Geometry & Number in Hilbert (1891–1905) — Part 2

0:00 The point, though, that I wanted to get to is this. that the natural interpretations, in other words, the interpretations of this geometrical setter given to us by contemplation of things in space, as Hilbert says, will be too imprecise to be of any use in the establishment of the dependence results. Moreover, the restriction to empirical or intuitional models is actually hampering, especially when one is considering results which seem very basic. So what Hilbert does is to look for abstract models translations which go counter to intuition. So what follows from this is that while the use of analytical models is not officially necessary, in practice it's essential because of the degree of control that these afford. to give one brief example of this, involving some independence results concerning Desargs Theorem. The spatial version of Desargs Theorem, sorry about the quality of this drawing, this is just badly scant, by me, badly scant, from actually from Hilbert and Cohen Rosson's book. the spatial version of De Sartre's theorem close to a triviality it says that if you've got two triangles I keep pointing at my screen but you can't see my screen if you've got two triangles A, B, and C arranged in space, so these triangles are in different planes generally arranged such that the lines going from A through A, A prime C, C prime, all run through a single point given here, then the lines of intersection

2:30 given where AB intersects A prime, B prime, that gives you one point, where AC, A prime, C prime intersects, that gives you a second point, and the same with BC, B prime, C prime, gives you a third point. Those three points all lie on a single line. Now, if you look at it like this in three space here, where these are set in different planes, it's, as I say, close to a triviality because the plane of ABC and the plane of A'B' have to intersect And those points of intersection, the R, S, and T here, must all lie on that intersection because they're both, they're both, sorry, they're all in both planes. So they can only be in both planes if they lie on the line between the planes of intersection. So the spatial version of Desartes' theorem doesn't need much proof. The spatial version also has a planar version, where the triangles, it's, oh, I want to see from this triangle, so there's one triangle, A, B, C, there's the second one, A prime, B prime, C prime, these lie in this one plane here. Now, this version is very, very much harder to prove. The standard proof of it consists of taking a point S outside the plane and then drawing various lines of connection in such a way that you reconstruct a spatial Dessart configuration. From the spatial Dessart theorem, that there must be a line where the corresponding side's intersection points for the lines of these triangles, they must all lie on some line wherever it is, up here in space. that line projected down onto this plane will actually then form a line of

5:00 a line where these corresponding lines all intersect so you can prove it but the proof is very much more elaborate strangely enough because and this fascinated Hilbert that the planar theorem is in some sense more elementary than the spatial theorem because it only involves planar concepts. You could state it in purely planar terms, right? But somehow you need space and spatial assumptions to prove it. So Hilbert first asked himself the question whether it's necessary? Is the involvement of spatial notions necessary? And the first thing he did then was construct a model in which, if you don't, if just in the plane, where the Desiret planar theorem would fail. And this is the model that he gives you, this strange looking thing. So you take the surface of the screen here, is a plane. You remove this line, oh, I keep doing that. You remove this line, so zero and all the points, this is a standard analytic plane here. You remove 0 and all the points above 0 on the x-axis. Then you say that if you have straight lines in the lower half plane that, as it were, run into the line that's being removed here, you regard those straight lines, of course, with no end point here, as straight lines in the new model. In the upper half plane, you only consider arcs of circles. What's interesting is what happens over here where this line has not been removed. Here you regard, you have arcs of circles which are then continued in straight lines, but the straight lines are tangent to the circle at that point So this is a really weird construction, a really, really weird construction that's not really given to you by intuition.

7:30 I mean, it's the kind of thing logicians are very used to these days. It's the justification for it is that it works, but there isn't much more of a justification for it. Now, of course, I mean, it's fair enough sketching, as I've just done, this, you know, informally sketching this model. But what you have to show is that it actually is a model for the axioms. So one question, do axioms one, the incidence axioms, the planar incidence axioms hold? For instance, here's a question. Is it the case that there's a straight line in the new sense, joining a point A in the upper half plane with a point B in the lower half plane? Is there always such a point? Sorry, is there always such a line? Well, that question becomes this. Is it the case that there's always a circle passing through 0 and A, so the origin and the given point A, which cuts the interval somewhere between 0 and minus infinity, such that the tangent to that circle at that point is a straight line which passes through B. So it's a complicated question, right? And the answer is yes, but it's only really the analytic nature of the model that lets you see that. And here is the calculation. You can actually carry through the calculation. So, you know, assume that M is the center of a circle you're seeking, then what the existence of that center amounts to is the solubility of that fairly simple equation, and you can go through the conditions for the solubility of that, and it turns out there are two possible solutions to that equation. One has to be ruled out because it would involve, wouldn't give you a straight line of the right kind in the model, but the other solution would. So the answer to this question is, of course, yes, but it's only this, it's not difficult, this calculation, but it's somewhat involved.

10:00 It's only that calculation that shows you that the model is actually a model. So the point about this is that here the calculating r, as Hilbert calls it, comes into its own. And it's only the use of this plane and the accompanying simple algebra that gives Hilbert the control over the pieces and how to glue them together in the right way, even though the result, when you translate it to the intuitive level, is fairly easy to describe and fairly easy to see visually. So the last point about this, of course, is that this model allows Hilbert to construct a design configuration. Here you've got two triangles. Wow, we're on the A, B, and C. you know where they're this is actually the converse of design theory which is equivalent to the other version where the intersection points all align you have to show that the other intersection points all go through a common point so this is what you're given here the intersections of the corresponding sides of the triangle all y on this line and these so a a prime b b prime c c prime all should run through a common point and of course in the ordinary plane they would and it would be up here but in this model once they hit that line here they go off into circles right and those circles Hilbert shows intersect in three distinct points not a common point so Desartes examples like this show what I call the weak dependence of geometry on that analytic models supplement intuition they refine it, instruct it but there's a stronger kind of dependence which is what I want to finish with The analytic models are the ones that show that these various geometries that Hilbert considers via these independence proofs are all possible.

12:30 This is what Hilbert thinks of as mathematical existence. So these geometries exist because the numbers do. So this is actually the stronger kind of difference of geometry. It puts geometry firmly into the domain of pure thought. if we're going to stop here, either for you, any of the questions. Thank you very much. Questions, comments? Doug? I think it's a very simple, very quick question that I actually noticed because I've been in the end of the year, and I've been talking about it very early. Arithmetical concepts are required to do that. So I'm not thinking of that. That means that I wonder what this is susceptible to prayer come to? Is the thesis that you could have the arithmetical concept without having the geometric concept, but you couldn't have the geometrical without the arithmetical concept? No, I don't think it's that. I mean, there is a sense in which the geometrical must be prior to the arithmetical in the, I mean, in the history of mathematics, this comes first, right? And there's a sense in which it's more basic, because it has something more direct for all of us, with more and more visual equipment and so on. So it's not that. I think part of the point here is that And there is a, with this thesis, what you do with the Naxiom system, what's essential, what you have to do essentially with the Naxiom system, once you've reached a maturing stage, is divorce it from its natural interpretations. there's a sense in which the notion of what constitutes a mathematical theorem so in this sense geometry changes you're not now solely interested in deriving theorems within a framework what you're interested in

15:00 as a mature mathematician is such questions as So what's the precise logical relationship between this and this? Okay, you can prove this from this, but do you have to use all of this to prove this? Can you use just some weakening of this to prove this? If you weaken it in a natural way, can you still prove it? and so on. These things are not, would not normally be regarded as geometrical questions. They're sort of logical questions based on the geometrical framework. So now to answer that kind of question, Hilbert thinks that number is and in a way it goes back to in a way it goes back to traditional problems the problem of squaring the circle Euclid of course in book one he has a raft of theories about if you have a triangle with a certain area how can you construct a parallelogram with the same area how can you construct this kind of figure with the same areas? Now, if you ask that question about the square and the circle, given a circle in a certain area, can you construct a square with that area? That's a perfectly natural question in the Euclidean framework. But Hilbert would say you can only, it turns out, you can only answer this by calling on some higher analytical framework. where you get some analytic treatment of what's constructible by what means and so on and then show that there are there's a certain class of numbers which you can't construct by this and that somehow the pi is is going to fall in that class so you can't do the construction so it's answering that kind of question uh so geometry is dependent on number for that sort of reason

17:30 marco sebastian you say that in a certain model I think that you are telling me the restructuration, the synthetic restructuration, part of that is the use of groups of axioms one and four in order to prove so much as possible. So the first remark is that this view from Jessica that has gone to you is saying as something quite strange that the properties of continuity or connected to the intermediate properties of continuity in general sense that are formulated in express the connection with the part are in a sense not something essential on the synthetic. as such, it's not to do with continuity. Whenever, in a classical point of view, the use of synthesis was exactly connected with the essential part of the current continuity, is the first one. So essentially there is quite a big change of things that is independent of continuity of something like that. And the second question is that it seems, from what you say, that in other forms of view, where analyticity is essentially connected with numbers, so you need essentially numbers. But the question now, I have a question, is really essential to use the reference to a numerical field in order to define the axiom, like you need an axiom and the axiom of continuity, It's necessary in order to make an algebraic geometry, in a sense, to prefer numbers. We cannot really think about, for example, as a property of a system of parts.

20:00 Or the parts in a certain way. So why number essentially? Once we have defined the central operation on the two different entities, and who is A, that does that, why necessarily numbers? What is essential in numbers that make it an antithesis? I don't have any answers to either of those questions. But the first point is one that I had not thought about before. That's actually interesting. I mean, if that's true, that continuity was thought of as being essentially a synthetic notion. anymore. I mean, what do you mean by that? It's something like this, that to give a correct adequate account of congruence, for instance, one appeals to the notion of movement of a segment to somewhere else in comparison. Do you mean something like that? For example, yes, classically, the use of scientific geometry was necessary in order to solve problems connected with commensurability. The idea that in the field of numbers you cannot really treat this continuous magnet, and you cannot only treat this certain magnet to commensurability. So the idea, the fact that only the probably geometric treatment can allow you to feel it. It seems to me that here, the perspective is completely worse. Now you see, I think what Hilbert is, I think he's less concerned about continuity than he is about the nature, the logical nature of the Archimedean. Because that involves an explicit appeal to ordinary whole numbers, right? Because it's got to do with, you know, you take a segment and a point outside that segment on the same line,

22:30 then there is some n-fold iteration of the segment that leads you beyond that point, right? So it's an explicit use, explicit quantifier running over the natural numbers. There exists an end such that. That, I think, is of much more concern to him than he calls that a continuity. But it's that use, that appeal to natural numbers that are explicitly, in the statement of the exit, that concerns him more than he'll say to continuity in something like confidence. Can I say one thing? But I think there are two different things with respect to the use of numbers. One thing is, you are saying that we need numbers in order to formulate Alchemian property, that, in a sense, a very, very weak use of positive numbers that also perfect the case in all sorts of classical geometry. You have what I call external quantification. So use number numbers to count the iterations. But what another thing is, a completely different thing, is to say that the bond is a couple on a certain set. It's a pair on a certain set. It's not the same thing, essentially. So that the idea, the continuity on the side, Analytically, on the side of numbers, it's not simply the first thing, it's the second thing. You think, or see the part, as a pair, is a number itself. It's not. The two things are not equivalent in the situation. It's a good point. I think Hilbert very clearly, once he's sorted out these field properties with segments, He's preferably happy to co-ordinatize using those. And in fact, he does some very, very clever things. So the use of coordinate pairs, number of coordinate pairs, is not so essential once one has that analysis. That's true. But it's true that these things, I have to think much more carefully about these things.

25:00 Do you have a good reference to appeals to continuity in geometry before Hilderstein? Anyway, we can talk about this one. I'm interested in turning back to one of your slides, when you said that you that it's much more difficult to have the experimentation, like an experimentation. So my question would be, what does it mean by experimentation? It's related with, I don't know, the slide, when you talk about literature, natural science, and also with the fact that I am thinking to the lecture given by Hubert on the Anthony when he says that infinite never exists in small or in large. So from what sense do you think? Think of how... I mean, I guess he must think of something like this. Think of how complicated it would be to set up an experiment to see whether the parallel axiom holds or not in the physical world. Gauss apparently considered this question, not with the famous in this story. There's this story, I think I first learned just from you. There's a story that Gauss considered these three mountain tops, the triangle formed by these mountain tops. Could you get a deviation from 180 degrees in the angle summertime? But I think you told me, I think you, Jeremy, told me that that's ridiculous because Gauss, of course, knew that this is far too small to produce...

27:30 I argue this backwards and forwards, and I argue this backwards and forwards. Just to get another word, it's complicated about what a surveying industry is. It doesn't matter what you think you're going to measure. Presently, I think they must agree that perhaps we can see from the observations on that, I don't know, but it's not the case that the survey was conducted with that aid in mind. If you look at the map, the triangle are up north and northwest of the Big Mountain. This is something like a back of the old world calculation of someone like Dallas. And the board in measuring will not allow you to determine. This is given the equipment you've got. It doesn't mean it couldn't be done, but it supports it. What I think we're just making now is to ask you that it's going to be a very complicated... We're in the real resolution, whereas all of the stuff prior to this, as I was talking about, So I'm out of the common sense experience. So all those axioms are impossible to be made in Gauss-Newman. As for the Archimedean axiom, I think Hilbert takes this as in some ways fundamental for doing any kind of science because it's the Archimedean axiom that tells you that measurements you make in the small, say, intra-atomic measurements are commensurable with, you know, cosmo-logical measurements. It's the Archimedean principle that tells you you're dealing with the same measurement. So Hilbert thinks that that's, you know, a fundamental sine qua non scientific investigation. but that's but can you do experiments that actually show you that that must be the case oh sure you said on the other end that

30:00 infinite never exists in small You can't say that she is a flag. You can't say that she is a flag. And say that that was all told. You cannot say that she is a flag. You cannot say that she is a flag. You cannot explain it. Explain it. Yeah. Yeah. But, you know, would Hilbert have known at that time any experiments that would actually lead you to believe that the universe was finite in extent? This is a principle that follows from Einsteinian models of Einsteinian mechanics. The field of infinity does not exist. Yeah, but the claim there must be that, you know, the best physics we have tells us that, right? We have to go with the best physics we have. It's not that I can look out of the window and conclude that the universe is finite, both in the small and the large. The best physics we have tells us that that must be the case. physics we have. We've got to go to the next question that we have. Who's power? I'd like to go back to the purity of the method question. Somehow that's what you think, and geometry is suited for the position of number, and this question is answered by the distribution of the instruction. And I would like to know if there is, because there is a similarity to another question and he's actually asking why are magnitudes suitable for a description by numbers? Why can they be represented by numbers? And the same was made by Veronese, but they arrived to very different results in the sense that Keywords in the presentation, even he discussed the norm-to-medium geometry that's not allowed for an norm-to-medium continuous or not a norm-to-medium or complete structures or so to do it.

32:30 So the idea is, did he ever consider these possibilities in other lectures, back from the monograph, and did he maybe consider the necessity of representation of theorem that concerns different classes of magnitudes and different classes of numbers, or the sort of reduction in the idea that he had in the beginning? this somehow implied in the fact that he used real numbers of special subsystem of real numbers I'm not sure go back to the first part of the question now about non-Archimedean fields and so on did Hilbert ever consider these? is that what you're asking? well especially if you consider these as part I mean, he certainly did consider non-cognitive fields very, I mean, very seriously. Actually, I wanted to ask you a question about whether now is not the time, whether he could have taken things from their native work. But he uses them as if this is fairly standard. He uses infinite expansions and then defines an archipelian field taking these expansions as points in that field and then imposes a certain ordering on them which gives even the decimal elements. This is what he takes always as the basis of a non-partnerian view. Now, did he try to answer any purity of method questions involving that? I don't know, I'd have to think about that. I'd have to think about that. But it's an interesting question. but I think he certainly takes the view that these structures so this is part of the divorce from natural interpretation that you get once this axiom system is given you cut yourself off

35:00 from the natural reading of point, line, and plane in the geometrical context and then consider models And when you consider models, you start to consider models in which key things fail. And these for him are natural mathematical structures, I'll call them that for want of a better term, that the mathematician will consider. These are just as important, or they might turn out to be just as important as the classical structures. But it's important that you divorce yourself from these natural interpretations. Otherwise, you wouldn't allow yourself to consider these. So a non-Archimedean field is a perfectly good magnitude field. Fine. I mean, call it a magnitude field if you like. have anything like the prejudice that one finds, say, in Kant, which somehow or other, once you drop this principle these aren't proper magnitudes anymore because they don't satisfy this or that property. Hilbert doesn't have that view. He thinks they're perfectly good objects to investigate. And in fact, you can often, by investigating them, get very important information these other classical structures. The only one is that it cannot be continuous in the case, because So we cannot get continuous . You're right. Right. It's concerning this notion of translation. In his reply, He speaks in similar talks about the schema and so on, but then he also mentions explicitly this reversibility of this translation. And in my understanding, actually, it cannot be otherwise, because this notion of scheme wouldn't make sense. It only makes sense, it's a kind of claim, if you can refute it, that it would be very good. because only reversible translation isomorphism would give you like the equivalence relation and that's all we may caution by this relation and it's like fregius obstruction and just translations it just doesn't work in

37:30 that way at all so we wouldn't have anything like scheme so it's one point and now question to this time is how it's a laser logic actually because of kind of scheme is not part of logic right it's it's kind of axioms of geometry of something so what is exactly in this story relation between say logic this particular theory I don't think I I don't think Hilbert had this view of logic as a theory of everything. I mean, that was very much Frege's view. Yeah. Right. I mean, you have these people, I'm not sure this is right, but there's some truth in it. You have these people like Warren Goldfarb and Drebin and so on claim that Frege could not have had any, that there are no meta-mathematical considerations in Frege because there is a logic presented with a framework that you can't get outside to consider other things. And Hilbert, I think, had very much the modern, and this is the way I see it, very much the modern kind of view. I mean, what does a mathematician do? What does a mathematician do studying set, they would say? You look at the axioms, somehow you stand outside it, right? And then as a mathematician you say what tools can i bring to construct you know new models of this or models of that or whatever so i'm doing mathematics on this axiom system right and what and i feel quite free as to what mathematics i appeal to when i do that but also i have some logical calculus as you do that, right, and I have Right, but this is yeah, so so foundational study in this sense is very much the kind of foundational study that Hilbert started I mean of course at the time when he was doing all this he didn't have any logic or logical calculus, but he did He mentions this quotation, he mentions logic He mentions unalterable laws

40:00 not too long. And a puzzle by various things, the synthetic, geometrical, ambitious is to say there are some concepts like point, line, plane, reason with that. A great example of course with Euclid, and it makes a kind of conversation more sense to do it, and so you look in say he disdains the idea of rigorizing Euclid by passing through the Arab Republic, but somehow extremes. But nonetheless, the synthetic approach. The inspiration comes from the imperial .. So you do a whole multi-number. You walk around, forms of facts, and then you rigorize them, and so on. We noticed. system. But it seems to me that that philosophy was somehow aimed at making sense of geometry. It was one something, or maybe two or three, maybe there's an outline, maybe there's a projection . But, you know, there's a kind of sophistication of true geometry, And suddenly with Hilbert we have the idea, no, you'll do something with maybe some primitive concerns. You're not at all expecting a unique geometry. You're going to get lots of different geometry, and that's part of the fun. And at the same time, it's been known for ages that there's lots of strange kind of numbers out there, any algebraic, number zeroes piece, number fields, various kinds. And he'll, of course, exactly in this period is becoming an algebraic mountain series. So it's not difficult for him somehow to say, well, there's lots of strange fields out there that could be the qualifications of major strange geometries. He's in a very rich, but very unnatural world. Is that a reasonable way to put it? There are lots of things you can make geometry do. Well, it's not unnatural, but it's not the empirical world. So what his geography is going to be is some kind of axiomatic structure of some of the key words at the exact point, but as you rightly say, a polysemic view, and there's many different meanings.

42:30 If possible, that's crucial. Now, I'm just going to get at my personal concern is somehow about the decisive movement of originality. But he's not the first person to think of several different geometries, and the only thing in the case, I mean, quite in the Hebrew of our century. He's clearly not the first number theory to think there are many different number fields than the real. But almost everybody before him who wondered about why geometry turns into a piece of geometry had no alternative. What else could geometry do? It couldn't fail to be co-ordatized, then it had to be co-ordatized with the real, so there was nothing else to land on it. Now, somewhere in all of this, Helford says that's just wrong. He pulls these many different possibilities together, and he constructs a new aim for geometry, to have multiple pieces. Do you have any sense from your study of this whole period, from 1891 onwards, but when this new constellation of ideas began to form in his mind, does it come in one go? I mean, you give a lovely talk, and it's all part of identifying the moment. The point of heart, no, that's true. What was the point? I'll ask it like that. I just want an oversimplification. What was the Pauline conversion? I mean, one thing that one can say about this is that I think what Hilbert is striving for is some sort of unity, right? There are these different approaches. I mean, you, in a way, put your finger on this by saying that there is this implicit polemic in people like that. in these earlier projected georgias that say somehow all of this stuff is unnatural and we shouldn't do it that way. What Hilbert wants to do is show, yes, you can do it the way Pash and these other people want to do it, and I will do it better than they can and show and extend this to Euclid and show that you can get an awful lot of Euclid out by doing this.

45:00 But at the same time, this is unified. This is just a part of mathematics. And these other parts come into it too. Moreover, not only do they come into it, but they yield very important geometrical information. and of course I, Hilbert, have extended the notion of geometrical information from just merely what one can prove in this particular framework to questions of what is necessary to prove what and that actually he would say ties up with intuition too these axiom groups as expressing, as he says this in the monograph, various basic facts of our intuition, right? The analysis of intuition, which he says is part of the purpose of this project, will boil down to exactly the study of these logical relationships, right? Which, what intuition is called on for proving what right and an important part of his work is showing that you can do these things, you can do a certain thing in various ways, you could do it by calling on spatial assumptions right, spatial axioms of this or you can drop these but then you would have to call on the Archimedean axiom to get this through so you have a choice if your intuition one way, then you can do it one way. If your intuition goes another way, you can do it another way. But anyway, the basic point is some sort of, there's some sort of unification in the background. That this is all part of mathematics. And it's not that one is privileged over another. But that can bring all this together. Yvonne? In your view, is it just a question of some sort of

47:30 Göttingen stuff, Göttingen stuff, connected to some sort of planning and notion of Arabization, or is it something more powerful, deeply connected to his work, You mentioned that this ridiculous expanse was around the 90s, about 95, 97. So, what is it, how is this understood? You are just hollow buildings. And let me come back to that point in a minute. Why I made this distinction between the 1891 remarks and the Salbericht remarks is because in the 1891 thing, who states this reductionist or thesis by saying every theorem of pure mathematics has to be blah, blah, blah. In the Salbericht version, he says, every theorem of higher function theory has to be reducible to relations between commoners. And there is a big difference between those two things, right? The first one says that whatever you do in mathematics it has to reduce to this. The second one just says whatever you do in higher function theory has to be reducible. That's one thing. So I don't know he'd had deeper or more considered thoughts about this, but I suspect he probably had. As for what's going on in the first quote, I don't know. I mean, I'm tempted to say, I'm tempted to dismiss it and say that Hilbert was just accepting certain beliefs about the priority of analysis that were fairly common time. A lot of that passage I quoted could be regarded as taken from the introduction to Dedekind's monograph on the natural elements from three years before. You know, in particular,

50:00 this passage about, you know, doing this reduction often involves a long and difficult way, but But nevertheless, that is almost directly taken from the Dadekin's passage in 1888. And obviously carries echoes of the Verstrasse and Chronicle. Sorry? And obviously carries echoes of the Verstrasse and Chronicle. Right. Right. So I'm tempted to say that Hilbert was just repeating what he just took over as a convention there. And that what comes later is a rather more considered reflection of this, and a more sophisticated view. Yeah, that was during the talk, there was a question, and you've answered it by saying that the question was whether superimposing the reals on geometry might not lead to contradiction, and that rather surprised me. I thought the problem was a different one, was to know whether the numbers come out of the geometry. In other words, the danger, I didn't think the danger was that geometry with the numbers might be contradictory, but rather that geometry was too weak to give you the numbers. In other words, the traditional way of looking at number, lengths and so on superimposed on them was unwarranted by the geometric facts, that this was to reach a theory. Right. Now, what I meant by the contradiction was that wasn't a serious claim. What it was was more of a provocative remark to say that it's not a priori obvious. I mean, of course it was known by, you know, very early on, that in some sense the geometry is compatible with the use of numbers. I mean, very early on in the Greek period.

52:30 So that wasn't a serious, that's not a serious historical claim, but it's more of a, you experiment in place. Why should it have turned out to be that way? But I think you're right that it's the other question about the weakness that's the important one. And what Hilder tries to show is that, you know, actually an awful lot could be gone out of the geometrical framework. So it's really more of a reply to that sort of question. I have a question, Mike. So I'm interested in this idea of Hilbert's, that when he constructed models, he should intentionally try to fix something that was not into it. And I think he had an idea of something like Hobbes, that if I deal with things that are familiar to me, I'm likely to be making all kinds of assumptions or have all kinds of beliefs that are playing a role that I'm not aware of. And that's the exact contrary to Riefer's reasoning. Okay, so he wanted something odd to kind of block the tacit presuppositions. All right? What I don't really see is why it had to be number at all. Why couldn't it be, I see no essential reason why it had to be anything arithmetical whatsoever. Okay, and yet it seems that if it's not strict necessity, there's some pretty potent reason for why, according to your account, for why Hilbert chose arithmetic. All right, and it's that that I'm unclear about. we'd go past it you've read many times but for the sake of those the audience so this is from the 1900 problems lecture it says in the further development of a branch of mathematics the human mind encouraged by the success of its solution becomes conscious of its independence it evolves from itself alone without appreciable influence from without okay and he says then he goes he

55:00 goes on to say then that it seems that the numerous and surprising analogies and that apparently pre-arranged harmony which the mathematicians so often perceives in his questions methods and ideas have their origin in this ever-recurring interplay between thought and experience so he seems to think now that there's it's important to have not just stuff you make up create whatever which you might have some sort of complete knowledge of but somehow this has to be tested by or nourished by stuff you don't make up and it's it's actually that question that seems to me to need answering here uh if we're to really understand whether hilbert could be saying that geometries exist because numbers do okay if you're taking pure products of thought to be the essential characterization of the numbers and you pure products of thought, being these things we think up without consultation or influence from experience, then it seems to me he really wouldn't want to say that geometries exist because the numbers do, because he seems to be saying, is there any way that you couldn't do without this influence from experience? so that this is too simple story so So, why models using numbers, that was part of your first one. Uh, one answer to that is simply practical, that we have, we, Hilbert and the mathematicians of the 19th century, of course, have an enormous amount of experience with, uh, analytic, geometry,

57:30 all sorts of weird many-dimensional we have this rich fund of mathematics let's draw on that if we can if it's a rich fund if it's something we receive why doesn't it pose the same kind of threat that receiving nature represents in other words we're going to be making use the identity which we don't recognize and hence violate rigor yeah that's a thing so you're asking why are numbers more consistent or why do we believe more in their consistency than the consistency of the exit of the outcome right, I mean another way of putting it is there something like an arithmetical nature that's a question of course that comes to consider It's exactly the other way around. It's precisely because it's the objects of arithmetic that for Hilbert to have the absolute clear-cut identity conditions, that they're in a position to supply the ultimate ingredients of definition of these structuralised and therefore being enormously generalised geometrical notions. It seems to me that there are two questions here, and Hilbert's very interestingly moved from that 1891 statement, which seems to carry quite a strong ontological undertone. Your question, you know, is Hilbert claiming that geometries exist because numbers do, which does seem to be a question which carries a strong general metaphysical component, to a much more internal to the methodological underpinnings of mathematics question, which is what is it that makes the unity of mathematical methods and the conceptual organization of mathematics possible? And it's just the kind of structuralization via this program of arithmetization that provides the key, it seems to me, in his later position to the answer to that question. But it seems to me that there's been a very definite move broadly ontological questions who are more broadly methodological questions in the course of this. I didn't mean to make it to be shared. No, but I think Hilbert did. That's my claim. But was there an alternative about which was I don't know. Well, I take it that that's partly Michael's point,

1:00:00 is that just practically speaking there weren't other ways, known ways, and that's a good enough answer I mean historically that's it that's a perfectly good answer I think but it's it's not a very powerful theoretical consideration seems to me I mean that's like saying like because I don't know how else to do it all right but if my imagination were a little wilder zippier I could come up with a geometrical model what is essential in these models it's not that they are concerned with the field so the point is what's the number is i think that in a sense what we are suggesting the number is an element of a field point so we need failure so then numbers are element of fields if you define your fear is on that i mean in a sense the question is not numbers as such but what is essential it seems to me that you have Marko actually touches on a point which doesn't come out in what I was talking about, but at some point in preparation for this, in so far as my control of preparation, I did think of bringing out, which is that in doing all this, Hilbert actually transforms our notion of what a number is. That's right. Because then a number becomes just what satisfies this notion of ordered fields. Actually, Dedekind's already done that. Yeah, Dedekind's already done that. Hilbert just redoes it in a different way. Yeah. In this one I can recall from what I decided elsewhere and the children did all these terrible things on analysis and whatnot.