Cardinality, Schanuel, Euler

0:00 Don't like me, because I show to them that they do not exist. Oh dear! Oh, well, well, well, I hate to say it, but the English have this expression about the little Argentinian in your soul, which is, you know, a little egotism that is always, you know, however much you may pretend to false modesty is always there to make it out. To see that with this I could prove some of, one of the existence, well, something that helped to prove the existence here. Yeah. One of these axioms could be proved by using this axiom of choice. Sorry, when you say the action of choice for atoms, you mean atoms as in Bill's sense, A-T-O-M-S, you know, amazingly tiny objective motions, if I think I've got the right answer. So the idea, if you have a bunch like that, then there's a function that chooses to... Uh-huh. If for every H in the atom then something happens, then there is, for every H there is something, and then there is a function, for every H, right? So this is the active of choice. Yeah, that's right. That the universal quantifier has to quantify over atoms, has to range over atoms, over an atom. It's quite strange. And the function always has an interest. In any case, I would say... What do you call atom? Well, there are... It means that it has a right... It has a brighter joint. Oh, it's exponential. It's the additional adjoint to exponentiation that Bill talks about. That, for instance, is the source of his whole theory of fractional exponentiation. That's right. And, in fact, he's got some very interesting ideas of how that might connect with number theory as well, with some of these ideas that Mabry and his group are working on, on simply infinite systems of different lengths. he was talking a little bit about in Bristol but in this connection it's obviously to do with the first neighborhood of the diagonal and SDGs

2:30 conjecture may be proved you never know whether Bill has conjectured something or proved something but anyway from my point of view he had conjecture because we never saw a proof of that first order differential equations differential equations, and a lot of things like that on the top of it. Yes, yes. But this somehow discussed? Yes, yes. And then we proved this with another, using this precisely, fractional exponents. Yes, yes. That's what he was suggested by. Yes, yes. Bill suggested. Well, that idea of the additional right adjoined to exponentiation is tremendously important. I think it's one of his... Fractional. But he called now fractional exponents. Yeah. ...which have the right adjoined to form. But the fractional exponents are only one instance of this construction of the additional right adjoint to exponentiation because it's a very powerful general idea, it seems to me, of the situations in which it exists are not properly classified or studied yet. Then we produce an actual proof. After it is done, he wrote his proof of that. Actually, I have to say that somehow it's in this lecture one of the few lectures which actually are available. I mean, topos and low motion, something like this. This is one of three lectures. laws of motion, I think you said that's fine, yes, yes, yes, that's, uh, and he discusses a little bit also of this, but to some extent, I've never been geometrically convinced to this idea, I mean, that I never had a good visualization of this, of this jointness, I mean, maybe I haven't seen it just working in practice, but... Yeah, but you may look at that, at his paper, Bill, for instance, about the first order. It's not the first order, but that's very easy. The second order and other things, or our paper with Andersen, you will see how it is used there. Actually, I think that they... but in which paper you use it?

5:00 It's called... I don't really know if I remember the name of the paper. How can you tell me to remember? Sorry, Paul. I'm sorry. Thank you. Thanks again. Cheers. Thank you. But by the way, it was for me quite striking that you don't use the possibilities given by inversions or infinitesimals in order to approach differential equations. I mean, you know that... Well, it's not surprising because we never talked about differential equations at that time. So it's only now that it's the first time that we think about differential equations because of these work that we're doing, but at that time we didn't think about differential equations. And we thought that about, you know... Oh, hi. Well, and also because in the case of the coplovia ring, you don't have invertibility, so I suppose it was no other natural not to think of the invertible. but you're right obviously you are going to need it for differential equations the other thing while we're still together because I have to go off this evening after the winery trip well at least I'm not quite sure when I've got to go off because Panagis is going to find out what the train times are but I really wanted to talk to you a little bit about that philosophical work with McNamara back in 1992 This whole business of entity and of the metaphysical understanding of domains of variation. You should decide on a time now? Just tell me when you want to.

7:30 Maybe when we come back from the winery? Fine. Right, let's make an hour or so. That would be terrific. Yes, and then I will give you that time. Yeah, I can copy the papers at the same time. That's brilliant. That's a good idea. Yeah, because I don't have to go until late this evening or even small hours in the morning. Okay. I should be. Almost there. ...of length one centimeter. So I realize, looking at this slide, that my one centimeter is kind of big here and a bit smaller there, so you know, sorry. Suppose you want a ruler like this, then a half-open interval is a good ruler, because if you put two half-open intervals of length one centimeter each together, you get a half-open interval of length two centimeters, which is great, perfect. what walker do you want? On the other hand, a closed interval is not so good, because if you put two of them together, then you get the spare point in the middle. So what you end up with is a 2cm long interval together with one point. So we can think of a closed interval as consisting of a half open interval and then another point on the end. So we can say that the measure, I'm going to use the word measure in a very loose way, the measure of 0, 1 is 1 centimetre plus 1 point, or 1 centimetre plus 1 centimetre to 0, or simply one centimeter plus one. And for exactly the same reasons, if a is any non-negative real number, you can regard the measure of an integral of a as being eight centimeters plus one. Well, there's no precise statement here, but nevertheless you can somehow take the idea So you can try and push this thought and see where it leads you. So let's do that. So first of all, we have a closed rectangle. So this is the product of 0a and 0b. And to calculate this measure, we simply do some kind of formal manipulation.

10:00 We do A centimeters plus 1 times B centimeters plus 1, and you multiply it out, and then you have to try and see what the significance of the terms is, well, and read it what the significance of the terms is. It's clear enough what AB is. The 1, well, it's not particularly clear that the significance of 1 is the other characteristic, but we'll see this soon. And A plus B is part of the perimeter. Okay. Now, why should this be? Well, what Chaniel says is, he said, imagine painting this rectangle. Imagine painting this closed rectangle. You certainly have to paint the inside. Imagine the inside uses up A, B, centimeters squared, You also have to paint the perimeter. Well, you only have to paint the inside of the perimeter. Right? So it only takes part of the perimeter. And so you get this turn here, and by some similar reasoning, you get one left over here. So the one is still a bit mysterious, but But maybe this paint thing is going to explain something. So let's try another one. So this is a hollow triangle. It's not solid. There are various ways you can calculate it, so-called measure. One of them is to say, well, it consists of three sides. Three sides. But you've counted each of these three vertices twice. So this vertex here, you count it once on this side and once on this side. So you subtract off 3 here. And so you end up with a plus b plus c centimeters, and then you might add plus 0. So that's the Euler characteristic here. The Euler characteristic of the circle is 0. And you can use the same kind of reasoning with a few extra thoughts to calculate measures of various other things here.

12:30 And if you want to read about that, then I recommend Daniel's paper. Well, that's nice and it's a good idea, but it's still not a precise statement. So let me show you a precise statement. I'm going to explain Pathé's theorem, which is from this nice book by K, Gorkind, and Roto. So it's about fully convex sex. So the subset of RNs, fully convex, is the finite union of compact convex sex. If you can express it as the finite union of compact convex sex. So I've drawn two polycombex subsets of R2. This one is a union of rectangles and triangles. This one is a union of closed disks, rather small closed disks. So the point of the second one is that you can approximate anything by polycombex sets. So it's, in some sense, rather a large class of subsects. It's a slightly strange class of subsects, but yeah, it's quite big. I'm going to use this word measure again. Perhaps I'm using it for too many different things. This use of measure is different from both the standard use of measure in material inspiration and in different forms of use of measure that I was just using in the previous slide. So by definition, a measure is a function that's assigned to each polyconvex set, a real number. And it has to classify some axioms. The most important of these axioms is this finite additivity one here. So it says that the measure behaves like cardinality. For instance, the area of the union is the area of the first thing, but the area of the second thing, minus the part that you counted twice. So it's not necessarily the A, because it is . So you can ask yourself, what measures are there? What measures can you think of on the polycombex subsets of the plane?

15:00 Well, there's one obvious one, which is area, right? And it's also clear that if you take a measure and multiply it by a constant, that's still a measure. And with a bit more thought, if you take two measures and add them together, that's also still a measure. So the measures will affect the space, but at the third time, it's going to be one-dimensional. Well, maybe it doesn't look that way to you. But the challenge now is to see if you can think of other measures on polycombeck tests. I think this theorem is going to classify the measures completely. So, would anyone like to suggest any other measures, except for you two, we've seen this talk before? Sorry. Would anyone like to suggest any other measures, do you think of any other measures on what you call the excess? So, wait. Sorry? Wait. Wait. So, in fact, in this book here, Plano wrote a later point in the introduction that this is a missing part of mathematical culture that I think Rota wrote this, that most mathematicians bask in the illusion that there's only one measure, namely volume. So, there are, in fact, n plus 1 different measures. The dimension of this basic space of measures is n plus 1, so, and moreover, there's a more or less canonical basis given by one d-dimensional measure for each d between 0 and n. So I haven't said what d-dimensional is, but it's kind of clear, you know, areas of two-dimensional measure, poly is a three-dimensional measure, and there's an easy damage. So, how does this look in the case of two? N equals two, well, Euler characteristic perimeter and area for all measures. So, it's useful maybe to think about why perimeter is a measure.

17:30 Let's say, a union of two sets like this, you have to convince yourself that the perimeter of the outside is the sum of these two perimeters minus that perimeter there. So that tells That's what happens in dimension two. Then in arbitrary dimension, the zero measure is always Euler characteristic. The n measure is always the Bay measure. And then the ones in between are a bit more thought-promoting and not things that we may be so familiar with, but they're nevertheless natural. And these d-dimensional measures are well-defined up to scale and multiplication. So you can talk about the dimensional measure. OK, so that's the geometry part of the background. And now we come on to the category theory part. There are invariants of finite categories that could be called either cardinality or Euler characteristic. So the philosophy behind this is that for every mathematical object, there is a basic invariant, a basic dimensionless invariant. So that's a very sweeping statement, but it encourages you to think of Euler characteristics as a generalization of cardinality. So these are, these are synonyms in this context of cardinality and the Euler-Cardinian thing. So, I make an obvious remark here that to each finite set there is assigned a number. So the basic number to assign the finite set is the number of elements of cardinality, which I'm writing with bars. Okay, so, how do you define the Euler characteristic or cardinality of a finite category? Well, you put its object in some order, A1 after AN.

20:00 Then you write down the matrix Z, whose entries are the cardinalities of the onsets. And I'm going to assume this matrix is inversible. is not always invertible, and there's another way to express this definition that doesn't assume it's invertible, but this is the quickest way to say it. So we have this matrix, and the cardinality, or Euler-Country, the category, is the sum of all n-squared entries of the inverse matrix. So it's a rational number, an inverting over Q. From the definition, it's completely unclear why this definition is appropriate or useful or interesting or anything like that. And because this isn't a talk about the cardinality of categories, I'm not going to justify it. You can find the information here. In brief, you can show that it behaves like polycharacteristic of topological spaces, So it's compatible with products and vibrations and this kind of thing. It's also compatible with Euler characteristic of other kinds of mathematical objects. So for instance, if you take a topological space, you could either calculate its Euler characteristic, or you could... I'm sorry, I said that the long way around. If you take a finite category, you could either calculate its Euler characteristic or cardinality, or you can take its classifying space the geometric realisation of its nerve and captivate its solid characteristic and those two things are the same so it's compatible with other kinds of solid characteristics so let's do a couple of examples of this thing so here's a category with two objects to matrix Z, saying how big the concepts are. Well, there's one map from the first object to itself, namely the identity, which I didn't draw. And similarly to the second object, there are two maps from the first to the second,

22:30 and there are none from the second to the first. I now have to convert this matrix to the sum of all four entries, which is zero. And, look, this category looks like a circle, and that's the Euler characteristic of the circle. That's an instance of the fact that this construction, this definition is compatible with a classifying space of this category itself. How about the discrete category on n objects? Well, its matrix, Z, is simply the identity, the n by n identity matrix. And so, Z inverse, same thing. What's the sum of the entries of the n by n identity matrix? Well, it's n. So that's good, because a discrete category is just a set. So this shows that cardinality of categories is compatible with cardinality of sets. So I've recorded here just one of the good properties of cardinality of categories, which is multiplicativity. So it's fairly easy to show that this is the case And to do it, you crucially use the fact that cardinality of sets is multiplicative So in this line, the vertical bars mean cardinality of categories In this line, they mean cardinality of sets Well, that's fine And as I said, I'm not going to go very much into this one thing you can notice is that this works perfectly well for enriched categories. So suppose we're enriching in some category V where we know what the cardinality of the objects of V are. I mean, just what we think we know. We have some idea

25:00 of what the cardinality of objects of V might be. You can simply write down this definition and very what has changed I said that V be a monoidal category finite text becomes objects of V this is now calling this cardinality is now a definition of what cardinality means We take a V category with only finitely many objects. We do exactly the same thing here. This thing, well I've used a vague word here, I've said a number. And a number can mean anything you want really. This thing here is now a number. So, a typical context for this is V might be finitely dimensional vector spaces. So, then V category is a linear category. And, oh, I'm sorry, I should have added the crucial thing, which is cardinality. We probably need dimension, right? And then... We have the same multiplicity, so this is now the tensor product of V categories. So this goes through as long as the original cardinality of objects of V respects multiplications. applications. So in this example, this is the case, the dimension of x tends to y, this is the dimension of x times the dimension of y. So I said that the background was going to consist of one piece of geometry, one piece of Cashbury theory, and one connection between the two. So we've now done the geometry and the Cashbury theory, and it's time for the connection. So this is something that many of you know, which is how to view a metric space as an enriched category. So if you type the words in blue into a search engine, you

27:30 you will find a copy of this paper on where Lorbeer has to be using it. So, the point is this. There is a funny little monoidal category made out of the real numbers, or rather the non-negative real numbers. I'm including infinity here, which is what Lorbeer does. There are good reasons for doing this, although in this book, you could exclude it if you wanted to. It wouldn't make much difference. So this is a post-set, and therefore a category. But notice that this is greater than, not less than. So the arrows go the opposite direction from usual. So there's one arrow from x to y, here are y real numbers. There's one arrow from x to y, there's no bigger than y, and there are none otherwise. So it's a category, and in fact it's a monoidal category, because you can add real numbers together and this respects the order. The order category and the addition with unit 0. So what's a V category? Well, you just go through the definition. So, like any V category, it's got a set of objects which I would like to call points. for each pair of objects or points it's got a POM object, now POM AB is an object of B, in other words it's a number it wants to be a real number, and I'm going to call it B of AB so of course we're thinking of this as distance composition becomes an inequality so composition is usually an arrow in B, the arrows in B are assertions of inequality. So the composition becomes this inequality here, the triangle of inequality. And similarly, identity, the identity is usually a map from the unit object, so it becomes an inequality like this. And because by definition, DAA is non-negative, So this means that VAA must be zero. So this means you can regard any metric space as a V category, a V-enriched category. But there goes further.

30:00 You notice that we're not using all of the classical metric space axioms in here. We're not using symmetry, and we're not using the fact that, or the axiom, that points distance zero apart has to be equal. This doesn't matter. It's still true that it's any metric space for the Hatchery. Actually, Loebier argues that those extra two axioms were maybe a historical mistake, and I think he's quite convincing about this, but I'm going to stick to metric space in a classical sense for this talk. Okay, so that's all of the background. Let me just summarize it. So, first of all, there was Shannon's point that the measure of a quotient full of length A should be A centimeters plus 1. We had Hadley's theorem that there are essentially N plus 1 different measures on the polycomvex subsets of Rn. More precisely, every measure is a scalar combination of B's in a different way. Here's the definition of the cardinality of a finite category. Here's the observation that you can do the cardinality of enriched categories just as long as in your enriching category B, you have a notion of what the cardinality of the And then finally, Lorbea's observations of metric spaces are enriched categories. Well, see what to do with the last two steps, right? So it's clear what's happening next. We're going to work out what the cardinality of metric spaces is by viewing them as enriched categories. So, the only place that has been written about this is here, and again, you can type these words if you want to play with that sentence, you can take into a long discussion of this.

32:30 One thing that's not obvious, we have to know what the cardinality of V-objects is in order to do this construction. So we have to find a way of associating, did each x in B a number? Well, x is already a number, so you might think you can do nothing. But remember, we wanted this axiom that the cardinality of x tends to y is the cardinality of x times the cardinality of y. We can't just do nothing. We have to define these. We have to define in such a way that this is true. And there's more or less a unique solution to this equation. The cardinality of X has to be alpha to the power of X, the sum constant alpha. So the only question is, what's your alpha going to be? And it turns out that the best choice is to take output speed, e to the minus 2. And I'll explain that choice later, but for now, I hope you can just live with it. If you chose a different number than e to the minus 2, or a different number less than 1 anyway, it would just amount to rescaling the metric. So clearly these bars do not mean absolute value for what is. So now we really have to solve the ingredients and we can just write down the definition of the cardinality of the metric space. It's just a special case of the previous, the definition of the risk categories. So, we take our finite B category, or finite metric space, with points A1, so A1 up to An. We define the matrix whose entries are the cardinalities of the complex. So that means that zij is this expression here, where d is the distance. We invert this matrix, and just assume it's invertible for now, I'll say something about that in a minute, and

35:00 we sum up all the entries of the inverse matrix, and that's the part of the elementary space So, I mean, I realized that you could do this a couple of years ago, but I didn't understand what it meant until a couple of months ago, so that was when I got excited. I couldn't make anything of the formula that came out of it. So let's look at the formula for a very, very, very simple metric space, the two-point metric space. So, here's the Z-matrix. A small calculation tells you that the cardinality is one plus the hyperbolic tangents of the distance between the two points. Okay, and Z is graphed. So, what's The way I like to think of this is the cardinality is how many points would there appear to be if you're not wearing your glasses or those of us don't wear glasses if you do this in your eyes. So if D is very small, then you can't see that there are two different points, right? It looks like there's just one point. As D grows, it becomes more and more clear that there really are two points. when the points are very far apart, you can clearly see that there are two separate points. So as the distance increases, it goes from being one point to almost two separate points. We're going to be using this definition for the rest of the talk, so I'm just going to write it on board. That's the definition of this, n-by-n matrix, and the cardinality of the sum of the n-case

37:30 So first of all, something unpleasant. You can't always do this. If you work a little bit, you can find metric spaces which do not have cardinality. In other words, where z is not inversible. So you do have to work with it because the determinant of z is a real number, and generically it's non-zero. So generically, metric spaces do have cardinality. However, there are examples that don't. So the first example of a metric space that doesn't was found by Terence Pau, and it was a six-point metric space. So that's all there is to know that there are some that are, in a sense, bad. Now, for this reason and other reasons, I want to introduce a slightly, a variance on the notion of probability. the ancient Greeks knew that ratios were more important than lengths so lengths depend on whether you use centimetres or inches or whatever, ratios don't so when you consider a metric space, it's often helpful to think of all the ways of rescaling it, so don't just consider this metric space, imagine all its brothers and sisters arise from scaling by some positive factor That's the idea behind the definition here. You take the finite metric space, and you put together all the cardinalities of its rescaling. So TA here means A with the metric scale by a factor of T, and the cardinality function of A tells you not only the cardinality of A, which is the value of this function of 1, but also the cardinality of all its brothers and sisters. And typically what happens with a metric space when you graph its cardinality function is that there's been a bad behavior at the beginning

40:00 so there's a finite number of points where the cardinality is undefined but then at some point it settles down and it becomes monotone increasing and as it tends to infinity, the cardinality tends to the number of points. So again, think of it as if you've got bad eyesight. You've got, say, three points as they drift apart. They're just three separate points with no connection between them, and the cardinality becomes close to three. Well, it's still not particularly clear what's going on bad eyesight. The thing that really makes this interesting, I think, is what happens when you try and do infinite metric spaces. And the way that you can get infinite spaces is by noticing that every compact metric space can be approximated by finite ones. So, let me say this more precisely. Given a compact metric space, you can always take some nested sequence of finite subsets that converges to A in the sense that their union is dense in A. And when you've got a situation like this, you can attempt to define the cardinality of A as the limit of the cardinality of the finite subset. Now, clearly there's a question of consistency here. Maybe you have some subset, you have some chain, and I have some chain, and our limits give different results, or the limits don't exist. And I suspect that it's not for all of the paxometric spaces, there is a consistent path again. But we know that even some finite metric spaces give trouble. Nevertheless, you can try and do this. And I also suspect there are large classes of paxometric spaces when it is small. And here's one of them in this theorem. Let's take any closed interval and let's take a sequence like this. then this limit is the same, no matter what sequence you take.

42:30 What is this? Please guess. What would you like it to be? the cardinality of this closed-digital speed. This A here, that's one possibility. A plus 1, indeed. And it's A plus 1. So this is the link back to the first part of the talk, the geometric measure theory, this notion of the cardinality of the category automatically gives you this invariant from geometric measure theory. And, well, among other things, this explains the reason why I took the constant C e to the minus 2. So if you took some other, So, if you replace e to the minus two by some other constant k, then on the right hand side we get some constant times a plus one. So it would just be like rescaling the metric. Take e to the minus two, then you get exactly the length of 51. it's a cardinality function remember that by definition the cardinality function of a space at t is the cardinality of the space we scaled by an anchor of t so this by definition of that which is a t plus 1 which looks rather like a centimeters plus 1 unit of measure, right? The whole point of T is to allow you to rescale the metric. Okay, now everything I've said so far is rigorous and I hope correct. But I'm now going to say and things that are slightly more speculative.

45:00 So I'm going to assume that all the spaces I'm talking about have well-defined cardinality. So I don't mean to assume that every space has well-defined cardinality, but I just assume that I've worked in some context where everything works nicely. So let's try and do products. So let's take a pair of packed metric spaces. Well, remember we chose, we chose this function e to the minus 2x deliberately so that it would be compatible with products. So we also get the result that the part of the product space is a product of the cardinality, and we do. But there's a catch, because with metric spaces, there are lots of ways of putting a metric on a product of two metric spaces. You can do something in Euclidean. You can take a maximum of the distance in this direction and the distance in that direction. You can do all sorts of other things. And in particular, you can do this one where you add them. So this is sometimes called the D1 metric, which is the Euclidean one type of D2 metric because you use squares and square roots to be one metric. The reason why this works is that taking the product with the theme of one metric is the tensor productive of rich categories. So it's the product of metric spaces that arises from viewing them as in rich categories. So, for instance, if you take the cardinality of a rectangle with this metric, exactly what you want them to be. Again, t is like centimetres, so you would have exactly the formula that Shannon discussed. It would, of course, be nice to understand other product metrics, such as the Euclidean one. And all I can say at the moment is I wish I did. And I'm trying, but we do have something extremely convenient for the D1 metric.

47:30 Let me do another example. so circles C A is the circle of circumference A so the point of view here is that the circle is a space in its own right, I'm not viewing the circle as a subset of the plane, so that's why I'm using circumference as the parameter rather than radius. So the circle of circumference and the metric is the length of the shortest arc. So the distance between P and Q here, well, P and Q are almost opposite each other, so this distance is a bit less than half of A. Half of the constant. And you can work out what the cardinality of the circle is by approximating it by finite subtext. and you get this function here which is rather an interesting function because its exponential generating function here is the one corresponding to the Voluli numbers and the Voluli numbers arise in all sorts of places, they're particularly important in combinatorics but I can't explain their significance here but it's nice to see them. You can do a few little calculations with this and in particular you can think about what happens as A becomes large or small. So as A tends to 0, well, if A tends to 0, you get 0 over 0 here, so it's not totally obvious what happens, but the fact is that the probability tends to 1. Your circle is becoming rather like a point, and so you'd expect its cardinality to become like the cardinality of a point in itself. On the other hand, suppose A becomes large, then the cardinality is atom-topic to A. In other words, the left-hand side minus the right-hand side tends to 0, and P tends to infinity. So when the circle's large, it looks quite like a line, because it's flat.

50:00 It has length A. On the other hand, it has Euler characteristic 0. A is length 0. So, that's also good. You see the Euler characteristic of the constant term, which is what we saw in Hadling's theorem. with two conjectures. They're conjectures of different kinds. So the first one, I would almost dare to call a theorem, but I might ruin my reputation. The second one is much more wild. So, in the first one, the first one is, the context is rather similar to the context for Haddegard's theorem. So, Haddegard's theorem was about polyconvex steps, finite units of packed convex steps. This is about polyhuboids, finite units of huboids. Here are two of them. Again, anyone who's looked at a computer screen knows that you can approximate any shape like polycuboids. This is meant to be a connected fusion. So the connection is that if you take a polycuboid shape like this, then its cardinality function is a polynomial made up by all the measures in Hadley-Gisteric. And this is when you view Rn having the D1 mechanism, so you add up all the distances in the various directions. So, we already know this is true for cuboids. I might show you the case for a rectangle. Just ducking down to this blue passage at the bottom, the conjecture, the first conjecture says that in this restricted context of polycuboids, the cardinality includes all the invariants, all these invariants from Hadley and theorem. If you know the cardinality function, then

52:30 The second concept, as I've said is, I've got no very good reason for thinking it's true except for certain examples but that's simply that this works for all polycontact tests as long as you work asymptotically and works in the definition. So what this that the cardinality function includes, first of all, all the invariants from Hadziger's theorem, because that's what it is, near infinity. But secondly, some other rather mysterious stuff to do with smaller differences. And I have almost no idea of what that other stuff fine, signify. Okay, that's it. Questions? I like very much your talk, and I'd like to add a few remarks. You think that you had the point, you know, we take the length of the interval plus the point. Yeah. This type of thinking is also in Karate O'Durin's book, Arturic Animator Theory. Oh, okay. It's very nicely expressed in this. And of course, leads also to his thinking to algebraic major theory. Right. Okay. When is this book from? What year, I see? which I can probably show you. Then, the second remark is this, if you have additivity for your cardinality,

55:00 Then we can prove all the fields of combinatorial analysis. These are the two actions. Usually we suppose not to prove all the fields in combinatorial analysis. So we can do all of this in Norway. Thirdly, in order to go from finite metric spaces, of course, compactness is one way because it is qualitative, finite thing. But the other thing is the most obvious is infinitesimal, hyperfinite. I mean, you can do all you have done for hyperfinite metric spaces, and then try to make a shadow of the infinite metric spaces approaching from below from finite and from above from hyperfinite, then grab the thing. Hyperfinite is formally finite, but for non-stattar models, say, for natural nature. So there's an infinite large natural number, which, from the point of view of the non-starter, it's just a finite number. And if you take the star open, then every hyperfinite number actually has the same formal properties of finite natural numbers. So you can, using this, and of course if you have all this stuff, you can do probabilities. Taking the sub-objects of things and then instead of having the cardinality of the sub-objects, So you have a kind of probability of this, and this probability might be very interesting in respects like biology and a lot of strange things happen.

57:30 It's not very easy to do. And, of course, the other thing that you have done, very important, at least from my point of view, is that when you have two points, you can either discriminate depending on the distance or... So if you look to two points, say, with some glasses of a bottom of a board group, then you can see a continuum of two points. Right. So, right there. We know that the cardinality is not absolute from model to model, but glasses, glasses. So, one other thing, if you consider the carbonality of subsets as a real line, there's an integral formula for this. So, this somewhat makes precisely this subdivision idea. So, you can imagine that if you have a point here it's kind of spread out like this, if you have two points, then you get these two distributions like this, now you consider taking the area under the maximum, that would be sort of the amount of stuff that you've got, and Yeah, right. So if A is a subset of R, finite subset, I think, of the fact of the subset, then the primality of A is given by this, and we have to label these curves up, and it's a rather strange probability distribution. It's taking the maximum of the degree of overall AA, so it's this hyperbolic sequence of... So, it's not the same thing as bearing your vision for that. Question? Well, I also like this very nice idea that given the point A and the point B on the line, and then the cardinalcy is really a function that depends on the distance by just calculating.

1:00:00 Suppose the two points are a distance, little d a part where d is square zero. And then the distance I just calculate is one plus d half. Or the cardinalcy is one plus d half, that's what we see. You just take a Taylor expansion of the function of the Bay, and then the square is 0. And can you see why that's the answer? Well, if e is 0, there's suddenly just 1. Right. Right. So, 1 plus t half. Right. So, t half is like painting one time. Other questions? Maybe I have a comment, a question. The comment is that I have a much like the idea that it can be a different space. It's between one and seven. Now far apart, then there are two persons. But if they get close, there are less than two persons. I can become one, if the company is very close. And the question is, is the cardinality additive somehow? Is there some kind of additive? If you take a metric space and you split it into pieces, and try to compute the cardinality of the union, I mean, if it's a co-product of the distance between all of these and all of these in infinity, then, yeah, it's just a sum. But I don't know any I don't know any best formulas. I think there are two problems. One is that these things as enriched categories, they're non-symmetric. So if you're trying to do some kind of collage of programs with some kind of lax co-limit then everything's not symmetric. So my intuition about symmetric spaces is no use. The other thing is that because they're really categories, it's maybe an actual thing to do is take two-dimensional co-limits. And so somehow things don't work so easily.

1:02:30 I mean, we were discussing what should the vibration of metric spaces be, and this is, I think, the first thing that you did. Yeah, I don't know.