Continuity & Logical Completeness in Topos Theory

5:00 And I want to emphasize that that's not the case. This is something new. The semantics that I'm going to be talking about, what I'll call topological semantics, this is the semantics with respect to continuously variable models, permitted by this algebraic formulation, is not standard in the theory of mathematics. When we have an algebraic structure... I want to focus on just the structure of the reals that's expressible in the language of rings. This is the arithmetic of real numbers. And among the different properties of the reals that can be formulated in this language is the property of being a field. The reals have this further property. Every element is either zero or it has a field. That's the special property of the ring of real numbers. And finally, pairs of real numbers, which we can do just by taking the elements of pairs like this of reals and we extend the algebraic operations just one to the sum of pairs, in the usual way, just two-component-wise in the pairing, and now, because the operations are all given by equations, Because the axioms for a ring are given by equations in the operations, this structure given by pairs, the operations are still associative, commutative, distributive, and so the pairs of rings still form a ring.

7:30 But now let's just look and see why it is that the pairs of rings no longer form a field. We can find an element that's not zero, namely this one here, 1, 0. So it's not zero, and yet it's not invertible, because if it were invertible, it would have to be component-wise invertible in the pair, but this zero here doesn't have an inverse. And so r cross r, although it's a ring, is not a field. So, in a sense, we've abstracted away from the property of being a field by passing from r to r cross r. So, let's just take it step by step and see where we get with it. We can do the same thing for any r to the n, right? It doesn't matter that we're just taking pairs. Or, indeed, we could do it for i for anything that's so high. We could take any i to the power of r. We could do the same thing, we could take... I hold tuples here of real numbers, and we could extend the ring operations to these things in just this way, and we would still get a ring, but again, it would not be a field, for the same reason that our crossbody is not. Now, what I want to say is that we can violate the, these rings, these archetypes, these product rings, they do all share one property with the real numbers, beyond being a ring. A ring is when one on regular, anytime you take an element out of it, you can find another one, y, such that x, y, x is x. And it's not hard to see that these product rings all have that property. Let's just, let's take some tuple, or some x, and then let's find a y that has that property. Well, what we can do is, we can just look at the x component logs. If it's not zero, we take its inverse, and if it is zero, we just take zero, and then that y that we find in that way will satisfy this equation, simply because then we can consider the cases where the eigencomponents are the same. So now the main point of the example is this. Additional properties, like the property of being non-regular, by passing from product rings to rings of continuously variable real numbers.

10:00 Instead of considering eyeful fuels of real numbers, we let the real number vary continuously over some parameter space, and that is why there is this condition of regularity. Now what do I mean by continuously variable real number? Spooky, is this where the category theory part comes in? No, it's nothing like that. It's just a real value continuous function. That's what a continuously variable function is. So, if we have a topological space, a parameter space, then a real number of r sub x varying continuously over x is just a real value function on x. And now we can find the ring operations on the continuous functions. Just as a point-wise operation, let me give us some ring of the product ring raised to the power of just the points, the set of points in the space-time. So that's a ring. And now let's check that that ring is no longer one-numbered. I'm trying to see it already because the way we did the, the way we violated that one-numbered, that our definition was not continuous. We're going to violate this condition here. So let's take a specific example. For our x, we'll take the reals themselves, and for our f, we'll take x squared. Here's the example here. We're just taking the plane here, and here's our f. And now we want to find some g, which will have this property, g times f times g. Again, if x is not zero, there's only one possible choice for g, because then x is not zero, then f of x is invertible, and so g of x will have to be one over x squared. So, what does g have to look like? Well, g has to look like this, right, because if you approach zero, it's got to be 1 over f, 1 over x squared, but of course now we're in the case where g has to also be continuous, so g at zero is going to have to be the limit of g as x goes to zero, so it's going to have to explode there. So there's no such g, there's no such g given this f that will satisfy us, because it's a continuity. Alright, that was the example.

12:30 Let me just summarize the argument, because now I'm going to try to generalize this time. So, the continuously varying reals, then, have even fewer properties that would fill the constant reals than the product brings, and they are therefore closer to the general equation of the quantum. The passing from the constants to continuous variation abstracts away some of the properties of the constants. That's the argument. So, now we're going to proceed by analogy. And the analogy is supposed to be between the real numbers, in the case that it's looked at, and sets on the other hand. So you say, what is the analogy between real numbers and sets? Well, the analogy is that the real numbers measure as a way of measuring length, and sets, you think of it as a way of measuring extent, perhaps cardinality, that's the analogy. Sets are a way of measuring, or another way of measuring length. Now the algebraic operations here on the reels that we look at are the ring operations, and the algebraic operations on sets are just going to be these logical ones that I told you about before, these algebraic logical operations. And then we can consider conditions like this condition of regularity. And similarly, we can consider algebraic conditions here on sets. For example, this one here, which is formally similar to the condition of non-regularity, and in fact is a form of the axiom of choice. I don't have time to go into that now, but I'd be happy to talk about that later. So, this also, I think, is an observation of uniform regularity. The axiom of choice is actually non-regularity on sets. So by analogy, we proceed from variable real numbers to variable sets, and here a variable real number we said was something easy, it's a continuous R-validated function, a variable set is just a sheet. I'll try to explain that to you. So, how did we proceed before? Well, we started by pairing, and we went through two things. So, let's pair up our sets and show how to interpret the algebraic operations on pairs of sets. Well, I'll again just do it component-wise. So, we're going to look at our objects and how we're going to consider our pairs of sets. That's an object. The product of a pair of sets is just a pair of products. A power set is a pair of power sets, and so on. Element joint is like this. An element of a pair is a pair of elements, and so on, okay?

15:00 And now it turns out that if we model the operations in this way, that all of the equations that I mentioned for ideal algebra logic are satisfied. So in that sense, we have a model of ideal algebra logic in this more general algebraic structure. Now, these models have lots of the properties of sets but they don't have all the properties of sets. Just like pairs of reals form a ring but not a field, a condition that's formally similar to the condition of being a field is this one. In sets, every set is either empty, that's what I mean by isomorphic to zero, every set is either empty or it has an element. That's true in sets. It's not true of pairs of sets, though, right? Because here's a pair of sets, one by one, I just mean any single set. So take a pair like this, well, that's not the zero in this model here, but it doesn't have any elements either, because if it had an element, it would have to be a pair of elements, like this, a pair of this, that's okay, but there are no pairs of this. So this field-like property is violated by pairs. Now, before we cast from pairs to tuples and then on to i-indexed families, we can do the same thing with sets. We pick here an i-indexed family of sets, brain set i, and we'll just tuple the operations just as we did before. And again, we model all of the equations by our logic and divide that by some property of sets. Now, what property of sets is satisfied in all of these product type models? Is that one of those formally similar to the bottom line, namely the axiom of choice? Families of sets like this always satisfy the axiom. How are we going to violate the axiom of choice now? Well, as you've seen that before, we'll consider not just discrete families like this, but continuously varying families of sets in the parameter x. Now we have a little bit of work to do. We have to say, what do we mean by a family of sets varying continuously in a parameter? Over a space. That's what we have to answer. It was easy before to say that continuously varying real r of x is just a continuous real value function on a space.

17:30 You see here, you don't really know what a continuously varying set is. You can't just say it's a continuous function like that. That doesn't make any sense. Because sets is not a topological space. So we've got to figure out what this is. Now, fortunately we don't have to start from scratch here. That is, mathematicians have already figured this out long ago. They need to deal with continuously varying structures all the time. They have continuously varying anthropological spaces over a grammar space. That's a fighter model. Or they have continuously varying algebraic structures over a space. That's a sheet of groups. It's already been done. All we have to do is recognize that that's what it is and then make use of it. So a continuously varying set is simply a sheaf of sets. Let me try to describe for you with simply a picture what a sheaf of sets is, okay? We want to describe a continuously varying family of sets, f sub x, indexed by the points of the topological space x. So what we'll do is this. We'll say, we'll take all of those sets that we want to We'll sum them all up, take the disjoint sum of all the broad points in x of the f of x here, and we'll make a big space out of this disjoint sum, and then we'll project it here by continuous function pi down to base space x, so that the fibers here of pi over each set My pi inverse x will be the varying set that we're looking for. These will all be discrete, and now this space here is to be topologized in such a way that pi, the projection, is a local homomorphism. This is the relevant condition that describes what a sheaf is. What that says is that if I have my space here, x of parameters, the sheaf up here, f, and the projection here, pi, Each point x here has a free image up here. That's my set f of x. The condition of being a sheath says that that set f of x is going to vary continuously as I move the x down here. I have to have a continuous way of getting from one spike to a nearby one. The Low Compromising Condition describes that by saying, if I take some y up here, then I can find a small neighborhood here, u. Which will be mapped, homeomorphically, under pi, through this image down here, where pi and pi are.

20:00 So that's the condition of it. Just to give you kind of a sense of how that continuous variability is supposed to work. I make this up and build it here to make it out for people who use this all over the place in mathematics when they need these kinds of constructions. Continuous and variable structures. Okay, now... The point is that these continuously variable sets also model the logical operations, just like the product universes did. So, for example, the product is taken as one unit, but not all of the operations are complies, otherwise we wouldn't get anything out of it. For example, this function space here is not complies. That's what's called a sheath of components, done in terms of the germs of the two functions. You've seen that construction. And now the fact is, this is a model of the higher order of operating operations which violates some further properties of sex. In particular, it violates the axiom of choice. Well, actually it's not hard to see what it does, so I'll just show you what it does. Rather than taking that condition of unknown irregularity, we could take the condition that says Well, if you have any set X and another set Y, and a mapping here which is surjective, then you can find some G, such that F composed of G is the identity of X. And I'll leave that as an exercise for you to show that that's equivalent to the Von Munden regularity, which is what I gave you. It's not hard, just take it. If you have an arbitrary one that's not surjective, just take the image. So now what we have to do is violate the answer in the choice of given examples of space X and a surjection like this, a topological surjection, which is a local homomorphism, which doesn't have a section like this. Something like this that's not a retraction. That's not hard. Take the circle and the plane. I want to do it right here. Take the circle and the plane, S1, and then take the twisted double covering. That certainly is a surjection.

22:30 It's a local homeomorphism because any point here has a pretty good neighborhood here where you can take a, in fact it's a branch of the logarithm where we use complex logarithms. So this is locally homeomorphic, but there's no section because you can't take the entire circle and put it back up here into the twisted. Simply put, why maximum choice is violated in machines, because you can't take arbitrary sections of a section, because the topology interferes in just the way that it did in the case of the real numbers. I've played kind of fast and loose with this notion of the universe. I told you before that the universe, that is the model of the algebraic operations, is just a topos. Let me be just a slight bit more considerate about that. In fact, the topos is a category equipped with the adjoint structure sufficient to model these operations. That's essentially what the topos is, in my view. You can think of it, then, as a universal abstract set to the extent that it models those set operations. And now a theorem is that the deductive... Conception on classical higher-order logic, formulated equivalently to the algebraic theory, is complete with respect to monodys and topos. Now, that's not the theorem that I wanted to tell you about, but that's more of a sanity check that we've got the definition of the topos right. If this wasn't the case, we might come up with a new kind of algebraic structure that would... The theory is that we can specialize in tokuses of a very particular kind and still get this kind of completeness. Namely, the tokuses that I was describing before are continuously variable sets over topological space. That is, if we look at the collection of all schemes on top of fixed topological space, then we think of those things as continuously variable sets in the parameter.

25:00 These concepts have a logic which is much more general, or much weaker than the constant sets, of course. I've shown you that now, that they violate a lot of the properties of the constant sets. Fewer things are true of variable sets than are true of constant sets. Just as fewer things are true of continuous real value functions than are true of reals. And now we can reasonably ask, well, what is exactly the logic of continuously variable sets? Continuously variable sets over stages that we can have. What, what logical do they have? What sentences of hierarchical logic, classical hierarchical logic, are true in all such continuously variable models? And that is the theorem that tells you about the global logic that is of arbitrary continuously variable sets is exactly the system that we know logically as classically deductible. I'd like to point out to you that, unlike the theorem that I mentioned before, which I said was a kind of sanity check, there's no reason for this one to be true, at least not rhetorically. This is a notion defined in terms of continuous variability in topological spaces. It has nothing to do with deduction. This is a notion, going back to Frege and Russell, of a logical system that has something to do with... Now, the proof, let me say, it's hard to deny. It's a simple application of a more fundamental result in topos theory due to Ikemorda and Carson-Woods and coming out of the geometric school. But it does have this logical consequence that classical hieroglyphic logic, it turns out, is exactly the logic of continuous variability. Summarize that by saying that in general,

27:30 Well, in particular here, fewer things are true of continuously variable sets than are true of constant ones, just like, in general, fewer things are true of continuously variable quantities than are true of constant quantities. In the logical case, when we specialize that, we learn that classical higher-order logic is exactly the logic of the continuously variable tree. It is indeed, there's a very interesting relation between the hand-to-hand business results. This one, I described before a sheet in terms of the fibers of the projection mapping. These fibers are called the stalks of the sheet. And it's a general fact about sheaves that the global sections of a given sheave embed into the product of the stocks. And in fact, it turns out that the stocks of the models that we are looking at here are Hinken models. So the Hinken will lead us here and follow us from this one by sort of taking the sheaf models and then taking all the stocks. So the stock and sheaf model will be a Hinken model. And the difference is that in the Hinken model is, in a sense, Destroying the higher order stock of sheep is a way of collapsing it into a certain perspective on the model, or on the sheep, which throws away some of the information and preserves just a certain fragment of the shorter information, the geometric fragment of the information itself. I guess if you think of a Hinken model as a stock of a sheet, then that's something like a continuously variable thing frozen at a particular point from a certain point of view.

30:00 In the beginning of your talk, you said that topos is to hierarchy or the building algebra is to propositional. More traditionally, what I would expect would be that something like a symmetric algebra is to predict algebra, is to view algebra, is to proposition algebra, so is there any kind of relationship between symmetric algebra and cohomology? While certainly cylindrical algebra was one of the attempts by a certain school of algebraic logic to formulate the notion of predicate logic, I think one very simple difference between the two is that topos is capturing the notion of a higher order system of logic, of higher types of functions and relations and functions of relations and relations of functions of relations. So there is a categorical description also of first-order logic and that's what we want to compare with the notion of cylindrical algebra. Now the exact correspondence between cylindrical algebras and first-order models I think is given quite mutually by another notion called aversive hyperalgebra, Faithfully represents both aspects of the idea of solute algebra, which are germane to the concept of a first order system. But I'm not as familiar with solute algorithms as other people here are, so maybe there are people who know more exactly what they are. Precisely, what is a solute algorithm? A categorical description of a first order system. It's the word classical in your theorem.

32:30 How does it arise from the proof and why do you want it? I need to have it because I need to interpret the type of true values as the object 2 in the sheath. You see, the sheath on space, as you all know, is generally intuitionistic. And I'm interpreting classical logic into an intuitionistic background system. So I interpret the true value object as the object 2, the simple double covering of space. So they're taking delegation from each other. A way of thinking of it is as the double negation of classical into intuitionistic logic, but in a higher order setting. Another way of thinking of it is as the letting of a Boolean algebra into the Boolean space. So if you remove the word classical, you still have a theorem. I have a theorem, but it's a little bit delicate to formulate because you need a sheet of truth value objects. We need a sheet of truth studies that's not the sort of on-the-fly-fire. I would like to ask you about where you were, because when you showed us the example of the ring of ring-wound interactions, was it just because it's something for the people here in the room who are not used to sheets, or was this analogy really a bad idea for you? Both of the things, the real value functions and the sheets as models of higher order logic, are examples of a general idea that is a guideline for my work and it's a guideline for lots of people working in category theory, and that's this idea of variability, of ordinal variability, of continuous variability, as a way of abstracting relationships. I think there is a general kind of principle at work there. It's not a matter of limiting from the drama of continuous functions to that of sheaths, but both of these exemplify a certain kind of principle. That's the point that I was trying to make about logical abstraction being modeled mathematically. Yes, but there's a certain analogy between precisely those two things. Yes, yes, yes. Between the wheels and the ring operator, you have the sets and the operator operator and the set.

35:00 And there is in fact, in that case, a very close one. For example, you have the category of cohomons, which is the category of the range, and what the extent of the category is. And there's a lot of people in that context that actually agree with that. Right, so there is a specific analogy in that case. A lot of these are abstract measurements of the actual sets, so it's a little bit of a struggle. I'm following up, I think, on David's question. The theorem says that global logic of continuously varying sets is classical deductible. H.O.L., if you make that local, do you get the intuitionistic? The word global there was just a spicing thing. General, logical, arbitrary, or any of these sets. So I would want to try to specialize in that instead of probably not the most precise conversation with you. Thank you. So before I give this talk in English, I want to say a few words in French because I think that it's in French that I should thank the organisers of this conference.

37:30 Thank you very much for having opened the door. Not only contemporary mathematics, ancient mathematics of what we call the West, but also of mathematics that have been done elsewhere, and in a way it will be in China. It turned out that a certain number of you knew that I speak English, so ask if you could... In the history and philosophy of mathematics, the term foundation usually refers to a very specific type of endeavor, associated with very specific periods of time in history. And two, a specific concern vis-à-vis mathematics. As you may guess, it is not in this sense that I shall use the term when dealing with mathematics in ancient China. I suggest an extension for the meaning of the term. An extension that could allow us to deal with that are contested in historical sources. Mathematical knowledge, for which, one recently argued, a body of statements, a body of operations, a given body of statements, is designated as the basis of mathematical knowledge.

40:00 There is a wide variety of mathematical knowledge and it seems really interesting that a wider class of foundational exercises might discuss one such attempt, or I should say one last of such attempts, that was carried out in ancient China. Now, it is not by chance that my topic is about ancient China. It's not only because I've been working on ancient China for 20 years now, but it is because we have a very specific kind of sources, like baryonyl clay tablets or Egyptian pyrolysis. We do not have many second-order statements concerning mathematics. Many second order remarks on physics in which you could develop an analysis of how people conceive of an organization of mathematical knowledge. This is not the case in either ancient China or ancient India where we have commentaries in which people express views as to the organization of mathematical knowledge. And this is because we have such sources that I can rely on them. To develop and present ideas about ways in which the conceit of an organization of mathematics can emerge. We orient my description of this Chinese case theory that we have the foundation for mathematics.

42:30 Concerns the ways in which the basis for mathematics is given. In the actors' eyes, how was this basis for mathematical knowledge given? This will be my second set of questions and we will see that there is variety, according to our historical sources, there is variety in the ways in which Foundations, bases for mathematical knowledge can be expressed and can be given. My third question relates to the kind of knowledge actors believe they should be in order to bring this basis, mathematical knowledge, which kind of words they carry out to exhibit this basis, this source of mathematical knowledge. With statements concerning mathematics, at first sight, quite obscure to me, but I'm going to analyze it, so just follow me. Of mathematical knowledge, yes, yes, because I'm a little bit far away from them, and I should probably, let us have... As for what is named Lü, I come back immediately to that Lü at the moment, there are nine types of arrays that flow from them.

45:00 So we have Lü, which is in fact numbers of a certain type. Lü refers to a concept of number. I'm going to show you more precisely in the corner. Let us have a look at the picture. We have numbers of a certain type. From which flows a water-in-a-tree mathematical knowledge. In fact, we are not yet done with the mathematical knowledge. From it, from these numbers, nine parts of mathematics, and in the list that I skit on the transparency, in the list that follows the nine parts of mathematics, one recognises how they cannot, in which I come back in a moment, The nine chapters of the book flow from this concept of number. Then the statement goes on, these nine chapters, these nine branches of mathematics, all, and then we have a list of Fundamental operations on Lü that these nine chatters bring into play. They all multiply to disaggregate Lü, divide to as a group, homogenize and disguise to make them communicate, apply the procedure of support, which is a strange but interesting way of referring to the group of three in Chinese, to link them together. These are the fundamental procedures and from the other deductions, hence the methods of mathematical procedures are exhausted. The fact that the nine chapters make use of four fundamental operations is the fact that all of mathematical, specifically the methods of mathematical procedures are exhausted by these operations.

47:30 But to sum up this vision of the architecture of mathematics or mathematical knowledge, we have a concept of number at the source. From this we have a differentiation of mathematics into nine branches. On the basis of these nine branches, we notice that they only imply four fundamental operations And this leads in to a statement that these operations exhaust all the methods of mathematical procedures. Here is conceived of as made of mathematical procedures. The mathematical principle here is that the methods of all these can be reduced to four commandments. What we call the rule of three. The rule of three is not conceived of as rule of three, but is conceived in another way in ancient China. The rule of what one has to what one sees. The rule of what one has to the rule of what one sees. So we have a pair of, a couple of numbers that rule the end of one entity into the other. In fact, that you can multiply the two of them or divide the two of them by the same number, this will not affect what they represent, which is the relationship between the two quantities that can be exchanged one into the other.

50:00 In fact, this is the occurrence of the term blue, the concept of blue in the nine chapters. But it's going to be extended to designating any set of numbers that are defined in relationship to each other. Any set of numbers that can be all multiplied or all divided by the same number and still express the relationship of the quantities they refer to will be called U and the first occur in the rule of three. So the rule of three with these terms and it is prescribed to multiply open hands by the value of open hands and then one gets what one sees. Please remember these terms because they will recur later on in my talk. So this is this concept of the rule. Numbers that can be all multiplied or all divided by the same number without their meaning as a set to be affected. It is this concept and the fundamental operations that rely on such knowledge that are said to be a basis of mathematical knowledge. And, first of all, what is this book on the basis of which one can work in order to bring to life the source of mathematical knowledge? In fact, these nine chapters on mathematical procedures, the book on the basis of which Richard Kahn and the major researchers have been commenting on, this book, which was completed in the first century, Before the common era or after the common era, this book was believed to be a canon, and this implies that all along Chinese history, mathematicians went on commenting on it or went on referring to it.

52:30 Up until the beginning of the 14th century, nearly every mathematician in ancient China referred to the book, commented on it. So you see here a list of commentators. So let's define the statement that we've been reading is a commentator, and I shall also be mentioning another commentator of yours in this session. How in the book it is that it is supposed to contain, to give us this basis of mathematical combination, the 9 chapters is composed very simply of problems. Two of my students have an idea of finding a book to use. We have 246 problems. The first one is a problem where you're given the face and the height of a white man with a triangle and it's asked to find the hypotenuse, from which you have the procedure which amounts to isomeric theorem. But you have also other kinds of problems where you have similar situations, whether it's a good measure of force, why don't you poop, why don't you buy water, why don't you stick to the pizza, why don't you have a first course, why don't you poop, before the good measure and so on and so on. I would say abstract in the situation they deal with but provide the particular values of the magnitudes with which they are raising the question and the procedure of the human in abstract terms as well and they have other kinds of problems that are particular both in the sequence of problems and in the numerical values. There are also procedures that are given outside the context of any problem, such as the rules we just read, but in the canon itself there is no second order statement relating to mathematics and the organization of mathematical knowledge, so it will be the commentators that are going to make the statements of the kind I've been discussing. But this is the book, again, 2004, on the basis of which commentators believe that the basis for mathematical knowledge can be found, and in this sense, we can consider that they believe this book to be foundational in this sense.

55:00 Now, the question is, how did they read it to get to such a conclusion? How did they read this set of problems and algorithms to solve these problems? To get to the conclusion that it was deliberately a basis for mathematical knowledge. Before I dig into this question, I would like just very briefly that you, me, the third century commentator, shares with Lichtenstein the basic assumptions as to the architecture of mathematical knowledge. I'm going to read with you a very brief summary. He makes about the nine chapters of the canon in which he writes for his comments on the book, and he stressed only some points, so he said, as a child I studied the nine chapters when I was an adult and again looked at it in detail, I observed the divisibility in the book, which synthesized the stories of mathematical precision. So again we find this idea that there is a source of mathematical procedures, and please note that it is not a source of mathematical procedures that are in the literature, but it is the source of mathematical procedures, and his commentary is linked with finding this source of mathematical procedures. Let me simply stress what I underlined. The reason why, although they think of the algorithms, the reason why they divide into branches, they share the same standard, is that they emerge from only one of their hands. So there is again this picture that there is a standard to a medical knowledge. From these 10 mathematical procedures diverged, differentiated each other and his claim is that there is one of the anti-mathematics from which the stem emerged and this is the source that he wants to find out on the basis of his work.

57:30 I'm going to stress one more point and only one more point from his book. The main aim of his commentary, which is going to be quite important for us, he says, having spent much time to follow its depth, I managed to understand its meaning, its intention. So, the aim of the commentary captures this meaning of a set of problems and procedures. There is one very specific passage in the commentary where we have the square meaning intentionally occurred and where we are going to see how a given piece of commentary relates the fundamental ratios found in the statement about the architecture of mathematical knowledge at the start of the book. In fact, in his commentaries that are given on Sunday, the point is There seems to be a correlation, and this is my main point, there seems to be a correlation between the willingness of algorithms and the fact that they bring to light fundamental operations that lie at the basis of mathematical knowledge. So my focus will be on the relationship between the greatness of the algorithms as this is carried out in the commentaries.

1:00:00 And exhibiting also mathematical knowledge as fundamental operations in the list of which we read. In the French corner of my talk, because since I can use the two augmented vectors, I'm going to use the transparency I had in French because I would like to compare two pieces of commentary. So I'm going to translate them already into English, but I'm going to show the text in French. So this goes back to the problem I read at the beginning. A good walker walks 100 feet while a bad walker walks 60 feet. If not, the bad walker first walks 100 feet here before the good walker, which is the line below, starts pursuing him. In fact, he is going to show that this procedure amounts to And he is going to show this by making use of the set of terms that lie at the basis of the chemistry and by knowing what they are, what they have, what they do, what they can see, what they can't do, what they can't see. Now, he is going to interpret the fact that one subtracts 60 from 100 in terms of the situation. If you perform the first operation, which is described by a canon, you get the lieu of what the bar-Mocker first wrote.

1:02:30 This expression is extremely interesting because it gives the saturation of the couple of numbers what the bar-Mocker wrote. And 100 minus 60, which is 1 1⁄2, and the loop of 1 1⁄2, so it's the beginning of the statement of... mainly that this is 2, the 1 1⁄2 that the bad water goes, what is 2, is 2, and then the second 1 1⁄2... The term which is multiplied is interpreted as the lieu of, the person and the system which the good or the person enriches the bad or the good. So, in giving an example to the terms of the procedure, Brink's lines, in terms of the situation of the problem, the meaning of Brink's lines, It seems like the fact that these amounts need to play a role too, with the chemistry to be the next problem, which I'm not going to leave you with, I'm going to just talk about the problem now. The bell marker first starts with making 10 units, another measure, it's not mine, it's another measure for the length. And it's only at that point that the good walker pursues Hilbert, and then when he's asked to find out the length around which the good walker had walked at the moment when he joined the bad walker, he prescribes to add the delay to the distance that had been walked beyond the bad walker, to turn this into devising.

1:05:00 And to multiply the 10, the delay, by the 100, the distance that I will enter into the detailed description of the commentary and also into how he accounts for the duration of the procedure. The element of stress is doubt by accounting for what the procedure does, by accounting for the fact that the procedure amounts to using a rule of thumb. The commentator brings to light the fact that the two procedures, the one we just saw and this one, have the same intention, have the same meaning. At the level of accounting for the correctness of procedures, the two procedures that appear to be distinct, that appear to be two branches of mathematics, All understood are the same. The way in which he accounts for the correctness brings to light that it is the same reasoning that is used in physics. And this is what he comments by saying that the structure of this procedure is the same as in the previous procedure. We find here this word intention, reminded of the purpose of the commentary to bring to light the intention of the book. It's very interesting that if we compare both commentaries, we are able to say what is the same and what is intentional. And we come up, and this can be drawn as a conclusion from the commentary, we come up with the idea that this word intention can be understood as true. The intention of an operation, which this operation obtains, yields, interpreted in terms of the situation of the problem, semantics for the operations and the procedure.

1:07:30 But if you put together the intentions of all the operations to get the intention of the whole procedure, which is the reasoning that lies at the basis of it, and the reasoning that accounts for its correctness. This is why they are correct. So you see that the proofs are carried out and finding a level at which they are the same. They are the same here at the level of the reasoning, but they are the same also at a higher, more formal level, which is the fact that... Both instantiation of the general feature of the account for the correctness of the procedures. The feature that they will try to bring to light the fundamental operations that appear in the various procedures. This is where they are aware of the fact that the correctness of the procedure brings to light the same fundamental operations that are at play in all the procedures. So they are aware of the consequences of the procedure. The correctness of the procedure brings to mind the fundamental operations that are common to all the procedures in the camp. We find one other term which can be translated as mean for a procedure that is expressed by the fact that it involves a fundamental procedure.

1:10:00 So, the fundamental procedure which is found to be at the basis of a given procedure, and I use basis now in the sense of the statements that they start with, that constitute a stem for this procedure, if you wish, is written by a word for meaning which is different from the word intention that refers more specifically to The kind of reasoning we make to account for the correctness of our procedures. I sketched this on the basis of the Hoover stream, but all the operations that are in the statement of Hitchin's plan we started with, all appear as procedures that are fundamentally lying at the basis of... What does it mean that we can look at the canon, find out all the fundamental procedures that lie at the base of these procedures and make a statement about all mathematical procedures? It means that the commentators believe that in the variety of procedures they give, fundamental procedures are needed to build procedures. It contains all building blocks for making procedures and in fact this way of understanding why the procedures are correct, you get to the fundamental procedures that Lyme and the business of mathematics.

1:12:30 Bob Krugievich is one of the last editors of the current CELISA. We have thus a book called Methods and Procedures that only the nine chapters can be taken as the first. With the methods of its nine parts of mathematics, and now the expression of all methods, because it's not the methods that are used to create the procedure, there is nothing which is not complete. Although the procedures established by the various schools express a variation, you can find other procedures, but when we look for the original meaning, and this meaning is the same as the one I've been discussing, which refers to the meaning you get when you account for the regularity of the procedure, when one looks for the original meaning, they always come from the extremely complex of the procedures. All of these will always be found in the canon, even though we can check other procedures upon which we can analyze them, and analyzing the units with which they were built, we will always go back to the canon and find out what we find there, the building blocks or the fields. So, it was very brief and probably not explicit enough. A group of people to identify the basis as the set of operations with which we could produce all the small variables and variables. The communicators provide new knowledge about the inside of a method and they also apply new knowledge about the inside of a method. Thank you for reading this question because that allows me to make the point as you will.

1:15:00 In my understanding of the project, and I wrote about this in a target course, the projects are not statements to be solved. They are kinds of paradigms, and one aim of the commentary is to determine The extent to which one problem stands for a class of problems. There is evidence in the commentaries that through the commentaries they try to understand the extent to which the algorithm can be solved in a class of problems. This is the first thing. A problem stands for a problem. The examples I have discussed very briefly, they are a kind of situation within which you can interpret the operations, and this appears very clearly in one piece where the mathematician in order to develop a proof of the correctness of the algorithm changes the problem because the first problem does not allow him to develop interpretations for the operations that are needed not for the procedure but for the proof. So, the problems internally not only have statements to be solved, but have fields in which you can interpret what the operations are doing, which is quite important. Now, with this in mind, you understand that the committee must plan. Develop new methods. And in fact, there is a link between looking for the origin of the procedure, looking for the standard of the procedure, and developing new methods. There is a link between the two activities. I have no time to comment and try to expand on that. There are many examples of Lü in respect to each other. Then you can simplify the method by simplifying the Lü because the Lü can be divided by the same number without their meaning as a pair of numbers being privileged.

1:17:30 So the point is that we suggest that we can simplify the method and simplify the tool before planning it. On the basis of looking for the source of the methods in relation to other methods, because looking for the source is never for the source of one method, but for the common source of various procedures, that gives clarity for unifying procedures and in fact devising new ways of covering one procedure. He is also an advocate for something we should have said, but I haven't said it because I still didn't think about it. I want to stress the relevance of the history of algebra for this kind of work and operations. Because you are interested in the more fundamental operations that are applied in different contexts. There are other cases where we can show how procedures take on different meanings, but since from the form of saying, then they can be brought to life as one common form of procedure. So there is a relevance for the history of algebra for this kind of work. And it's not translated because I never found a good translation because it should designate one number and I've been thinking a lot and I couldn't find if you have the same perhaps range that it covers. For instance, coefficients in the linear equation are designated as mu because you can multiply or divide all the coefficients without damaging the equation in the linear equation.

1:20:00 They represent the diameter of the circle and the circumference, so they are also review because you are going to use, in this case, the approximate value meter to express them up to... So it turns up in the context, in the concept. But why do you have to integrate them? Because how would you integrate them when you are dealing with a linear equation? They have the same ratio, not the other way round, the ratio we use to simplify the question, but look, designate the set of numbers as a set. I mean, it's not the same structure. Question. Is there any common theme or common premise about the fact that the mathematics of the original class The second question is, is there any feedback on the text after a while, do you find, for instance, Mathematical effects were viscerally integrated inside the text or, for instance, a reorganization of the text, not with the program but with the operation, and with no downstream zones, from which it was automatically added. Okay, so the first question, again thank you because that allowed me to say something I should have said but I had nothing to say, namely that in this case, the foundation is given as a book, as a statement as such, so we do not have a set of statements in the book that say, here is the basis, and now we are going to see how it is differentiated into various pieces, no, the basis is in the sense that the branches are given.

1:22:30 In the commentator's world, stems and branches are given together and the commentator understands that they have to struggle to organize this writing into a shape. What is given in the book, in which foundations are understood to be given, is a forest. So this is one way in which foundations can be given or can be understood by the actors to be given. Now, you remember I mentioned a list of commentators up until the 13th century. In fact, the commentator of the 13th century adds the last chapter to the canon in which he reorganizes the matters on the basis of an organization of the operations. Well, in stressing the mathematical aspects of these books, would you go so far to say that we have in these nine chapters an example of mathematics as science, in terms of theoretical science, or is it still more the aspect of mathematics as art, in the sense of collection of problems and collection of paradigm solutions? It's wonderful that I get to do all my talk thanks to the questions. In fact, there was a quotation that I didn't mention but I did tell you to mention, which was a piece of the commentary where you explain why the problem in which we are asked for the equation is a right integral triangle.

1:25:00 When one has the data and the facts to find the why, it is even at the beginning of a chapter where one has all kinds of applications of why can we travel. We have 24 problems in chapter 9, in which, according to the analysis of the commentator, They amount to when you give any sum or any difference of the base, the height, the width, the total erosive. And if there are such things, you are asked to provide the dimension of the triangle. And the commentator always shows how this amounts to applying the Heidelberg theorem or something that derives from the Heidelberg theorem. At the beginning of the chapter, I can show you, they are about to be extended. This is a term that goes around its paradigm. I use paradigm for a problem, but the procedure is something that extends rather than is applied to. Thereabouts we extend it to all the algorithms in the chapter. This is why this procedure is set out first, so as to make the origin key. And here you have the clear use of the term origin to refer to a general procedure. That can be found lying at the basis of all the procedures that are dealing with one kind of time. So there is a very interesting connection in this statement between the origin, something which is designated as the origin of other procedures, and the connection between and the fact that it extends to the other situation. This is the conception of mathematical knowledge. We took the wisdom and we get something which is much better than other situations.

1:27:30 Thank you very much. You know, no, you insert one arrow into the, as one of the two things in the pair.

1:32:30 So the con-extension to that, so that's obviously a trivial operation. Now, you take the con-extension to that, it'll be composition. Or you insert the arrow in the other place and take the other composition, you get the same, it's adjoining in two ways. You're going from C to C, the arrow category of C, C to C2, is that the idea? Yeah, to C. C to C2, and then you're coming back on both sides. Another beautiful fact is that in the category of... I'll show you that, don't you? You could say, well, I would actually do it to a rather different,

1:52:30 because then you'd want to be able to also plot it. Algebraically reflect that, so you're not going to...