Examples of visual-spatial reasoning in algebra & analysis / Giuseppe Longo: discussion

0:00 ...and who will talk about the role of images in mathematics, algebra, and analysis. Okay, so I will repeat in English. So, I open the afternoon session and the first speaker will be Bernard Tessy, who will speak on the role of representations in mathematics, mathematical proofs, analysis and algebra. Representation in pictorial representations. Okay. Yes, thank you. Not only that. Would you like to translate it to Japan? Thank you. Not only that. In fact, this is going to be partially an answer to the question which was raised this morning by Daniel Andler about the proto-mathematical abilities of our own terms when they make knots. But I will begin by a somewhat philosophical introduction, so according to the rule in philosophy talks, I will sit down. This is the introduction. Assuming that mathematical logic is, among other things, a mathematical model of reasoning, why is it that when a mathematician understands a theorem, his mental activity seems to him to have so little in common with the logical exposition of the proof? The feeling of understanding is often, at least for part of the proof, of the nature brought about by the feeling that the mechanisms proposed are right in the light of our intuition and experience of how things work. It's more as if someone explained to you a climbing route with a way to surmount each difficulty to get to the top, and at the end you realize that, yes, this will take you to the top. It relies very much on your intuition, experience of the difficulties. In this metaphor, the logic of the proof would correspond to the programming of a robot to make the climb. And the mathematician is pretty much convinced that this is possible only once he has been there.

2:30 And then the programmer ceased to interest him. There is a fundamental difference in that the robot does not know what is significant. And in mathematics, the choice of statements is crucial. So I emphasize that usually a model of reasoning or a model of anything is supposed to correspond quite closely to the experiments, at least in its structure. And in this case, it does not seem to be the case. Why is it so? I think we have to look for explanation in the fact that a part of the mathematical construction is of a pre-verbal nature. And this gets me closer to Watana, that it's very interesting to explore in the higher manners what can be a trace or an indication of something which looks like what we think of as mathematical reasoning. So I'm going to propose in a route which has been explored in the last few years by and myself, that one should try to look at cognitive phases, at the way we understand our environment through our perceptual system as a possible place where to look for proto-mathematical reasoning. And the thing which I want to say in a very provocative way is that one should not look for explanation of the meaning of words I propose that, at least in some instances, meaning is of a pre-verbal nature. And I claim that I can give some indications that this may be the case using what we begin to know about the neurophysiology of perception. So my starting point is the mathematical line. And I claim that the mathematical line comes from our perception of space and movement. How? Well, our body knows of two lines, at least. One, which I call the visual line, exists in a non-trivial manner, because in our visual perceptual system, there is an equipment, there is a neuron on equipment in the visual area V1 in the cortex, which makes neurons are sensitive to directions.

5:00 cortex are sensitive to directions. And there are connections between the neurons corresponding to different points in space. So when I say directions, there are two things. There is the direction in space at which you point, and given that direction, which I mark as a point, so that you can imagine a point on the retina, there is also a direction transversal, so to speak. And so in our cortex, there is an image of the retina, what is known as the peak phenomenon, which is a logarithmic map, actually. And then if you have two points corresponding to two visual directions, and neurons, so these points, these directions, this is a very simplified way, but given a direction of vision, the directions to which the neurons are sensitive are organized along columns. And as you know, along the column, the direction turns around. And the same thing is true here. And then there is a connection between the two columns, which is done by axons in the cortex, which can be quite long compared to other neurons, which connect what happens at this point with what happens at that point. And then if it connects in a very strong manner, the neurons which correspond to the direction of alignments of the two points. So this is a privileged direction. So here and here, they will be strongly connected. And this point will be much more weakly connected to the others so weakly that we don't even know if it is connected. this allows us to know what a line is. According to Jacques Lignot, a line is a car which everywhere has the same direction. And this is a non-trivial statement, because our perceptual system can detect what a curve is, and secondly, it can detect that the direction is constant, thanks to this property. I'm not explaining how our visual system can detect what a curve is, because this is complicated and not completely understood yet. Anyway, you must believe me that our perceptual system gives us a motion of line in a non-trivial way, and I call this the visual line. We have another definition of what a line is, which is due to our vestibular system.

7:30 When we move, or even when we stay fixed, our vestibular system is a very sensitive system, mostly located in the inner ear, which detects all kinds of acceleration. It's an inertia, a central for inertial navigation. So our body detects movements, accelerations in a straight line, and changes of orientation, and also, of course, inclination of the head in a very sensitive manner, as I said. And this is essentially due to the fact that there are small stones which are sensitive to inertia and which brush against syria, which themselves emerge from neural cells, which transmit information to the brain where it is integrated. it. So, whenever we move, our brain, our vestibular system turns our brain how we move. And when we turn, it turns us and memorizes it. Now, what is a straight line? Essentially, a straight line is what happens when we move, and our vestibular system detects nothing. Because this corresponds to uniform translation. The only time when our vestibular system detects thing is when we are in uniform translation. No acceleration. Then another minor miracle happens because our body has a biological clock. In fact, it has several, but let us say it has a biological clock. And it turns out that the fact the inertial movement, what is known as inertial, without acceleration, corresponds to an uniform translation for the time of physics. It also corresponds to something for our internal or subjective clock. So this gives us a connection between biological time and physical time. So our body knows also what is a straight line on which we live at constant speed. So our body has two lines, the vestibular line and the visual line. And I claim that the mathematical line is obtained by identification of these two. Poincaré said a long time ago that he thought that the position of a point in space could

10:00 be identified with the movement we have to make, in fact he thought of the movement of our hand, to seize an object which is situated at that point. And I think this is a deep thought, and it has been taken up already a number of years ago by Alain Berthaud at the College de France, who has made a number of experiments to, first of all, give a meaning to that. For example, the gesture to seize an object is appearing not unique. You could seize like that. But Berthaud has shown that there is a unique, well-defined natural gesture, which is submitted to a number of rules. And then there is made very large number of experiments to understand this correspondence between the coordinates which you get by the tension of the muscles and the orientation of the articulations and so on with cartesian coordinates in space so for that reason the identification of the line of the vestibular line with the visual line which i think of as a generalization of Poincaré Statement, I call the Poincaré-Bertot isomorphism. So there is an isomorphism between the vestibular line and the visual line. And this has an advantage. The vestibular line does not come alone. It comes equipped with time. It's parametrized by time because it's It's a line of inertial movement. So this one has time as a parameter. The visual line has no natural parameter. The visual line is the definition of new. It's a curve which is everywhere in the same direction. So this identification produces an object which is parametrized by time. And this for me also has a question which has been on my mind for many, many years, which is why do people accept so easily that time is represented by points on a real line? has always puzzled me. And I think maybe the psychologists have a good answer to that, but I've never met it. So this Poincariotos isomorphism allows me, gives me something which I find satisfactory to understand why. Also, the usual line, the line which we see, can be thought of as representing time.

12:30 Now, of course, it's a matter of representation. But here we meet an idea which is dear to the heart of, you say, Pelongo, is that mathematical objects, like our platonism, is essentially the constatation that some things are independent of the coding. And that gives existence. Whenever something is sort of independent of the representation, we tend to think of it as an object. And I think this is a basic property of our mind. Maybe not of our mind as a primates, but of our mind as human beings. So if you think about it, this isomorphism is a way to produce an object which does not exist in nature, which is again a mathematical line, but which is somehow independent of the coding. So this is one object which has two codings. One is the visual coding and the other is the inertial coding, the vestibular coding. And we produce the object which is independent. You can check that an object exists and is independent of the coding or you can also given two objects which are very similar, decide to identify them, and produce an object which by construction is represented by two different queries and therefore is independent in itself. We just force it to be independent. So this is the basic vision of a mathematical line, which I want to propose. and now I make pronouncements which may seem outrageous I say this is a pre-verbal operation this has nothing to do with language it's an operation of the mind identification it's a kind of metaphor a very strong metaphor but it has nothing to do with language so it pre-exists to the linguistic abilities Secondly, the object which it creates is of a mathematical nature in the sense that it is a construct which is forced to be independent of the coding which I think is one trait of mathematical constructions And finally it gives us intuition which is of basic importance to us That's whenever we think of mathematical constructions. Because whenever I think of a line, I can think that I'm working around it.

15:00 This is really, really important. Or if I choose, I can view it as a visual line in my field of vision. And little by little, the two representations coalesce and merge by this point carrier vector isomorphism. And when, as a mathematician, I think of a line, I have a complete choice. line, I can code it as a visual line, I can parallelize it by time, and this is all the same. These are all attributes of this mathematical line. So, I think this gives you the gist of what I want to say, and let me just quote among the many experiments of the laboratory. You can generalize this, of course, to more complicated curves, you can think of curvature, and let me just say that the experiment has been made of verifying that the sum of the angles of a triangle is equal to pi. In the following way, you blindfold the subject, you ask him to start from a point, then move straight ahead for a certain length of time, because when you walk at a uniform pace, of course, you cannot measure distance other than using time. Then you ask the subject to turn of angle, which he may choose, and to walk again for some time, then he's here, and you ask him to point to his point of departure. He's blindfolded. You ask him, point towards the point from which you started. And with great accuracy, he will point to his starting point. So this proves that the sum of the angles of a triangle is equal to pi, because your brain, in a non-verbal way, has computed the angle beta, knowing the angle alpha, and the two length. This works. So, and again, I insist, this, of course, you have to explain to the subject, but this capacity has nothing to do with linguistics, I think. It's a purely vestibular inertial statement. So in this proto-mathematics, I say yes. Without any doubt. And that's why I think this is partly inspired by the lecture last Monday of Professor Okada, who talked to us about the intuition of objects, according to Husserl and to Hilbert. I think there is a way for me, neither of those, however brilliant and intelligent they are, is completely satisfactory.

17:30 I'm more satisfied with things like this. All right. Now, I promised some examples. So let me give you some examples. First, so I claim that this generalizes, for example, the vestibular line comes equipped with an addition because when you walk, you can stop. imagine this is completely brutal then start again and then you can add up the movements so to speak and this is what gave so much trouble to Zeno this sort of contradiction because the stop and start and the sum of the distances this really brought him into some deep meditation which we know as Zeno's paradox and it's not really a paradox it's an extraordinarily intelligent comment on this isomorphism, I think. It's an extraordinarily fruitful comment. Now, I've talked to you about lines and curves. You said mathematics does not stop there. That's true. So, for example, the next big invention, I think, is ARIA. And ARIA does not belong to proto-mathematics. This belongs to proto-mathematics, but ARIA definitely And how do I, I don't want to say how do I know that, but why do I suggest that? First of all, because in historical times, people did not know what area was. Our perceptual system does not give us any kind of reasonable measure of area. If you show people two lakes, which are of widely different shapes, they are completely unable to say which one has the greater area. this is attested by the fact that in Omer the size of the cities, for example Troy is measured by their perimeter because Aura is too complicated, you cannot say a city is bigger than another city other than by saying that its perimeter is bigger, because length belongs to proto-mathematics and is something of which we have a direct intuition Aura is not the case, so it's perhaps one of the first 3D mathematical constructs, that it was a necessity for agricultural people to have a concept similar to it. It's an economic necessity, but it does not belong to the necessities

20:00 of life for a primate. So it does not belong in our perceptual system. This is a very simplified But let me go back to how this kind of viewpoint can help us to understand more elaborate constructions. Another thing which the vestibular line carries is an order, because time has an ordering. There is before and after. There is a total order. and now we have already spoken today of orderings and linear orderings and not very long ago I had a great surprise because I want to talk to you about ordinances and I claim that I'm not moving too far away from the subject I say that the vestibular line helps you to understand ordinances what are ordinances? ordinances are well ordered sets it means any has to stop, and some people call them linear orderings. When a set is well ordered, some people say it's a linear order. I heard the great surprise talking to another theorist a little long ago to realize that he thought of a linear order as an order on a line. But of course he knew that this is not a real line, this is not the usual line, because some things have to happen. Let me explain to you what happens. I never thought of a well-ordered set as a linear order. For me, a well-ordered set is like this. Take two-dimensional space, and then look at the integral points. So every point is determined by two coordinates, So this is i, and this is j. Now, say that ij is less than or equal i prime j prime. This is taken to mean that either i is less than i prime, or i is equal to i prime, and j is less than j prime. So, for example, this point is smaller than this point. this one here is smaller than this one or this one

22:30 but this point is smaller than that point so this is a total order it is not a well ordered set so this is for a mathematician this is d2 with the lexicographic order now it's well known but this is not a well ordered set many infinite decreasing sequences, when we take any sequence decreasing like this, it never stops. So it is not a well-ordered set. But inside it, many well-ordered sets live. All I have to do is to put a stop on every vertical line. So I put a stop here, I put a stop there, put a stop there, and I put a stop here, and so on, for every vertical line. Then, I no longer have infinite sequences of decreasing points, and so I look only at, let's put a stop here, so here, and that, and that, and this is a well-ordered set. So it's linearly ordered, but you see it's not on a line. Now what is the advantage? How do I have an intuition of that? When I walk, I walk up from here, one step, two steps, three steps, and so on, then I go to infinity. Okay, I have to accept that I go to infinity. But then I know where to go next. I cannot go anywhere but here. So this point, so I start from here, notice it says zero. This point is called omega. Okay, this is the first infinite ordinal. Then I have omega plus one, omega plus two, omega plus three, and so on. And here I have two omega. then 2 omega plus 1, and so on, and then 3 omega, and so on. So this is, for me, a perfectly readable representation of the ordinary omega square. And as you see, if you try to represent it on a line, then you're like my magicist friend the other day. You make a picture like this, then you take a point, and then I have many points which are bigger than this one but smaller than all the others then I take another point and then I have many, many here

25:00 each one of these is actually one of my stopping points here but the picture makes it completely clear what happened let's start with the small one close In this way, I can go up to omega. And there is even worse. But then, of course, I cannot go to epsilon zero using this representation. I agree. Anyway, when I was talking about what makes the proof intuitive, this kind of representation makes the proof intuitive. If you want to prove something about omega square or omega to the n, and you look at it like that, immediately you understand if the proof is correct or not. This is what I wanted you to see. And basically, I claim that basically it is a juxtaposition of vestibular lines. Okay? And this is what builds my intuition. Okay, now I have to continue. How much time do I have? No, but it's all right. One minute. One minute. Okay, so this is one thing. The other thing is, I showed you the line. Of course, now I'll make a big jump. I'll go to the finite dimensional space. And thanks to Descartes, we know how to represent finite dimensional space as a product of lives. principle, conceptually, we have a tool, which is not, of course, not completely true, but I don't have much time, and I want to talk about function spaces. So just to make a summary what I talked about so far this contains a parametrization and is equal to the visual line just one direction it's not surprising in some way

27:30 that this part one could call it almost Galileo's package because vestibular lines as I explained to you is connected with inertial movement inertial movement is deemed to be an invention of Galileo-Galilei the group of inertial movement is called the Galileo's group and at the same time Galileo was probably the first in a completely coherent manner parametrized trajectories he is the person i think who saw trajectories as movement on a curve parametrized by time okay so it's the same mind the same mindset which is at work and this mindset i think is really in front of our eyes so to speak making this triple appear at the same time. Even though I claim that it is proto-mathematics, somebody had to transform it into mathematics at some point. And I think this is essentially the work of Galileo. Now I want to go to... Maybe a rescue. Not so much. Now I want to make a... Then we have many other things which I will talk about. And I want to talk a little bit about This is another, I promise to talk to you about geometric illustration of a mathematical idea. I think one of the next big idea is function spaces. What is the idea? The idea is due to Lito-Volterra, perhaps he had the recesses, but anyway, the important idea is that you think of functions, and for a long time, analysis, mathematics was connected with trying to find functions with specific properties like parametrizations of curves movement for example but at some point someone had to try to think of many functions together when you start doing approximation and Vito Volterra is recorded as being the first person to say that we should think of a function as a point in some space and this is I think a great leap and this is another case of

30:00 mathematization I'm not claiming that this is proton mathematics, I'm claiming this is a beautiful mathematical idea, so what happens? He said we can look at vector spaces that means only you can add and multiply and subtract multiply by constant vector spaces of functions plus a topology because as we had this morning on at least two occasions it's important to be able to say that two functions are close to one another and in fact in the study of topological vector spaces like this, the choice of the topology is always extremely important so what happens then in these spaces I'm just going to go very superficially but what happens is that in these spaces which are infinite dimensional you carry over your intuition of finite dimensional space and practically all the main theorems of the theory of topological vector spaces or functional analysis are in the first period in the period when this theory was born that means roughly the beginning of the 20th century Valterra's idea is I think 1890 or something like that and then Diodenay says that he could not develop it because the theory of topological vector spaces was not invented yet he forgets to say that it was invented because of that idea anyway this is associated with the names of Hilbert, of Banach of Fréchet beginning 19th century, 20th century, sorry. The idea is always that you carry over to these spaces your intuition of finite dimensional spaces. For example, if you have a topology, you know what it means to be a closed set. It means if you take a limit, it's still in the set. And then a hyperplane can be found which separates two disjoint closed sets. this is a an easy theorem in finite dimensions oh convex, sorry, yes, convex

32:30 otherwise we run into trouble this is a very important theorem in the theory of topological vector spaces because it allows one to find linear functionals which have certain properties with respect to subsets so many of the proofs of that era consist many of the theorems and also many of the proofs but not always consist of understanding that some basic fact of life in finite dimensions gives you a deep result in spaces of functions it gives you the existence of the measures the existence of the functions with such and such property Another one which looks very simple, which is that if you have a convex set, once again, it's the convex hull of its extreme points. The extreme points are those which I've drawn here, and the convex hull is this one's convex This means that any point in here can be written as the barycenter of these points with weights. So you give these points some weights, positive weights, and then you take the barycenter, the center of gravity, and you find a point inside the convex hull. And every point in the convex hull can be described in this way. If you generalize this to infinite dimensions, you find, I think it's known as Scherke's theorem, which is a very powerful theorem in the measure theory. So this is to, and how do you move? I hope you can hear I made a big jump, but you move fairly continuously from the intuition of the vestibular line. Vector spaces perhaps have more to do with the visual line. And then you proceed with your visual intuition in front of dimensions, and then you push it to infinite dimensions. Of course, you have proofs to make. It's not clear that these theorems are always true in infinite dimensions every time you have to give a proof. But the basic that motivates you is to generalize to infinite dimensional spaces, the theorem which you know to be true in dimensions 2 and 3. Yet another example, and I will stop about function spaces and this, oh no, one more example, is Hilbert space. Since Fourier, it was very important to develop functions into sums of complex exponentials

35:00 or series of sines and cosines enics. And Hilbert's idea was to create a theory in which this was exactly the correspondence of the decomposition of a point, the projection of a point on the axis of an orthogonal basis in some finite dimensional space. So you have a notion of orthogonality, and every point then has Cartesian coordinates like this, and it's given by its projection on the coordinate axis and then this is exactly the same thing except that you have infinitely many dimensions but you still have a notion of orthogonality and this is exactly the same thing so all your intuition about orthogonal projections in final dimensional spaces is carried over to these infinite spaces and helps you to find and to prove theorems. Now final example of function space. This time it's not a vector space. It's because this is in honor of our old friends, the space of knots. So from the mathematical viewpoint I said this morning, the theory is not at all terminated, but at least from the mathematical viewpoint, this is an embedding, a knot is an embedding, that means an injective map which is differentiable from the circle to three-dimensional space. Now, of course, if you look at all the maps, all the differentiable maps from S1 to R3, There are many degenerate knots. For example, knots where the curve meets itself. But you see where you have an ambiguity. Then you can lift the ambiguity by, for example, deciding that things would go like that. This is not a knot either. I mean, it's a trivial knot. But still it is a knot because the knot is no injective. and one of the other ways in which we can try to understand Mars is to say this is again a function space, this is infinite dimensional then it lives inside the space

37:30 of all the maps from S1 to R3 and inside that big space it's the complement of the maps which we call degenerate where the curve self-intersects the maps which are not objective big space of all the maps from S1 to R3. This is just a parametrized knot. Inside there we have a hypersurface which is going to be extremely complicated. So this is a space of infinite dimension. Here we have this complicated hypersurface where corresponding to the knots which are not injective. And then the complement of this hypersurface is the space of knots. So we have a geometric representation of the space of knots as the complement of the hypersurface in an infinite dimensional space. Now from that representation alone, we can find a lot of information about the classification of knots. This is the idea of Tom and Vasiliev. You start from a point which is somewhere in one of the connected components, And then you see what changes when you move, and you can make a whole theory of invariants of knots by deciding how you're going to measure what moves, what changes when you move across the boundary. This is a theory which is very much alive now. now finally I'll go back to final dimensions for a little while I want to show you how again intuition proto-mathematical intuition helps us to prove Florence we know that there is a projection the plane couples of real numbers with the mind. We know that they have the same cardinality. There's a bijection. This is very embarrassing because it means that every function of two variables is a function of one variable, which is sort of contrary to our basic intuition of the world. Why? Well, this bijection is very bad. So we want to prove that there cannot be a continuous bijection between the two. this just reassess

40:00 otherwise there is no meaning to dimension otherwise R2 is just like R so what is it when there is no continuous projection when I think we know that if again we take the vestibular line and we remove one point if we remove one point it becomes disconnected you can no longer pass from before to after if you decide that the present does not exist you can no longer go after in a continuous manner. So this is true in R. But suppose we are in R2, and suppose that R2 is topologically is homomorphic to R. That means there is a continuous bijection whose inverse is continuous. Well, it cannot be, because in R2, if I remove one point, what remains is still connected. I can go from any point here to any other point continuously. So this proves that like this cannot be a homeomorphism, because I've built a contradiction. If there was a homeomorphism, then the complement of a point would be connected and non-connected. And this is a very basic, primitive intuition of space. I think you won't agree with me. but then the theorem is not trivial the theorem has been annoying to people of the time of Kantor so in conclusion I think it's time to stop I'm trying to convince you that it's interesting to try to understand how pre-verbal notions and pre-verbal constraints like logical Logical constructions like successor, like order, like comparison, all these are, I think, of a prerable nature, and they intervene all the time when we do mathematics. I claim that even the existence of some mathematical objects, like the line, rely ultimately on some kind of need to identify various perceptions that we have of the world. That mathematics is actually a process by which we try to digest the world by constructing simple objects, and maybe the simplest is the line, but I think there are many more complicated objects which are also of a preferable nature, and then of course what we call mathematics is certainly not of a preferable nature, and there is some kind of jump, as I tried to explain, with the concept of Aya.

42:30 The concept of aria, I think, was a source of incomprehension at the time when it was born. If you look at the elements of Euclide, a large part of the elements is devoted to comparison of arias of simple shapes, like rectangles, like elementary geometric figures. But this was taking place at least five centuries before Christ. On the other hand, we have the traces of judiciary processes in Greece in the first century after Christ, that means 600 years later at least, where people were fighting because they had decided to share the land equally. And in the rural Greece, sharing the land equally the same perimeter, six centuries after Greek philosophers knew about era. And so at the time of harvest, this created some clever people took round fields and left to the others those fields that were long and narrow. And at the time of harvest, of course, there were some disappointments, frustrations, which led to judicial processes. And thanks to these judicial processes, we have the trace. So this, again, emphasizes the fact that ERA is not a proto-mathematical concept. And I leave you with that because I don't have any more precise idea. But I still think that it would be very interesting to look at the history of mathematics from the viewpoint of how mathematical concepts developed in a pre-verbal environment. Thank you. Well, it's time for discussing this talk, so, okay. Okay, I will think of the general lines of your presentation, which I have is very

45:00 in particular in focusing the overall project that we have in mind that we share in discussing foundations of mathematics in a context like this. The point is that we have to recover something which has been analyzed for one century in the discussion in foundations of mathematics between, partly between the formalists and the platonists, the debate and this kind of things which are really been taking us back of centuries. But the main issue here is to recover the relevance of proposing structures, which is a key part of the mathematical activity. Since Frege, or even the early Husserl, there has been a focus on logic, and logic, including arithmetic, according to Freges, has been taken as the paradigm for the foundation. Of course, and logic are relevant part of the mathematical activity we get to know by proofs of course there's no doubt but by this we've missed the point of what is the proposal of structures which is absolutely crucial in the mathematical construction from proposing a triangle add to a fiber on a manifold I mean all this is part an integral part of the mathematical activity and what happens that it is also part of the proof my insistence since a few years on concrete incompleteness is exactly to point out something that cannot be seen in Gödel's incompleteness because it is a diagonal argument but in concrete incompleteness we see that even in arithmetic there are same proofs where in order to go from one line to another one has to use a construction in particular, as part of the passage from one line to another, and it is provable, unavoidable. So along the proof, there is an embedded meaningful construction. This is why I consider the major failure of the logicist and formalist program in exactly missing, even in proof, the component of the proposing concept and structures, the key activity in mathematics. Of course then, the foundational discussion is going back to a genealogy of concept and structures. How we get

47:30 to constitute along a path which is phylogenetic and historical. A friction of a reality which is made by our species and our collective human activity. At core of it, of course, there is constitution of invariance. And now I get to your main constructions. Constitution of really the key point. And a way to constitute invariance is, of course, independence from specific activities, specific praxis. And in a sense, you've been telling us a crucial one. The different praxis, which is the visual praxis for a line, and the action and movement which is behind the idea of inertial line. The blending of the two is a way of constituting an invariant. And from this, of course, we get to a concept. But that's much earlier than the concept. And I often mentioned another example, which is strictly linked to you, also derivated from the experience on saccades in Berto's book. We know that animals with fovea precede the prey with a saccade, an aqua saccade. They do not follow it. They precede it in order to get to it. And then they describe, in another saccade, the line reaching the animal, which is a very difficult mathematical line. In this case, by concrete action, in action by AIs, there is a practice of an axerotl notion, which will be ours, that of line. For example, the memory of these predictions, of this blend of prediction, which is a complex already mixture of action and vision. in animal practice the memory of such a prediction is very absurd because there is nothing there it is it is the second it precedes the movement that's why we can then propose and construct something as subtle in the language as a parameterized line of a cantor that it can continue for example not only we get to propose it by all this path but we can communicate the meaning of a linguistic expression, by gesture to other humans, because we share these common practices behind us. So that blend you mentioned is really at the core, and examples perhaps don't need to be many.

50:00 I mean, the counting and ordering will be conserved with numbers, and this kind of activities will be behind some basic geometric activities. It's possible that, after all, the grounding practices of the construction of concept and structure are not so many. I mean, bordering lines, you know, crossing a line, this kind of activities. Of course, then there is a mountain which has been constructed by language, and of course, logic and direction, and so on and so forth. But also concerning logic, it is clear that the peculiar status of mathematics between our practices for knowledge, linguistic and pre-linguistic and for action is that I would call it as a maximally stable and invariant from the point of a concept so anytime you have a concept which is maximally it's not a maximum I don't like absolute I say maximally stable the concept and the concept I'm sorry a concept which is maxially stable and invariant in a human practice, whether linguistic or even pre-linguistic, then we are doing mathematics. So not only mathematics as such, but it can really be taken as the many activities we have, as the one which by definition proposes conceptual stability and invariance. And in this frame, one can also single out the role of logic. Of course, logic is absolutely relevant in the mathematical activity logical structures are invariants of the discourse they begin on the Greek agora not by chance because by reasoning they got to have power a novelty in human history probably in the Greek agora they were written well yes I don't know very much in history I'm telling you of our history and it would be nice to compare, of course. In our history, when reasoning becomes a component of the human social activities in particular to get to power, then some invariants stabilize in the discourse to get to power. And this is the core of mathematical deduction and reasoning at the same time, not by chance. This is really a perspective that we are proposing in our discussions that I see

52:30 are progressing and these remarks of yours are seemingly interesting for that which is a reversing reversing the usual attitude in logicism and formalist which have been dominating for a century the foundation with lots of enormous fallouts of course not only an enormous technical amount of work in mathematical logic but machines which are logic and formal of course and these machines are the most important instrument we have invented in our life so of course it's very important but both if you want to understand more and if you want to think to the next machine because that's also important activity which may be radically different we have to broaden the perspective and the reversing of the paradigm namely to focus on the formation of invariants by a blend of different praxis and the cumulative structure of different praxis, it's really reversing the absolute on which it has been grounding the foundation of mathematics. This, I think, is a crucial point. I mean, with Frege in particular, there is this focus on absolute, which are before human negativity and which are the logical ones. and independent of human activities like also in other platonist views with this attitude there is a way of constructing science in that case mathematical logic which goes back to the way science has been constructed by galileo and newton of absolutes exactly in the moment by the end of the 19th century beginning of 20th century when it became to the limelight we had to construct relativizing science and there was the understanding of the dynamicity of the construction of knowledge in the summary we have to reverse the absolute of necessity and absolute even in deduction by saying something perhaps a surlain but very strong which i i care very much as to stretch this constitution of mathematics is contingent any constitution is contingent and by these it is effective and objective. That's the strength

55:00 of mathematics, of being contingent of our history, phylogenetic and human history, and because of that being effective and objective. The exact reverse of the usual paradigms. And I have necessity in the absolute. That's I think something we share and entirely derive from the approach I've been presenting to us today. Thank you. I cannot answer in the same length, but I just want to say, of course, we know I agree. It is true that what we propose is a vision of mathematics as contingent. I just want to make one comment. It is only in a purely formal view that one can be surprised of the effectivity of mathematics. If you take this contingent view, then the fact that the so-called unreasonable effectiveness of mathematics becomes totally reasonable. It's just because our perception brings us so close to the world that the objects we build starting from that perception also remain very close to the world. That it can lead to surprising computations. lack of imagination concerning computation, but it is certainly not due to anything unreasonable. I just would like to make a comment on the question. I think you're very much in the same line. Of course we know that line is a basic concept in geometry just in the sense that that's how geometry is constructed to explain geometry we start starting explaining things like points and lines and i just wonder what what is new which is brought by this sort of empirical research on perceptual apparatus what What is new is brought by the fact that we really make all this, you know, fine management about vestibular line and visual line and so on. And I guess it's an orthodox Kantian would find sort of methodological vicious circle in that, right?

57:30 because as far as I understand at least those measurements are made with some sort of Euclidean space-time physical space-time a sort of underlying geometry which is recovered in conceptual apparatus okay and if this is true okay I'm not Kantian and so my interpretation and I I agree that there is a circle but i don't think it's necessarily vicious circle right so we find sort of coherency between geometry which we apply as physical geometry of the world or geometry of our measurements and what we really find in perception but actually then the question is and why i say it's maybe similar to comment of giuseppe longa i just sort of resist this idea that this result mean that that sort of basic concepts really in built in our I don't know biological constitution or whatever because think for example about what Tarski suggested more as a logical game than really a mathematical concept he called it ball geometry right so we introduced ball and we think about balls and then some sort of primitive relation between ball and then we can build up or of geometry which at least partly recovering sort of standard euclidean geometry and it's not impossible or that's a question i don't know what do you think about that that if we apply this sort of different geometry or just say non-eclidean geometry as basic for measurements and for these experiments about vestibular apparatus or whatever isn't this impossible that we find this other concept just as in-built in our perceptual apparatus as we we find the lines and whatever and i i would i would be inclined to to reply in positive and in that sense i would just well agree with giuseppe that mathematics is contingent and not really inbuilt in in whatever physiological or psychological way. Well, if I can answer, I think we specifically agree on what it means to be contingent.

1:00:00 It's contingent to our human nature, including our perceptual apparatus. And then to answer what you say, I think perhaps conscience would agree there could be a vicious circle but not Kant himself I think that Kant would be happy if he could see that there is a sort of construction of space inside our perceptual system now what you suggest if I may say is quite coherent with what people have made out of Kant later but then it's more it's more a play of words I think than a real analysis Because if you look at what we begin to understand of what goes on in the visual area, for example, this has nothing to do with Euclidean geometry. This has to do with biology, with the way the retina is mapped onto the visual area. Of course, it takes place in Occlidean space, but that's precisely the kind of play which I find sometimes irritating in the followers of Kant. I think this is not what the problem is about. The problem is about what is the nature of the mathematics which we build. Kant, I think, from what I understand, tried to say we must start from somewhere, and I call this a priori, but obviously we did not find it satisfactory. this was covered by a layer of the philosophy which Giuseppe was denouncing, explaining words with other words, and even more words. And I don't think that's what Kant proposed that we should do. And so maybe Kantians did that later on, but not Kant himself. So in doing this, I don't think we contradict what Kant would like to have us do. I think contradicting some later interpretations which have a lot to do with not wanting to look at the problem itself but looking at the words which we think describe the problem and so that's my answer i think it's time to stop and after a week of a few minutes we have the next talk starting at

1:02:30 Thank you.