Visualisation in Mathematics — Roundtable

0:00 Continuous fraction can be noted like this, and if we start to calculate the continuous fraction of E, we come to the following conclusion. We come to the following 2-1, 2-1-1, 4-1-1, 6-1-1, 8-1-1, 10-1-1, 12-1-1, 14-1-1. Who is not convinced, from this point of view, of the following 3-7-1-1, 6-1-1, 14-1-1. Who is not convinced, who does not have the aptitude. If we could deduce that, indeed, the sequence of the development in a continuous fashion of e is given by the singularity that we have in the equation, it would be great because we would have a demonstration that e is a rational number since the rational numbers have a continuous development in a continuous fashion and even a little more, we would immediately have the proof that the number e cannot be written in the form p plus the root of r. p and r are rational numbers since rational numbers have continuous development that are permanent. Well, today, no mathematician claims to have the only view of the regularity of operations, but no one does. Yeah, that was a fun paper to write by now. I'd like to thank you for this opportunity. Oh, I was going to say, there's also a very interesting place back in the 70s, probably from music somewhere, where he quotes somebody else's musical of being confused for a musical idea, and then he goes on to get along with it, and then sort of re-informs it. Sure. That's, I think, the only difference between the two of them. Yeah. Yeah. Yeah, I think that's what you're looking at, you know, but then, oh, yes, as I recall, no, that's because you're saying every, oh, no, no, I don't know, it's where you're saying, but then you put in, you know, it's possible that even the visions of the world are possible, and, oh, yeah, but it's possible, it's possible, it's possible, it's possible, it's possible, it's possible, it's possible.

2:30 But I now actually have a much deeper understanding of how trigonometry is mathematically learned than I did, so I might just finish the book and go on to the next chapter. Thank you for your attention. I'm going to go back and forth.

5:00 I'll start with a little story that everyone knows, but it's for me a paradigm that I use all the time in this kind of discussion, which is a reference to a legend, controversial or unclear, namely the fact that little Gauss at the age of seven gave a remark about algebra, the question of its master who He computed the value of the addition of the first 100 numbers, and probably, the legend says, he gave a general proof concerning the sum of the first n digits. We all know how it is. He just put in order the first n digits and had the extraordinary conceptual courage to avert the problem. Which is really the step of the genius in order to obtain, as we all know, a very familiar problem.

7:30 This ordering that he was using is something which I would refer as a basic construction principle, the ordering of numbers, which steps in in this proof. Of course, we all know that we can give a formal proof by induction of the given program. Of course, you need to have the formula, so let's rethink a second what it may mean in mathematics to prove a theorem. I believe that in a discussion about proof, we first have to say that proving a theorem in mathematics is answering a question. The analysis of proof as only proving a normal thing in a given formula is a rather limited one. Of course, we do care and need to carry on such a kind of analysis, but it's impossible, let's say. It is important to distinguish as heuristic and non-interesting heuristic or as an interesting heuristic from the fact of the proof for several reasons. First of all, in a case like that, you see that the answer to the question here requires a reference to a construction which is simple but indeed is very complex. With the notion of ordering of numbers, which is rooted in a variety of practical and conceptual experience, and in this particular case, the idea of reversing it, which is the extraordinary idea of genius, which ends in the proof to give you the formula and the induction load. In the case of whatever important theorem you have, and you need for that theorem a proof by induction, very rarely... You really use that formula to prove the induction. It is a common experience that in order to perform a proof by induction, even of a given formula, you need to choose another formula, which is called a conduction load, which very often is much stronger. It may be a very heavy induction load. That implies the given formula, and on that induction load, in that extra formula, you perform the inductive step. So along the proof of even an already given statement, even an already given inductive formula, you make the use, the intrinsic use, of a choice of an induction load which is much heavier than the given.

10:00 Indeed, I learned the relevance of these facts exactly from the people that were mentioned before by Jean-Paul Delay, namely Gilles Daugherty and Lecontier, because in sheer improvement, proving this is really one of the key problems they are facing. In particular, in sheer improvement, What happens is that when you give a data, even in a ready for a live statement to the computer, if along the proof you need an induction, as most of the time you will never use exactly the formula, that's the moment where the interaction With the human, it depends on a variety of conceptual experience, making the choice of the load, which may be very different from the given formula, I will give you an example later on, and is the key point where the computer steps in later, after your choice of the induction load, and is a remarkable use then of propositions, during proving merely which was the leading idea of many things. This is a paradigmatic example where the game goes along a firm interaction between human and computer in many, many situations, and in particular in the choice of this action. It should be clear that using the analysis of proof to the fact of proving an already given formula, which contains the full information, I believe it's rather misleading. And the key point is that he steps in an essential way and in an undecidable way. There are other key difficulties in an entirely formalized manner of proving a result, like a computer. What exactly do the participants expect? So we are going even in heuristic in choosing which is the fragment that we can give to them, not less, of course, proof-assistance and proof-checking are changing proof, I entirely agree with that, but far away from the need of complicity, of formalism and complicity.

12:30 What do we refer to by saying that... A heuristic steps in essentially in. The fact that there is meaning and sense stepping in proves, and first of all, meaning and sense are present in accents, and now I will go back to the discussion of accents by referring to a remarkable sentence by Husserl in The Origin of Geometry. It's in French, but... Let me try to simplify what we may mean by this. First of all, I will make a very arbitrary Historically, during complete discussion about Euclid's axioms, that they want to see a principle of construction, they are conceptual principles of construction. Euclid is not a calculator, he is a ruler, and concepts are conceptual tools with direct results, with no algorithms. But they are performed, these constructions, with these conceptual tools, and in no way can be exchanged with the formal aspects. You see the action which is there, you draw a straight line and so on and so forth, you draw a circle, and the field axion of course is also a comparable construction with nature. The key point of course is that this treatment is very different from any formalization, which is on the inside very formal, For any couple of points, there exists one and unique line. It is conceptually different in the sense, for example, that the issue of unicity here is intended by the unicity of the ruler. It is the action and the gesture that the unit is doing in this construction that is also. The unit is not given, say, by a logical spelling out. It's a different attribute. In this case, of course, the translation is complete, in the sense that you can reconstruct geometry and algebra in a fully formal way, but in Uke's approach, there are principal constructions to this. These are the minimal ones on the grounds of which you can continue and support the rest of the course.

15:00 In particular, these constructions are grounded upon some underlying principles, those which are behind them, as Husserl would say. Symmetries are at the core of them. If you look at the axioms, they all represent symmetric situations, or in modern terms, geodetic, and they are grounded by the fact that it refers to... Translations and rotations given by ruler and compass on the tool of majors, again abstract majors. So the way you access the space are given by these tools and the construction follows these tools of major and access. Of course, the situation we have there, put it in modern terms, has two typical features that are related. In Riemannian terms, the curvature is zero, so there is no scale. This is the absolute, possible absolute, which will turn out essential to get to Newton's notion of space. And this is also understood by the fact that the group of automorphisms is not closed under homosapiens. So there is a symmetry by dilatation which is crucial to a UP geometry. Why is that crucial? Because by this symmetry, by dilatation, it's like a closure by a homosapiens. When you prove a theorem at the level of our space of senses, you can prove you have exactly the same theorems among the stars or over the atoms. So this is the identification of the space of senses with the physical space, which is only possible in the critical case when curvature is equal to zero. I'm trying to see what I mean by the epistemological relevance of asking these questions which precede axioms and those of course concern the proof.

17:30 First of all, what I want to make explicit is that in this discussion I'm trying to make a identification or a link between foundation and epistemology. I want to discuss the knowledge process. Exactly the opposite of Greger's proposal was to discard entirely the analysis of the knowledge process or, if you prefer, the phenomenal constitution. This is another beautiful quotation that I borrowed from The Origin of Geometry by Rousseau. The powerful dogma of the fundamental break between epistemological verification and historical explanation, as well as psychological explicitation in the frame of scientists of mind, The dogma of the break between its technological origin and genetic origin, this dogma, as long as the concepts of history, of historical analysis, and of genesis are not limited in as difficult ways as they usually are, this dogma must be reversed from top to bottom. In this sense, I am trying to focus here on what I call principle of constructions with respect to principle of proof. We know very well what principles of proofs are. The remarkable work in mathematical logic in the 20th century has been given as a crucial principle of proof. Principles of constructions are of different nature. They need an analysis, an epistemological one, which mixes up history, a phylogenetic path, and a historical path, in order to find the center which lies behind the axiom, as we tried to sketch before. Let me try now to see how this kind of issue of finding the sense behind and how mathematic approaches, in particular in geometry, do depend on our forms of access to space, tries to spell out very briefly how this is a concept that can be found in many moments of... The history of geometry. I will focus on three in a rather arbitrary way, but which are crucial moments, namely Euclidean construction that's already mentioned, Riemann and Allen Trump. I think one can understand the physiological issue or discuss it by observing that they answer to similar questions. In the three cases, the issue is how do I act on space?

20:00 Access and measure space. Of course, in an abstract manner, I'm not trying to reduce measure to compartmentalization. I already mentioned Euclid, with his own digit bodies, the ruler and compass, using translation and rotation in a structure, which is this fundamental property of being shown by an anatomothetist, symmetry, and salivation. And where the global structure of space is identical to the local one, which allows to identify spatial centers, physical states. Then, this is view of nowhere, which is typical of European structures, which will go through the chaos, geometry, and the view of space. With Riemann, the rigid body is generalized, and the first monumental theorem characterizes the spaces where the rigid bodies are preserved, and these are the spaces with constant curvature. In those spaces, in general, where coverage is different from zero, or in those where there is no constant coverage, the radical difference is that the local is no longer identical to the global. There is a topological treatment of the dimension at the global level, then there is a local level which is symmetric, and that's crucial to the physical constructions which have been built on the geometry. But again, the question is the same, and the approach, of course, and the answer is different one. How do we access these things? How do we measure them? This is only two little reasons, but this is what happens in modern approach to the state of microphysics due to Alain Connes. There, the starting point is again access, measure, action, and phase of microphysics, and in that phase is the quantum measure. The idea is to reproduce the passage from Gelfand to Jones, namely using an algebra, a C star algebra, and reconstructing space from the properties of the C star algebra. But in that case, Alan Kohn uses what? The access to microphysics, which is given by the tools of quantum mechanics, which are reproduced mathematically by the algebra matrices, with effect. However, it is really non-commutative measures, namely if you measure an impulsion and movement in a direction, in an order, or in an order, you get different results.

22:30 So non-commutativity gives you a different measure and a different geometric structure. The point is that in this third case, which is made evident, is that measurement is not a simple matter. It's based on the use of very complicated measuring instruments in each of robust theoretical approaches. There is a very long path to get to the construction of sense that gives you the axioms behind the axioms, behind the constructions. By referring to Poincaré, we observe that the term of geometry is to study the properties of our instruments, that is to say, of the body, and that geometry is the action of a group, and then continues by a new science in the future because it says that the act that covers a normal sensation in an abnormal order would make a different geometry from ours. Now, I don't want to say that eight years before he had an intuition on non-commutative geometry, but that's exactly what happens with the fact that the order of measure is made differently in non-commutative geometry. Let me try to exemplify a little more the distinction that I'm making between construction principles and proof principles, where construction principles are grounded on a variety. A simple example is given by the connection between a conjunction in proofs and the Cartesian notion of product, which generalizes, of course, the classical one in set theory. This is given diagrammatically. It is meaningful by the reference to structures. And we know that there is a beautiful, I would say, perfect correspondence between mathematics and geometry that guarantees that you can go back and forth.

25:00 So in this case, the proof principle and the construction principles are essentially equivalent. But inside, even by the construction principles, it's quite remarkable because you can handle by duality. This junction is very useful in categories, namely in categories you can very easily take the dual category and the dual notion or the dual notion and category, so in this case you can take the dual of the Cartesian process to produce the comprogram, which is immediately understood by this game of duality, which is a game of symmetries, a game as a construction principle. And this gives meaning, even though the equation is much more complex and slightly simplifying, this gives meaning to something which to many may seem mysterious, namely, intuitionistic disjunction. Structurally, there is an immediate passage between the two. At the level of proof principle, it is more complicated or less evident, even though, of course, the situation here is of a correspondence between the two. The point is that what I think is in the reflection on construction principles is that they may be elementary but very complex in a sense even what's behind the little poop of Gauss is very complex because there is this habit of an orchard which is grounded on rather complex cognitive experiences. In particular, if you go to infinity and the well-ordering of numbers, it's not an obvious passage if you hurry to this constitutive history, which goes back to the variety of construction of sense. I believe that this is the invariant of what Brauer would call the two-ness of time, namely the discrete sequence of time, but also it is formed by the... Experiences in space, of stepwise moving, of all those actions that organize space in a discrete fashion. And the conceptual environment may be understood and may be given because behind it, you have this cognitive experience of ordering and reloading.

27:30 The formalist means, in the proof in particular, is that, and it's a Cartesian tradition, is that the intelligibility goes by reducing There are two elementary and simple steps, namely that every time you have elementary steps, then it is simple. Now, with statistics, that's very often we have elementary steps in knowledge, presentations, nations, or people, which are very complex. So what really is changing, and I will make a couple of final remarks by observing that even proof you may have is questionable. With the idea that the understanding may only go by reduction to an elementary which is simple, we may have situations where we have an autonomous phenomenal level and phenomenal description, and a construction of knowledge where elementary is intrinsically complex. I think that today this is the major issue in the relation of mathematics to natural science, in particular in biology where the cell which is elementary. So if you cut the shell, the cell you kill is no longer alive. The living cell is elementary but very complex. Some say, for example, that it has the same objective complexity of an elephant, and there are two reasons to say that. And if you want to construct knowledge of the phenomenal level of life, you need to deal with an elementary level which stays very complex. In quantum mechanics it's the same, because what's elementary may be extremely complex, for example in the understanding in terms of space, in the understanding of non-locality, non-separability, conceptually that is very difficult, even though it is elementary, meaning you cannot decompose it. Now, I believe that the incompleteness of format systems suggests the same, namely that there are situations where the proof A certain moment finds a step which is elementary, you cannot decompose it any further, but it's very complex, and the understanding of it goes by using this variety of classical and conceptual experiences that give meaning, for example, to the reloading of numbers. Now, I don't think I have much time to give any details, but two examples may be given by concrete incompleteness of these schools, namely, theorems which may be stated in arithmetic, and one can prove that they are formally provable, but one can give a proof.

30:00 How does this code prove? Well, I will quote from the normalization, in particular the one which... These arguments may be used both for Gödel's system P or for Girard's system F. In the case of Girard's system F, then this is carried on as a property of second order arithmetic. Normalization is a result which states any passable term possesses a normal force. In those cases, what happens is that at certain points of the proof, there is a typical instance of what I was saying before. You have to prove that every term normalizes. You can state that in one line in your infinitive, as you all have done. But in order to prove it, you must take an enormous induction load. To put it in a typographic way, if you need only one line for the statement, every term normalizes, you need at least seven lines to give the full argument of the induction load, which is called candidates over disabilities. And this choice, of course, is part of the intuitive complexity of the proof. Then there's lots of steps with blend-up theory and meta-theory. Can we, for us, simplify that? Essentially, no, in the sense that if you want to formalize it fully, you have to go to third order, where quantification goes over sets of sets, and since this is a statement about consistency, of course the question comes up of the consistency for the second order, which may be answered by third order, and so on and so forth. The ordinal analysis, which is another approach, doesn't even work in that case. So again there is a step where while you may formalize everything else, but at a certain point induction loads requires an extremely complex statement and inside it what you really use, unless you want to formalize in theories whose consistency requires further and further climbing the borders of arithmetic, that statement refers to well-ordering.

32:30 Along with the proof, you observe that the generic, non-emphasized set of numbers has at least helped, which every mathematician accepts, but of course is very complex, because it is grounded on the point of this view. There is one onion meaning that we have no answer to. Another example may be even also by the very famous and I think very beautiful theorem by Friedman, which is a variant of Proust's theorem, which is fully formalizable as a statement in Proust's order arithmetic, is a statement on trees, perhaps I won't detail it, it's a matter of inclusion of trees, of good properties of fun and sequences of trees. It is provable, non-provable formally, because it implies consistency. But one can prove it. If one looks at the proof that's given by many, Friedman's is the less readable, but there is in so many others, what happens is that along the proof, you have a certain set, and it is written in that way, a certain set where you say, a set, which logically is very complex, is a sigma 1 1 statement, so a set of numbers is logically handled in a very complicated way. But, along the proof, you assume that this set is not empty, then you say, must have a least element, and you go on iterating this argument, and everybody is perplexed that it works. There is an alternative analysis, which is, by ordinal analysis, you need an enormous ordinal, an infinitive ordinal, larger than gamma naught, which, of course, allows to treat If you treat this step in any way, it remains very complex because making an induction add to an enormous ordinal is a very complex step, so that inductive step either you see from the point of the ordinal analysis and it's elementary because you cannot decompose it, it's very complex, but in fact, in the understanding of the proof which is used, the proof of the positive state in the standard model, This is given by reference, again, to a law in the implicit sense that, I see it as a dramatic statement, a generic subset of the natural numbers, which is not empty, is a list element, as any mathematician accepts, but again, in spite of being elementary, you cannot decompose it any farther formally, it is extremely complex because it refers to a certainty which is the meaning behind the conceptual construction, in this case, the order.

35:00 Okay, well then I think I have to stop here. Perhaps I can just conclude by recalling my first quotation of Husserl, which is for me paradigmatic, for the analysis of proof, of course, as a rhyme on axiom, and on the fact that even along the proof, this kind of meaning behind the step, which is, you see, realized, It's there and may be extremely complex as grounded epistemologically on a variety of conceptual theories. Now, I say that the dinner of this night will be at 8 o'clock. 8 o'clock. 8 o'clock. Drastery, law, 11 plus reverse, F-R-H-O-E-R. But in any case I think that we go together. But if you got lost, if you get lost, drastery, law, 11 plus reverse, limit. 11 and 8 plus 2. Much better. Technological problem.

37:30 So, we measure the importance of visualization. It's a practical test. No, we have to go from here. Is it that way? Go ahead, yes, go ahead. If you get lost, please do not try to prove it without me. Why isn't he doing it? Okay.

40:00 Okay. So, maths. Sorry. We have to take very seriously the very different roles that visual thinking can play. Here, I've listed just a few to give you an idea of what I have in mind. Visual thinking can help one for a proof, it can aid comprehension of a definition or a proof, it can give an idea for a proof, it can give an idea for conjecture. Visual thinking may possibly also be part of a proof or a proof process, by which I mean the process of following a proof or the process of constructing a proof. Visual thinking may help warrant a belief when proof is not available, and visual thinking may sometimes provide experiential evidence for a belief. I'm not saying that these are all the roles, these are just some of the roles that visual thinking made them in mathematics, so when we're faced with any particulars, ask ourselves, well, what is the role of visual thinking in this case? I've somewhat simple-mindedly divided these roles into classes of heuristic roles, roles of cognitive facilitation, the epistemic roles. A world where digital thinking contributes to rationality, and when you think of it this way, it automatically produces the scene for a space of different views that everybody agrees with digital thinking in mathematics.

42:30 Heuristically, I guess that everybody agrees with that, but there the agreement ends. Visual thinking mathematics has no epistemic role whatsoever. Then you can get more and more relaxed even though one could say that visual thinking has some epistemic roles but not in analysis, or you could say that it has some epistemic roles even in analysis, but when it comes to proofs in analysis, visual thinking has no epistemic role whatsoever. Or you could be more relaxed still and say that There are epistemic roles even in proofs and analysis, and then there will be room for more and more views depending on the scope of visual thinking that you will allow there to be for proofs and analysis. So we face the question, what epistemic roles, if any, can visual thinking play in analysis? Of course I'm not going to answer that question, but I know what the answer is. But it's the general setting of the particular case that I want to consider now. This is the difference of opinion between Bolsonaro and James Roberts of France on prudent intermediate value theorems. Here we start with the intermediate zero theorem. I'm going to read this once and I'm not going to continue, but just to remind you that is continuous on the closed interval a, b, and f of a is negative and f of b is positive, and for some x in the open interval a, b, f of x is zero. It's not very interesting in itself because the generalization of this is what in English textbooks is called the intermediate value theorem. I won't bother reading it to you because I think... And then there's a more generalized theorem still, which is the theorem that Bolzano actually proves in his paper of 1817, which discusses all these things and attempted proofs of them, but this is the one that he proves if f and g are continuous on the closed interval a, b, and f of a is less than g of a,

45:00 While f of b is greater than g of b, then for some x in the open interval a, b, f coincides with g of x. So those are the theorems, but in fact, the discussion can all be considered by focusing on the intermediate zero theorem. He wrote in his paper in the British Journal of Philosophy in 1997 about this. He says that the diagram is visual evidence for the intermediate theorem. And here I quote, he says, we have a continuous line running from below to above the x-axis. Clearly it must cross that axis in doing so. This isn't his diagram, but it's very similar to the diagram that is there in the paper. And then he says, using the picture alone, we can be certain of this result if we can be certain of anything. This is what Oksano said in 1817. He discussed attempted proofs given by several arguments, mathematicians including some great figures. He says, it's very evident. That a continuous line of simple curvature running from below to above the x-axis crosses the x-axis, but that geometrical plane is not the same as the analytic plane. Why not? Because the concepts of line continuity and function continuity are not the same. And then, in this considerable discussion, Bolzano makes a number of points, and here are two of the points that I just picked up. He says, first of all, the geometrical claim is an application of the analytic claim. The theorem is not proved by citing applications. And then he says, secondly, a proof of the geometrical claim would depend on the analytic claim. So no proof of the analytic claim can depend on the geometric one. So he's rejecting a whole class of attempted proofs very much along the lines.

47:30 You would expect there to be a response from Jim, because Jim was writing 180 years after this, and this is what he says. First of all, he says, the pencil and the epsilon-delta concepts of continuity are different, but they are so related that the diagram constitutes grounds for the analytic plane. By the pencil concept of continuity, he means what Boltzmann is calling the line concept, the epsilon-dot concept. And then, Guillaume also says, he makes these two points. First of all, with regard to the certainty, the ordering is as follows. The geometric plane is more certain than the analytic plane. And the analytic plane is more certain than Denning's axiom. So, the direction of justification must be from the geometric plane to the analytic plane, and from the analytic plane that is absolutely not the other way around. So, he certainly has a fair number of amazing points to give, and I'm not going to be able to... There's one tool in this, but what I do want to do is look very closely at the idea that we can go directly from the diagram, and Jim says from the diagram alone, from the picture alone, to the analytical plane. So it's some kind of inference, it looks like a big inference, but this kind of inference persuaded a lot of people. What is actually going on is, I think what's going on is that when we make this inference, or when somebody is tempted to make it, they are imploring to assumptions, and these assumptions link analytic concepts with what I'll call visual concepts, and so I call these link assumptions. And the first of these is number one. Any function f that is epsilon delta continuous on the closed interval a, b, with f of a negative and f of b positive, has a pencil continuous curve from below the x axis to above. This is a link assumption because it links epsilon delta continuity as a analytic concept with pencil continuity, which is a visual concept.

50:00 And also it links the notion of function here, which is the analytic notion, with the notion of a curve here, which is intended to be aligned, this is a visual notion, I'm using it in this sense, this is the sense in which it is relevant to you, by curve I mean visual curve, visualized curve, and then this is what we get from the diagram directly, any pencil continuous curve from below the x-axis or above, across the x-axis. Then there's a second link assumption. Any function whose curve crosses the x-axis has a zero value. This is a link assumption because the notion of crossing the x-axis is meant to be a visuospatial concept. Having a zero concept is having a zero value of x at that moment. And from that we can draw the conclusion, which is the intermediate zero here. Now, there are a number of ways you might... One thing you might do is you might say, well, let's look at Assumption 2 here from the diagram to these generalizations about any of the principles in this curve. What the diagram is, it presents you with one principle, and you're making a big generalization to anything. You put objectives on to that generalization. I'm not going to do that. I'm going to set that aside. Let's just grant that. Is the second linear function correct? This is the second linear function. Any function whose curve crosses the x axis has a zero value. Now I want you to consider these three functions. First of all, the function defined on e to the minus 1 to 1, which is the identical g of x to g of x to g of x to g. And now h of x is exactly the same as g of x, except it's undefined at zero. Finally, j of x is the identity of x from minus 1 to 0, and then for the rest of it, it's x plus a minute number. The point is that whatever your unit distance, you can choose this number so square that it raises the line less than a plankton.

52:30 So the curves, by which I mean visualizable curves, of all these functions, this is assuming that they have curves, All these functions are indistinguishable. They're the same. The curve of g crosses the x-axis, so the curves of h and j also cross it, but of course h and j lack a similar value. Okay, now, there are ways of responding to this. I'm not sure I've got time to consider this in detail. But one thing you can say is, look, we can ditch this link assumption and replace it with a more sophisticated link assumption to get rid of this h and this j, which are the kinds of things. And that is to say, any epsilon-delta continuous function whose curve crosses the x-axis has a zero value. And I agree that does get rid of this counter-reference, but it is not going to work dialectically in this context, because the point was that we want to be able to refer directly on the diagram of the conclusion. We need to have a reason to believe it, given that we only adopted it as an ad-hoc modification of something we came to recognize. We can't infer has a zero value from cross of the x axis. So the direct root from the diagram of the amygdala is blocked. Now I'm going to look at the first link assumption. The first link assumption is slightly more complicated in that I'm going to factor it into two parts. Any function f, which is epsilon-delta-continuous-on-the-potential-ab, has a curve that is hence the continuous-on-the-potential-ab, and if a is negative, that's being positive, that curve runs from below the attachment to above. Now I want to focus on Roman 1. The objection is that some epsilon-delta-continuous functions have no curve at all. The curve, I remind you, means visualizable curve. Every epsilon-delta continuous but nowhere differentiable function lacks a visualisable curve. So I reject this link assumption.

55:00 Now let's turn once more to James Blaine. First of all, he says that the pencil and epsilon-delta concepts of continuity are so related that the diagram constitutes grounds for the analytic theorem. I think this is wrong. An epsilon-delta continuous function may have no curve. So no pencil continuous curve. So the diagram gives no grounds to look for a plane above all epsilon delta continuous functions. The diagram just does not support such a big generalization. Number two, Graham James says, using the diagram alone, we can be certain of this result if we can be certain of anything. The route from the diagram to the theorem relies on all the functions linking the analytic with the linear, so I'd say no, this is also wrong. How might we prove the intermediate zero theorem? Well, I've seen several proofs of it, but something pretty typical is this. You do it like this. You consider the set S of all points in the closed interval AB for which S is negative. Then you know that A is in S, you know that B is an upper bound of S, that's given, the information is given, so we know that the supremum of S exists, called a sigma. That's step one. And stage two would be to show that the sigma falls strictly between A and B. And step three would be to show that F of sigma can't be negative, and it can't be positive. And then, from that, you know that F-sigma has to be zero, okay, so now you've got your, you've got the zero, you need to get zero value to get through. Now, I want to focus on how one might prove, construct a proof of 3-roman-1. I mean, if you just don't look at the book, but you try and do it yourself, you might come up with something like this. Suppose that the state of reduction over f sigma is negative, then make epsilon the distance between zero and f sigma. Clearly it's positive in the assumption, so by continuity there's a positive delta as below in the diagram, this is how I imagined it, such that f sigma plus half delta lies within epsilon of f sigma.

57:30 And that means that f of sigma plus half delta has got to be negative itself, and that's a problem. Why? Because sigma is less than sigma plus half delta, but we see that f of sigma plus half delta is negative, that s plus half delta must fall within s. That means that sigma is not the upper bound of s, right, and that contradicts the definition. Now, okay, this is all elementary, there's a bit to all this, I'm sure. But the question is, what is the role of visual thinking in a case like this? That's the question for us. Now, for me, I mean, my information, without having much to say about it, and I would be very interested to know if your information is different, The role is not to provide visual evidence at all. Rather, it's to help one to construct, in some cases recall perhaps, the proper epistemological reasoning. So, his role is heuristic in this case. And such heuristic uses of visual thinking I'm sure wouldn't have been rejected by Borsano. Why do I say that? Well, he didn't discuss visual thinking as such, but he did. He also discussed the use of concepts of space and time and motion in exposition and analysis. And this is what he said. It's an exaggerated realism to ban concepts of space, time and motion from exposition and analysis when they are used solely for clarification, but they can't be used in proof. So, I think he had it. Of course, in some cases I do find myself disagreeing with Boltzano, but I am way closer to him than I am to Gene Brown.

1:00:00 So here are my conclusions. The best case, the most convincing case, the maximum promising case for visual proof in analysis turns out to be Elucid. Agreed. We should reject the very strict view which says that there's no epistemic role for visual thinking in maths whatsoever. I would say more, I would say we should reject the view that there are no visual proofs in mathematics. But, I think it's very wrong to reject standards of analytic proof established by Boltz-Arnott and other mathematicians. In other words, don't throw the baby out with the blackboard. Thank you very much. Now I have another program. So now, the next speaker is Piotr, it will be Piotr for Organizing and the University of Lille for Organizing and the University of Lille. A really exciting conference I'm looking forward to. I have already more learning to watch. Now also, I sent in the title before I knew Marcus's title, so I wanted to add a subtitle for him.

1:02:30 I want to do this. A friend of mine, Mark Kingslip, at Reed College, who had a saying that I'm going to append. Since there are not any English speakers here, you might not. Understand the idioms. There are two idioms in English that have similar meanings. One of them is, don't throw the baby over the bathwater, right? And the other is, don't go from the frying pan into the fire. So I'm just going to have Hinchcliffe's version, don't throw the baby over the frying pan. Okay, so don't throw the baby over the frying pan. And what I'm getting at here is I'm going to say even when we're considering why visualization is helpful, where we're considering just visualization and not perhaps broader conceptual reasoning, we have to spend some time thinking about why visualization is helpful. I've spent some time thinking about why concepts generally in mathematics, even non-visualizable ones, might be more valuable than others in mathematical reasoning. Now, a really more basic point. Visualization is very complicated. There are many different ways to confer a cognitive advantage, and the different ways are often interesting for different reasons. It's not just that there can be an advantage derived from the ability to visualize, but the preference for visualization can mark a style of reasoning, different but not necessarily better or worse than some other style without the prevalence of visual representation. And here I have in mind things like this. Two colleagues of mine in the math department, say, Mel Hobsker of the University of Algebras, Bob Lazarsfeld, an algebraic geometer, will often address exceedingly similar questions from an abstract point of view, where they count as algebraic terms, they count as proper formulations, and can be strikingly different, at least in part because of a preference for a certain kind of diaphragmatic representation of an algebraic geometer. Now, there's also a remark we've separated. Bringing up, as it might be, a certain kind of cultural divide in regard to these group counts and stories.

1:05:00 One algebraic geometer asked me, excuse me, about algebraics. He said, isn't there sometimes when you visualize things and you deal with algebraic curves? I said, sometimes? He said, oh yeah, yeah, yeah, I visualize all the time. For example, I visualize this. Now, okay, so visualization is complicated. Now, I've got actually a bunch of examples. This is not a polished talk by any means, but in conformity with people who have their own situation, I'm going to have a bunch of examples that I'm struggling with, some of them that I'm thinking are going to have connections, that I think raise questions about what would you say would be visualization, and I'm going to struggle with them along with them. I want to give one way in which visualization gives us an advantage cognitively. And I want it mostly as a contrast. It's going to be an advantage, but one which turns out, well, even this is complicated, but as a first approximation, turns out not to be so interesting for the student of methodology. And then we'll maybe contrast, bring out by contrast some of what we did. Visualization is used in systems of memory right now. It's not just today, certainly if today, if you go to any bookstore and buy a book with a title like improve your memory in 30 days, you'll find some variation, almost surely find some variation on the following technique which has been used since medieval times, and it's also, it was used by, I don't know if it's a famous case study, a mnemonist. What you do is you make the lists of things you're trying to remember into coherent sequences of images and form a picture. There are these elaborate medieval memory palaces that were built on this principle. So what I want to do is just go through a single example, because we need something immediate, to illustrate the technique and then to draw some conclusions about what we can and cannot do. I can't find interesting about it. And also this has the advantage that unlike a lot of my talks, I'm pretty sure that everyone, except for Henrik it turns out, will go away with one genuine piece of new knowledge.

1:07:30 In Danish, this is a kofod. Now, I mean actually Henrik will also speak Danish sounds when he's spoken with a heavy Canadian accent. This is a genuine piece of knowledge too. That is a kofod. Now, let's say you want to remember what this, you want to remember this, right, because that's not the sort of thing that comes frequently in a telegraph, and, you know, you're trying to learn, you know, a day in front of a limited amount of data, so how are you going to remember this? Now, the medieval memory palace technique can really form a picture, which is going to be particularly vivid, and which will allow, in which you can incorporate, In ways that will allow you to remember the name. And so first we need a picture of a Poe. And so here's one. That is Edgar Allan Poe. You'll notice actually this is going to be handy in a little bit. There's a raven on his head, which is something I can't give away. So there we have Edgar Allan Poe. So we also need a fool. And so there's a fool. And so then you sort of think, well, what you need to do is do something like this, and I've got a couple of variations. One of them's going to be maybe more vivid, and the other one's going to be more helpful for the follow-up. So you say, well, here, here, you form a picture of something like this. Put this in your head, right? There we have, instead of the raven, we have a peacock on the head of a brown toe, and then, you know, also stuck in here, although I should say not very preparedly, is the fool, right? So, the whole fool. Or here's another one. This is Andy, right? There is the pole pool, Edgar Allan Poe and the Cap'n Bell. So a former picture of Edgar Allan Poe wearing a Cap'n Bell, right? And then, especially to a previously committed memory, a picture of Poe with the Raven. Thank you. I'll keep going. Now, you might say, well, okay, this is, oh, that may work, but it's kind of one-off. You can't really vary it or extend it, but really you can, and so if you'll bear with me, I'll just give one example to show how this can be extended. You can actually push this forward. You can use compositional principles and so on. Here's something else. This is a peacock feather. So let's say you want also, using this technique, to remember...

1:10:00 The word for peacock feather. Well, first you need a representation of a peacock. And so here is one. You know, these people fleeing in terror from a couple of giant monsters. That's a feral right. And so then you're sort of applying the technique, incorporate a picture, and so here we have Poku with Geo, a couple of residents of Tokyo, and then a peacock that are fleeing in terror from a couple of Pokus. Okay, so there you go. So anyway, that's the true thing. And I've got to say, you know, it may well be that you'll remember this word, right, having seen my little, for an awfully long time, right? Okay, so anyway, here you have just committed to memory. Okay, now... That's a cognitive advantage. There are lots of reasons. There's a vividness, a visualization, lots of things going on there. You know, but there's a case where you might say, okay, you do have a cognitive advantage. It's not one perhaps that's particularly interesting, although it's complicated and I'm prepared to take it back. It's not particularly interesting for the study of methodology. So now why is that? Now I sort of thought at the time, or there was a period where I thought, Well, that's just because it's somehow purely psychological and in a way that makes it uninteresting, but I think I have to refine the story a little bit. And the way I'd refine it is to say, in addition, if the prompting of, if the mnemonic effect is going to be interesting for methodology, It has to be that it arises, so to speak, from something, and I'll explain this in a bit, but something that is essential in the picture and in the connection between the picture and the concepts, not something which is accidental in the way that these representations of Pofu and Pofu Pio are. So, it's rather the idea is something like this, that Plato had this idea in the Mino of... Arguments and reasoning is somehow tying down concepts, allowing me to remember them, but there the arguments and reasoning are tying them down in perhaps a more direct way, more connected with the concepts themselves than this kind of visual one.

1:12:30 Now, next example, and in this I have a couple of remarks. First I should say, This is a warning of a false memory that has been assigned, but when I first saw this example in a textbook by Patterson and Cronin, I had a memory of having seen it before. It seemed to me that I could remember having used this when I was doing my third year algebra class as an undergraduate. And I went, wow, this is great. Man, that was so useful. And then I later found out that actually Cromen had invented it himself. And when I wrote a paper mentioning this example, I credited it to folklore. Standard, called the standard, whereas the fact that I should now take the opportunity to credit Cromann, when I rechecked my third year lecture notes, in fact, I was remembering something politically different. Okay, so here we have two ways of trying to represent the multiplication table for the units of the octonium triangle, one of them. And then the rest of them. You have the remaining seven, and you can multiply them like this. So these are how they multiply, right? E7 times E3. E7 times E5. Now, that's a lot to remember. Certainly more gifts, I'll say, than hopefully gifts. You might say, well, can I come up with a technique? This is a trick that Cromann came up with. It's really very cute. You can set the thing up as this kind of giant, this triangle, and when you do that, you say, well, look, each of these nodes can be labeled with one of the units. And then, it turns out that it's here, multiplying E5 times E6 gives you E2, multiplying E3 times E7 gives you E5, and E8 is in the middle there. Multiplying E6 times E7 gives you E4, now you have to, I'm not, sorry, I haven't given you the whole thing, but you have to make this thing counterclockwise to give you plus and minus, but, you know, that's that, we don't need that refinement here. So, you know, this is actually a great, great trick.

1:15:00 Now, I was inclined also to say that that's an example of visualization, which is accidental, and so I'm inclined to set it aside because it's psychological in a way that's maybe not psychologically interesting. But then something began to nag at me, a second confession. I suddenly realized I had overlooked something pretty obvious, much to my embarrassment, which is that, of course, this is visualized, too, right? That's what it looks like. There it is. I couldn't, this is the best representation of it I know, right? Form a picture of that, right? So the issue is not whether, it's not visualization as such, but rather some, this one sort of admits of being visualized more easily, and I think that it admits of being visualized more easily, and partly because it incorporates some of the structure of what you're trying to represent into the picture. In a way that makes the picture easier to take in. I'm afraid I can't be more precise than that, but I want to bring up that there are certain facts about the way the information is encoded into the picture that makes it easy to visualize, not just sort of bare data of sense, so to speak. So we might say, I mean, one thing we could ask about then is to get a clearer sense of What we're talking about when we talk about information coded in different ways into pictures, in ways that make them easier, or like these visualizations, you know, it might be that we find interesting interactions between the ways in which visualization can be helpful in some cases in reasoning and the ways in which choosing the right conceptual representations can be useful in reasoning. And here's one example I've studied in other connections. In connection, in particular, with mathematical explanation, but I think it raises the question of visualization, too. So, I'll go through it a little bit, and this is a case where I might say, among the things you might ask when you're assessing a mathematical proof, or more generally, some technique, some technique like, for example, the technique for remembering a poem, is, along with Plato, does it explain it?

1:17:30 And then you would certainly say, well, certainly, however helpful it may be to remember the word local, by drawing those pictures, you certainly wouldn't explain it by invoking pictures like that, whereas you might sort of hope to explain, for example, these new symmetries in the structure of Octonian's black line appealing to that picture. So maybe there is some hope that sometimes, at least... You can talk about explanations in a way that connects with visualization and that might allow you to bring out another dimension of what visualization can involve. Okay, now, in particular I want to consider a couple of opinions and then sort of unpack them a bit. One is from a recent textbook, one is from a recent textbook and one from a biography of Riemann. The first one, they're still all saying, the reason for the double periodicity becomes completely clear with the introduction of Riemann's services by Riemann. One does not ask for a more satisfying explanation of the mysteries of the holistic integral. The field of elliptic functions had grown rapidly for a quarter of a century, though their most fundamental property, double periodicity, had not been properly understood and had been discovered by Adler and Jacobi as an algebraic curiosity rather than a topological necessity. The more the field is expanded, the more was algorithmic skill required to compensate for a lack of fundamental understanding of the field. He says, as Cauchy even came, to understand the periods of elliptic and hyper-elliptic integrals, though not the reason for their existence. There is one thing he lacked, which was the surface. So here we have, you know, that's just a whole bunch of dynamite-thrown-in-the-philosopher things that we want to explain,

1:20:00 And so forth. So how does he do it? And it will turn out the visualization shows up in this story in a kind of an interesting way, although it's complicated to assess just how we should ultimately assess what role it is. Okay, so what Riemann did is he actually restructured the problem in a very far-reaching way. Very far-reaching reorientation. Now there's a contrast with Weierstrass that I won't go into in any detail, just for the sake of time, but I will mention that among the things that was distinctive of Weierstrass is that he rejected the core technique of using Riemann surfaces, didn't reject it, but he sort of just... He minimized its value by calling it merely a means of visualization. So let's see here. Yes, I'll also say, I mean, Riemann, when he describes what he was trying to do, he seeks formulations which will allow him to see answers, quote, nearly without computation, by describing the objects of study, quote, in terms of essential characteristics. And these expressions, like nearly without computation in terms of essential characteristics, are almost picked up and repeated by disciples like Gettikin to the point that they became recognized as buzz phrases among the various schools of mathematics in its style. And as I say, it was dismissed by Barclay, but it was merely an example. Okay, so what's the deal here? I'll just quickly tell you what these things are. Okay, and also I'll define them in a 19th century way. But here there are a couple of kinds of integrals you can consider. One is the integral of the square root of this fourth degree polynomial and then similarly for the third degree polynomial.

1:22:30 This is a subclass known in the trade as the holistic integrals. That presents y as a function of z, you can also think of z as a function of y, just by inverting the equation, take the inverse, and actually in some way, just as happens with sine and arcsine, the behavior of the inverse function is in important ways simpler than the original function. And so here we could say that if you have phi inverse of z integral, then phi would be the function you get phi. Converting that integral, and that is called an elliptic function. The basic fact about elliptic functions that is being referred to in these quotes, it turns out that every elliptic function is superior. If you have an elliptic function phi, general phi, there are going to be complex numbers omega 1 and omega 2, such that phi of z is sine of z plus n omega 1 plus n omega 2 for integers n and n, so something like a sine function has phi except two periods rather than one. Very interesting fact, very striking. So you might say, and as was mentioned in one of the earlier quotes, it could be regarded as just an algebraic curiosity. How might we explain why this happens? And here is up to a first approximation of the Riemann explanation of why it would become the 17th grade. First, you have to find a way of reinterpreting the original integral as an integral on a new surface. Now, I'm going to skip over some details just because, although they're quite interesting, I think the essence of the story can survive their loss, you're now taking and finding what you call a natural surface on which to integrate them, and because of the square root, for reasons I didn't use, the square root sign and the fact that...

1:25:00 The inverse of that would be ambiguous between the positive and negative square root. You need a surface with two sheets that gets connected together in a certain way. And just to show a picture of what that looks like, you'd need something like this. Right. Here would be the first thing. You need these two sheets. For the one part corresponding to the positive square root, the other part corresponding to the negative square root, and then you have to put them together somehow, and for that you need to sort of have these splices that you're going to glue together in a certain way, so here we see them being reshaped for purposes of gluing, and then what you get after you've done that is topologically a donut. So, there we have it. That's the natural surface to integrate on. Now, this is an integral, and So, considering the integral on that surface, you're getting a length. Now, it turns out that if you integrate from one point to another, and this is a detail that's fascinating, I'll skip over for a moment. If you are, I'd rather bypass, maybe one more time, maybe two more times. If you integrate around a circle on the surface of this site and come back to the same point you started from, Then the value of the integral is going to be zero, except in those cases where you're setting up a curve that isn't closed, which is something like your... Here are two ways that can be done on a doughnut. You're flipping around the other side of the doughnut, or you're flipping around like this. So then what integrating on this doughnut turns into is you can begin at a point... You can go to another point, or you can begin the point, sort of zip around here for a while, and then zip around here for a while, and then end up at that point. And traversing either of those distances is going to give you the same result. The first period is going to correspond to this length, the second period is going to correspond to that length. Then on a fan, you know, sort of jumping over a bunch of details, you can say we have an explanation. Holistic functions are doubly periodic.

1:27:30 And part of what makes the explanation vivid is the fact that you can form such a nice picture of it. Okay, now, there's the question that I'll pose, exactly how important is the fact that you can form that picture? I can give lots of reasons for thinking that The Riemann, the explanation in terms of Riemann surfaces gives an explanation, right, maybe for bad reasons, but there are reasons in the book that they don't involve fuels to the fact that there's visualization going on. Like here's one, and you look for the flavor, there's one you could just say, look, there's a higher order reason for requiring this setup, the property of being a donut is a topological invariant. And there are general reasons for thinking that answering questions in terms of topological invariance is generally mathematically productive, and so we formulate things in terms that we have independent reasons to think are mathematically valuable. This we should regard as a mathematically valuable way of answering the question. More generally, I won't go into all the details, but this formulation has turned out to be extremely fruitful. Mathematically, quite independent of its connections to visualization. It would appear, but here is where I reach the point where I'll just leave it for the open discussion. It's not clear to me that the right thing to say in this case might not be that one of the reasons our ability to visualize What I think is helpful is that when you're visualizing a particular piece of information encoded in the paper, and where that encoding is an essential part of the ability to visualize, not an accidental part, where you really have to understand the structure of what you're representing or why it looks the way it does, And those concepts are particularly mathematically valuable and fruitful, and so although there's this complicated set of interrelations between the bare psychological fact that being able to form the picture is useful for us, and the sort of deeper and more mathematically robust fact that things that are formulated in the way that correspond to formulating the picture are mathematically valuable.

1:30:00 And the fact that that picture, as Ronald in that way happens to represent things as formulated conceptually enough. So the question I end with is, is the mathematical fruitfulness and the unquestionably helpful psychological vividness connected in any way? So you distinguish between the mathematical utility of a certain representation and sort of the psychological utility of it. Yeah, I'm not sure that that's a deep and robust distinction for initial information. Well, but you express concern that you're a little bit uncomfortable about something that's too much important psychologically. So one reason that visualization is helpful is that we have very good visual capacity. We can imagine twisting things and turning animals. And so one story you can tell is that you can try to characterize it. In other words, so this will be a lot of examples of the diagrams that are useful. And then characterize what is it that we do, what are the manipulations and diagrams that we do that make them useful. And then you sort of have an abstract characterization of some capacity. And then the story you can tell is these representations are useful relative to these loose dimensions. So it just so happens that as human beings we, you know, we have these sort of capacities that we have. So that's a good explanation.

1:32:30 Abstracts away or factors out. You know, so these representations are useful for anything that has these capacities. And then you leave it to the psychologists to figure out, you know, why is it that we have these capacities. Well, you know, at least I do think there's a certain kind of distinction which is so hard to draw. You know, I want to have the resources to be able to make, I mean, there are such a ton of cases, like, and I meant these sort of examples of memory systems to serve as a benchmark for that. I mean, it may turn out for all I know, and let's stipulate for the sake of argument, because people discover that it's actually much better for mathematicians to work with yellow paper than white paper, right, because it reduces eye strain and fatigue levels down, right, so we then find, right, yeah, absolutely, there's no question that it was the wise mathematician who used yellow paper. And uniform, you know, absolutely. Now, we might say, well, there's an interesting fact about our capacities that people like us who are trying to do mathematics are well-involved to do it in this way, but it seems to have nothing to do with the content of what's being done. There, you might say, well, that really is just a completely accidental fact about our cognitive systems that has no interesting connection to the content of cognition. Now, the next step down the road is to say, well, okay, how about, it's not eye strain, it's not like yellow paper, and this is why I chose these focal examples, but here's a case where it really does seem as if something you're trying to remember, there is a fact about us which happens to be true, is when we represent things in terms of images, there's a certain kind of vivacity to that, an emergency to the representation, and that makes it easy to remember, and so it's a helpful technique. But I'm inclined to assimilate that case to the yellow paper case rather than the integration on the torus case, which I do think is a case where the visual representation has an interesting and important connection to the content.

1:35:00 And then the reason their visual representation is valuable is at least in part because it's a representation of that content. See, so that's why. I mean, that was a very helpful suggestion, and I've now explained why it is that I'm drawing the line. Does that sort of give you a sense of why? Yeah, I'll argue with you later. Okay, good. Actually, the whole panel is initially for Marcus, but also just to answer the same question, well, now I've got two big questions. First, Mr. Marcus, you mentioned that the animated values there are sort of our last step for making the Jim Brown case that we've created. Thank you for your attention. I don't remember if I read any more articles or keep articles, but what about the sum of the infinite series, one over two times, where you use the unit, so you want to sum that series, and you use the unit squared, and you divide it in half, and then divide the half in half, et cetera. Anyway, it seems like that's an excellent case in which, as you said in your article, it brings to mind a form of non-visual thinking. I'm going to get to those questions. It seems that that case and some other ones we've opened the idea that we can tell a more substantive explanatory story about what the connection is between the operations that we're performing on a picture and some formal proof. And I'm wondering if... I've got no idea about that. I'm not sure that it counts. Not within my paper.

1:37:30 I've presented it. We've also got quite a bit of reason to go with it. I have to look again to see what I now think about that. I also discussed Littlewood's example. He draws a picture. I think he's forgotten now what the picture means. And he says, for the professional, you need nothing more than this picture. I investigated that, and I thought, well, look, this isn't just plot, because I could work out how we could reach the conclusion. I could reach the theorem using the picture. But, I was doing essentially the same in that, and then I realized, as I did at the end, that Rubin, that basically, that segment of the, of the next year, really the diagrams are being used for the juristic, they're helping you, and you do actually do the proper, it's not about the engine, you find it, you discover it, it's not saying that the diagrams aren't, it's not a very important role, it's important that they're all in these cases. Just to be clear, I have reservations about the IP as a visual representation, which is a catering. Rather, that's why I'm concentrating on it.

1:40:00 Your talk suggested an issue that involved an explanation, and the question is this, or not, the talk itself and also your experience, to what extent is what counts as an explanation and asking the right question, tied to the kind of beings we are, as opposed to being, oh, it's like an explanation or an objective matter, or somehow an explanation tied to objectivity? I don't have an answer to that, do you? Yeah, I don't either. Because he's taller. Yeah, yeah, yeah, exactly. I'll wake up here tomorrow morning. Except that, yeah, except that... It's a good question, actually, for the world. I want to know the answer. I don't care who. Well, yeah, this part of the world is just a non-answer. On the one hand, I don't know what to say, except I find it very hearty. Can I comment on that? More reacting to your very final remark, but of course it is worth it then. I think we really have to go to justification. There are two levels where I would say that, looking at category theory, you can of course write those diagrams as equations. But category theory as a mathematical theory relies on the fact that you pull out same-piece information equations as symmetries and dualities by diagrams. By writing diagrams you say something that is, of course, in equations, but at the time it isn't. It's the job of mathematics to pull them out, and if we carry out diagrams like in physics, we have equations, but then, looking at the symmetries there, you keep conservation problems. Other symmetries give you conservation of momentum. So, the formal side of the computation is very important, but is intrinsically incomplete as far as mathematical structures of symmetries and dualities are there. That's the first step. And the other point is the one I was trying to make, to talk about incompleteness, where the justice system... It relies on what they call the geometric jargon, in which sense, of course, not good as incompleteness, which is a fundamental theorem, but it says nothing on how to prove a independent statement is consistent, because it is just a diagonal argument, just an exact diagonal argument.

1:42:30 But, if one looks at concrete incompleteness here, beginning with normalization, then Paris-Harrison, then Krigman-Kruscon, and then it would remain even bogged down. There are three constants, but essentially there are three constants. You look at a code there, beginning with Paris-Harris or normalization again, or even the most interesting one again is the Friedman-Cruz constant. At a certain moment you see that the proof, even by formalists like Simpson, that there is a line where it says, oh, this set is a subset of numbers, integer numbers. Is that a complete justification? Well, people say no. We need a final analysis. Then they go to ordinal analysis. And as the century has been living of induction, as all of mathematics was alive induction, People with such great talent, like Rajam and so on and the others, have been able to pull the neck of induction along the ordinals, and incredibly big ordinals, in order to prove that sigma-1 is one state that is indeed inductive. Is this a justification? Yes, perhaps. It is a weapon of hierarchy. I like it very much. Yet, the key issue being consistency. You just send consistency farther on over ordinals. Why this very one-line statement, which refers to the geometric statement? Don't you see the number line there? The discrete sequence of increasing numbers? If your set is not empty, then the set is empty. It's absolute. It's completely animatronic. Animatronic should do it everywhere. Why does this statement exist? Because it provides a game of world order. It's organizing it in space and time and then accepting each other. And you cannot avoid the concretely stated that you cannot avoid. And all the work done by ordinal analysis or by also getting up of order is very interesting for giving classification and we can adopt it. Each of these terms has an ad-hoc analysis that gives you an ad-hoc ordinal or an ad-hoc orza. You can do it, for example, in second orza, a systematic or ordinal analysis there.

1:45:00 Each time there's an ad-hoc construction, very informative because it gives you the relative consistency strength, lots of things. But the only reliable justification is in this very mountain to make this statement, which is the fact that the word order allows you to say, oh yeah, the research is better. Yes, we have the courage to say that this is foundational to these and justifying proofs. I wonder if I could just quickly answer, and just to say, it does seem to me that I don't understand how you could have a concept of explanation according to which getting the explanation right wouldn't be some kind of a success for people like us about merely getting the explanation right. And so, even though I don't have a deep rationale for that, that's still my explanation. That's just something that you mentioned, digital. That could be something that my memory palates. It illustrates, you're saying that the visual is in the diagram's list of two terms. The middle position is what theory does it novelize? And then the proofs, actually, given feedback of the time, will be omitted. And what he expected is he found the number of diagrams that contain points that aren't referred to in the geometry of the books, but are just the auxiliary points that you would use when you do analytics. And so that's how he introduced poetry to mathematics. The second point, about Jamie, he wrote about it. The example with the elliptic functions illustrates also very well the underlying tension we've been getting in the verandah, which is something which was lying also under the truth. Things that what I was touching on earlier that Stuart wanted to talk about, and there is a perfect watch for this that I want to give attention to, namely Hurwitz-Kuhlhoff, where in the first part Hurwitz left behind the second lecture notes, where this theory is set out beautiful alibi to us. Just epsilon, delta, h up and down, no figures, no nothing. Models. And then Cura took that and applied it, get the new theory of multiplication, and it's all there with Riemann's surface environment. So, that's the thing to read, to fill out the theory.

1:47:30 And now it's my, I've written the list, ten, the list of ten minutes. I have a question to Matthew. I think that your argument is very good and very clear in order to prove that the socialists act alone, but I'm not sure that it is complete with the fact that socialization cannot end in a monstrosity or in an animal. Please. I hope that you agree that liberalization has a democratic role in Okinawa, but it has not a democratic role in Okinawa, although it makes information from text, you cannot prove, it's not that you cannot prove. The generative theory of quotidian geometry only means visualization of visual tools. And in fact, the argument that you, your argument is that you cannot use a geographical code in order to understand about functions, because you can't use that app from which you can study geography. But in fact, what about diagrams with something together that says this is a diagram of this That is a square triangle, it's not an integral. This is the angle, for example, is also called the c-th. My real feeling about this is that my good friend Jim was just bending the stick way too far, and it really just made me think, well, the history of thought about this is just going to be an oscillation from one fashion to another.

1:50:00 Kantian views and anti-Kantian view, and then back again, and so I wanted to resist his most extreme claims. But you're absolutely right, I didn't provide an argument that visualization could not play a major as part of it, and if you allow yourself visualization together with it... If you have a lot of well-explained conventions of representation, then of course you can very easily distinguish visually the functions of G, H, and J, but then of course your argument must incorporate the rules governing these conventions that you're using as either, so that the argument gets a hell of a lot more complicated. Jim's idea was that... The question is, what's in the sense in the middle between the talk of Giuseppe Longo and Marcus Giacquinto, it's about the idea of generosity in proofs, so the proof of Gauss is okay because, or is a correct one, because n is a generic integer, in the sense that if you make a reasoning with n, it will offer all the integers. The same observation is true if you take a triangle, for example, and you do a triangle on a blackboard and you make a construction, the same construction will work for all triangles, just because you can move from one triangle to the other one using the affine loop, for example. So there's some idea of genericity in proofs. And the point with analysis is that there's no generic curve, generic...

1:52:30 Continuous curve in any sense. And so that's a difficult point, I think, to analyze from this point of view of the link between proofs and visualization. I don't know if you have any idea on that. No, I agree and I think there is no uniform treatment of what the generic element is for a prototype proof. Herbrand has the notion of prototype proof and it's beautiful. It's not being pursued. Again, why? Because proof theory is all concentrated about arithmetic. There is no proof, autonomous proof theory of geometry. That's it. Mania of the century. And of course we have to overcome that. And even there, what happens is that, of course, we use the generic statement I said before, and this is the real reason, the real reason of the conflict. Let me say that again. Gauss' theorem is a prototype proof, Gauss' proof is a prototype proof in respect to a generic answer. Okay, and then you can replace it by inductive proof. The idea of completeness of formal arithmetic is the fact that any prototype group with a generic element can be replaced by an inductive group. In mathematics, usually, you cannot do that because you have countable domains and so on and so forth. If you prove Pythagoras' theorem on a right triangle, you will not prove it for four right triangles, but you just draw one. And that's the fantastic of mathematics. You just draw one to say... I give this proof with specific relation between the sides, because I have been drawing one specific. At the end of the proof, the great mathematician who invented mathematics says, look, my proof does not depend on the specific drawing, but only on the right angle. And the justification is grounded on that. There is nothing else to say. The fact of identifying a generic triangle with an assertion for all triangles seems to be equivalent, but it is not, because there is behind it the myth of set theory, that there is a set of all right triangles, including those in the black hole, no black hole in the universe, they are compressible. There doesn't exist a set of right triangles, there is only one concept of the right triangle, it's a concept.

1:55:00 Like in categories, there's no set of objects. A category is one concept, the concept of objects and the concept of morphs that can be specified in various ways. It's absolutely misleading to go to brawl and do the set. In mathematics, one word is as much as possible, and analysis should be very careful in some cases, like you are, for the continuous function. Genetic objects and prototype tools. In the specific case of arithmetic, we can replace that with induction, with two tools for all. And the need to work with induction is complete and is full. In the rest of mathematics, we go all the time with one concept, the genetic structure and the prototype tools. Cannot, of course, be replaced always by induction. But the situation is very delicate in some cases, and there is no proof here about that, because we have been focusing only on induction for a century. Yeah, I have two comments. One on Marcus's presentation. I found your principle curves human-analytic for me, especially the way you defined age and age, and suggest that, I don't know if you know, Felix Klein's work... There is a couple of them in the paper, and that would be great if we bring it to their work, because for him, any kind of tensile curve or geometric curve would be something that we can write as thick and smooth, right? Thick and smooth. And then, of course, you cannot say that there is, for any function, for any virus-based function, there is a corresponding testing curve. Maybe it would argue that for any virus-based function, curves would be a testing curve somehow. I don't know. So I think the whole discussion would change to consider the kind of people. And then I have another comment, which maybe just emphasizes again something that is arising at this moment, which is that among the many different possibilities, of course, as you all know, there is a kind of view that visual elements must be in mathematical groups. And that is, of course, that has to do with historical research, one does not have to stick only to the hundred-century versions of things, and there has been a lot of work trying to understand better what is going on in geometry too, in motion geometry and so on, and there's a lot of dialogue there, and that reminds me of the whole issue of many different types of...

1:57:30 So, I don't know if it's going to be able to say something else, but it's fine, it's very, very, very, very, very, very, very, very, very, It does seem to me that there are there are contrasts between different sub-communities of mathematics where you can mark a quite healthy style of effect with a reasonable formulation of a problem and what they count as being an adequately answered problem and what they call a natural generalization and so forth. And that in some cases... The distinction in style seems to be bound up with the difference in representation, so I can use the example of algebra and geometry, I mean, in Belgium, two very similar subjects, but with a whole different set of standards and explanations and so on and so forth, a different stuff.

2:00:00 Robots. The role of diagraphs is explaining the stories of scholars. The role of the visual representation. And it seems to be clearly a crucial aspect of the reasoning of these concepts and a less important aspect for perhaps community health directors. Is that just an accident? Is that just a superficial social... So, is there a quirk, or is there something deeper there? My suggestion is that it could be that it corresponds to something that is deeper and more organized in the subject, so that corresponding to the preference for visualization in one case is also different conceptually, apart from that basis. No, no, we let you go. Thank you.