Reasoning with Infinite Diagrams

0:00 ...a number of times or possibly coming to a conclusion. And so already there you can illustrate the idea of the difference between the substantial infinite and the completed infinite. And so I don't have any illustrations of that, but that would certainly be something that should be included. Now, here's, um, some of these simpler things come from a book called Proves Without Words, edited by a man named Roger Nelson, and it's published by the Mathematical Association of America. this diagram on the book we'll come back to in a moment what's a little strange what he did is collect diagrams proofs based on diagrams that have been suggested by a number of people so it's not that he is the author of these particular diagrams and in his aren't really proofs. To have proofs, you have to have a rigorous demonstration. So just contrary to what... And the other is that he calls these proofs without words, but in fact, you cannot, as I would still argue, have the proof without words, except in extraordinarily rare circumstances. So here we have, again, the arithmetic regression of some of the first n integers is n times n plus 1 over 2, but now represented as n squared over 2 plus n over 2. So n, you take the square, It's area is n squared. Everything under the diagonal is n squared over 2. And you have any of these half squares, so it's n over 2. And the sum of these n things is, as you see, n squared over 2 plus n over 2.

2:30 This is the odd integers. Some of the odd integers, 1 plus 3 plus 5 and so on, equals n squared. Well, that's pretty clear. More interesting, um, some of the squares from one squared, two squared, up to n squared is a complicated formula. So you take, uh, let's get the cursor out of the way here. There you have n squared at the base, where you such 1 squared, 2 squared, 3 squared, n squared, right? Now you take three copies of that, and you fit them together. So when you fit them together, this one goes here, this one goes here, and this one goes here. Now you have this sitting on top, n of these, right? You cut them in half and lay it over. And what do you get? First of all, when you pick them together, this n has to go to n plus 1 on this side, n on this side, and you get n plus 1 half on this side. So, that's this, but the original thing is just one-third of them. Right? Do you all agree? Have I convinced you that the sum of the squares is equal to... Of course that's not a proof, but it's totally convincing. I'm not sure. I'm not sure it's not a proof. You're not sure it's not a proof? Now, I thought to do a psychological experiment here, an empirical experiment, asking for each one of these, how many of you think this is a proof? Just one? How many people understand the question? So, let's take this example. How many of you think that what I just explained is a proof?

5:00 I think it's a non-trivial part. Exactly. I think it's a non-trivial part, the fact that those three people actually couldn't get. Yeah, well of course, I'm not claiming it's trivial, but... You have to say something to convince you to go together. Of course, you're not a call, that's true. Well, that's just the issue. Here's a particular geometric theory, one half plus a half of a half plus a half of that. but you add those all together, you get one. Why? Because as you cut successively by half, no point can escape. Yeah, that's it next. 1 plus r plus r squared and so on. Here's how this person did it. You lay off on the line, first 1 in R, then you do a vertical R, and you construct a triangle like this. And so you see then that what's not important here is that these successive things are R squared and so on. But the base of the triangle is the sum, 1 plus R plus R squared and so on. Now you take this other triangle over here, which is similar to this one over here, and you say, well, this sum is to 1 as 1 is to, of course, the difference, 1 minus r. Quite different for geometric series, I think, again, you have to say something. I mean, it's not that you're not, but it's not a formal thing in any sense.

7:30 We could do more geometric series. These are some examples. But, um, yeah, look at this. Um, one-half, uh, one-fourth was one-fourth squared plus one-fourth two, and so on. So, one-half squared, that's one-third of the whole thing. This is one-third of this, this is one-third of this, and so on. Okay, here we have an infinite procedure, the formation of the Cantracet. Cantracet is supposed to be a set which has many interesting properties, but one of them is that it's uncountable, but it has measures zero. and described this way. You omit the middle third here, and then from each of those you omit the middle third, and from each of those you omit the middle third, and so on. And you make a calculation. So it's not that it's totally diagrammatic here. You make a calculation. What's the length of the segments that have been omitted? So, one-third plus two-nines plus four-twenty-seven, and so on. And factoring out one-third, you see that that's the sum of two-thirds to the eye, and you'd use geometric theory results, and that's the way to balance this one. So there's a little argument there, but without looking at this, you wouldn't really see, it seems to me, that the measure of the set described in this way is clearly one, or that the measure of the set points that's left over is clearly zero. And, of course, you have to make an argument about uncountability, a usual set-theoretic argument.

10:00 Schierpinski had an interesting higher dimensional version for this. So you leave out the middle square and then in each subsquare over here you leave out the middle square and then each subsquare and so on. and you consider that extended to infinity. You can imagine what that's going to lead to. I'm not stating any results about that. One can use that. This is what's called a snowflake. It's an example of a bounded curve of infinite length. How do you get it? triangle, on each edge you build an equilateral triangle on the middle third. And you keep doing that. So each time you add to one of the sides that little equilateral triangle all the way around, you're multiplying the length of the figure by 4 thirds. The length of the figure out here is 4 thirds on the previous one. And so it then goes to infinity, It's a continuous curve. Now, if you want to be rigorous, you have to say, well, what does it mean to be a continuous curve? You have to use the official definition of continuity. But in some intuitive sense of the word, the limit is a continuous curve, and it's clearly bounded. This I took from Lakato's book, Proofs and Recutations. So this is the first proof we looked at. And the rest of the book was designed to show this. To start with saying, this is no good as a proof, or there are problems about this, and we're going to go on

12:30 and ask various questions, what's the problem? But in fact, it is a proof. What's the proof? So what's to be shown is that for a convex polyhedron T, the number of vertices minus the number of edges plus the number of faces equal to two. The steps are as follows. You remove one face from the polyhedron, and then you spread the thing out on the plane, and it doesn't matter that these are exactly straight lines. The essential thing is that you can flatten it out topologically, and we'll call each of these, of course, these vertices as before, you can call each of these an edge, and you can call such things a face. And the second step is to cut each face into triangles. You triangulate, and the simplest way is to start with one vertex, doesn't matter which one, and head in a certain direction, cut it into triangles. So now we have a triangulated poly—well, it was a convex polyhedron with one face removed, so left being triangulated. And now we remove triangles with at least one edge, an outer edge, one at a time. So here's this whole thing, and we're concentrating on one of these triangles here. And in the case that our picture looks like this, we removed one edge, but we've also removed the face. So, we haven't changed D minus D plus F in this case. In this case, we have removed two edges and one face, so that stays the same. I should have, uh, there's still another possibility we should have considered.

15:00 We have a triangle sitting out here like this. Now, what do we do? We remove the triangle, so... We remove two vertices, three edges, and a face. two vertices, three edges, and one face. That's the removed part, and so it hasn't changed. So, in the end, we have to end up with one triangle. And for a single triangle, the number of vertices is 3, the number of edges is 3, the number of faces is 1, it's 1. So we wanted to get it equal to 2. What happened was, at the beginning, we removed one face. Is that a convincing proof? No, you're right. You need to show them to stay connected every time you take away something. Otherwise, you might have two pieces. Yeah, so we'll be getting into... We can... Just as in completely rigorous verbal proofs, we can, or what we try to have as completely rigorous verbal proof or symbolic proof, we can overlook something. And we may have the same phenomenon going on in this kind of truth and saying, oh, look here, you didn't take care of it. When I drew this picture over here, included that I had overlooked it, but I realized that I had overlooked it, and such things will come up. But, I should hope that this is pretty convincing. Yes, I like it, sir. Hmm? It's very nice, sir. What about you, sir? The Cauchy, sir. Here's the Cantor-Bernstein theorem. So, it has to do with the relation of

17:30 A is equivalent to a subset of B in a one-one way. there's just a mapping F, 1, 1, A into B, and A is equinumerous with B if there's a mapping 1, 1 onto B. And the theorem is that if A is equinumerous with a subset of B and B is equinumerous with a subset of A, then they are in 1, 1 part of the formula. So, how does this go? We look at a given F, which maps A, 1, 1 into B, and a G, which has B11 into A. So we think of F going this way and G going this way with A15. Now, we look at A0 as what G doesn't capture. And then we looked at B0 as what F doesn't count. So A0 is A minus B, B0 is B minus F. And then we started taking images. Take the image A0 under F, called FD1. Take the image of E0 under G, called A1, and we continue in that way. So A1 is G of B0, B1 is F of A0, A2 is G of B1, B2 F of A1, dot, dot, dot, et cetera. So what do we have? The even A's are equivalent to the odd B's, and the odd A's are equivalent to the previous even B's. So, we have a 1-1 correspondence then between the union of all these A's and the union of all these B's. Um, now, there might be things that aren't caught in this. One way to see that, uh, that's the kind of thing that might have been overlooked, So one way to see that is, suppose A is an uncountable set and D is an uncountable set,

20:00 and that A0 is countable and D0 is countable. Then, of course, all of these will be countable. We'll have a countable union of countable sets, and what's left over is uncountable. So there have to be things left over. But how do we see that what's left over is, in one one correspondence, already by F. Well, everything in here is sent by F into something in here because otherwise it would end up in one of these things over here, but the things that are end up come from one of the AIs. And by a similar argument, so we have F mapping this one one in here, And if anything were reasonable, does everything in here get caused by something in here? If it weren't, it would have to be caused by something in there. But then it would not be in here, but it would be up there. Now that's an actual infinite diagram, right? but with a finite representation. This is a little more technical. So I need some more explanation. Now we're getting into pictures with certain diagrammatic aspects, what's called the Montague Roth theorem in set theory. But I think this is the kind of thing that can be explained with just a little background understanding of set theory. So the language of set theory just has two basic symbols, equality and membership. And we use letters 5, psi, and so on for formulas of that language. And the picture, what's called the cumulative hierarchy, which starts with the empty set and is the iteration of adjoining the power set of each stage. So the power set is the set of all subsets of a given set.

22:30 And so we go through what are called all the ordinals, which we think of as going up this way. And at the successor stage, V-alpha plus 1 is V-alpha union, the set of all subsets are V-alpha. And at limit stages, we just take the union of everything we have before. And what we think of as the universe of set theory is simply the union of, in the cumulative hierarchy, It's simply the union of all the V alphas. And any X which lies in the cumulative hierarchy lies at the lowest stage, so it's the least alpha, but the X belongs to V alpha. It's called the rank of X. Now, if you have a subclass of V, I write epsilon M for the membership relation restricted to M. And if we have two subclubs of M and M prime, we say that M is an elementary substructure of M prime. If tested at an arbitrary formula in our language over here, at arbitrary elements of M, And this is satisfied in M, just in case it's satisfied in M prime. So M prime can't tell anything different about formulas, or any formulas, with parameters in M. And therefore, if you have a formula with no free variables, these will satisfy exactly the same first-order senses in the language of physics. Now, the aim of the Montague-Waft theorem is that there's going to be an ordinal stage, alpha, which the cumulative hierarchy of the alpha is an elementary substructure of the the entire universe. In other words, that you can't tell the difference. You can't, by first order logic means, have something that can only be satisfied by the universe itself. If it's satisfied at all, it's going to be satisfied at an ordinal stage.

25:00 And if you think of that theoretical picture of having something to do with truth about sets, anything that all the truths about the universe of all sets is already contained, all the first-order truths, is already obtained. So how do you prove that? So we need a lemma here, which is a little technical and not given by the diagram, but once we have the lemma, then we can use the diagram to vary out the truth. So if we want to show that M is an elementary substructure of an M prime, What we do is, it's sufficient to show that for every formula of psi with a bunch of free variables x and a single free variable y, and any element A of m, that if there is some solution in n prime of that formula, at parameters in M, then there's already a solution in M of that same thing. But now, notice that in both cases, we're testing at M prime. Thank you. So, this has what's called the scollum function picture. If we have a solution in M prime, we try to find a solution in M. So, to carry out our proof, we introduce what are called the scollum functions. So the squalm function associated with a given formula of psi, a parameter of A, is we have the accident of choice. We could say, pick a Y such as the universe satisfies that existential statement associated with psi.

27:30 There is just a Y, psi AY. If we don't have the accident choice, we simply take all the solutions, possible solutions y, of least rank. So it says, node z of smaller rank has size rank. And so that's what the strong function does. So this is our original picture. We have the stimulative priority, the stages, we go from V alpha to V alpha plus 1 by taking a set of all subsets to V alpha and adjoining that. But now, what we do is we're going to build up the alpha, which satisfies this theorem, in stages, alpha 0, alpha 1, alpha. So, suppose we're at stage alpha n, we take what's called the Stoll and Hall of that by taking, for each a belong to d alpha n, and each formula, psi xy, and each parameter that we're substituting for x in d alpha n, the value of these following functions, which give solutions towards the side of any possible existential statements that's realized in the universe. You take those all together, and now, by a little argument set there, you can, using a virtual replacement, And show that, well, there has to be a larger ordinal which contains all of those, B alpha n plus 1. And so if we know what alpha n is, we simply take alpha n plus 1 to be the least ordinal which contains this whole and whole of the alpha n. And now we take alpha n plus 1 as the soup of all those, uh, soup of, if you will, of, uh, sorry, we take alpha to be the soup of the alpha n, and that has the property required by the lemma that if we're testing, because we have closure on your phone functions at each stage in the next stage, uh, we're going to have this

30:00 property that if there's any B at all in the universe that satisfies a given sign, then there's going to be a B below level, B alpha, and that will show that B alpha is an elementary subscript. Now, here's an example of a proof that can be made completely rigorous. Don't leave the diagram for the picture. But if I'm teaching this in a class, I won't teach this truth in just this way. Because to do it without this picture, you don't really see what's going on. So this is kind of a borderline case where you would say, well, So you really see what's going on by having the picture before you, but it's not a test case for the question, do you have a proof that can't be formalized by no means, nor Or was the Kantor-Gernstein Theorem such a test case that also can be given in a completely formal way. These are not test cases, but I think I would argue that in both these cases, the understanding of what's going on makes a central use of the diagram. So that's just examples of the kind of data that I had in mind at the beginning. Um, and it would have been interesting to, um, give examples from computer science flow diagrams to show the way that the computer program works, um, exactly reflects if you If you have the right flow diagram, exactly shows that it's going to do what you want it to do.

32:30 And without the diagram, just writing down a series of lines of what the program is would not give that at all. You really have to see somewhere around it. So this is, again, I mentioned that infinite diagrams are very often used in model theory. This is from the book by Walter Haag is a shorter model theory. It's just a very difficult diagram. You build up extensions of models in a sequence of stages, And at the limit, you have two unions of models which are related to each other in a certain way. This is a proof of what's called the Robinson consistency theorem, which is that you have two overlapping languages, and you have models of each, and that the properties on the common language are consistent, then when you take the two languages together, you can get a model, which is a common model of roughly the idea. Here's the kind of diagram you see all over the place in what's called homological algebra, and I can't begin to go into an explanation of what's involved here, you see that this is a diagram which is dotted at both ends. There it's dotted at both ends. And each of these things is a rather complicated object and what it means to be an exact sequence, and what it means to be an exact sequence of homology is all rigorously explained. So, in principle, you wouldn't need these diagrams at all to verify that if this is an exact sequence of complexes, then the associated sequence of homology objects

35:00 is an exact sequence that's suitable for the work. But again, I would say that the experience is that if you didn't have these diagrams to refer to, you really wouldn't understand how the proof goes. ...experts in the test of them. So, um... I just want to read that you've given a certain amount of diagram what? Down there? Yes, this is an exact sequence of... No, no, no, the last time before the last one. A certain amount of diagram what? Chasing. Diagram Chasing, so what's common in this kind of argument and in category theory, commonatural qualities, and so on, diagram chasing is that two different ways of surviving That's the same thing. It has to be shown to be equal. So here you're going to chase this going this way, and another way you're going to chase it is going down this way. And this arrow over here is part of this argument in here, and it's going to be part of the argument as to... So to be what's called commutative means that all possible ways point to another point in the diagram are going to be equal. And showing that a diagram is commuted means chasing around various of those paths. And you look at typical paths So, here are thousands and thousands and thousands of proofs that look like this. can't use that to illustrate the use of infinite diagrams

37:30 without substantial technical background. Nevertheless, I think these simpler diagrams that I did as illustrations of the idea, I hope, were convincing to an extent that the use of these diagrams And by themselves, with side arguments, of course, give you convincing proof, and that Translating that proof into formal proof may not, in fact, may not be too easy or a weaker statement. To understand what's going on, the diagram is indispensable. End of story. I was very struck by this column picture. I was very struck by this coronal coronation. First of all, would you agree to compare this? If I reformed what you said, it is a kind of mass. It is a mass type which you can orient yourself in the text of the truth. So you see the diagram that has a completely different function, which compared to the diagram we saw previously, it plays a part in the way in which Sharlene used the text of the person. And this is why I would suggest the metaphor of the map. The map, yeah. It's a map for the sex of the food. Yeah, yeah, definitely.

40:00 So, it's not a diagram on which you're going to carry out operations, the result of which which I'm going to try to go further the proof. It's a map with respect to which you can do some operations on the text of the proof. You can see your corrections. Thank you. Right. I think you have to be able to tell what you mean by a map of a proof. So, crucial step here. What is the meaning of this? You know, if I just show you these diagrams, these three diagrams. First, cumulative hierarchy. Second, exposure on the swollen function. Third, take the limit. It doesn't have everything, unless you already have a clear idea of what colon functions are, how they're to be defined, and... What I mean is that the function of this diagram makes you really completely different from the function of, say, the diagram in which you started, when you started with semi-germatric series or whatever, it plays a different part. Yeah, it's closer to the, yeah, and as I said, this is an example of the Cantor-Bernstein theory, an example of something where you don't need a diagram at all to have an official proof. So here the diagram serves more the function of really helping you see why this thing works. You know, I have a very strange question. Do you have an idea when you can identify such a diagram in Netflix and literature? This seems to me to be very modern diagram. Very what? Modern diagram. Which one? This kind of diagram, I cannot think of, for instance, in Arabic.

42:30 Oh, thank goodness! I'm just wondering, this is a new kind of diagram with respect to Euclidean territory or whatever, and I was just wondering when and why did such diagram appear? In which kind of diagram? A diagram, a diagram that is a picture on which you can rely to follow a proof. The illustration of the proof. The problem is to define a function. You feel that the function of this diagram is not the same as the function of the diagram when you want to sum up a theoretical theory. when did this kind of diagram appear? Well, I think 19th century somewhere the beginnings of the combinatorial mastery of the influence. Okay, it's just a looking question. It was a very similar question, because it was a very similar question. I'm not sure if I got it right, but when you talked about the other column diagram, you said that this time it was a picture and not a diagram? So I was, the other one, the one which is this one. I don't know if I got it right. So I wanted to ask you as a clarification, if you were making a difference between pictures than diagram then if it goes in the direction that's telling me the question you know people talk about venn diagrams are diagrams but you can also say well they're pictures and so where do diagrams leave off and pictures begin i don't know i don't have a few I've given a nice example of a flow-drive diagram, an example of computer programming. But it seems to me that a flow-drive diagram in itself is a conceptual sense of computer programming,

45:00 because, for example, an instruction I emit generates is two-dimensional in itself. So we write a problem in a series of lines just for the convenience of writing, But there's a real sense of programming, just a picture, no charge. Just a picture? Just a picture. Just a picture, just a picture, no charge. No, but it's a picture that you used to indicate how the program is going to proceed through time. So it's the picture which allows you to grasp the dynamism of the program. And also, recursive aspects of... Of course, but I am saying that the problem... It is a problem itself that it relies on some kind of simplification, because an if-then-else in itself has a two direction. So when you write an if-then-else as a single line, you have made some sort of simplification, which, consequently, is made up of a little simplification program. So, I'm sorry, I don't really get the point. I think that if you write down like a problem, you are naturally conducted to write in the flow chart. In certain sense, it is a natural project of the problem itself. But the experience, I think, is that if, yes, if you have the program before you, you can construct the flow diagram. But you won't know what the program ought to be until you already have a picture of what the flow of computations is supposed to be. And somehow or other, you have to have that faux diagram, if not directly on the page in the back of your mind, before you put down those lines. So going from the lines to the diagram clearly shifts. And also, arguments about termination of programs. Again, if you have the lines in the program,

47:30 it's very hard to carry out a formal proof that, in fact, it's going to terminate, and that when it terminates, it's going to compute what you want it to compute. But having a flow diagram is an enormous help and allowing you to see that there are different ways of representing computer programs, but that's one way that's particularly important. When you were discussing that Kendall's Bernstein in the month of New War... I'm sorry, I'm tired of hearing... I'm sorry. When you were discussing the Cantor-Bendick, Cantor-Bernstein, Cantor-Bendickson is another yeah, yeah, and the Montague book, you know, you said that the pictures are dispensable, But you also said that the proof could be formalized. And I wonder whether you're really complaining to things here because it seems to be at least possible that you can have purely formal proofs where the syntactic objects really are pictures or diagrams. In fact, John is in the room here to use the formal system with diagrams of a portion of Euclidean geometry. So it may be that you can have a formal system where the diagrams are essential. So it means that you can't equate formalizable with the diagram of inocentious. I'm not sure I have this comment clearly in my opinion. You can have a formal system in which diagrams are sanctioned. There is no position between the fact that the diagram is sanctioned and the fact that the system is formal. You can have formal systems that include sanction in the use of diagrams.

50:00 And the example for the joint system that is a representation of the Euterium John, the formalization of the Euterium John using diagrams. Yes, but that's a different kind of formal system. It's a formal system that we think of as a formal system about diagrams. But it's not an informal use of diagram for personal conviction as the truth. John's system is a marvelous system, but it's not a formal system. It's just another kind of formal system. Just as there are formal systems for DEM diagrams or more complicated first diagrams, traditional diagrams, very similar. But that's just changing what the formal system is the formal system of. It's not addressing what I take to be essential here, and that is, what is it about these kinds of diagrams that allows us, at least to gain understanding of what's going on and even more together to be kind of convinced of the truth of something without going through one of proof. Does that make that a difference? Yes, let me do quite a bit of my question. I think that I grasp your point, but first, what is a formal system in the graphic of science, in the sense in which a general system is not a formal system? What do you mean of which formal system is a system where you only advance by substitutions of the finite number of symbols composing, in a certain way, formula with other, where the only compare to the rule of constitution. Is that what you call a formal system? What's a formal system? Formal system is a system in which you have a specified language, which is on the screen, something called

52:30 You may think of it as a formal system of Euclidean diagrams, you may think of it as being without Euclidean diagrams. That's our interpretation of what that formal system is. Can I suggest making it clear that I can imagine that in John's formal system about diagrams, he draws diagrams to do something on these formal systems. And he can have an informal use of diagrams when he's working with these formal systems. I should hope he does. That's the difference, I think. I think that this is the point you're asking. Well, I mean, we could go into the specific definition of what diagrams are in the system. And you could tell me if they are truly diagrammatic or not. I mean, the intention was if they're actually diagrams. But they, I mean, as formal objects, they're discrete arrays in which you pick out elements. So you could just list it as a, it's all discrete, so you could code it up into arithmetic. I mean, there's a structure to it that's supposed to give you what a diagram is like, a physical diagram. What I was hearing you say, though, is that because it's all discrete, it's really just a formal object. And it's not, you can't really equate it with, it's an interpretation to think of it as a, I mean, that's the point, I guess, what you're saying. Well, maybe it's not a direct interpretation. What I take it as an issue there, I suppose there are several things, but the basic thing of the issue is when you read Euclid, or modern exhibitions, you're worried about. You've just shown one particular example by a figure of an infinity of possible figures that could be. And so the question as I understood it was, how can you treat diagrams in a formal way so that you would know that, in fact, there's no specialization

55:00 after actual figure involved, but that you have a truly representative or representational a representation of all possible configurations that would fit to the given data of a given graph. And I take that as being a, I think it's been a very difficult thing to do. But Take this guy here. This is somehow typical, in a different way, of a typical illustration of that arithmetical identity. But the typicality of it is not of the same kind as the typicality that we're after when we're talking about pre-terrified equivalence. And so what is it about this kind of thing that allows us to say, on the basis of n equals 4, calling it n instead of 4, that we see that it's a proof of the sum of the squares up to n. And I think it would be marvelous if somebody could set of the system, which shows that what it means to be a typical diagram in that sort, and that that, on the basis of that, it gives us proof. But my guess, and that's, although these are so different from the ones that I ended up with, the truly infinite diagram, But the curious thing here is that n is not arbitrary here in this diagram. n is actually four. And yet we see the whole thing.

57:30 If there are specific more points on this, I'll let you have a go. I think there are a number of people who want to say something on this specific point, but then there is Marco's question, not Marco's, but Marco's question, that's still hanging in the air and then I have Robin so no Michael and Robin so please if you have a specific point on this make it very quickly and we'll then proceed otherwise we won't be able to get to the end of it. So, Murphy and I just want to say something specifically. Okay, specifically on this very question, there is actually work by a computer scientist in Cambridge called Matea Yamnich, a computer programmer, and these generic diagrams. They're just talking about diagrams for proving theorems like this, 501. And her idea is that these diagrams enable us to extract a schema which we can apply uniformly to all and whatsoever. What's that step of abstraction? Well, I mean, you have to read the book. I can't say, but it seems very important. even psychologically that this is what's going on when we're convinced by seeing the specific number and we draw the general conclusion. It's something general that we... I'd be very interested. So, I think, you know, we asked, Marco asked the professor to hang where does the higher mathematics come from? Starting with this is too left to make for questions of answers. And, you know, I think the first talk we heard today is is aiming to address some of these issues here.

1:00:00 And so I'll be interested to find out. But, so to speak, the actual cognitive sign, at some level, of what it is we're doing when we say, oh, I've understood this, So, oh, I see why it's not true, and yet, formally, doesn't do anything like that. In this case, why do I have no problem with the infinity? Why do I have no problem? Well, there are infinitely many cases to consider, but each of them is a finite. That infinity is given. N is given. For you, in this case, this is my question. You need the N. I need N, yes. The potential N. And some properties have N. Properties. Properties. Properties, yes. Oh, so earlier I commented on the various thoughts that I have to, I can't state a theorem without understanding the concepts involved. A statement of a theorem cannot be diagrammatic by itself. And so I have to understand that I'm talking about positive images, I have to have a conception of positive images, have the conception of addition, multiplication, iterated addition, so on. Truth within within. And true for every end, yes. All that has to be understood that that's what we're after. And then, only then do we come to the diagram. It's just the difference between the actual and the potential. Yes, yes. So, do the question. In fact, the question is not so far from this one. There is a very frequent discussion between historians,

1:02:30 of mathematics, of modern age, concerning the use of induction in mathematics. And there are different topics that argue for a position that I do not agree with. That is the fact that when you use paradigmatic proof, for example, in order to speak about n, you speak about time, this inductive proof. And I think to me that it's not true that in fact what 18th century and 17th century mathematicians are, they simply use the specific number, the state of numbers, as a substitution of what we use in place of certain symbols and or those. So, I don't think, in fact, that the use of a paradigmatic example in order to make an infinite proof make the proof not satisfactorily decode particle-wise. It is, in fact, a general proof used made with a notation that is not our notation. It isn't a paradigmatic notation. So, here you are accepted by this. You do not accept the idea of my argument. But it is acceptable here that paradigmatic proofs are not human national, just correct proofs. Then why it is not a proof? Because, in fact, you can continue the diagram that you want, and you see in the diagram that it is a paradigmatic one. And of course it's any sport, but it's not the fact that it is sport has absolutely nothing to do in your argument. We didn't appeal to induction at all. We did not appeal to induction at all. We didn't say, oh, I see how I did this step to go on to the next step. It's my fault. You are not saying if n is 1, then it's 2, then it's 2, then it's 3, then it's 2, etc. You say n is anyone. So it works for anyone. It's a parity. n is a parity. The fact that it's four is not a sanction. How do we see that? How do we see that? That n equals four is not a sanction. What is it about our understanding of the situation that allows us to think of this as n where in fact n is four?

1:05:00 Yes. What is it about our way of thinking about this that makes it generic or pragmatic? I don't have an answer to that, but I think... I want an answer. I see the remark that 2,000 tiers of mathematics is not a good way. So if you say that this way of doing it is not a good way to make it a magic, you say that in a sense that 2,000 theorem of mathematics is not a magic. So that 2,000 theorem of mathematics is done in a sense that we know that we can, that n is not a sanction. We've seen that we were completing two things. Okay. One is taking n into the control, and the second thing is induction in g. Exactly. I say this is completely different. So it's not induction. My point is not induction. So it's not induction. It's a defect. It has to prove. So I don't see why it has to prove. I don't think that the point... I don't think that the point... Sorry? Paul never said it was an agreement. Paul decided. No, no, no. The point about whether it's a proof or not is not because there is N instead of 4. I think the point, whether it's difficult or not, and whether there's nothing new to any audience, even what I'm told. So what I'm saying is, and I don't think it's just me, that we can go into our classroom and show this to prove something to our students. Thank you. Okay, my thoughts, especially fascinating was the end of your talk about the actually infinite diagrams in set theory and then in homology theory and all of that. And I just want to kind of make a historical remark. Now, when Russell emphasized what was so good about modern logic, Often, he was opposing the idea that mathematical reasoning would have to rely on spatial intuition. And instead, if you have a formal proof using logical symbol from cell lines,

1:07:30 you can do it without spatial intuition, and the background is common. Now, what you seem to be suggesting is, well, there's a limitation to that. Even when we go to the most sophisticated use of modern, the kind of reasoning associated with modern logic and modern mathematics and set theories, because I think if the pictures and the diagrams, what they introduce is some kind of spatial intuition within which to embed, let's just call it, aspects of the truth in a very vague way. And what you're saying is, and then those dot dots that go off to the two sides, I mean, we're using kind of the fact that space, we think of space as somehow infinite, and we put something in space, we realize we can go on, and we can also get smaller and smaller, and the infinite will be visible, and so those dots are in a two-dimensional space. And we have the ideas of the infinite infinity of two-dimensional space that are guiding us in this, you know, even more sophisticated in Timothy, like, of the high record test. Yeah. So what you seem to be saying is that, I mean, Russell's view, there are limits to what Russell has said. Even embracing, you know, the idea of a formal proof that's made possible by modern logic, at the very least, there are proofs that you get in that tradition of modern formal logic such that, well, at the very least, can't really understand what they mean and how they work without embedding aspects of the truth into the special intuition. So I think it's interesting historically. Robin, without questions. So what I thought was interesting about your talk was the entire spectrum of proofs, the proofs without words at the beginning, were almost entirely done by the And on the other hand, you have proofs that are done entirely with words where the diagram helps. So I see a whole spectrum. Right. And the question to ask then is how much does one have to say about the diagram, add to a diagram to make a proof? And I think that goes for the one you had there about the three cubes, the N-cube thing. You know, at first sight you say, oh, of course, it's obvious. Do you stop thinking, wait a minute, how do I know these three blocks actually fit together if you don't compolite them? Yeah, right.

1:10:00 with your food from the oil it's one more. You say, oh, that's wonderful. You say, well, wait a minute. I have to make sure it's connected, otherwise I won't get the right answer. So to each diagram, I would suggest the question, how much do we have to ask or add to that diagram to make it real proof? Good question. Okay, great. It's always nice to end with a good question. Thank you. I would like to take advantage of what someone said a month ago. There will be, in February or March, a new issue of the French periodical La Revue de a very good theoretical and big papers in it one by me on diagrams so I mentioned this the title is not so much evident because it is in French and the title is there is not the flesh this is a Western vibration right here if you started with random yellow and so this paper is a quite big paper with many many kind of diagrams presented including those seven mentioned flow diagrams not only in computer sense but also in the analog computation team where they played a very important role So, you can have it in Dumont. And if you want to have it sooner, you just send me your mail and I will send you. This is my own mail. Not very good. That's it. Thank you very much. Thank you.

1:12:30 Thank you. Thank you.