The relationship of foundations & applications (contd.)

0:00 What is the relationship between, then again, across the various fields, there are certain common features which need to be, I think the analogy of inverse limit, or projective limit, or even call it now just limit, is not out of place, because we look for the things which could have something common in each of several fields.

2:30 The so-called applied calculus techniques in the U.S. which emaciated and incorrect applications replaced a definition of the possibility to develop even the smallest pieces. But similarly anti-dialectical is the theory that working on difficult problems includes work on unification. The limits of dollars are now being spent, in fact, on that theory that these two things are mutually exclusive. And the Urbaki are being published on a very large scale, saying that unification is impossible, I think ultimately means that education is impossible. Perhaps the Urbaki did err in one direction, or were extreme anti-Urbaktists even more in the other direction. But must we really go on in this anti-conscious manner, a shallow interpretation of the world, and as no more than oscillations by the will of God?

5:00 I don't think so. So what is more exactly this concentration? What is the result of this concentration involved in the axiomatic method? For example, Cantor concentrated the concept of isomorphism, which by the way he himself says he extracted from the work of Jakob Steiner on algebraic geometry. Kantor also introduced something which Zermelo didn't understand and therefore it's not been passed down very clearly, but Kantor introduced a striking refinement of the universe of discourse principle, which was basically the principle that structure must be carried by the abstract sense. And that direction was developed by Dedekind, Hausdorff, Hadamard, Frechet, and others. and others into mathematics as we know it today. But meanwhile, this different foundationalist trend and speculate that membership chains might forever pile up one another if only the ordinals existed and so on and so forth. But aren't these abstract sets of cantor themselves problematic? In addressing this question of whether they're problematic or not, one has to I think make explicit the fact that there is a lot of confusion around between an inconsistency, advocate inconsistent systems, very dialectical, aggressive, but it's actually nonsense. As Tarski recognized in the 1930s, an inconsistent form of system is one in which such a system is used.

7:30 In the same system, it depends on the precise rules of inference, or if you like, on the meaning which is attached to. Of course, you know that one of the principles of dialectics is that contradictions are the key to moral development. For example, an analysis a hundred years ago needed to become explicit, a notion which is basic to general topology. It could be taken and sold. The principle of general topology. And that is a very basic contradiction in the Cartesian closed category of closed subsets of a space. The A and not A is precisely the boundary of A. The boundary is one of the notions. It means that, for example, to move from the room to the non-room, we have to pass through the threshold. It's a very simple idea. So even in static, we have to pass through the threshold. The notion of abstract set, or as Kantor called them, karganautsalen, he used the word karganautsalen in a way that almost none of his followers did, he used the word karganautsalen to mean exactly abstract set, so that notion is, I think, not inconsistent, but it's a very strong contradiction, because it is precisely the point, the notion of abstract set is that the points are completely distinct, and yet... Completely indistinguishable recursional features of an abstract set. So how is this contradiction resolved in mathematical practice, and indeed where do the abstract sets come from? So as Hegel argued, the only thing we can reasonably begin with is being. And I think that to find a foundation for mathematics, we must start by recognizing that mathematics is already in being and in becoming without asking our permission.

10:00 Now we can describe the process of the ongoing mathematics, the algebraic topology and functional analysis, to both of which both Voltaire and Horowitz, for example, made fundamental contributions, and of course also related to algebraic geometry and indeed to Godendieck's work on all three of those fields. Several branches of ongoing mathematics can be viewed as taking place in categories of the sort which I will presently describe. For example, the category of categories or the category of spaces, each of which are relatively discrete and isolated, as explained below, play this role that stands for dimension. Having isolated a relatively discrete subcategory of spaces, we can proceed to the study of the more interesting spaces in between the particular concentration of cohesion on the one hand and the discrete on the other. I'll come back to that in a moment. But first, let me say that we're also studying, via the usual method of discovery versus axiomatization, The extent to which these relatively discrete spaces are really discrete, in the purest sense, is desired by Cantwell. Now, of course, in the mathematical practice, for all kinds of reasons guided, for example, by Grothendieck, we have, we emphasize, I think, correctly, the fact that this difference is relative.

12:30 The less structured, the more structured. It's something that we think we understand relatively speaking. So, for example, if we want to study the topos of second order differential equations, we might say, well, that's relative to the topos of the smooth basis. Of course, we don't really fully understand either, but we think it's relatively. But this discreteness desired by Cantor does have its, I think, and as I say, One of these is that it is a way of improvement on the simple idea of the universe of discourse, which is necessary for any sort of systematic meaning that ever makes sense in the universe. The fact that the, let's say, sets out of which we might make models of structures are themselves devoid of any structure at all is... For the, what you might call the physical consistency, that is to say, of course it's to speak about the relative concordance of reality and a theory is very problematic, I'm not exactly talking about that, I'm saying rather that if we claim as a, if I claim as a physicist that such and such a phenomenon is due to such and such, I make a formulation of what I think is the role of the concept. I make this claim. Well, this claim is usually based on drawing some conclusion. If I draw a conclusion in a way that's non-rigorous, emphasized by Maxwell, for example, or if somehow the material out of which I'm making my models might intervene, then I don't really have, I'm really still not even justified in saying... And my theory explains that, just because I don't really know that physics and computing is really a hypothesis, either of those in further detail.

15:00 But I think there is a reason to, not that I'm doing it myself, but I think that there is a reason for somebody to pursue this idea of the pure species. But the fact that the map from any space to its discrete space of components has a section. This is actually the general axiom of choice because the general space may actually include an arbitrary, a roomfully arbitrary. And of course it implies the Boolean character. There are some other more general terms on the logic, moving character of the logic of the discrete spaces themselves. Now a stronger axiom, by the idea that pure sets would not have any variation, would not have any room to move, is Cantor's that there are no isomorphism types between x and 2 to the x. It's called generalized continuum, general continuum. And I claim that Gödel already proved this. Although I realize that decades later many foundations still consider it to be an open problem, but you see, the experience is that most of the sets come up in mathematics, they have cohesion, they have variation, and for these it's a true reality that there is variation. Cantor only put forth his continuum hypothesis in the context of these pristine sets. Of course, this constant subcategory extracted from a particular mathematical category will have special properties depending on what mathematical category we build it inside of.

17:30 But Gödel, in effect, showed that one can always find a still more constant L within any V. Thus, describing the direction of constancy or discreteness will probably never reach the ideal because of Gödel's input, but certainly it passes the GCH barrier. There are probably still more surprises. Not many people, a few people have worked on it. I think the set theorists in the last 40 years did marvelous work, but I think many of them drew in a way a wrong philosophical conclusion from these results, namely the conclusion that somehow power sets should be big. This is really a misapprehension because they were, of course, they're big. But the basic concern really was about, they were first order approximations to the theory of this, and what they showed is that, well, you can have variation, and then you can make it look almost constant, and find out that it refutes some other axiom of constancy, even though it certainly doesn't look satisfied, working in this small neighborhood, so to speak. ...of constancy, and noting, well, also thinking can vary a little bit, one got a little bit the wrong idea and forgot that Cantor's actual goal was to download an air-to-space instrument. What we see if we look at the various branches of mathematics is that there are variable quantities, one sort of one quantity or another. So this should be a basic part of the dialectical concentration, a basic part of the... For a long time, we have resolved this into a contrast between space and quantity, that is to say, we introduce spaces which serve as domains of variation for the variable quantity.

20:00 There are many categories of spaces, common and plural spaces, simple spaces, etc. And also many kinds of quantities, tensors, truth values, quantities. But there's another fundamental dialectic within that, which is not treated that way, which is the extensive versus intensive. This has nothing to do with extensional and intentional. It's something that was... A long time talked about in philosophy, in Maxwell for example, in physics, in thermodynamics, and it's still used by chemists in that narrow context, but actually it is a general, both extensive and intensive quantities abound in all the branches of mathematics. Although, as already Grossman complained, there is a certain overemphasis on the intensive as opposed to the extensive. Accompanied by a not necessarily appropriate attempt to, in some sense, reduce extensive quantities to extensive quantities. And of course, that tendency, as described by Grassmann, was only further aggravated by the use of the misleading term generalized function. An excellent function of distribution. I want to, extensive quantity, find an extensive quantity.

22:30 Co-variant come to main spaces, so E of X should be the system or space or whatever of extensive quantities varying over X. Now the category V is what you might call a linear category, which just means that, as pointed out already in 1950 by McLean, this condition of the co-products, finite co-products of products exist and are moreover isomorphic. This implies that maps can be added, and this is an actual setting for interpreting linear algebra. There are some matrices, block entries, that can be interpreted in any such way, so I won't really hone well on that at all. I just wanted to point out that just from the category alone, there is quite an extreme amount of notions can be, We can say, well, E of 1, this is the constant for extensive quantities. There's no such thing as quantity on any other space except for various things.

25:00 By definition, if we apply the functorality, we get a math from, well, what's that? That's the total. Any extensive quantity variable of x has a total value. Of course, in general, the fundamental reality often goes by the name of push-forward, which extends to quantities, probabilities, which probability has the property that the total value is always 1, e to the 1 equals 1. But push-forward is the probability of this reality. But now, let's take, in sometimes the opposite case, once the 1 has been singled out by understanding the maps toward it, Once it exists, there's the possibility of having maps from it. So, of course, maps from 1 to x are the points of x. If we apply the simple functorality to that, we see that for each point x, this map from constant quantity to the extensive quantity. Well, this is the Dirac delta. Dirac delta is the intensive quantity concentrated at the point x. Of course, what the actual value of that is could be any constant value, but it's concentrated on the point x. All that follows just from punctuality, as in fact, as even also another notion, very important about extensive quantities, is the notion of support. So, if we have an extensive quantity of i, b varying over x, we call it mu.

27:30 What it is, suppose we have, think of it, for example, as a sub. Well, then there's the use map, the way we have actually pushed forward along that inclusion. And so to say that mu is supported on this inclusion map, right, is extremely important. For example, the population of the world is concentrated on the land for the most part. It simply means that there should exist mu prime, let's say. So that's the definition of support. These are extension quantities. What are intensive quantities? Constant, contrarian. Contrarian functions, say F, could be a type of intensive quantity. So once again, what sort of particularities can we see automatically just from the functionalities and the fact that they're in the terminal space? It goes from f of 1 into f of x. So now we say, oh, for intensive quantities, we do have, among the variable ones, always the constant ones. This is the inclusion of constants as a special case of preparedness, not just the evaluation.

30:00 The idea of lifting and extending maps along the functor in terms of extending, then we have this. If this deserves a name of conclusion, then the functorality is often called not pullback in general, but restriction. We can ask, for example, whether an intensive quantity varying over A could be extended to one on X. And again, that's the existence of the diagram. This is still not the whole story about intensive and extensive, not at all, because in all these important examples, There are two further features that the intensive quantities, unlike the extensive ones, have also a multiplicative character, not just a linear one. And secondly, that in fact the intensive quantities act on the extensive ones. You may have come across this concretely in a more abstract setting, namely which is an example of cohomology and cohomology. Cohomology acts on cohomology and multiplicative. Also for distributions and functions. So how can this relationship between intensive and extensive be made more specific? Well, one could make some categorical two-dimensional axiomization of this, but in fact there are two different standard ways of constructing, both of which will... All of these terms automatically give rise to these other features.

32:30 Again, because of this, for some unexplained historical reason, I don't have a clear explanation over extents, the construction we're most familiar with is extents from intents, and this is usually associated with the name of Greeks. So the Greeks' idea is that we can construct a needed extents of one. If we have some things going on over there in V. So in other words, essentially the idea was to define E of X to be Hom in V from F of X into, let's say, F of 1. X in turn was basically defined as, as your example, the power of X, function space, the type of L. The basic idea is that whatever the intensive function is, we can define. Well now, of course, if we assume that f is multiplicative, in other words, that there is some kind of multiplication of the f values and the functoriality of pullback preserves that. This is the standard situation of logic, by the way, the truth value of the pullback and and, for example, on the order, but also and is there and is preserved by the pullback. Then the extensive quantities are still, in themselves, really adequate. It's still, it's like Machiavelli's definition of pronoun. Actually, that's a special case. You can multiply inside and then integrate, and that defines extensive quantities in the reach sense, whose value on any g of u on the product and analysis, if we have a notion of multiplication inside the intensive quantities, we can define extensive ones in the reach style, but in particular we have a module structure as in the module of reference.

35:00 You were more than just an object of the category B, whatever that was, but it has this additional feature of being a module over a multiplicative set of X. Now, whenever you have a module structure, there is posed a problem with division. That is, can I get, given two extensive quantities, can I get from one to the other by applying some intensive quantities? So that some nu, which I like, is equal to mu dot f. I've given those nu and mu. I ask for the f. Of course, in general, like any division problem, it's harder than multiplication. It may not exist, may not be unique, etc., etc., but it's important, it's a very important study. But on a more particular nature, we can solve that. Again, we learned in two parts of the measure theory that... Goddard and Nicodemus basically solved this in the case where x is a category of Borel spaces and Borel mass in a very particular context of this sort. It's called absolute continuity, actually. It says that nu is absolutely continuous with respect to mu. It just means divisibility, but not divisibility in the usual sense of multiplying functions,

37:30 but rather divisibility in the sense that though, of course, this This app is called a density. Again, often in logic, one extensive quantity, one extensive truth value, and another extensive truth value might happen to be related by a density which is a property of the usual. A traditional example, of course, is in thermodynamics you have mass, volume, energy, entropy. All these are basically extensive quantities, not intensive quantities. But one relates them to, let's say, volume, or sometimes to mass, by trying to find density, which may or may not be of interest. So, for example, trying to find the density for the Dirac distribution, a very simple, extensive one, led to all kinds of metaphysical worries about whether it had the density with respect to, basically, that's the absurdity of what, of course, they're approximation. There is a very interesting idea about relating the two, which is that we, observationally, in physics, it seems that the basic things are really extensive, one, measurement of space, measurement of space. We just accept, in some sense, coming from physics oscillating in the direction, because we can't quite construct them on an issue, such programming clusters. We can then construct by naturality. All of this is not very well studied. I've seen it proposed once and then not developed very much in some papers of Fulton and McPherson.

40:00 Perhaps someone has seen it in other ones. Of course, we may want to consider more than one type of extensive quantity at the same time, say, tensor-valued mass and scalar-valued volume, and that doesn't make sense exactly, but those different kinds of dimensionalities and different kinds of physical dimensions, so to speak, give rise to different keys, so we really want to consider several, and the... These are all intents and quantities, and actually they act as natural transformations between these. The ones that I talked about here were just endomorphisms of one, in a sense. But be careful now, because if we look just at endomorphisms of the function E, then as such, then typically we just get constants. These endomorphisms are natural, hence they are this sort of... So basically, if you select from each space in such a way as being compatible with all possible maps, you can only somehow extract the constant operation. So when I say natural, it means very slight, but in any case, the sort of idea that we're aiming for is that the extensive quantity types should form objects in a category. And the intensive quantities are just maps. There is such a category for each space x. If we apply it to the whole space x, and not just on the one,

42:30 but in the following very simple way, sorry, of objects over x, vibrations over x, and then a very stupid thing, just the simple functor back to x. And now we were postulating on this x that we had some e, well, maybe an e1 or an e2. Natural transformations between the composites for any space over x you want me to do natural with respect to these commutative triangles only. I will call that, you know, this is the density which transforms a particular element here, volume maybe, into a particular element here, mass. It's obvious formally that all these things can be composed. There are a number of random morphisms I can multiply, and if not, E1, E2, E3, I can just compose them, so you certainly have the multiplicative structure, and it depends, it really does depend on x, because this naturality with respect to a particular unit of triangles is looser than naturality in general, but also, that's what I define. Now, I want this to be a category, really, to have a condition in the model. The objects are the globally defined extensive functors, but this is just the natural transformations on the category u over x from, well, roughly speaking, from e1 to e2. That's very precise because it means the composites would just forget the functors and then take the natural transformations.

45:00 I should have said there is a relationship between intensive, extensive, and their actions. It occurs in all branches of mathematics. Of course, it gets different names when it's not recognized between the same phenomenon. So in logic, it has some name about how you should avoid non-variable or something. And in algebraic topology, it's sometimes called the projection formula. In other fields, it might be called the homogeneous reciprocity. It has lots of different names. But it really just is exactly just naturalities. Which is a consequence, I didn't write it down here, but it's a consequence of each style. But it's simply built in, it is the definition here, in a way, of the naturalities described here in this square. Well, this category of intensive quantities f of x is supposed to be a contravariant of x. That's also trivial, because if I were given the change of x, again, in no stupid way, I could forget it. You know, given things over x prime, I compose with that math, I get things over x. Natural transformations defined over x pull back along this function and become a special feature of time, and so I get that this additive category probably intends to the quantity of . So these are very simple general things. As I said, they occur in physics, algebraic topology, and logic. They should be made explicit to the core students, one by one. Now, do I have any time left? I wanted to say something more explicit about what kind of category is this script X.

47:30 That is, a category of spaces which could serve as domains for the variable quantities. We'll typically have a lot of linear categories, not one in which we thought up three points. That's something quite different, namely one which, again, both products and products do exist, but instead they distribute. Okay, so from the universal property of sums and products separately, you see right away that something like this will canonically map across, say, in any category where you have products and products, there's a canonical map like that. The modernizing work is in the linear category, vector space, and so on and so forth. We want to consider categories in which this is true, that these are, this is certainly a property of most of those categories of different domains in all the fields that we have in another example, category spaces. And by the way, a particular case of this is that The A times the initial object, zero, is an isomorphism. Now, why is this true? To analyze this properly, there are really two kinds of reasons why distributive law might hold. And you can find examples of categories where one reason is valid, but not the other, and vice versa. The example I wanted to discuss was both. Well, one is good products meets Cartesian code.

50:00 Not only do I have multiplication by A, but I have a right adjoint for the math A times X into Y into mapping X into Y to the power A. There's a uniquely defined notion of the function space, as long as the points of this function space are just the maps and so forth. So I guess the next time you rehearse it, there's a lot that can be extracted just from the notion of those categories, and I assume you know all that. Why is this important? It's important, actually, both for, oh, I should say, well, as far as the general principle of agglomeration, the punter has a right argument to infer zero sums, and that's what the distribution law says, and certainly that's one reason why the distribution law might be important, because of the existence of function spaces. Apparently, the concept which was... Proposed first by Horowitz. I don't know the actual reference. If anyone does, I would be very grateful, but I have to concede that the reference is from the 1950s to the fact that Horowitz was the first to actually propose, or he just proposed, in fact, the way of continuous basis, which is interesting because Horowitz, like Volterra, contributed to both algebraic anthropology and optical analysis, which is very good. Special fields which were concentrated into the category theory. We should mention something about Volterra's notion of element.

52:30 In the 1880s, Volterra pointed out that elements which are not only points, surface elements, are also elements. He used the word element for that. Now, the course of prelude is saying, well, compress a whole bunch of history here, but... Again, I think it was Eisenberg who pointed out that the elements or the figures would be allowed to be singular, because if they're not singular, you'd have trouble with punctuality. Really, so really an element is just a map of the shape A, A to X. Now, of course, we may have a few kinds of A's that are preferred, even a very few. A equals 1. A equals 1, of course, is a case. We have also functions of these other elements. Now, that, again, has been area of supplement. This has been resolved into some steps, objectified, one reason why it's tricky.

55:00 Now, there's actually another, almost independent, which is extensivity. This is the same thing which Grotendieck thought was universal in this joint. We're extensive for it because it turned out that it was worth studying just in its own right, not lots of other things. But also because of its connection with what we call objective number theory or the tradition of Cantor, Burnside, Rodendy, and so forth to try to realize quantities purely abstractly as they've been so far, but actually to make these quantities come from objects. So it's rather well known the idea that a vibration It's sort of like an intensive quantity because you can pull back the fibers, if you have something which is varying over one space, you can pull it back along a map and get something varying over another space. There's this kind of intensive contrariety that's involved in quantities that are defined in that way by vibration. On the other hand, but by abstraction, there's vibration.

57:30 There should be objective, whether it's co-hearing, k-theoretical, or contradictory, or nonlinear situations. So basically, the kind of categories for which the objects over a given object form a reasonable notion of extensive property are exactly the following. In the following sense, that if we consider objects over a plus b over a, and then there's always obviously a functor in this way. Which is, given something over A and something over B, we just add them together and that will come out over A plus B. You see, sometimes this is an isomorph. In other words, every map into A plus B just splits into something over A. Additivity of measures. Additivity of measures of objective measures. Well, it turns out, you see, this doesn't even mention products. I should have mentioned sums, but it has a consequence. Likewise here, you don't even assume that X has products. But if products happen to exist, let me just summarize. So suppose we look at the infinity, or how do we do it? There are graphs. All these things are both Cartesian flows and extent.

1:00:00 Inside, depends on which look discrete to it. So this says that X is A discrete. Then you could say, well, do these? Well, yes, if, namely that our These numbers of objects in mathematics come about just like this. Let's assume that our category has foot. Well, then it turns out we can define the discrete space of a big supply, actually, to the actual inclusion of these.

1:02:30 And in turn, there's a further adjoint, which is to take the set of points of any space. Now, as Cantor's movement wants the size of everything, even though he knows it has cohesion, he wants but a moment to abstract from that and just look at the abstract set.

1:05:00 As cardinality is nothing else, so to speak, you can define homotopy classes of math between two objects as just the components of the function space. This is why he was led to define the notion of function space. Know or assume that I0 preserves finite products. To be exact, it shouldn't preserve equalizers, but it may preserve finite products. So with this definition, you can define composition. In other words, we have a new category of homotopy types, so the same objects in new math. Now notice that this one is again, these are closed and extensive. The puncture precisely preserves the multiplication, the creation, and the one. What is the qualitative achievement of the puncture basis? Well, these are closed and extensive, but it has one thing. New property, namely, yes, it still has, and points, the maps from the terminal object, from the terminal object in the homotopy category, the same as the distribution of discrete spaces, we still have the adjointness, if you like any one of these things, if you think you understand pi zero, if you think you understand the conclusion, if you think you understand all of the maps from one, they all determine one another by this, but now unlike the original case,

1:07:30 Where you thought, my god, of course the number of points is different from the number of cells. Everyone has millions of cells, so the number of points is quite different from the number of points. Well, I think this is something that the logicians never dreamed of, but it's a common statement that the lambda calculus is basically just presenting the Cartesian closed category, and this is true. And, of course, there is the underlying strategy, the math can go on and so forth over the extension, but the fact that the math can also be left adjoint to the inclusion, this seems to be quite different. I shall be very brief. I just want to say some words about how I understand this kind of approach to the foundations of mathematics, namely that the foundations... This means seeking what is universal in mathematics in its place at several times. So the process of finding what is universal is sometimes called the objective logic in contrast to the subjective logic which deals with the syntax and deduction rules. So this view on foundations, finding one is universal in mathematics, is in contrast to the reductionist or monolithic view where you seek to reduce all of mathematics to one single, first of all, severely sane, similar framework or something like that, and to put the kinetistic analysis of the subjective logic of that.

1:10:00 David Hilbert is often demonized or deified as the proponent of this reductionist mathematics, because after all it was the proof theory. I want to defend Hilbert, not just you, but rather the objective logic, the precursor in Hilbert. Well, now for why did Hilbert launch, that was also part of his endeavor. Because there was the problem of preamble arithmetic. Was it consistent? The argument why this was consistent, hence, he put the proof theory on sufficient clear and sharp terms so that the question could be seriously clear.

1:12:30 To say that Hitler's time was an endeavor till 1932, i.e. just more than two years after a legend of Hitler's service. And Schaule's geometry, something like intuitive geometry. The core of deep geometric considerations, theories, and pictures is this second sentence. The foundation is concentrating the mathematical theories.

1:15:00 Are then the foundations of these mathematical theories? Hilbert talks about vast, intuitively given material. So my question is, have you any place for that in your logic? So, mathematics, geometry, that's what I, oh yeah, I think so, yes, but it's hard to formalize that. But I did say application. I am not very sure about what he said about...

1:17:30 Subjective logic can come from the monology of proof theory of Hilbert, because usually people consider Hilbert's proof theory is also another type of objective logic, and also he only, you know, made some type of objective intuition in the case of optimist framework. For him, he has a distinction between the ideal of mathematics and the more real of mathematics, like numbers and everything, and when he talks about a very small framework of physics, he thinks he talks about something, you know, something, I guess, but can't be an infusion into the universe. I don't know if you have met the following situation or if for technical reasons there is some kind of organization or discrete category, maybe in the category which we call mathematics, is... As such that the canonical factor from this category into the discrete set of components has a right atonement, which means that each connected component has a terminal component, and in a sense, and they turn out to be very, very few of things which I did, was hanky or fractal.

1:20:00 Yes, we meet with him occasionally, but enough to believe that it's important. The other question I think wasn't actually an answer. First of all, I'd like to emphasize, contrary to the encyclopedia entries that say Hilbert equaled formalists. Hilbert was a mathematician, so he actually worked on algebra, analysis, physics, various things. And the foremost program, the proof-hearing program, is a particular program, a particular conjecture. So, certainly, the non-schauliche geometries, the kind of intuition being a particular, is nothing but a kind of mystic geometrical problem. When I speak about objective and subjective, I typically mean subjective referring to the processes of thought as such, as opposed to the striving for content without it. So, of course, the standard method proof theory, as a theory, consists of attempting to objectify this objective, make an objective model of this process. It's not bad about subjectivity, it's just as well. The fact that we say, oh, that's a subjectivity, it's a computer thing like that, it's suggesting that the person who reached that position reached it because of some chemical function of his brain and not because of the correct meaning of his thinking. So it's really the same.

1:22:30 There is only one law of the clock, which are specifically completely independent of any concept in that sense. It is such a model logical theory that it could work very well. It's an objective model of the type of research that you want. Thank you very much.