously politically. I'm not involved politically in it. I just find at the moment there's nothing that somebody in my position really can do effectively in this country. Maybe that's faulty analysis. I was quite, I mean, George Rousseau said to Potomac that something to the effect, you know, that what I'm doing is the best thing that I could do, you know what I mean? In other words, because I've always felt this... In the back of my mind, I should be doing something else, namely political organizations in a country which totally lacks it. Theoretical work is of the utmost importance, though, and particularly at this juncture, and particularly given what is coming, I think, in the way of... Well, what we saw from those Georgians, what the authority of mathematics is going to be used for in bourgeois ideology in the next generation, it's going to be on a qualitatively higher level, it's going to be on a qualitatively altogether different level, and that has to be confronted. So what you're doing in publishing your position in Foundations of Mathematics and... Outlining it more exactly and in a more integrated form, I think, is of the utmost ideological importance, well, intellectual importance to the whole culture, anyway, of the most profound intellectual importance to the whole culture, but just looking at it from the point of view of concrete political tasks, I think it's also of immediate ideological importance.

1:40:00 This is what I have felt, even... With my very limited ability to understand the mathematical issues in any depth, I was at stake, and that's why I've done what I did by organizing this workshop. Thank you. Okay, so speech over. I must say that the sector of Marxist-Leninist parties in Canada has always encouraged George and Anders and I. I know nothing of the Marxist-Leninist party in Canada, I'm afraid. The Marxist-Leninist party in Britain is split into three factions, and I've never, I mean, I know people, I know and respect the people in one of those factions particularly, and perhaps I should have made the commitment to join them. What I wanted to say was that they always encouraged scientific, historical, and philosophical work from those people who are interested in doing this. I meant to be honest, the only, the main reason I had to not... It was against their orders that I participated. Oh. No, I'm afraid not. Was this something that the Canadians were doing to support American imperialism in Vietnam? Well, I suppose indirectly, but I mean, it was 1970. Shortly after the Marxist-Leninist Party had been founded, they declared a martial law, a suspension of... Oh, yes, yes, yes, on the pretext of a, of a, of a, yes, on the pretext of somebody letting off a letter bomb in, in, in, in Quebec, yeah, but that was, uh, uh, as well as a, uh, Quebec labor minister, Quebec labor, labor minister was actually killed, yeah, I'm sorry, I hadn't forgotten that, but, uh, in, in, even in the hearings in the bourgeois courts later, it emerged that these very terrorists were agents of the RCMP.

1:42:30 Yes, yes. It would come as no surprise to me. According to Gonzalo Reyes' wife, who comes from a bourgeois family and was friends with the personal friends of this labor minister who was murdered, the wife of the man who was murdered and this whole circle have always assumed that the labor minister was murdered on the orders of Bourassa, who later became prime minister of Quebec. He was a kind of rival for that position. So the terrorists were RCMP agents. British diplomats were later released. But the point is that the police pretended to be searching for these terrorists for many months, even though the imprisoned labor minister was sending messages to his wife, which, according to the wife, was clearly, there was a clear code that only she could understand, which told exactly where. He was, and she told this to the police, but they still pretended they couldn't find him. And she herself was imprisoned, in fact, in a luxurious hotel. Oh, yes, sure, sure. In order to be debriefed, is the phrase. Oh, I have had some experience of this myself. The actual target of the operation was our title. They burned down bookstores. They arrested 400 people. I'm going to ask... Really? That's very interesting. I should have known this because I... Ah, really? I'd love to hear that. In English? Was this a foreign language mock-up? So Anders is a comrade, is he? Anders is a Marxist-Leninist. I knew he was obviously a materialist from his talk and a good... That's very good. I'd kind of... Very good. Well, I guess he was obviously a materialist, but I've never got to talk to him about political questions.

1:45:00 In a few days of our arrival, we saw signs advertising anti-imperialist meetings. The attitude of the administration. ...suddenly changed, even though I'm sure, you know, you and George Rousseau were still producing very good mathematics, weren't you? But you must have been looking, I mean, the fact that you were so receptive to that meeting, you must obviously have been radicalized. And was Anders already active politically before then, or was that also something which came about through...? I was hung up with this contradiction between what you called the two principles of anti-science, or the two maxims of anti-science, that in order to do mathematics of any kind, obviously, I thought you were right, category theory, but in order to do any mathematics of any kind, one had to resist absolutely the kind of sloppy thinking.

1:50:00 Embodied in those two principles because without it goes wrong and particularly domains and co-domains in the case of category theory or just ordinary domains of function muddied up and yet physics, the physics of the 20th century, the physics that is the key to the laboratory of metaphysics, the laboratory of all true metaphysics, particularly quantum field theories, appear to be based on having to accept the second of these principles of anti-science, hence...

1:52:30 Now, you could simply ignore this problem by adopting the tranquilizing philosophy and say, well, just accept it because it works. Here is the toolkit that you need to do mathematics, accept it and use it. Here are the guiding ideas that you need to use for physics. Don't ask how they fit together. I could never accept that attitude. The other attitude was to go towards some form of objective idealism, like Platonism, mathematics, or to some form of... Subjective idealism, like Wittgenstein's ideas about process and physics. Terminal 2, I think, for Italia.