Elementary Geometry, Groups & Fields

0:00 The School of Interdisciplinary Arts and Science. Okay, giving an interdisciplinary talk on elementary geography groups and fields. Thanks, Victor. 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. So here is a view not from somebody looking from the outside at geometry, if it's from a practitioner who philosophizes a bit. Very little. 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 so I 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. And if one is a practitioner of geometry today, by which I mean the mathematical reviews 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. If one is just 51, just plain geometry, well, the question is, why is one still doing it? But for the question of thinking geometrically, maybe this is the best vantage point to give an answer,

2:30 because there is nothing else. We just have geometry here. So, the attack usually comes for geometry from either the pathological 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 archaic vocabulary for really saying algebraic things. That's the prevalent idea, I mean, the idea of Euclid must go, of Diodonet, 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 the first algebraic structure that will come to mind that you would think, well, geometry is that, would be field theory. Because once you have an elementary geometry that you can embed in a projective plane that satisfies Papus, you know that you will have a field there, a commutative field, in fact. If you're having, if the geometry that you can embed it in, if the projective geometry is at least of dimension 3, that at least you get a skew field, you still get something. If you assume that it satisfies some form of elementary dedicating, so some form of continuity, the first one who wrote down actually what that means is Lustaski, then 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 to answer questions that are formulated in its own language,

5:00 some kind of a geometric question, then this solves it. 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, and you're done. Question for real clause, because, of course, in terms of, like, field extension, this construction of real encompasses, it wouldn't give you all real numbers. No, but I said if you assume Dedeck in continuity, right? In other words, you take Dedeck's axiom, but your cuts have to be defined by 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 an axiom saying that every odd degree polynomial has a root. And that's it. You don't need to have a real close field. Yeah, no, this is clear, real close, so if you don't like those axioms, you can use this, and you get this real close field, right? So if you think that that's geometry, 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, so, in other words, all the 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. Why not? Now, if you're elementary... You can reduce, but it does not mean that there is no... 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. Now, if your geometry has a finite axiom system,

7:30 because the stuff with the real closed requires infinitely many axioms, right? As a finite axiom system, say, such as the geometry of ruler and compass they were just mentioned earlier, right? Then the field theories associated with it are known to be undecidable. This is what Siegler proved in 1982. He proved that every field theory that has R as a model and is finitely axiomitizable be hereditarily undecidable. So in this case, the reduction of your geometric question to an algebraic one is an academic exercise because you won't do anything with the the resulting algebra. It's as bad as the geometry. So you probably have much better, you 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 algebrization, 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 a poke kind of holes into the attempt at a reduction, right? So, now, this stuff, the fact 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. But you know, they both axiomotize, say, Euclidean geometry. How would you say they're equal? Well, there is this notion of mutual interpretability. You can say in the universe of these people with lines, well, the point isn't vice versa. And you can express, you can give definitions for all the relations with the line people and the point people,

10:00 and you get a translation where everything in one direction is true and the other as well. 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 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 right now you would think that well maybe all theories are mutually interpretable no it doesn't matter it doesn't matter whether they're constructible or not, because I'm saying the corresponding field theory. So if you have the ruler and compass, you'd have an Euclidean-ordered field. And the theory of Euclidean-ordered fields and the geometry of ruler and compass are mutually interpretable. Right? I think much depends on how, say, properties, precise properties of this interpretation, because the mere existence of this kind of interpretation is very weak. On the other hand, if you say one interpretation which is it cancels the backward in the sense it composes to identity. That's very strong. So you have the whole spectrum of possibility. No, no. The concept of this interpretation has been worked out by several authors in the 60s. They try to figure this stuff out. And it's quite easy what it is. You have to be able to define your individuals. Your individuals will be defined by a predicate. And then you have to be your primitive notions from here in terms of the other one, right? So, you would, you know, it's... and then, with these interpretations, all the axioms from here should be provable here and vice versa. So, but the problem is that... and if you think now that everything becomes mutually interpretable, it's not true. For example, field theory and piano arithmetic are not mutually interpretable. So it's not really that everything becomes the same. So, now, with this view that it is this, it's kind of, for a geometer, you'll say, but 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 a decidability machine, right? Even then, 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 a different part Yes, yes. And this means that also they themselves, Euclidean and hyperbolic geometry, are mutually interpretable even without going over the fields. Inside one you can define places, you can define universes in one where the other happens. So, and there is no better, I have not found anywhere, a concept of mutual interpretability where this story line will be preserved. Right? You cannot get something like that, that you would preserve the story line. So, I don't know, so there is this stuff that we feel is there, but logic really doesn't allow you to express it in that manner. 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 first-order sentences true in this structure, or in this class of structures.

15:00 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 accent 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 axiolotizable 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, story is irrelevant. That somehow 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 axiom system to figure out what's a minimal one and so on all of these questions are inter i mean intra-linguistic things of these theory of this theory and and that is i think the geometric part of it okay but that's what i'm wondering about because your sentence says

17:30 is that you can read it this way. For every axiomitization of this theory, you have to have geometrical thinking in order to find that axiomitization. That's a strong claim, because it could be lots of axiomitization. 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. You believe in geometry isn't the case. We're given a model here, and now we're interested in organizing the truths about these class of models. The theory of the model. And the question is, is this algebraic description of the model gonna be telling me what the axioms Right. I see. So it's a negative claim. It's a negative claim. The answer is no. 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. 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. What are the relations that hold with this notion? 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 of how certain theorems that are 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.

20:00 I'm talking about this area that is called characterizations of geometric transformations under weak hypothesis. Now the origin of this may be going back to maybe Darwin. Things like a map that preserves circles have to be in this manner. Or, later on, the Mazur-Ulam theorem, right? It 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 the special relativity, for example, Zeeman has this article on causality implies the Lorentz group. So, so that yet you're, the miracle is here that the mapping that just preserves causality, 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. The miraculously preserved things 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.

22:30 No implies either. So, that's what, who tells us this? A combination of Beth's definability theorem and Linden's preservation theorem. basically, tells us this story. 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 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. And here I'm going to give two or three examples of such things that you can get a better idea. Okay, here I say that this is equivalent by Webb's 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 say people say that it goes back to Darbu and so on versions of this Cara Theodori was on his way to the United States to a lecture, and on the ship he didn't have a problem to think about and he thought about this problem 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, because they kind of miss some points. On to circles, so they just preserve circles. either a Moebius transformation or a conjugate Moebius transformation.

25:00 Now, what are these things for us geometrists? So this is proved with, you know, all the machinery of C and so on. We geometrists don't have the luxury of these powerful structures, and so all we can do is think of what is the underlying geometry, the skeleton of these things. 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, , and so on, which were all done in the real context. He did it in abstract contexts over any field of characteristic different from two in general is the requirement. schematization, just completely elementary. There the Mikkel theorem about some circles plays the role of the Pappus theorem. But fine. Now the question is this, what does this say? Now this is equivalent by Bette's definability theorem, together with Linden's preservation theorem, with the statement that circle orthogonality is positively existentially Negated equality is allowed, because this is a one-to-one map, in terms of point-circle incidents in Moebius geometry. In fact, in Mequeline Moebius geometry, but I spare you the details of these things. So, Moody'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, you take things on a sphere over field, and you have your circles just as they are here, and here is the definition. Now, nowhere in the theorem is there a... So, the two circles, k1 is this, and k2 is this one. And the definition says exactly what you see on the picture.

27:30 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. If they do hold, then the two circles, k1 and k2, which I was just pointing at, k1 is this one, and k2 is this one, 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 internal like substation? 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 a thought. What is the definition that amounts to? 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? And there you have the definition of circles. So instead of saying Karateodorist's theorem, instead you're saying this. Karateodorist's theorem follows immediately from this. I'm going to precise, I mean, you said instead. but so you you prove the theorem according to which you can have this definition with this requirement right 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 if i had the two the following two statements are equivalent there is a positive existential definition with negated equality allowed of the notion preserved namely circle orthogonality because if you look at what Karateodori says it says this if point circle incidence is preserved then so is circle orthogonality because this moebiusness is the same as some kind of conformality the angles, and if it keeps the 90 degree angle, you would believe that it keeps the other ones as well. Right? So it's preserving the angle that's the miracle. Right? Now the

30:00 miracle here is defined by saying that all of these things exist, and it's defined positively existentially in terms of point, circle, incidence. And 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 requires the Beth, the Linden to prove these things 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. These people did prove that theorem, the model theoretic one. 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. Another example is that Roland Höfer 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 which map 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. So 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. So all this theorem says is that 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. So there must be a definition of point difference that must be positive in terms of collinearity because it doesn't say that it preserves non-collinearity as well. And this definition must be by necessity 1 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-Linden theorem works in omega-1 omega as well, so there you have 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 a few 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. And, I could do this only for n is equal to 2. I don't know how to do it for higher dimensions. A more recent theorem, this here, tells us that a self-map of the affine plane over any subfield of the reals, that preserves both the between us and the non-between us relation.

35:00 So between us means this point is between these two, right? Must be an affine mapping. Again, the equivalent is that line parallality, 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 CD, is positively existentially definable in L omega-1 omega. in terms of negated equality because you're allowed to do that and the term of relation B of betweenness. So I was able to show more than that that it preserves B and not B. All you need from not B is negated equality. And this works over all our Archimedean-ordered affine fields, so in Archimedean-ordered affine geometry. Again, there is, their proof was using heavily analysis, so they used the structure of R. This uses none of that, it's purely geometric. You have to actually produce a definition with lines and things, and you know, there has to be a picture there that is understandable. Now, how about another way in which geometry can be seen to be something else is that it can be seen as a group here. We know from Plyde that 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's a certain determination of it. Now, in this elementary setting that I'm interested in, Arnold Schmidt and later Friedrich Bachmann in 59 have presented a very simple axon system for metric planes expressed entirely inside group theory. Now, 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, know what they are and others which are some some distinguished elements that you know are line reflections and these are I get a unity predicate symbol to

37:30 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 you want to get a group. What are metric planes really? The idea is to get the skeleton, the metric skeleton of Euclidean hyperbolic and 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 ideometric 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. Orthogonal lines intersect and through 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. And line here is primitive? It's a primitive yes it's a primitive in this in this setting it's a primitive yes yes their points are capital letters the other so there we have two as two sorted uh language so now we have the notion of reflection in a line this is not really an axiomatics this is more of a model axiomatics but i but to give you the idea i put it this way there is a reflection in that line now reflections are rigid here we define a rigid motion as something that preserves point-line incidence, and orthogonality. And it's one-to-one and on. 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 that metric planes are. This is all you're asking. And you get a whole lot out of that.

40:00 Now, non-elliptic metric plates, in other words, those that are not for elliptic 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 will put it, right? The group is acting on this set. This was shown by a student of Bachmann in 1966 by Müller. They can also be axiomatized with one sort of variables that are points and 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. place to go. The accent system can be expressed by means of universal existential accents, so this gives it even more interest. And to give you an idea of how these look like, 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 Papus, so you would have a, you would have as a coordinatizing 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 because that's what it is. Now, is this 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 Now, I'm going over to a different subject geometric spirit is best described, which is reverse geometry. And to justify 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, and I thought that, and I have a poet here that you could also say. 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 in the world of the lived experience, we experienced 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, nor mathematical time with all of its forms. And here I have a poem that inspires me in this sense. The two-thirds poem. 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 je suis parvier oh, I missed her in 1938 So, what does this have to do with geometry? Well, I think it's just the secret sea, you see.

45:00 This sea that we were looking at all the time was this geometry which needed to have all the things that you thought it needed to have. It needed to be algebraically representable. but space maybe doesn't want to be that, maybe it's which has other poissons other waves you should allow it to be what it is not what you put in it why should one 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 Nobody knows why it has to be all that. Why does the C have to be the observed C? So, when you recognize that the traditionally received geometries are the C 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 yourself to the task of, you know, to 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. In other words, start with a task rather than going with your theorem to this oracle that knows everything. to this regarded scene. Try to ask yourself, what does the scene need to have so that the task you have can be accomplished? This goes back to Papus, we know, right? The analysis, right? 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 Papus, 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 the Papus

47:30 thing, right? So I can see it. Except that in the Papus case, you are already in a geometrical theory. You are not a thousand from... No, no, no, not quite. Papus says that you start with something, and you're asking yourself what you need, but it's not what you need what you actually need I mean what the situation tells you you would need not what the other people think you need and here is the the the relancing this relaunching this program is Hilbert in 1902 with his paper on the on the isosceles triangle, where he says that under the analysis, the axiomatic analysis of a mathematical truth, I understand an investigation which has its task in finding 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 clarify their logical connection in such a way that you can certainly indicate what hypotheses are needed for the justification, are necessary and sufficient for the justification of that truth, which is a very nice way of rephrasing pathos. And 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 seven 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? So 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

50:00 triangles this given triangle. And it turns out to be seven. In other words, it can always be done by seven, and it can never be done by field work, right? Okay. 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 off C, but the C that is given already. Now, 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 orthogonal. I need a concept of order to know what's bigger, right? Okay, so here is what I just said. You need these two things. Now, what 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 Hilberton axioms of order are there. We need them. Collinearity can be defined in the sense you say that three points are collinear if they're in some kind of order relation, you know, 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? But between us, we can say that. Now, another axiom must be, I mean, we assume another axiom that says, you can raise a perpendicular on any line in any given point on it. Okay? Here is the axiom, if you care about it. The next one. There are no isotropic lines. So perpendicular lines, they're not. Lines I'm talking obviously, figuratively, there are no lines here, we have only points, but we understand. That's what it looks like here now. Okay, one can drop a perpendicular from a point C not on the

52:30 line to that line. Autogonality is symmetric, and the perpendicular from a point not on a line to that line is unique. these are the two axons that we're talking about perpendicularity is a relation between lines what does that say well if I have things like this a B C and here U and I know that a B is perpendicular to AC but a B is also perpendicular to AU then it must mean that these three are called in here right and the second one says that if i have this ac here and a okay this one said that if okay the the 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 a u is perpendicular I know 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, acute as well you see what it means that that you need to know that ab is inside this angle and so if you don't have an axiom you don't know that so you need an axiom to that and that's the formal thing and here's the last axiom what it says is well you have it there but But I'm going to draw it here again. Maybe it turns out to be better. So, I have some points here, A, B, and here's some, the feet of the perpendiculars. This axon 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, this angle here is acute.

55:00 So, I have a line, two points in the same semi-plane of this line, and I draw the perpendiculars. Then the accent says there is a point here, maybe too kind of close, we should think of it close to U, 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. The triangles are EZA, ZAG, and so on. The altitudes down is not dividing that triangle. That's just one triangle, Z, W, S, right? Everything else is an actual triangle of the triangulation. So, the point is here what? The axioms that you've seen so far are so weak that they do not allow any kind of algebraization. In other words, to show that there is a field at all, there is no field whatsoever, order axioms are way too weak to tell you anything, you don't have a desire, 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 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. And 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 In elliptic geometry, this is not true. We introduced the elliptic part by saying that the perpendicular needs to be unique, point outside. So, I have other examples of the same thing, but if you've got the idea, then I could spare you those. I can just tell you what they were about, but not go into details. Yes, but the interesting thing is what else you can prove within the theory that you construct in order to solve a certain problem. Yes, I know. But your point is this, that you need a theory, you need a C 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 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 that there's 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 plane. 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 axiom systems incomparable, extremely weak where you can get absolutely no structure where this can be proved now I skip all that and I'm trying to get an answer to what this is all about. That geometric thinking, from what we've seen,

1:00:00 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. 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. So, now the possible critiques that could come from these is, why, when the actual space satisfies so much more, do we need to go to such lengths to devise contorted proofs of geometric truth from very weak actions? the stuff with the seven acute triangulation takes about five and a half pages compared to the three lines of the proof in Euclidean geometry. So, what can you say to such critiques? That, you know, there is a reality out there and this is what we're out there to study, some kind of a pragmatic critique, and here I have an answer from Simone Bale, a kind of definition of pragmatism that I've taken in my defense. And then I got one from what Tom Sawyer says to Huck Finn, which is, I think, the essence of any civilization, which says, why grab it, Huggins? It's the stupidest arrangement I ever see. This is in the context of trying to liberate Jim. You've got to invent all the difficulties. 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 describe them all out of your own head. And then it's the defense of all people who do useless things. They go back to this

1:02:30 letter of Jacobi to Legendre after the death of Fourier, which tells us that it's all for 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. And I guess that was my talk about the geometric thinking. Okay, very cool. Thank you. I think a general remark is that there is this big difficulty about any kind of this formalizations of mathematics is it was, I thought about that one time, but also for the quotation of paper, he sent me, I don't know, I don't know, the author, who just says that when we kind of formalize things and we are thinking we get some, they're more precise, right? We make them, we gain rigor, actually where we lose it normally, somehow just out of control is exactly how this works. translation, how we translate one to the other, something that not formalized, we call formalized like the end of the process, and that, you say, goes just somehow, and I think in your talk exactly, you just, in many places, pointed to importance on this thing, and I also, I would say, the most natural apparatus to analyze that is, of course, category theory, Yes, yes, yes. Because, of course, even this point-wise 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. If you can, you don't need to go in the air, you know. No, it's not in the air. Well, to me, category theory is seeing the world from high up, from the stratosphere.

1:05:00 There's not much of category theory, but just a simple fact that exactly you have this situation, when you have two mappings going in a 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. It was some kind of hope in the beginning, but when people did all homology theory, they discovered, like, plenty of groups, and that's a very rich theory, which, of course, makes relevance between group theory and, say, differential geometry, but in no sense it reduces one to another, as people might think. Yes, this is why I restricted my attention to the elementary case, so everything I'm saying could be put exactly, 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, you know. By sameness. By sameness, by what is, and moreover, this kind of metaphysical argument, because people would say germity really is that, right? That's absolutely, it's critical, completely. I have a related question. Okay. The question is how you identify a formal acting system as geometric, because you refer to the Bands theory, and you presented that as more geometric than just C. Yeah, let's say 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. I say, by looking at this 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.

1:07:30 If you express hyperbolic geometry with this operation of midpoint, then your axioms will have I mean the matrix of the axiom is this and that and that or this implies there's something long thing in algebra if you scolamize the bright you end up with equation of theories in other words there's no end that n implies there's just this is that things like you know associativity commutativity inverse and all of this thing it's just there it's 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. Then it's not algebra. I'll put it this way. It's not algebra. That we can accept. 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 so you can't you wouldn't want to say 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 by a group theory. You would say, well, who preserves all these things? What group? and you know if i 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 a rather rich group theory there involved again it's just that geometry likes that's all you can say 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 there. All right, Marco. Yes, you know the story of this man that was reading the

1:10:00 cat's geometry and writing in England. This is not a geometry. It's not a geometry. And it was Newton. I think that he was thinking more or less like you. Yes, in fact he has musings on Papus' analysis. No, no, now, the serious thing. Since I understand your point, it's exactly as mathematically, one wonder how much existence, or existential supposition is necessary in order to get some results. You are wondering how much algebra is necessary in order to have some results. So, my question is much algebra better, much structure. You know, algebra is a very, very vague name. How much structure is necessary in order to have that? Yes. 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, that you don't have simple results 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 with the system and say what in system happens. And so, in a sense, the question that I put during your talk, essentially that you make it with a single theorem, you take this theorem and see that this asymptomatic system is completely not algebraic there is no structure there now the question very natural question is what you do with this uh nothing automatic system nothing was really interesting yeah but reverse mathematics i see the interest because in a sense when you have a certain system is a weak one but look this is a very weak system but even if it is a weak system we can make a respectable part mathematics but that is a very good enterprise system but i can do this here but what respect other parts of germany you can do sure this is 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 but what i'm what i would ask you is why do you think that why why do you have a bias for the

1:12:30 going forward for the synthesis against the analysis. Why is it worth proving a theorem at all? I spent all my life to understand history, to study history of analysis, so I'm not pious 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, people who are claiming that that that the uh that the first book of euclid's elements is all done so he proves the last two results so that he's a the first reverse geometer yeah yeah yes that's yes yeah so you you you can 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 I 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 a theorem there. Yes, but all of them are needed to prove the Pythagorean theorem, and it's converse. So I have many results until I get to prove this. and then the only question is do you think that this is of the same the question is how to choose the theorem 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 proof for all the geometries that were ever discovered this type of analysis and then all you need to check is whether these geometries satisfy the axioms that we have yes but in fact it is not the question that you are what is your question if i understand well it's not this one it's not which geometry is necessary in order to have a question how much structure i need for yes yes not exactly the same yes so the second question okay is very important historically yes but i have no doubt about that my question is really interesting question to ask how much that do i need for that is a social question yes absolutely

1:15:00 right it's a social question this is decided by a social contract between mathematicians Okay, now we'll move to Rousseau. Jean-Jacques. Maybe my mistake. It seems to me that for you, elementality geometry is something as purely geometry. Well, in a certain sense, yes. If you somehow believe in logicists, believe that first-order logic is all there is. So everything else is masquerading as logic. 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 the 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 Shalas' books and papers and wooden 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. 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? higher order variables and then I'm in trouble because your logic is no longer good any you know it's it's it loses all its power it has a very bad proof theory has a very bad model theory it's just to put it in a textbook so you

1:17:30 impress the students it's no good from a foundational point 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 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 sets, 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 The foundational exercise is empty. So this is what I'm saying. I'm saying if you want to do this, I'm not saying that you want one ought to do that. 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. But 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 magnetics. Is this your

1:20:00 term, borrowed from...? It's my term, kind of, yes. The inspiration comes from there, though the enterprise is totally different. Yeah. 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 zero. Exactly. That's what I said, it's different. that everybody recognizes. Yeah, ACA0 or something, and you take. Or the Heine-Barel theorem or something. Something that plays an absolutely central role. And then you're asking, what in addition do I need? Yeah. What in addition? It's a completely different thing. Particularly, then, a very specific question about what kind of comprehension axioms do I have to get them. Right. So, in other words, how much can I restrict the concept of set and still get the result? Yes, yes. Now, it seems to me that's not what you're doing here, right? No, no. First of all, for two reasons. One, 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 Dessart theorem or the Pabst. analysis. When he did his 1902 thing, he didn't take anything that he didn't need. He just took what he needed to put. Well, he wanted to analyze the isosceles. Yes, exactly. And see whether you still get it if you weaken the congruence. So, but in his earlier work on the say the Desartes theorem, wants to look at what you can I mean, this is where reverse mathematics begins, right? What, you know, if you take the Dessart, planar Dessart theorem as an axiom, what do you have to add to this to get things out? Exactly, exactly. You can phrase it. Certainly, your point is true. Whereas you're not doing that. Let me finish this. It seems to me what you were doing is just, it's just a, it's a logical exercise. essentially. What do you need, what minimal assumptions do you need to prove this particular

1:22:30 theorem that you've grasped? 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 in... It does? Yes, there is a whole literature on acute triangulations. There is 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, n acute triangles. And it's very hard to find what this minimal number is, even if you go to things like Pentagon, 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 of acute triangulations. When? Yes, now. Okay. But I just have one more comment shortly about set theory. I mean, why, it seems to me a bizarre, I mean, maybe it is correct, but it seems to me bizarre are to say somehow if you allow some set theory and you get you know you can't keep everything you can't keep anything out it 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 parasit principle maybe even countable iterations and you get some l-o-set theory and you certainly don't get it I allowed a bit a bit more because I went into l-omega-1-omega but that's as far as I would venture want to go... But even if you stopped, even if you restricted yourself to first order logic, why couldn't you have some sort of axiom that says... You could, you could have... You have one or two iterations of parasitic, 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 a 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

1:25:00 when you go in one direction in the synthesis direction it's clear what's to be done here are the axioms, here is what's to be proved get a proof of this in reverse, 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? So there is a psychological moment, because if you're asking, from what can I prove this? Well, from itself, certainly, right? So logically, there is 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 OK. I'll leave it to ask Victor afterwards. Oh, OK. And then Sebastian. Yeah, thanks for your talk. It's a question of a scholar in only modern mathematics, I would say. You gave the two examples you gave, are examples dealing with theorems. And in the 17th century, you have discussions about the relationship between algebra and geometry, but it's related, for instance, between people like Descartes and Pascal. And these discussions deal not with theorems, they deal with problems. The problem is not to prove that something is true, but to find another. And so what I was wondering is, is it for you anecdotique, or do you think that if I take your general background of your lecture with your lovely metaphor of redundancy, etc., etc., It would be more natural in a sense to consider problems as a type of anti-grams. Or is it for you, I would say, because as you know, in the Papusian, we stress this

1:27:30 problematic analysis rather than... Yes, this is definitely my reverse analysis is a problematic analysis, yes. So, I was wondering if, for you, this kind of choice between, I would say, problem and theorem is anecdotic, regarding your aim, your purpose on your talk, or if you do that. I think in the case of reverse, you... And you see, 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 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 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 then it's trivial, there's nothing to be asked 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 in our formal setting the 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 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 there. Namely, you say, 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, you're right. This example, just I give you three circles, and I ask you, well, give me a circle, show me a circle, which is tangent with three circles, and so you'll take out and say, Yeah, well, there is an equation of degree 2, so this is the circle. And, well, I'm you, or I'm science, no, this is not the C. It's not an equation. I want the circle, I want the construction. So you have exactly the same, I would say, 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. So I thought that 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. Todd? I think in the course of the discussion, and you've actually already answered 96% of the questions I had. It was a natural development of what you were saying. I mean, you can imagine that we had some result that we were interested in, and the analysis reveals two rather different ways in which it could be demonstrated. There may be some common core that those two axiomizations have, but they're clearly different. And I take it that, on your view, there's no geometric reason to prefer A1 to A2. That's a fantastic question. It 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 finite too many points in a plane, and then there are either two for which there is a... 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 the 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 hear, the Reinheit der Methode that Hilbert wanted, which is, well, this proof talks, there's a statement talks about only collinearity, right? so a proof should be based only on collinearity well there is no 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 in the theorem