What is the true relation of logic and set theory?

0:00 So, I'm going to define what I want to call for the moment a model topos. Now, a model topos must have three properties. First of all, composition should not be true in the topos. Secondly, if P or Q in the topos, then... So, remember, brief discussion on the fact that... Yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, I wonder if, Jim, whether your proselytizing for this particular resolution of philosophical differences has actually had any, or any proof among philosophers of mathematics so far. I think part of the difficulty is, of course, that you have to convince many of them that, first of all, long-term, of course, our understanding of the language here, of course, part of it is category theory and part of it is, of course, floor-line. Also, of course, type theories themselves have had rather a mixed reception because they were quite formatively recognized when they were first introduced by Russell and Whitehead, so I was wondering whether you had...

2:30 Hardcore philosophers, some people in the room, and Saunders-McClain. Saunders-McClain said, the fatalism is obviously wrong. He also said that you can't pick one particular language. All I'm claiming is that you can reconcile this, whether that particular claim is the real world. I should warn you that this accounts only for elementary mathematics. And so on. And that's not a counterpoint. It may have mathematics, category theory, but you've got it on the policy. But there is a kind of formal parallel, I don't know how strong the analogy really is, but when the, there's a sense in which, if you compare it with the set theoretic idea of a foundation, for example, really there's more of a parallel with B equals L, I mean, where you really do have a kind of A kind of anomalous element there, to the absurdity of the sense of somehow amenable, it's true that you have to have the idea of modulo, the idea of the opposite of the originals. What is really close is the minimal model set theory in the classical sense, as invented by Cohen. And I think the real, I think as you pointed out in your book, which of course I've studied,

5:00 That the difficulty with the free Boolean, I mean, why it would be, for instance, that the free Boolean topics would be, of course, the problem is not 5a. There is this other problem that arises which is connected with the realist order. But it's often called the realist or a plagiarism interpretation of mathematics is the idea of bivalence. I mean statements are either true or false and this is somehow part of the package that you buy when you apply realism or plagiarism in whatever you call it. And of course it's quite true that the incompleteness phenomena show up in this scheme of things, of course show up precisely in the fact that the pre-movie autopsies are bivalent. And I've never... This has been a problem that I have thought about but never really resolved. I don't know how you... I counted it out, but you have excluded middle in the free Boolean complex, but you don't have bivalence, and the role of bivalence has always struck me as a... it's a very important one, it's underpinned a lot of philosophy, Wittgenstein and the Tractatus, for example, and many others. And this has been something that has brought that whole concept to me, to really... There's no natural way. What it shows in one sense is that biomechanics is a rather unnatural phenomenon. About 25 years ago, 30 years ago, I actually had Raoul listen to me. We were talking about Wittgenstein, and Raoul said to me, you know, Wittgenstein doesn't make any sense. Well, I thought he had invented those things. Actually, this was not correct. But anyway, Raoul said to me, what are two tables? At that time, I didn't know anything about intuitionism, so I sat down with a piece of paper and I tried to explain it to him. I realized later that he was putting my name to the piece of paper, and he wouldn't believe it. In conjunction with a philosopher, we published a paper in attendance on this, saying that we had absolutely no people.

7:30 I don't know if this has been said all the time, saying this, but contrary to what Russell, for having defined, taken a long time to define the number two, and in fact labeling his statements one to thirty-five in the process. There's something like that going on here if you're thinking of this foundationally, because you've, in effect, used something essentially the same as the natural number, defining what you mean by a type or a term. Correct. But I mean, this is true about any text on foundations. I mean, you have to use logic to argue and you want to talk about logic. Well, that's true, but it's not true, for example, that Denikin's quick theory of quantum I mean, I don't know what the actual number is. Well, I don't know what it is. I mean, it may not be very illuminating, but at least it's not certain. I mean, if he wants to define the number three as something like this, then you have to be able to count it for three. Yeah, but he can define the natural numbers without doing an analysis. Yeah, it'd be a natural number, but not a particular natural number. No, I agree. He probably has to take three steps to define three like everybody else. Well, I didn't want to define natural numbers at all. I think in this framework I can't even do it. I would have to quantify all the types that causes some problems. So, as I said, the logicists are certainly not going to be happy about this. The others, they all have to make some sacrifices. Freudenists and reformists both have to sacrifice classical mathematics. These are just small factors. They are.

10:00 It strikes me that it's too bad, of course, that this scheme didn't appear in the 1930s Certainly, logicians and sort of philosophers of mathematics would have been, in some sense, much more receptive to this at the time when they really were concerned with formalisms in various ways, and all these notions were actually . . . This particular form of type theory is, of course, Witten had not been for category theory on this particular form. Nobody would have seen, nobody had any idea if somebody had said, oh, well, let's invent, type theory isn't complicated enough, you know, it's only classical. Let's make it intuitionistic for it. I mean, there were people who might have considered that, but of course, it was hardly likely that, and yet the curious thing is that the increased generality. Which actually arose in the main theory as the result of a big simplification of the idea of interpretation, mainly the idea of category theory. Virtue. Well, certainly in our book we are trying to represent the various categorical structures in an equational way. In a technical sense, it's equation or not over sets, but over graphs. And we were certainly able to handle a weak natural numbers object, an equation. But a strong natural numbers object where the thing defined here is unique, we couldn't handle an equation.

12:30 In the case of topos, it doesn't matter. In the case of topos, it depends. But for arbitrage, the category is something that we couldn't do. And also, the computer scientists were interested in things like the lambda calculus, the international numbers of it. And they also skipped the uniqueness part because they can't handle it. Now they can handle it, but unfortunately they can't do what the lambda calculus people want. All of these things are equal by reducing them to a formal form. This problem is being looked at in the CADA, who have shown that by picking different things for M, you can satisfy one of these two conditions but not the other, but you can't satisfy them both. That would make an aneurysmic design. Geriales suggests that this might have some practical use. For instance, if you want to To check by calculation whether Fermat's theorem is safe by exploiting n equal to 4, I'm picking n equal to 4 because here I know that an introduction to the work is going to be Fermat-Loyle, and it's quite conceivable that if you use this method on Fermat's... You see, the Fermat's theorem by n equal to 4, by any exponent of that kind, can be expressed by saying that a certain parameter has a function... So, it's quite conceivable that in this particular case, this procedure would work. I don't think all equations work.

15:00 So, there is a limited computational procedure used to simplify an equation. You know what an equation is? You know it can't be always. Good work to some extent, and it could solve problems soon. Thank you for your attention.

17:30 Only the accumulated concepts, new concepts, can be interpreted. Usually individually, in particular, have two aspects, the two which are called abstract general and concrete general. The concrete general is usually the use of and listening of all possible interpretations of the abstract general in what? In the background of already accumulated concepts. I think that in a general way, the set theory is not as important as the middle kind of formulations of the background of the already accumulated into which the concepts may be interpreted. So, my dream for a long time, and I've always seemed to be getting closer to the goal, but it still requires a lot of serious work. Namely, that part of the process of using physics and mathematics should consist of describing directly the category of objects.

20:00 You make many steps along that, but few people have yet taken this up as an attitude when you depart from it. The view, in a way, of the very standard categorical way of analyzing a given subject, so here's a given subject, you know, it forms a category, because it consists of lots of concrete, there's a concrete general, it consists of all these concretes, and by the very virtue of... In fact, if these are representations of the same aspect, then there is a new notion of comparison between them, and so there are more versions, and so there's a category. Now, it consists, if you wanted to say, of operations. What are the collections on which these operations are operating? That comes out of the thing itself. It comes out in the following way, that by studying, in particular, what M is all about, we isolate a very small subcategory, A, and then we say that for each object X, arbitrary object X, we can consider the maps and the algorithms in the category from N to X, from the objects in our small reference point, and often the term figure is appropriate. In the extreme case of the category of abstract sets, we just take the one-point set, which is very small compared to the whole.

22:30 On the other hand, practically any non-trivial branch of mathematics, in contrast with the constant degree sets, It's distinguished between the general pair. So the operation F is operating on the collection of figures of X and giving as a result the figures of Y. So this is the beginning, and in many cases it is, whether it is explicit or not, a method of analysis. We have to take into account the fact But the transformation of figures into figures is compatible with change in figures. So in the small category itself, and you don't see this in the extreme case of abstract sets, but in general, in the small category itself, there are in particular maps, and these will operate on the figure on the right, and so the transformation F operates on a figure F, it will automatically satisfy The following naturalities in this is to say, transforming the figure x by means of alpha and then applying the operation f will give the same figure in y as would first apply the operation f to x and then the other side. So this naturality property becomes a way of introducing incidence relations among the figures. Because different figures might compose different algorithms in different ways, and that's the interest of the figure. And so, every operation, every map in the category, represents a transformation of the figures that can do what they were intending to do.

25:00 So the idea is, the general strategy of analysis then is that... If we only understand him well enough to take a large enough notion of basic figures down here, then a complete analysis, in some sense, a deep reflection of the objects in the map will be attained by considering figures of that shape and abstract transformations on them, which are out of the abstractness, however, we call it. All of which are restricted or constrained by this condition of naturalness. More exactly, we consider sets, abstract sets in most cases. So the idea is that the totality of transformation from X to Y, while it has probably an infinitely rich structure, we could apply the Cantorian abstraction process to that and just consider it as a cardinality. A bag of Lauter-Einsen for each x and y. So that's an object in this category. This category is defined as existent though. That much would be already. That much would be accumulated already. And so what we then, what we, but then, but then of course we were saying that to each x we assigned not just the, not just the one cardinal. The cardinal would match an a to x. But the cardinal maps from A to X, each A in our class is scripted. So what this mouse did was constructing the functor into the functor category, X to the A-Op. So what this functor does is it's going to make it work. It's going to say script F for figure. So script F of X is that functor from A-Op to abstract cardinal with the signs of every A.

27:30 The cardinal of the in-maps, domain X, but which also assigns to every map in fifth A, this tiny little category of fifth A, it assigns the corresponding transformation of the map between cardinals, and automatically will do that then in a way that satisfies this sensitivity of translation in that realm. The same equation. Eventually it looks like the special case of the associativity, why a special case, because we're only talking about math that started in A, that goes over into the naturality, the A-naturality, of the script that implies the naturality of this specific, the naturality derived from this specific word, the consequence of the associativity. But on the other hand, we can then consider this as a condition on a transformation. Independent is whether the F really comes in or not. So what we do have is the abstract idea of mathematical transformations, we're talking about A, and all the constitutions of that imaginable, which we can compare with those that actually rose from here. So just another way of expressing this information from the... The idea of the function f, the figure interpretation of an object, is that when applied to maps between objects, it gives rise to a transformation between these cardinals, and then the cardinal maps from x to y, and the cardinal natural maps from script f to x, to script f. So now, so far, Adam assumed that he thinks about A. The idea that the A is big enough to somehow describe it can be discovered through investigation.

30:00 It is most directly expressed by requiring that this math be a bijection for all x and y, or in which in other words, that's a technical kind of a language that's saying that the counter f is a full and faithful. And that's usually not true. It would trivially be true if we were allowed to take a to all f. On the other hand, if we take a to only a single point, something like that, it's most likely not true. In this specific case, it would be something in between. So this statement, this F I can call F subscript A, that's what it depends on, and that has been given. So, we say that A is adequate if and only if the subscript of A is faithful. This definition was first given by Isabel in 1906, a long time ago. Now, I emphasize that this mode of analysis can be, if you define the category gap, it consists of all the multi-sorted structures of some particular theory, in other words, the abstract general has been specified And then if you're looking at quantum mechanics, concrete general, category of structures, for short, I don't know if you can see it, then you can apply this method.

32:30 And often in many examples it's applied in the simplest sense, in geometry and algebra. Or you could also imagine that this M is something that you're trying to achieve. Maybe it's the category of all possible elementary particles, or all possible smooth spaces, in the sense that we haven't yet said exactly what that is. Arriving at an over-determined description of what it really is is approached by many different modes of study, some experimental, some historical, and some by various kinds of mathematical calculations. In particular to what extent M can be analyzed in this way. So in other words, even though in principle one should consider such categories in a quote given, unquote, whether or not they're given a structure, what emerges is if you can find a category A, some category is adequate, then you better represent them as structures. Because this is a pretty general notion of what a structure is. All of these are examples of shape age, but made up of arbitrary triangles and arbitrary maps between them, and a map of such structures being a natural transformation. That's a pretty, not the most general possibility, but a pretty general notion of structure. So, if you manage to find such an image, and you have represented it that way as structure, I think structures basically should be thought of in that way. They arise from the analysis. They need to be more precise. Now, of course, if you should be clear, here again we have this, the same type of ambiguity that we have with Macy's and Hecht's cases, etc., etc., mainly that at least unless we put much more conditions on A than I've said so far, you might find one script A and another script A prime quite different.

35:00 Which are both adequate, right? So if they're both adequate, we have two ways of representing this one category, M, that's consisting of abstract structures. So to say that the objects are abstract structures may be, may, in the acute case, be less than inaccurate. To say that the point of three-dimensional space is a triple of zero number. You can put, you can put more conditions on it so that it becomes less true. But that is an aspect that has to be taken into account. Okay, so to analyzing, in other words, when we speak of abstract general and concrete general, it means that they become, the A itself serves as, in the category of all possible abstract, primal, primal structures, serve as a concrete general. That itself, that itself arises from. So now, this method of analysis, of course, like many things in category theory, can be utilized. I think of this as the geometrical analysis in category theory. More specifically, it's a structure which basically consists of figures and instances, and the idea of a map is that it should preserve the map figures that it can preserve. An important example of this, of course, as you know, is the synthetic differential geometry.

37:30 The idea of the synthetic, what the synthetic basically means is that we at least partly take this stance that there is some idea to be specified, namely a category of two spaces. And we've analyzed more closely what that might mean through such, or through the possibilities that there is such a category, and we're going in a little more closely at it. So in that case, to give a specific example, the types of figures that are needed in order to be adequate are not just points, but also two paths, and also tangent vectors. So basically points, two paths, and tangent vectors are adequate in some conceptions. Also the combinatorial categories, these categories, combinatorial categories, are often really intrinsically described directly in this way already. Another one is called a triangle, for example, and so on. And then that instance really needs to convene inside the general phase. So this is a pretty common way of analyzing geometric categories, or a geometric way of analyzing any category. The new way is algebraic. The algebraic way of analyzing M is to choose again a small category, maybe a different one, maybe the same one. And so now the idea is to use instead functions rather than figures on an arbitrary object X in order to try to understand the specificity of X.

40:00 So functions on X are mapped from X into the objects in some chosen small category of B. There are maps inside B. The beta, the beta f, is another function that has this function. This is typically, this is called an algebraic operation. Instead of an incidence relation, we have an algebraic operation. You can apply the algebraic operation, beta, to the functions and get other. Very often these functions will deserve the name of quantities, because the quantity is, we thought it was variable quantities on x. So, we rely on the way of operation. Can I just ask, what do you call it? Is there a significant way to call it function? Well, it's just a choice of words, but I think that that is at least one of the traceable traditions to use the word function. There are no special kinds of maps which happen to have a real or complex or truth value. So logic, in the narrow sense, is often objectified in this way. If you take the two elements set here, then they're called propositional functions. It's a small variant. The truth is a quantity, so assuming that the propositional function, the real function is a complex function. They all have a significance. You certainly contemplate general domains very strictly. The word function you might use in some other way. You know, complex functions. It's that sort of thing. The complex manifold here is not necessarily one of these special ones, it might be, but it's narrowing down into totally.

42:30 In the same way, the word figure is chosen arbitrarily from that same spirit, except it's the legal spirit. Within the general space, we could consider triangles. And this, of course, includes the, it's gift B. Has Cartesian products. This includes everything that's normal and called. Math might be going to be two to the power of three, triple that to two, et cetera, so the possibility that Swift D has finite products is very important, both technically as well as for understanding specific situations, but plays no great role in that. I just dug that in to reassure you that by construing algebraic operations, what looks like a unary map, I'm not really restricting myself because the B might be the third power of the B track, though it's the multi-argument operator's special case in the univariate theory, not more than the univariate. So, with any given small category, I can make this a ten. If you look at covariance, that value comes from one of these, but the opposite of that, we call this, or argue it as a figure, and you could say that A is there, and that's the algebra. Here's the quantity. So, Q of X now is the algebra, the structure which deserves to be called an algebra. All of these value quantities on the variable space q. And the off here signifies the fact that obviously if you change x now along some general map of p, that the q of x will get mapped to the q of x times, the opposite of that itself.

45:00 On the other hand, the operation of the operator beta is what is decoding. And here it was concordant. In abstract algebraic terms, the alphas operate on the right, on the figure, whereas nouns really operate on the left, in the moral way of summing. I think there is probably a moral way of summing. Well, I mean, the notational bollocks, the fact that we have these two dimensions to writing compositions is no doubt needed in this dialectic that I'm describing. There's even one author named Lee Hairstein who actually said, you see, in algebra we write composition one way and in other parts of mathematics we write it the opposite way, and it seems to feel that's... That's convention is the opposite of theory. That's right, yeah. Well, I mean, it's a pure convention. I mean, globally it's a pure convention. But within a given context, you see, the fact that one can really be the opposite, even though you're talking about the same object X, that perhaps deserves to be... Sometimes an effect is, for example, when you write the interval of x and y, you have to really, I didn't want to get into it, but just to point out that these two variances are really two aspects of the same thing, of the same space x, of the same chain from one space to another. All right, so again, Isabel, of course, didn't fail to note that there's also the co-concept, where we say that B is co-adjective in M, if and only with Q.

47:30 Now, there's a very concrete, well-known example of an example, which should be known to everybody, namely, where M is the category of abstract, finite sets, and where B is taken to be a two-element set. Sorry, the three elements there. And then it's true. Because what in effect we're getting here is the truth tables, and the truth tables are just these vectors, and the point is that by looking at math, which are natural with respect to the truth tables, also known in the jargon and even out in the Morphism, but math, which are natural with respect to the truth tables, thought of as operations on the... Propositional quantities. We do achieve this fullness. Payfulness, of course, also comes with it. Structure is fullness. Namely, any map between different linear quantities in the opposite direction to the direction contemplated, which is in fact natural as expected to treat the tables, really does come from the feet. That's what this bijection means, right? The map of the abstract structures I actually found some real math. The real means in the M, the same thing over here. Well, the category of finite sets is fully converted into the opposite of the analogies. Now the way the two tables are usually understood, spirit B is actually the category of all those finite sets that happen to be powered by 2. 2, 2 squared, 2 cubed, and all the maps between those in M.

50:00 However, it's an interesting, I find, remark that it suffices to take one set, maybe a three-element set, and all the endomaps of that, and you still get, you just take two.