Lunch conversations

0:00 Interdisciplinary arts and sciences. Okay, giving an interdisciplinary talk on elementary geometry groups and fields. Thanks, Victor. Thank you. I think I'm here as a specimen of an unrepentant and irreducible geometer who probably has given some thought of why one does geometry. From somebody looking from the outside at geometry, but simply from a practitioner who philosophizes a bit. So, I was to, you know, when I got the question about thinking geometrically, I was thinking, okay, let's see, let's narrow down the geometry and ask ourselves within that narrow view, Whether we can give an answer to the existence of some kind of a geometric thinking and set myself this framework of elementary geometry by which I mean any kind of geometry that can be axiomatized in first order logic and we're wondering whether thinking geometrically makes sense inside this structure. If one is a practitioner of geometry today, by which I mean the mathematical reduced classification 51, one is on the defensive, because today geometry, when one says geometry, one expects that it's some kind of a topological, differential, algebraic, some kind of a respectable form of geometry. But for the question of thinking geometrically, maybe this is the best vantage point to give an answer because there is nothing else. We just have geometry here.

2:30 So the attack usually comes for geometry from either the topological point of view or from the algebraic point of view. And I'm going to present here the reasons why some would think that there is no geometry. And I'm going to try to refute them. So the first arguments will be whether elementary geometry is algebra in disguise. In other words, people would say, well, geometry is some kind of a funny way of talking about algebraic things. In other words, using some kind of an arcade vocabulary for really saying algebraic things. That's the prevalent idea, I mean, the idea of Euclid must go, of Diodone and so on, was that, you know, this is some kind of archaic thing that we have to get rid of, you know, you can present that in some kind of a modern fashion, just algebra. And a structure that will come to mind that you would think, well, geometry is that, would be field theory. Once you have elementary geometry that you can embed in a projective plane, If you have a field that satisfies Papus, you will have a field there, a commutative field. If the geometry that you can embed it in, if the projective geometry is at least of dimension 3, then at least you get a skew field, you still get something. If you assume that it satisfies some form of elementary dedekind, so some form of continuity, the first one who wrote down actually what that means was Tarski, you can ensure that the coordinate-izing field is a real closed field. And interestingly enough, real closed fields are decidable. So if you think that the task of geometry is, that are formulated in its own language, some kind of a geometric question, then this solves it.

5:00 There is no geometry. You ask a question that is geometric, you feed it into this machine, it transforms it into some real closed field sentence, you use the algorithm. Just a question for a real close. Of course, in terms of field extension, this contraction of real encompasses, it wouldn't give you all real numbers. No, but I said if you assume Dedeckin continuity, right? In other words, you take Dedeckin's axiom, but your cuts have to be defined. Buy some formulas in your language. So they're first order cuts, put it this way. If all first order cuts give you something... Then this is the same as, and if you don't like continuity, don't like it, do it this way, make an axiom that says every positive has a square root, and put the axiom saying that every odd degree polynomial has a root. And that's it. You don't need it to have a real closed field. Yeah, no, this is, this is, this is clear, real closed, so if you don't like those axioms you can use these and you get this real closed field, right? So if you think that that's geometry, that geometry is some kind of an oracle where you go with your little question and the oracle needs to answer it, there is no need for geometry at all. You have now the method to answer your questions. That's it. Now, stuff that is about geometric thinking in this setting, if you think that that's what geometry is, can be reduced to algebraic one, so there is no geometric thinking proper. Now, if you're elementary, yes, but if you can reduce it, then we can be allowed to rest, right? This is the prevalent view, right? And it is justified by these theorems. It's not just a bias against geometry. The bias can be justified in this manner. Geometry has a finite axiom system because the stuff with the real closed requires infinitely many axioms, right?

7:30 It has a finite axiom system, say, such as the geometry of ruler and compass they were just mentioned earlier. The field theories associated with it are known to be undecidable. This is what Ziegler proved in 1982. So in this case, your geometric question to an algebraic one is an academic exercise because you won't do anything with the resulting algebra. It's bad as the geometry. So you probably have a much better chance to get an answer if you're pursuing a geometric strategy. So reducing it to its algebra, if the algebra is not decidable, is not an interesting exercise. The idea that there is no geometry comes from the idea that via algebraization you get rid of the geometry, it becomes a computation, and the computation is doable. But in this case, the computation isn't doable. So here is, I poke kind of holes into the attempt at the reduction, right? So, that you get a field with these geometries is called mutual interpretability. In other words, if you have two different theories, and they have different interpretations for the individual variables, and you still think they represent the same thing. For example, you have geometry based on points, and another geometry based only on lines. They both axiomatize, say, Euclidean geometry. How would you say they are equal? Well, there is this notion of mutual interpretability. You can say in the universe of these people with lines what the point is and vice versa, and you can express, you can give definitions for all the relations with the line people and the point people, and you get a translation where everything in one direction is true and the other as well.

10:00 Now, the same thing is true with geometry and field theory. Euclidean geometry, for example, and the corresponding field theory are mutually interpretable. In other words, in geometry, I can find that these points, I can identify them with two variables from the field theory and vice versa, give definitions of everything, and they're the same. Now you would think that, well, maybe all theories are mutually interpretable, but no, they're not. There are a lot of points in the real closed field that are not constructible by the . That doesn't matter. It doesn't matter if they're constructible or not, because I'm saying the corresponding field theory. So, if you have the ruler and compass, you would have an Euclidean ordered field, and the theory of Euclidean ordered fields and the geometry of ruler and compass are mutually interpretable. Much depends on how the properties, precise properties of this interpretation, the mere existence of... This kind of interpretation is very weak. On the other hand, if you say one interpretation cancels the backward in this asset, composes to identity, that's very strong. So you have the whole spectrum of possibilities. No, no. The concept of this interpretation has been worked out by several authors in the 60s. They tried to figure this stuff out, right? And it's quite easy what it is. And then you have to be able to define your primitive notions from here in terms of the other one. And then with these interpretations all the axioms from here should be provable here and vice versa. And if you think now that everything becomes mutually interpretable, it's not true. For example, view theory and piano arithmetic are not mutually interpretable. It's not really that everything becomes the same. So, with this view that it is this, it's kind of, for a geometer, you would say that something is being lost in this identification.

12:30 Because, look, complete Euclidean hyperbolic and projected geometry all lead to real closed fields. Yet it seems that they all say a different story, and the storyline has been completely lost. So by doing this identification and saying geometry is algebra, even when it is possible, like in this case, and not only possible, even when it is efficient, because you get this decidability machine, right? You're losing really all the poetry that is geometry because the stuff that it says is not there. So is it the case though that if you take, if you break up theory of real closed fields into parts that... The real closed field interpretation of projected geometry corresponds to a part. The other kind of geometry corresponds to the same part. Yes, yes. And this means that also they themselves, Euclidean and hyperbolic geometry, are mutually interpretable even without going over the fields. You can define universes in one where the other happens, and there is no better, I have not found anywhere, a concept of mutual interpretability where this storyline will be preserved. You cannot get something like that, that you would preserve the storyline. I don't know, so there is this stuff that we feel is there, but logic really doesn't allow you to express it. So, now, usually the people who think there is no geometry present geometries in a model-theoretic manner, so algebraically. They tell you what the universe is. They say Euclidean geometry is R squared, and this is the norm, and this is it, and I'm done. And they tell you, after they tell you what the norm is, they tell you what is congruence of segments, in terms of the norm. And then they say, well, elementary geometry, this thing that used to exist, is in fact the theory containing all...

15:00 First-order sentence is true in this structure or in this class of structures, because I may say, well, Euclidean geometry is not really r squared, but it's k squared. And then, well, k is some kind of field, and then it's the stuff that is true in all these structures. Now, this is a very nice platonic thing, but geometry really is interested in organizing that structure, in axiomatizing it, in finding, well, What are the sentences of that structure? And this algebra I can do nothing about. Absolutely nothing. It cannot tell me what are the sentences, what is an axiom system. It cannot help me in any of these syntactic questions. So, if I ask, well, I'd like to define a geometric notion in terms of others, would it tell me anything? Nothing. Would it tell me, is this axiomatizable in terms of this notion? There is no algebraic machine that would do that. So, in fact, when one presents geometry, as is often done in this manner, you present the model, you explain what the relations are in that model, and you think you're gone and you have solved the problem. One kind of thinks that the relations that hold there, the whole geometric story is irrelevant. By just having named it, by just having said who this structure is, itself and all its truth is already included in the fact that, well, you just said what it is. Is that true for every axiomatization? Yes. I mean, I don't think that any axiomatization can be gotten from just saying, describing models. You see what I'm saying? Right, yeah. So, in other words, when one describes the model and says the geometry is the theory that is true in here, This is certainly a set of sentences, but to figure out what's an axion system, to figure out what's a minimal one, and so on, all of these questions are inter-linguistic things of this theory, and that is, I think, the geometric part of it.

17:30 Okay, but that's what I'm wondering about, because your sentence says that You can read it this way. For every axiomatization of this theory, you have to have geometrical thinking in order to find that axiomatization. That's a strong claim, because it could be lots of axiomatization. Correct, but what I'm saying is when I have introduced, suppose, which is historically not true, that we have introduced a theory like this, right? In other words, we've given a model first. Sometimes it's true for the more modern ones, if leading geometry isn't the case. We've given a model here. And now we're interested in organizing the truths about this class of models. Exactly, the theory of the model. And the question is, is this algebraic description of the model gonna be telling me what the axioms are? Right, I see. So it's a negative way. It's a negative way. The answer is no. Yes. So I think that here I see only because the algebra is usually telling me something about certain operations that hold here. But the geometry that we're talking about is about certain relations that have an algebraic description. So what you want to say is that even if you present the geometry as a model of an algebraic theory, in a sense the geometry is in the model. It is in the model. It is in the model, not in the algebra. In the algebra, that's what I'm saying. Because the algebraic structure is a model of other axioms that we know are, and they're just a way to describe the things, but the relations themselves are still mysterious. How can it self-describe itself, which is the question of an axiom system? That is still open, wide open. Now, I'm going to give a few examples found in mathematics that is not geometrical, as I would call it, have a geometric equivalent that you cannot find by any method, but that exists and that in a few cases I was able to find it. I'm talking about these, this area that is called characterizations of geometric transformations under weak hypothesis.

20:00 Now the origin of this may be going back to maybe Darwin. Things like that preserved circles have to be in this manner. The Mazur Ulam theory says that an isometry of normed spaces must be linear. So, in other words, I also call these geometric miracles. That a certain mapping that you think does only this, miraculously does other things. In special relativity, for example, Zeman has this article on causality implies the Lorentz group. The miracle is here that the mapping that just preserves causality and special relativity must be linear in and out of itself. Now, all of these, if you look at them with the geometer's eye, they speak about certain definabilities. Must be definable in terms of those things that you know the mapping preserves. And the definition itself has to have certain nice properties. For example, if your mapping is not assumed to be onto, and yet you get the miracle, it means that the definition must be purely existential. If you know that if the preservation is only of the notion, but not of its, it's assumed only of the notion, but not of its negation as well, then you know that the definition must be positive. In other words, none of these, the notion that is preserved, is allowed to occur negated. So you're allowed only to use and, ors, any quantifiers. But no negation at all, no implies either. So that's what, who tells us this? A combination of Beth's definability theorem and Lyndon's preservation theorem basically tells us this story.

22:30 Now both of these two theorems, when you prove them, the proof is highly non-constructive. So out of this theory that somebody proved, you cannot get the definition. The definition is a geometrically creative act. You have to get it out of thin air. Sometimes the proof of the theorem itself tells you what to look for, because the proof is in fact a hidden definition of this. But at other times the proof gives you, uses things that are first of all not first order because real mathematicians work with the real numbers and they use the properties of the real numbers that you don't have in these settings. And then you have to actually come up with something that has nothing to do with the proof. Get a better idea. Okay, here I say that this is equivalent by this definability theorem to these definitions, right, and that you have certain conditions on these definitions. And here I say that it's not constructive, so stuff that I've already said. Here is the theorem. I put it Cara Theodori, although as I said, people say that it goes back to Darbu and so on. He didn't have a problem to think about, and he thought about what happens to mappings that preserve circles. Well, in modern version you would put it this way. Let's take the completed complex plane. If you don't like the completed complex plane, think of a sphere, the circumference of a sphere. And you have a one-to-one map that maps circles. These complex people believe that the real lines are circles as well, On to circles, so they just preserve circles. And then F is either a Moebius transformation or a conjugate Moebius transformation.

25:00 What are these things for us geometers? So this is proved with, you know, all the machinery of C and so on. We geometers don't have the luxury of these powerful structures. So, all we can do is think of what is the underlying geometry, the skeleton. And the skeleton of these things has been quite well researched by Walter Bentz. He wrote this where you have all of these geometries that used to be, you know, Möbius, Laguerre, and so on. Which were all done in the real context. He did it in abstract context over any field of characteristic different from two in general is the requirement. And he has a nice axiomatization, just completely elementary. There the Miquel theorem about some circles plays the role of the papacy. But fine. Now the question is this. What does this say? Now, this is equivalent by Bente's definability theorem together with Linden's preservation theorem, with the statement that circle orthogonality is positively existentially definable, negated equality is allowed, because this is a one-to-one map, in terms of point-circle incidence in Moebius geometry. In fact, in Miquelian Moebius geometry, but I spare you the details of it. So, Muender's geometry is what? Is the generalization of, you don't have now C with a hat, but you take any field and you hat that if you want to, you take things on a sphere over a field. You have your circles just as they are here. Here is the definition. Nowhere in the theorem is there any, so the two circles, K1 is this and K2 is this one. And the definition says exactly what you see on the picture. Namely, it says there is a point A, P, M, N, B, C, Q, and there are all these other circles that you see there, so that all the incidences that are seen on the picture actually hold.

27:30 If they do hold, then the two circles K1 and K2, which I was just pointing at, K1 is this, and K2 is this. Must be orthogonal. So if they are orthogonal, then you can always find these points. And if you can't find these points, then they are orthogonal. What is your notion of orthogonal like systems? Well, yeah, for intuitively you can think of these two circles intersect, right? At the point of intersection you can think each has a tangent. Well, the tangents there ought to be orthogonal. And what is the definition that amounts to? Well, what it says is this. So, is this existential? Yes. Is this positive? Yes. It doesn't speak about anything negative. The only negative thing it speaks about is K1 is different from K2. It's the only place where you find a negated equality, right? The definition of circle, so instead of saying Caratheodoris theorem, instead you're saying this. Caratheodoris theorem follows immediately from this, but from its... I thought it precisely, I mean, you're saying instead, but, so you prove the theorem according to which you can have this definition. Is this required? No, no, no, no, no, no. It says something else. What I'm saying is this. There is a theorem which is a mixture of Bethe's definability with Linden's preservation theorem that says this. The following two statements are equivalent. Is a positive existential definition with negated equality allowed? Of the notion preserved, namely, circle orthogonality, because if you look at what Kara Theodori says, it says this, if point-circle incidence is preserved, then so is circle orthogonality, because this mobiusness is the same as some kind of conformality. It keeps the angles. And if it keeps the 90 degree angle, you would believe that it keeps the other ones as well.

30:00 So it's preserving the angle that's the miracle. Now the miracle here is defined by saying that all of these things exist and it's defined positively existentially in terms of point, circle, incidence. What I'm saying is that from the theorem itself, from the proof, from the existing literature of the proofs, you cannot distill this one. And what I'm saying also is that the syntactic equivalent is preferable to the model-theoretic one. The reason why it's preferable is that one direction is trivial. If you have this positive existential definability, then the theorem is yours in one step. Vice versa, to get this from this, this is what gives these theorems names. It required the best of Linden to prove these things. And it's sophisticated and it's non-constructive. So in other words, here is what I think is the geometrical thinking in providing the syntactic equivalent to these characterization theorems, which show you the different ways of thinking. Yet they couldn't get this. So it requires a different kind of thinking, although they're so-called equivalent, equivalent via some non-constructive proof. ...is that, proved in 1998, that surjective self-maps of the n-dimensional hyperbolic space, with n greater or equal than 2, over the field of real numbers, Collinear points into collinear points must be hyperbolic motions. So to put it differently, to get out of this sophistication, let's think two-dimensionally what it says. I have the inside of a circle and a map that moves this inside of a circle to itself. And all it does is it maps three collinear points into three collinear points.

32:30 Collinearity doesn't mean that they have to be different. It can theoretically move three points through like this, and to three points where two coincided, right? That would be preserving collinearity. Now, it is known that in hyperbolic geometry, the metric is definable in terms of collinearity. These maps are one-to-one as well. So the miracle these proclaim is that if you preserve collinearity, then you're going to preserve point difference as well. Now the actual proofs here use the real numbers and a bunch of algebra. So you can't get really the definition. Positive in terms of collinearity, because it doesn't say that it preserves non-collinearity as well. And this definition must be, by necessity, one in L omega 1 omega. So it must be in infinitary logic, because this will no longer be true unless the field, the base field, is Archimedean. So here is the equivalent definition. By the way, the Beth-Lindon theorem works in L omega 1 omega as well. The equivalence between the two notions as well. So, it can be positively defined, and there is a definition, I spare you the definition, it would take you pages. But this gives you the geometric reason why things are happening. Again, the two are equivalent. To get from one way to the other, there is no method. I could do this only for n is equal to 2. I don't know how to do it for higher dimensions. It tells us that a self-map of the affine plane over any subfield of the reals that preserves both the betweenness and the non-betweenness relation.

35:00 So betweenness means this point is between these two, right? Must be an affine mapping. Again, the equivalent is that line parallelity. So if two points are parallel, you know, the line formed by two points AB is parallel to the points. There's a line formed by two points CB. All of this is positively existentially definable in L omega 1 omega. In terms of negated equality, because you're allowed to use that, and the Turner relation B of betweenness. So I was able to show more than that it preserves B and not B. All you need from not B is negated equality. F-line fields, so in Archimedean order, F-line geometry. Again, there is, proof was using heavily analysis, so they used the structure bar. This uses none of that, it's purely geometric. You have to actually produce a definition with lines and things, you know, there has to be a picture there that is understandable. Now, if geometry can be seen to be something else, is that it can be seen as root field. There is a connection between geometries and their groups of transformations, and there is some belief that if you know the group, you know the geometry, so there is a certain determination of it. Now, in this elementary setting that I'm interested in, Arnold Schmitt and later Friedrich Bachmann in 1959 have presented a very simple action system for metric planes, expressed entirely inside group theory. You can express this in a one-sorted language. The variables will be interpreted as rigid motions, and it contains a unary predicate for lines, so you need to, you have basically elements that you don't, they're rigid motions, right, you don't know what they are. And others, which are some, some distinguished elements that you know are line reflections, and these are

37:30 I get a unary predicate symbol to just not say that I have two species of things. So I have rigid motions, and the others are line reflections. And I have a constant symbol, one which is the identity, and I have a binary operation which gives me the composition of two rigid motions, as you would expect if you want to get a group. Elliptic geometry. What can be said purely in metric terms if you forget about order and you forget about free mobility? What is left then? This is the idea of metric place. What it says is there are at least two points. For every two points there is exactly one line incident with them. There is an orthogonality relation on lines. Lines will be denoted by lowercase letters. The orthogonality relation is symmetric. To every point, to every line there is a perpendicular, which is unique if P is on the line, else it's not necessarily unique because you want to include elliptic geometry where you have more of these. It's a primitive. Yes, it's a primitive. In this setting, it's a primitive. Yes, points are capital letters, so we have two sorted letters. Now we have the notion of reflection in a line. This is not really an axiomatics, this is more of a model axiomatics. But to give you the idea, I put it this way. There is a reflection in that line. Here we define a rigid motion as something that preserves point-line incidence and orthogonality. And it's one-to-one and not, right? So, to every line there is at least a reflection in that line, and then the famous three-reflection theorem, which says that the composition of reflections in three lines that either share a common perpendicular or a common point must be a reflection in a line. This is all you're asking. And again, a whole lot. Now, in other words, those that are not for geometry, so where you know that the perpendicular is unique even if the point is not on the line, can be axiomatized also in a language with two sorts of variables. One for motions and the other for points. These are group actions, as you would put it, right? The group is acting on this set. This was shown by a student.

40:00 Bachmann in 1966 by Müller. It can also be axiomatized with one sort of variables that are points, the perpendicularity relation. So all you need to say is that there are points and A, B, C are perpendicular means that here you have a right angle at A. Now this again is something very geometric. If you give the description in other terms, say in group theoretic terms, it's not clear at all how this can be done. So to produce any axiomatization of a theory that you think is there, requires some kind of coming up with what are the axioms going to be. And in this case, the axioms weren't there. We didn't have a nuclear place to go. The axiom system can be expressed by means of universal existential axioms, so this gives it even more interest. All models can be embedded in projective metric planes, so projective planes where you have some kind of a metric which is given you by, which is described by a quadratic form. The projective plane here will satisfy Pappus, so you would have a, you would have as coordinate-izing field, a field of characteristic different from two. Now, as far as mutual interpretability is concerned, the three different versions are all mutually interpretable. So you can think that geometry in this setting turns out to be group theory, because that's what it is. Is there any advantage to getting rid of geometry? In other words, if you're doing the group theory, are you going to get farther ahead? And the answer is no, because this group theory is undecidable, just as the geometry is, so we don't get too far with that.

42:30 I'm going over to a different subject where I think that the geometric spirit is best described, which is reverse geometry. Just if I a bit reverse geometry, I thought of coming up with some kind of motivation for doing so. One would be going back to Papus and the other would be going back to, well, philosophers. Well, Husserl thought that this, in his He tells us that this space that we think of today is in fact Galilei's creation, that it's this mathematization of space that we believe is actual space, but this is not the actual space that the, that in the, in the world of the, of the We live the experience, we experience the intuition, we live according to our personal bodily way of being, and in these, in this, we find nothing of the geometric idealities, not geometric space or mathematical time with all of its forms. Here I have a poem that inspires me in this sense. No, I'll give you the poem, the poet after it's done, I'll read the translation. When no one looks at her, the sea is no longer the sea. She is what we are when no one looks at us. Indeed, she has other fish and other waves besides. She is the sea for the sea and for those who dream of her, as I do here.

45:00 The secret C, you see. This C that we were talking about was this geometry which needed to have all the things that you thought it needed to have. It needed to be algebraically representable. Space maybe doesn't want to be that. Maybe it's La Mer, it's okay, which has other poissons and other waves and you should allow it to be what it is, not what you put in it. Why should this geometry satisfy the axioms we want it to be? You see, one approach to geometry is to say, oh, I want geometry to be this and this and this and this and this. Nobody knows why it has to be all that. Why does the sea have to be the observed sea? So, when you recognize that the traditional received geometries are the sea we have observed for a long time, And to which we have superimposed the structures of our mind, basically we have imposed this algebraic structure on it, rather than the C itself, you can propose the task of, ask yourself not whether a geometry can perform a certain task for you, but rather what minimal properties a geometry should satisfy so it can perform a task you're interested in. Start with the task, your theorem to this oracle that knows everything, the disregarded C. You ask yourself, what does the C need to have so that the task you have can be accomplished? This goes back to Pappus, you know, right? The analysis. You start with the stuff you want to prove and you go back and see what you're having. And here is your quotation of Pappus, right? And that's more of the quotation of Papus. So, if you've read these things already, then you probably, I mean, I think that you have read them a few hundred times by now. And this is the synthesis part of things, right? So I could say...

47:30 Except that in the Papus case you are already in the geometrical theory, you are not at Saudi for me. No, no, no, no, not quite. Papu says that you start with something and you're asking yourself what you need, but it's not what you need out of this list of things. What you actually need. I mean, what the situation tells you you would need, not what the other people think you need. Launching this program is Hilbert in 1902 with his paper on the isosceles triangle. Well, he says that under the analysis, the axiomatic analysis of a mathematical truth, I understand an investigation which has its task. To find a connection with that truth, not more general theorems and not new theorems, but moreover to find the position of that theorem inside the system of known truths and to verify their logical connection in such a way that you can... They certainly indicate that hypotheses are needed for the justification, are necessary and sufficient for the justification of that truth. Which is a very nice way of rephrasing... Well, with this introduction I can give you two examples of what I'm thinking of. I'm thinking of a reverse analysis of this theorem here. The acute triangulation with 7 triangles. This is a theorem that came about in 1960, that you can divide any obtuse triangle. It started with a problem by Martin Gardner in Scientific American and so on, right? You have an obtuse triangle and you want to know what is the minimal amount of acute triangles you need, so you can triangulate by acute triangles this given triangle.

50:00 And it turns out to be seven. In other words, it can always be done by seven, and it can never be done by fewer. Now, the proof there uses all kinds of things. I mean, you start with a triangle, you get your center of the inscribed circle, and so on. In other words, you assume that you're in this, not the dreamed-of sea, but the sea that is given already. In a reverse order thinking, you're thinking, okay, what do I need for this theorem even to express it? Well, the notions of what turn out? Obtuse. Okay, obtuse means what? It means bigger than 90. So I need a concept of 90, of solvonality. I need a concept of order to know what's bigger, right? And, okay, so here is what I just said. You need these two things. There are axioms that we would need to ask of this so that we can prove the theorem. Between us need to satisfy the usual axioms that give us a dense unending order together with the posh axiom. So all the plain Hilbertian axioms of order are there. Collinearity can be defined in the sense you say that three points are collinear if they're in some kind of order relation with each other. You can also define the notion of acuteness. You would say, well, this thing is acute if when I raise here, so we say ABC is acute, if when I raise here the perpendicular, this one turns out to be in between, inside this angle, right? With between us, we can say that. Now, another axiom must be that, I mean, we assume another axiom that says you can raise the perpendicular on any line in any given point on it. Okay? Here is the axiom, if you care. Well, it's the next one. There are no isotropic lines. So perpendicular lines, there are none. Lines I'm talking, obviously, theoretically. There are no lines here. We have only points, but you understand. That's what it looks like in, okay? One can drop a perpendicular from a point C not on the line to that line.

52:30 Orthogonality is symmetric, and the perpendicular from a point not on the line to that line is unique. Perpendicularity is a relation between lines. What does that say? Well, if I have things like this, ABC, and here U, and I know that AB is perpendicular to AC, but AB is also perpendicular to AU, then it must mean that these three are collinear, right? This one said that if, okay, the converse, I don't need to draw another thing. The converse means if I know that U is on the line AC, right, then I know also that AU is perpendicular to AB. So I knew that AB and AC are perpendicular. I know that U is on the line AC, that means that AU is perpendicular to AB. So, which I have just synthesized by saying perpendicularity is a line relation, right? If BAC is acute, then CAB is acute. You see, in the way I defined this, I wrote the perpendicular on AB. Now the question is, if you raise it on AC, does it turn out to be acute as well? You see what it means? That you need to know that AB is inside this angle. And see, if you don't have an axiom, you don't know that. So you need an axiom for that. That's the formal thing. And here's the last axiom. What it says is, well, you have it there, but I'm going to draw it here again. Maybe it turns out to be better. I have some points here, A, B, and here's the feet of the perpendiculars. This axiom says the following. It says there is a point here, say C, such that for every point that lies between U and C, including C,

55:00 This angle here is acute. Two points in the same semi-plane of this line. And I draw the perpendiculars. Then the axon says there is a point here, maybe too kind of close, we should think of it close to you, such that for this whole segment here, every point I pick here, I get that the angle here will be acute. The reason for believing it is what? If this point were U, then by definition this is acute already, right? Because it's inside the angle here of 90 degrees, right? Well, by continuity you would think that this wouldn't immediately go to becoming non-acute. It would stay for a while acute. That's what this axiom says. And this is all we need for the acute triangulation. Here is the axiom itself, you don't care about it. And here is kind of the idea of how you would get the triangulation. Triangles are EZA, ZAG, you know, and so on. This, the magnitude down is not dividing that triangle. That's just one triangle, ZWS, right? Everything else is an actual triangle of the triangulation. So, the point is here what? Axioms that you've seen so far are so weak that they do not allow any kind of algebraization. To show that there is a field at all, there is no field whatsoever, because the order axioms are way too weak to tell you anything. You don't have a discharge, you don't have anything. And the orthogonality notion, you need at least the altitude theorem, the three altitudes meeting a point to get any hope of getting some kind of a A nice orthogonality that would allow you to introduce a field or something. With this, you can get nowhere. So there is no algebra that can help you in this thing. So this proof comes from much, much fewer things, incomparably fewer things than the existing ones that were. And these axioms are obviously satisfied by all the classical geometries you can think of that are not elliptic.

57:30 We introduced the elliptic thought by saying that the perpendicular needs to be unique. So, if you got the idea that I could spare you those, I can just tell you what they were about. But not go into details as... Yes, but the interesting thing is what else you can prove within the theory that you construct in order to solve a certain problem. Yeah, the... The only interesting thing to solve this problem... Yes, I know. But your point is this, that you need a theory, you need a sea that would be good for every person to look at it at every time. I'm saying the poet needs to look only in their dreams once. So in other words, the way we present geometry may have a didactical point that you've said once, the axioms, and now people can solve all the things in the world, but why wouldn't you be interested for a specific truth to know how much is needed for that one? You want to know, and this is what we're doing in general, you want to know how much truth is needed for a huge class of theorems to be true, right? But in each of those theorems, you may need dramatically less. So when you're using all that, isn't that overkill? Why would it be better? So the other case that I wanted to show, but I would cut it short, is a theorem that says that if you have a pentagon, a space pentagon in three dimensions, And it has all sides congruent and all diagonals congruent, then it must be plain. This does not work in higher dimensions. And I did the same analysis of this thing, a reverse analysis, what you need. I came up with two different action systems, incomparable, extremely weak, where you can get absolutely no structure, where this can be proved. Now I skip all that. I'm trying to get an answer to what this is all about. The geometric thinking from what we've seen is that of more finding deductions. So I think that if you want to characterize what it is, it's this deductive activity. Finding axioms that are strong enough to prove a theorem. So the reverse process.

1:00:00 And that it cannot be replaced by any kind of algebraic thinking. In the last instances, I hope to have convinced you of that. Previously, there was always an algebra connected with them. Here, there was none in the last example. What can you say to such critiques, you know? There is a reality out there, and this is what we are out there to study, some kind of a pragmatic critique, and here I have an answer from Simone Weil, a kind of definition of pragmatism that I take it in my defense. Well, we can't help it. We've got to do the best we can with the materials we've got. Anyhow, there's one thing. There's more honor in getting him out through a lot of difficulties and dangers where there weren't one of them furnished to you by the people who it was their duty to furnish them. And you have to keep running them all out of your own head. It's the defense of all people who do useless things.

1:02:30 Jacobi to Legendre, after the death of... which tells us that it's all for the honneur de l'esprit humain that we're doing it, right? That here was the defense of number theory, but you can use it for all defense of useless things. I guess that was my talk about the geometric... Very cool, thank you. I think a general remark is that this big difficulty about any kind of this formalizations of mathematics is it was I thought about that but also for the quotation of paper he sent me I forgot the author who just says that when we kind of formalize thing and we are thinking we get They're more precise, right? We make them, we gain a rigor. Actually, where we lose a normal is somehow just out of control. It's exactly how this works, this translation, how we translate one to that. It's something that's not formalized. We call formalizing the end of the process. And that, we say, goes just somehow. And I think in your talk, exactly, you just, in many places, pointed to importance of this thing. I would say the most natural apparatus to analyze that is, of course, category theory. Yes, yes, yes. Because, of course, even this pointwise analysis, when we say, like, onto, into, it's just more precise. Yes, absolutely. You're right. Instead of talking about mutual interpretability, I could have gone over to that. But I always think it's preferable to stay on the ground. Don't, if you can, you don't need to go to in the air, you know. No, it's not in the air, it's in my view.

1:05:00 Well, to me, to me, category theory is seeing the world from high up, from the stratosphere. You don't need much of category theory, but just simple fact that exactly you have this situation, when you have two mappings going, like, different direction. It's very, very weak and still for some people it would say, okay, geometry... Is like group theory, by the way, that's for group theory, I think it's failed historically, some kind of hope in the beginning, but when people did all homology theory, they discovered like plenty of groups, and that's where very rich theory, which of course makes relevance between group theory and, say, differential geometry, but in no sense it reduces. Yes, this is why I restricted my attention to the elementary case, so everything I'm saying could be put exactly, because if I went to geometry in general, then everything can be said. So I think just things like that should be developed and really analyzed more precisely what we mean by... By sameness. The question is how do you identify a formal acting system as geometric because you refer to the band spirit and you said that it's more geometric than it is... Yeah, let's see, I was thinking of this thing that you came up with, because when I was preparing this, I was thinking, well... Maybe you can describe in some formal way that geometry is not algebra, say by looking at its language, you know, does it contain relations is the first thing, you know, but then you know that I have done stuff with ternary operations, right? So it doesn't work. Then I was thinking, well, maybe binary operations would do it. Well, they don't do either because I was able to find an axiom system for hyperbolic geometry, which is expressed with just one binary operation, that of midpoint. Now, there still is, and here I came up with what the difference needs to be. If you express hyperbolic geometry with this operation of midpoint,

1:07:30 Then your axioms will have, I mean the matrix of the axiom is this and that and that or this implies there's something wrong thing. In algebra, if you scolomize them right, you end up with equational theory. In other words, there's no and and implies. There's just this and that. Things like, you know, associativity, commutativity, inverse, all of these things. It's just there. It's equal to. So, for me, if I went to look completely formally, what's the difference? This is the difference. Is it equational or not? If it's equational, it's algebra. If it's not, it's geometry. Whoa. It's not equational, it's geometry. No, I'm not saying it's geometry. If it's not algebra, I'll put it this way. It's not algebra. That we can accept. Yeah. It's not algebra. So, the non-algebraic nature of geometry, because this is what we want here. We want to show whether it is algebra or not. Can you separate it from algebra anyway? And formally, this is the only hope there is, that you would say it's not an algebraic theory. No, you wouldn't say, because it's, I mean, there are tons of things that are not algebra and not geometry either. Right, so there's formally no hope of identifying that subset of all relational theories. No, in fact, the only way to define a geometry that I know right now is still via group theory. You would say, well, who preserves all these things? What group? You know, if I get a group that looks like a group that geometry likes, I would say, well, it looks like a geometry. In other words, it needs to be a rather rich involved. That's true, that's true, because else I don't have a means to say that set theory isn't geometry. I don't know how to do it. Just that there wouldn't be an interesting group there.

1:10:00 Reading the cat's geometry and writing it in English, he is not a geometer, he is not a geometer, maybe he was Newton. I think that he was thinking more or less like you. Yes, in fact he has musings on Papous's analysis. No, no, now the serious thing. See, I understand well your point, it is exactly as in reverse mathematics. One wonders how much existence or existential suppositions are necessary in order to get some results. You are wondering how much algebra is necessary in order to have some geometrical results. So, my question is, much algebra better, much structure. Algebra is a very, very vague name. How much structure is necessary in order to have that? Now, it seems to me that what is relevant in reverse mathematics is, in fact, that when you have an answer for a certain theorem, You have in fact an answer for a big class of theorems, and in fact you don't have simply the result necessary for this theorem, you have different systems, and then when you have a system that you have constructed in this way, then you work within this system and say, what in this system happens? And so it was in a sense the question that I... So, during your talk, essentially that you make it with a single theorem. You take this theorem and you need this axiomatic system. The axiomatic system is completely not algebraic, there is no structure there. Now, the question, very natural question is, what do you do with this axiomatic system? Nothing. Nothing, what's worth the interest. Reverse mathematics, I see the interest, because in a sense, when you have a certain system, it's a weak one. Look, this is a very weak system, but even if it is a weak system, we can make a respectable part of mathematics. That is a very good enterprise, this is. It's a weak system, but I can do this theory, but what respectable part of geometry can you do? Sure, this is precisely what Bachmann does. You're absolutely right. Metric planes can do a whole lot. That's one way of justifying an axiom system. Why do you think that, why do you have a bias for the going forward, for the synthesis, against the analysis? Why is it worth proving a theorem at all?

1:12:30 I spent all my life to understand the history, to study the history of analysis, so I have no bias for synthesis. So then, so then, okay, so then, because if we spend time to prove a theorem from some axioms, then the same time should be as well justified by going backwards. If you think the theorem itself is dear to you. I mean, there are people who are claiming that the first book of Euclid's elements is all done so he proves the last two results. So that he's the first reverse geometer. Yeah, he understands, yes. So you can phrase it that way. I mean, there are people who disclaim that, but I don't know what he wanted to do. But the question is, if a result is dear to you, Obviously, you shouldn't choose any kind of nonsense, but something that you really care about. You would be interested to know, well, what do I need for this? In the first book of Euclidean geometry, an important part of this book is absolute geometry, in a sense. So you have not simply a system of action for a certain theorem. There are theorems there. Yes, but all of them are needed to prove the Pythagorean theorem and its converse. So I have many results until I get to prove this. And the only question is, do you think that this is of the same... The question is how to choose the theorem and how to apply it. Yes, the question is one of rather psychological or social interest, if you want, for the type of reverse analysis you have started. And you certainly would want to know, say, is this theorem true in hyperbolic geometry as well? Now, do you want to do separate proofs for all the geometries that were ever discovered, or would you want to have rather this type of analysis and then all you need to check is whether these geometries satisfy the axioms that we have here? Yes, but in fact it is not the question that you are... your question is to understand whether or not this one is not the... Which geometry is necessary in order to have a thesis? How much structure I need for this? It's not exactly the same. So the second question, okay, is very important, historically important, I have no doubt about that. My question is, really is it mathematically interesting to ask how much structure I need for this?

1:15:00 That is a social question. Absolutely right, it's a social question. This is decided by a social contract between mathematicians. Okay, now we'll move to Rousseau, Jean-Jacques. It may be my mistake. It seems to me that for you, elementary geometry is purely geometry. Well, in a certain sense, yes. If you, believing logicists, believe that first order logic is all there is. If you believe that, then certainly elementary geometry is all we have, in all honesty. For example, only a simple example, the concept of... You don't have it, I'm sorry. Because all of this, let me tell you why. Let me tell you why it would be completely unacceptable from a foundational point of view. Because all of these things, concept of area, I don't know what kind of area, curvature, length of a circumference of a circle, all of these questions can be meaningfully asked only if you invite all of set theory into geometry. And we want to keep that door shut. Firmly shut. Because you don't want to call geometry everything that has ever been produced in set theory or will be produced. Would you want to invite all of Shellaf's books and papers and Woodin and all of that and call it geometry? Because all of that would enter. You can't say, well, I don't want it. I just want my little r that is from geometry. You know, this little continuous thing, but we don't know these cardinals. Now, you can't do that. Why couldn't I say it's geometry's environment? Yeah, but I don't know how to do that foundationally. Foundationally, how do I speak about the circumference of the circle? Either I invite higher order variables, and then I'm in trouble because your logic is no longer good any. It loses all its power. It has a very bad proof theory, has a very bad model theory.

1:17:30 Just to put it in a textbook so you impress the students. It's no good. I actually can't understand that point. Excuse me? From a foundational point of view, you know what foundational means this. You are creating out of nothing a story, a poem. This is genesis. So you have to exclude things you cannot because there is no nothing outside of you You you have nothing outside of you you're creating up a poem here There is no I feel that this is how the curvature is and so on This is something where somebody reading it and not knowing what it is should be able to get caught in the action Yes, but I'm saying you assume then something. You assume that you have as blocks of construction set stuff that is somewhere. This is not a poem out of nothing. This is its own narrative. It's not a narrative about something that is outside or that the community of mathematicians thinks exists and we can take here a set and there a set and there a functor and there a function. These don't exist here, because we have made it all on paper. I mean, these are the constraints of formalism. I'm not saying that mathematicians should accept them, but I'm saying in this enterprise, if you want to start a foundational enterprise, there is no point in inviting sets. Then you can just introduce them the way textbooks introduce them. You say what the structure is and so on, else the foundational exercise is empty. It's just a game. This is how chess is played. I mean, we don't have to all play chess. We can play soccer. I wonder if in playing soccer we really are playing chess. Michael? Yeah, I had a question which initially I thought was Marco's question, but first a clarificatory point. This term reverse geometry, he mentioned reverse mathematics. Is this your term, borrowed form?

1:20:00 It's my term, kind of, yes, the inspiration comes from there, though the enterprise is totally different. Well, about that enterprise, so I got confused about that in the interchange between you and Marco, but I would have thought initially that the enterprise was different. My understanding in reverse mathematics is that You take a really central theory. Exactly. That's why I said it's different. Everybody recognizes it. You take the ACA zero or something and you take... Or the Heine-Varel theorem or something. Something that plays an absolutely central role. And then you're asking, what in addition do I need? What in addition do I need? It's a completely different... Particularly then a very specific question about what... What kind of comprehension axioms do I have to get them? So, in other words, how much can I restrict the concept of set and still get the result? Now, it seems to me that's not what you're doing here, right? First of all, for two reasons. You're not taking a central theorem, right? No, not at all. You're not saying, let's identify a theorem like Hilbert did that plays a central role in all of this, like the Desart theorem or the Pabst theorem. Yeah, but Hilbert, when he did his 1902 thing, he didn't take anything that he didn't need. He just took what he needed to. Well, he wanted to analyze the isotope. Yes, exactly. And see. Whether you still get it if you weaken the congruence of the series. So, but in his earlier work on the, say, the Desart theory, you know, he wants to look at what you can, I mean, this is where reverse mathematics begins, right? If you take the Des-Arc, Planar-Des-Arc theorem as an axiom, what do you have to add to this to get... Exactly, exactly, you can phrase it, certainly, your point is true, I mean, the, no... Whereas you're not doing that, let me finish, let me finish this, it seems to me what you were doing is just... It's just a logical exercise, actually. What do you need, what minimal assumptions do you need to prove this particular theorem that you grasped?

1:22:30 Well, I don't know, maybe there is some reason for it, but why did you take this covering theorem? Does that play a central role? It does? A conjecture that every surface, that there is a natural number N such that every surface of any genus can be triangulated by at most N acute triangles. It's very hard to find what this minimal number is, even if you go to things like pentadons, they found it for the platonic bodies, what the minimal number is, but it's very, very complicated and there's a lot of work going on in this area for Q-triangulations. I mean, maybe it is correct, but it seems to me bizarre to say, somehow if you allow some set theory in, you get, you know, you can't keep everything, you can't keep anything out. That seems to me strange. I mean, why couldn't you just... You have some axioms that say you're allowed one or two iterations of the Paracet Principle, maybe even countable iterations, and you get some L-O-C-T-O-R-I-E, and you certainly don't care. I allowed a bit more because I went to L-Omega-1-Omega, but that's as far as I would venture. But even if you restricted yourself to first-order logic, why couldn't you have some sort of axiom that says you could have one or two iterations of Paris and that's it? No, I don't know how that would work. I know that it would work in so-called weak second-order logic where you have second-order variables that vary over finite sets. But you can't get very far with that either. I was thinking of the question whether you can get stuff like, you know, What's the circumference of the circle, curvature, stuff like this? This is pretty much hopeless. In this setting, I mean. But I think your critique to this is clear. When you go in one direction, in the synthesis direction, it's clear what's to be done.

1:25:00 Here are the axioms, here is what's to be proved, get a proof of this. In reverse... Well, it's not clear. First of all, you end up, as the second example that you haven't seen is, you may end up with two different axiom systems that are incompatible and give you the same thing. So it's not clear at all what you're looking for, first. Second, what kind of things would you stop at? According to Papus, you stop at things that are kind of clear, right? Right. Well, so it's, there is a psychological moment, because if you're asking, from what can I prove this? Well, from itself, certainly, right? So logically there's nothing there. There's a psychological moment. There's an aesthetic moment. What would I accept here as an axiom and so on? It's not something as well defined as the proof. The finding from what can I prove it? So this critique is for any kind of enterprise that is like this would go the same way because it's simply in the nature of the enterprise that it's not as well defined. Okay, Michael had a... No, it's okay, I'll leave it to, ask Victor afterwards. Okay, and then Sebastian. Algebra and geometry, obviously, and the discussions deal not, the problem is not to prove that something is true, but to find another. And so, why I was wondering, is it for you anecdotic? Do you think that by... Do you think that if I take your general background of your lecture with your lovely metaphor of hidden C, etc., etc., it would be more natural, in a sense, or is it for you, I would say, because, as you know, in the Papusian, we stress problematic analysis.

1:27:30 Yes, this is definitely my reverse analysis is a problematic analysis, yes. I think in the case of reverse, a problem asks you to do something, so you don't know yet how to do it. That is very hard. You know, I was thinking, the first thing I thought, you know, the most simple thing you would think of justifying geometry for, you know, if you're, suppose you start teaching geometry and everybody thinks it's something that shouldn't be taught at all. And you want to justify it. The simplest justification is to say something like, well, how about if I have Here are two points and I'm asking myself where should a ray be directed here so the reflected ray goes here, right? And you have certainly the people there who will try to compute this and this and get some minimum by means of calculus and be very happy that they found the point. And they would have wasted some pages and after they found the point they would have no idea why it was there and not somewhere else. And then you would show them the simple trick of reflecting this and joining these two points, and you would say, here it is, right? But in this setting, you can't say much about this, because the question that we started with, where is this point so that that gets reflected, is not a sentence. It's not something I can write down. When I write it down, I already write it down the way it is, and it's trivial, there's nothing to be asked. I showed them already, I gave away the secret. See? So in a certain sense, this is not something you can say, well, here I cannot use algebra, but I have to use geometry because there is no sentence. It's a problem. And in our formal setting, it has to be a completed sentence. It can't be a question. It can't be open. No, where is it? Well, but that's maybe a reason then not to address the question in the formal setting.

1:30:00 Yes, but I was saying, well, in this talk I couldn't go past that. I mean, I was trying to stay with the imbalance. There's a very beautiful, almost magic sentence, namely saying, assume it's solved. Then you start the analysis. Then you can make that drawing. Certainly, certainly, but what I was saying is, if we assume it is solved, then the theorem itself can be proved geometrically quite simple, right? And you're right, then the algebras still have a problem figuring out, because they still need to do their own work. Yes, you're right. In this example, I give you three circles, and I ask you, well, give me a circle, show me a circle, which is a circle, and so your decadence is, yeah, well, there is an equation of degree 2, so... This is the circle. And, well, I'm you, or I'm Skyler, and say, no, this is not the C equation. I want the circle, I want the construction. So you have exactly the same, I would say, the same discussion. Absolutely. In problem solving, it's clear that the geometric point, but this was already the point of Newton and of everyone who has looked at this and who said that the geometric way is way preferable to the algebraic one. You all knew this and it wasn't worth mentioning that I'd rather tell you about these more intricate ways in which to try to come up with an answer to what is geometrical thinking. I think in the course of the discussion you've actually already answered 96% of the questions I had. It was a natural development of what you were saying. And the analysis reveals two rather different ways to demonstrate it. There may be some common core that those two maximizations have, but they are clearly different. And I take it that, on your view, there's no geometric reason. Is it possible to prefer A1 to A2? That's a fantastic question. There might be a sociological reason. Yeah, no, there are, in fact. Your question is very well taken. I have another analysis where I couldn't get all this stuff here.

1:32:30 For example, if you look at Sylvester's problem on collinear points. Well, I analyzed that. The Sylvester's problem said if you have finitely many points in a plane, And you look at the lines that are formed by these. That one of these lines must contain only two points, or all the points are on a line. The question is how much geometry is needed to get this. And I ended up with three different proofs, all getting to incomparable stuff. And then the question comes, there is also the purity of method. Concern, right, that I didn't come to here, the Reinheit der Methode that Hilbert wanted, which is, well, this statement talks about only collinearity, right? So a proof should be based only on collinearity. Well, there is no such proof so far. So all of them use either some kind of metric term or some kind of betweenness, which wasn't in the theorem.