Geometry & Number in Hilbert (1891–1905) — Part 1

0:00 Well, at the end of the month, essentially. And Gilbert's shifting conception of the relationship between geometry and number is positive. And his paper, which is found, is a tall part. It's supposed to be the world, and you don't have a place for the presentation of the Polish There are three guiding theses which are commonly included and intertwined in Hilbert's foundation of the thought. And all of these have to be taken seriously, you know, I don't mean like that. What I'm going to do here is not to address any of these theses directly, but, you know, I want to remind you that these are in the background. The first thesis is that geometry is a natural science, and its laws are in some sense dictated by the way The stress here is on the term in some sense. Arithmetic, though, is what he often called a product of the mind. So arithmetic, in its broadest sense, is a product of the mind. The second thesis is that number concepts are conceptually prior to geometrical concepts. They're partly tied up with the first thesis. And the third thesis is that intuition is important in the acquisition of geometrical knowledge, but it's guided by higher mathematics. And this ties up very much to what Jeremy was talking about this morning. And by the same token, intuition is only foundationally important in a very limited sense. And this higher mathematics, although it is not actually bound by the intuition. These theses, theses, of course, change in content, weight, and consequently

2:30 effect over time a part of the purpose of the examination is to of which this is a preliminary part is to see exactly how it's changed so so much for the background assumptions. These are the contents. This is the sections of this tool. And you can see by the highlighted, say, on the top there, which sections. So I wanted to explicitly mention that we've changed sections, but you can see by the elimination of that. we have changed sections. So what I've just gone through is my introduction. Here is a section called preliminaries. Writing in the introduction to his lectures in Konigsberg on Projective Geometry in 1991, these lectures are actually a fairly very conventional presentation somewhat limited even at that point, and in that respect, they're not particularly unusual or interesting, but one of the reasons which makes them interesting is that there is a fairly long introduction in which Hilbert makes various interesting, fascinating statements, And here's one of them. A statement of what could be called an arithematization principle or a principle of reductionism. So he says this, any proposition of pure mathematics, no matter how deep and complicated, must be capable of being reduced to relations between the whole numbers. Since this way is long and difficult, mathematics has thought out means of making the part smoother and shorter, or has made it more certain by the provision of intermediate steps, et cetera. Nevertheless, this reduction is not only possible, but is required. Today, a proposition only holds it proved when in the last instance it expresses a relationship, a relation between whole numbers. Thus, the whole number is the element.

5:00 It is to be examined, and I haven't actually carried out the examination, but it's questionable whether Hilbert still holds this 1891 position by the time that 1895 comes around. One finds similar but arguably weaker remarks than other places, for example, in the famous I've got 1895 up here too because there is an unpublished lecture that Gilbert gave to the meeting of the German Mathematical Society in 1895 which is a preliminary report and actually is a, the lecture is some, very similar anyway to what becomes the introduction to the published lecture. for example, to 1897. So you get similar statements in both of those places. But he does add this to these statements. This is new as compared to the statement of 1891. Arithmetization of geometry, he says, is completed by the modern investigations on non-including geometry, in which, in order to achieve a fully rigorous construction of such geometries, it's a matter of introducing number into geometry in the most direct and least objectable way. As I say, it's arguable whether Hiltz-Kir is actually endorsing the strong reductionist principle that he seems to state in 1891. one. In any case, reductionism is given up officially later. One finds this statement in 1921-22. I won't read all of it, but he says, right at the end, this is a bit read, that this principle, so the reductionist principle, which was regarded as embodying a fundamental epistemological principle must now be abandoned as a prejudice.

7:30 The reasons for the rejection of this principle once one adopts what he calls here the viewpoint of finite logic are, I suspect, complicated do with the involvement of set theory, theory of infinite sets, and so on, in the reduction itself. I don't want to go into any of this here. What I want to maintain is that, in In fact, Hilbert gives up the reductionist principle of 1891 at a very much earlier stage, but we only find this explicit rejection of it at a much later point. He actually gives it up rather earlier. But the situation is somewhat complicated, as I hope I will make clear, at least partially clear. Some of the complication involves the convoluted way in which mathematical theories bound up with one another another thing which makes it complicated is the question of whether or not geometry is to be considered part of pure mathematics and with this the extent also to which geometry relies on intuition or is dependent on intuition which would make it beyond outside of the what I want to try to show is that although Hilbert might later on in the 1890s have supported some pieces of reductionism about say higher function theory he did not support it about geometry although So that's basically what I'm trying to set out in this tool. Let me turn now to Hilbert's work on geometry. The strong reductionist statement comes, as you recall, from this paper of this lecture series of 1891 on predictive geometry. things about the introduction to that series of lectures

10:00 is that Hilbert sets out a interesting classification of geometry. Geometry, according to him, consists of three areas. One is the geometry of intuition. The second is what he calls the axiomatic, what today we would call the axiomatic treatment of geometry he has this heading axiomatic geometry and the third is what he calls analytical geometry the geometry of intuition is itself subclassified into three parts and this actually what you see on the screen here is a direct quote from those the first part consists of what he calls school geometry congruence theorems triangle polygon circle etc the second is what he calls projective geometry conic sections focal points curves and space the third is what he calls analysis sitters it's not there's no further elaboration there of what actually means by that in 1891, some time before the development of what we now think of as topology. And he makes it very clear in his discussion here that what he calls Euclidean geometry falls very much under geometry to Anishal. and usually he talks as if Euclidean geometry, the core of Euclidean geometry is what falls under the heading A what he calls school geometry so it deals with the congruence theory the theory of the triangle polygons, circles, etc. Euclidean geometry is Euclidean geometry or Euclidean geometry? He just calls it Euclidean geometry I mean I don't think it would be fair to say Euclid's geometry. It's Euclid's geometry as it's presented in schools, say, or in 19th

12:30 century textbooks. I mean, until Heiberg's, when was Heiberg's work? 1885? I mean, that not yet made its way through to the curriculum. I mean even if even the geometry I was taught at school was called Euclidean geometry but it was not it was not a textbook treatment of Euclid's geometry he says this about geometry that geometry is the theory of the properties of space I can never discover the properties of space through mere reflection, just as I cannot discover in this way the basic laws of mechanics or any other physical law. Sorry about all these italics in this, but this is actually how Hilbert does it. Space is not a product of my reflection, but is given to me through my senses. Therefore, I require my senses for working out its properties. I require intuition and experimentation, just as with the discovery of physical laws, where also the subject matter is given through the senses. Indeed, the oldest geometry arises from the contemplation of things in space as they are given in daily life and like all science at the beginning and pose problems of practical reports. It also rests on the simplest kind of experimentation that one can perform namely on drawing. And he contrasts this many of you might know this famous passage in a letter from Gauss to Bessel of 1880 when he says something very similar and what gas contrasts this view of geometry as well I'm not being able to prescribe laws to geometry cut somehow being derived from what we understand in the space around us So it cannot be arbitrary as he contrasts the theory of arithmetic with that, that this is just a product of pure thinking. Gilbert stresses this too in his introduction of these lectures.

15:00 He then talks about the development of Euclid's geometry and Euclid's geometry into the modern era, and then says this, that as marvelous as it was, he said, what it lacked, it had an essential defect. it had no general method without which a fruitful further development of the science possible this defect was rectified through the invention of Cartesian analytic geometry which provided a powerful unified method but But this, in turn, brought its own disadvantages, and he says this, this is the passage you have on the screen, in front of you, as important as this step forward was, of the introduction of analytic geometry, and as wonderful as the successes were, nevertheless, geometry as such, in the end, suffered under the one-sided development of this method. One calculated exclusively, without having any intuition of what was calculated. One lost the sense for the geometrical figure and for the geometrical construction. What one then finds in this introduction, Hilbert then goes on to say that the movement in synthetic geometry that one finds towards the beginning of the 19th century, particularly clear in the figures that he cites here, particularly clear in the work of Monsch and von Stout, is partly a reaction to this. It's an attempt to go back to the geometrical, the presentation of pure geometry We doubt the influence of analytic geometry. So this becomes, in some Hilbert's time,

17:30 one of the standard ways in which to present projective geometry. its own lecture notes, which follow this introduction, are more or less, as he said, a standard presentation of this. But what's interesting is that he takes this as one of the guiding principles of his subsequent investigations in the foundations of geometry, generally, and particularly the foundations of Euclidean geometry. One of the purposes of this work, then, was to reformulate and restructure full Slidian geometry itself as far as possible in a synthetic way. So in doing this, he developed geometry in a modern axiom system, which builds up from the simplest possible projective framework of some incidents in order geometries. And as congruent axioms develops what theorems, important theorems, we can get by adding that. Euclidean parallel axiom, so arriving at full Euclidean geometry with the standard results from the Euclidean theory of congruence proportions, area, surface measurement, parallels, all of which falls under the heading of what he'll look for as school geometry. So this being developed before the introduction of continuity, I would say. Is it the case that in this treatment, projective geometry is kind of just more general, serial geometry rather than a particular chapter of this? I'm not sure what your question is, I mean, for him, projective geometry is first of all which you add just on the basis of incidence of order and relationships. Then you add the theory of conquerors to that, and then see how you can start to understand. I mean, so it's like the most general general. Now, what's somewhat misleading about this, let me just give this, this is a table showing Hildebrandt's lectures, a series of lectures on geometry in the 1890s. So this first one, 1891, is these lectures on projection geometry.

20:00 In 1893-94, there is a course called the Foundations of Geometry, which begins to develop the axiomatic system that doesn't complete it. In 1898, we get this very short course of health. There was a tradition in German university that called courses during the Easter vacation for school teachers and some sort of refresher courses for the people who have been through university education before. They would give them short refresher courses in the Eastern world. And Hilbert gave several courses. but this one in 1898 was actually so it wasn't it was actually called but it really concentrated in one junction so there you find a much more modern elaborate than modern Hilbert but there's the book elaboration of the answers Then in 1898-1999, a little bit later than this, you get this marvelous course on the foundations of Euclidean doctrine. That appears here twice because in the... So this is, this one and this here are the same course, it's just that this, what exists of this are lecture notes in Hilter's own head. What you find, what this is, is a, an outside article, a suspense, by the name of Herms von Schatten, of this course, where many things, many details are given, which are not given here, and it's much more cleanly presented with beautiful diagrams and in a very readable pen, probably not on Shabbos, but probably by a professional, I'm sure the word you'd use is a professional scribe.

22:30 This was mimeographed, I think 17, over 17 copies were made, and this was a student's And then, very shortly after, this was published in that form, in March of 1899. By June, Hilbert had published his famous moniker, The Signation of the Church. Now, though the content of this lecture course, of course, covered by these two names, is very similar to the content of the monograph. The style of presentation is actually very different. So these, the lectures here, make the synthetic restructuring of Jungtree much clearer than the monograph does. In the monograph, what you find is all the axioms are given at the beginning. And then Hilbert begins to concentrate on some of the consequences. And what you find, though, in this lecture course is the axioms are presented in this piecemeal fashion. So some axioms are presented. Consequences are drawn. There's lots of philosophical discussion about what you can do with this, and why you have this and not that, all of which is dropped for the monoclon. So, although I think it's fair to say that the results are actually very, very close, the presentation is made. So, let me just finish with this table. In 1900, you get the French translation of the Monaco, in which Hilbert adds some sections, particularly a presenting what was now the complete section for Snedekites, and also the section describing Dane's work on the genre and theorems. And then in 1902, you actually get a much more sophisticated re-treatment of some material that comes out together with some new material.

25:00 Then the second edition, I missed this off, but the second edition of the monograph that he gives in my team is from the audience, which incorporates the language and partially dissects the audience. So these, if you're interested, propaganda and marketing. We collected these and put them together into an edition. I think it helped all the introductions. Published in 2004. Anyway, by 1898, actually by the time of this Ferium course, the Holodick course, and then of course in the lectures later that year, Hilbert had settled on what are now considered The only thing that should be remarked about this is that in the monograph, groups three and four strangely are turned around. So the parallel axiom is group three, and the congruence axiom is group four. But in the second edition, he returns to this. This is the ordering he has in the lecture notes. In the second edition, he returns to this. Why he changed it? The first edition of the lecture notes. So these four axioms are axioms for dealing with political school geometry, then you get continuity axioms, numerical axioms. There are two of those. The Archimedean axiom is is the first one, and then the Walsh-Tendekind's axiom, which only is later, is the second one. And it's really, really what he focuses on in all of his work is with the Archimedean axiom,

27:30 not with B and x. So the first four are three of numerical assumptions, and these only begin to come in with number five. So I should have said, which I don't say here, but I should is that this axiom system is based on three primitives and some primitive, so primitives point-line plane and some primitive relations that come across and so on, which are developed in this axiom system. Okay, so as I said before, the synthetic restructuring of Euclidean geometry is clear as in the 1898-1999 lectures. The groups of all axioms are designed by Hilbert to correspond to certain intuitive facts. He says this in the lectures, the general remark on the character of axioms 1 to 5 might be pertinent. The axioms 1 to 3, so incidence, order, congruent, state, very simple, one can even say original fact, that validity in nature can easily be demonstrated through experiment. Against this, however, the validity of 4 and 5, so parallelism continuity in the form of the Archimedean axiom, that's the only continuity of the axiom 5 at this point, is not so immediately clear. The experimental confirmation of these demands experiments. But at least there is still a link to intuitive empirical models. So, why do I say that what he wants to engage in here is a synthetic restructuring using these axiom groups want to form. There are two reasons. The first is this. There is what I call here the Euclidean project. Show that some proposition P or some theoretical development T can be deduced solely using some specified axiom sigma or generally some axiom sigma with explicit

30:00 exclusion of some other x equals gamma, where often where sigma, I missed out a font of this sentence, sigma and gamma factor naturally into some of these groups, right? And very often gamma is some sort of continuity of something. So then the task would be to show that you can do certain things or prove certain theorems without appeal some continuity why do I call it why do I call it Euclidean well I think there are two reasons one is that you find the idea in the first part anyway to try to do things by restrictive means. And it's not just that there are axioms underlying this. Of course, this provides a certain constraint or restriction on what you can do, because in the end, you have to be able to appeal to these axioms. I don't just mean that, but I mean, for instance, the way that Euclid refuses to introduce the principle of parallels until where he tries to do as much as possible before it's necessary to introduce them. Things like that. The fact that he only relies on flipping of figures, turning figures over in space at some limited point, even though you could give much clearer arguments for many things by using this much more widely. So, I mean restriction more in that sense. You find a similar kind of exercise going The second reason I call this the Euclidean project is that because part of what Hilbert wanted to do is to reconstruct the central results that one finds in Euclidean's book. You know, the things you could do with congruence assumptions. The way you can find figures with the same area,

32:30 different configurations with the same area as a given area. So a theory of theories of proportion, theories of plain polygamy areas, all of which one find say in the first book of Euclid's elements. Hilbert wants to reproduce the central results that one finds. So what I don't mean by this is that what Hilbert is trying to do is to mimic Euclid in a narrow historical sense, but he does want to mimic Euclid in a broader themselves. So let me cite some examples of what he does here. So he wants to develop the Euclidean theory of proportion and surface content for area, without any assumptions that the length of the areas are themselves magnitudes, an assumption which he criticizes So there's some background assumption that what one is dealing with here are general magnitudes and that the Euclidean common notions will apply to these. Hilbert thinks that this is a way of smuggling in numerical assumptions into the geometry. He wants to show that you can get the same results without using assumptions like this. Another example is proving his version of the Papus Pascal theorem from these axioms in groups 1 and 4 without using the Archimedean. That was the standard way of proving it in textbooks So he wants to show that you don't have to make this assumption. So you don't have to bring in numerical claims to do this. This here, 1 with a subscript p, what I mean by this is the

35:00 So, just the axioms that concern lines of points, not the axioms that concern, say, the intersectional claims. So, this is essential, we'll see this later, this is an essential restriction of the first group. The second way in which he tries to carry out a synthetic restructuring using the axiom groups, one to four, is very modern. That is, what he tries to give, or what he succeeds in giving, which he should say try, is a synthetic analysis of the field properties. So he does this by setting out axioms for a commutative ordered field. These are all fairly, these are all standard axioms. A lot of them. And then he sets up a calculus of segments, geometrical segments, with natural addition geometrically given, I mean geometrically defined as you should say, given, geometrically defined addition and multiple operations, and then shows that the purely synthetic axioms, so 1 to 4, will allow us to demonstrate that these segments satisfy the field values. And crucial in this is probably fits on the slide, but the DASARC theorem is responsible for showing the commutative, commutativity and associated, commutative and associative laws of addition, the associative law of multiplication showing that multiplication distributive over addition. The Pascal theorem, as Hilbert calls it, calls it, it's really the, I'm not talking about this with Jerry and I earlier, it's really the, what is known as the Pappas theorem, the restricted case of the Pascal theorem. responsible for showing the commutativity of multiplication. So what this shows is that the numbers in some sense are present on purely synthetic reasons because the Dezart

37:30 theorem can be proved from x in groups one and two or alternatively it can be proved from the plane part of one together with two if we add all of the congruent axioms. So in any way you get, in any case you get that this follows from these what are called synthetic axioms that proves one to four. And Pascal theorem or Pappas theorem can be proved from one to three. Alternatively it can be proved from the plane part of one together with so if you drop the spatial part of one so what if you recall what seemed to be at first sight the rather negative remark about analytic geometry from the 1891 introduction. In the 1894 lectures, you find a much more interesting remark, which you have on the screen in front of you. In all exact sciences, what first gains precise results when number is introduced. It is always of the highest epistemological importance to trace how this measuring occurs. In the 1898 lectures, one finds a similar kind of statement, but actually more elaborate. With these premises, he's actually talking about, this is in the introduction, he's talking here about what would happen if you just introduced analytic geometry says you would have one axiom essentially that namely a point is given to you by a pair of real numbers and then all of analytic geometry more or less follows with these premises geometry immediately becomes a calculating thus one embarks on a path by which one advances as quickly as possible to the introduction number into geometry and at any price now in fact in every exact science goal, and one can measure progress of a natural science or branch of a natural science precisely

40:00 by the degree to which number has been introduced. But if the science is not to fall victim to an unfruitful formalism, then at a later stage of development it will have to reflect upon itself again and at least examine the foundations on which it has arrived at the introduction of number. He goes on, geometry should not carry the rich means of analysis like chains. the means of analysis should be for it consciously sought out and applied sources of new knowledge and says accordingly in our lectures the introduction of number into geometry will appear right at the end of the goal crowning the whole structure of geometry built up before then so this actually puts the negative remark from 1891 about analytic geometry is much clear effective instead of a rejection what we have is what one can think of as a kind of purity of method question why is it that Euclidean geometry allows or is suited for the imposition of number often put purity of method questions to himself in this kind of form it's not so suppose you have the intrusion so to alien form of mathematics into another one which allows you to prove a theorem so one response to this might be let's try to get rid of this alien form and do everything it allows us to do but do it with the native pure principles the other response might be to say this was the one that Hilbert took. Why is it that that happens? Why is it that this so-called apparently alien thing has this effect? Let's look at that question. So if you pose that kind of framework here, you get this question, why is it the Euclidean geometry synthetically construed now allows or is suited for the imposition of So the synthetic reconstruction using Desaugs' theorem and Pascal's theorem and so on gives us a partial answer. I'll get what's on the next one.

42:30 Oh, well, yes. and forms at least, I think, some important part of the answer as to why Changshui does not have to be reduced to number one. Okay. So... Sorry. Well, you can think of, I mean, you can think of think of the use of I mean, if you start to think of the use of analytic geometry in Euclidean geometry I mean, think of this synthetically think of the use of analytic geometry by way of an imperial metaphor imperial military metaphor imposing itself on this debate right and taking it over so what what about Euclidean geometry allows it allows the use of the consistent use of analytic geometry I mean it could be it could just be it couldn't sorry it could just have been that somehow some principle within the geometry conflicts with the It's in direct contradiction with the use of, say, the view of real numbers, but that's not the case. Why is it not the case? What is it about the geometrical structure that allows us to co-ordinatize the point system? Hilbert, I think, gives a partial answer to this segment. You'll notice that we've, by looking at the top here, we've moved to a different section. This is synthetic restructuring, what I've been talking about now.

45:00 And now moving on to the consideration of the dependence of geometry. So far what I was stressing in this section on the, one of the things I was stressing in this section on the synthetic restructuring was the Euclidean nature, a part of this project. So to defend, the attempt to defend Euclid synthetically, and to show that what Euclid calls a theory of magnitudes arises intrinsically. But one shouldn't make the mistake of thinking that this exhausts Spielberg's work on geometry. him was the consideration of the phenomenon of independence, that the general form of this is, can we show that a given proposition P or some theoretical development cannot be digested or carried out solely using the restrictive methods given to you in some collection of principle, sigma. So the Euclidean project, as I stressed it, was show that a given proposition P can be deduced, just as important to Hilbert with consideration of the complementary question, show that it cannot be deduced, given some restrictive. So refinements on this, which amount to, as we would now call it, assessing the logical way of how several points is used. There are two refinements that show that P can be deduced using sigma, but not using sigma minus, where sigma minus is some slight The other refinement is the conference of that, if you like, show that he cannot be deduced using signal, but can if you use a slight strengthening of signal. So these are general forms of investigation that Pilburt was fascinated by, which make up actually the bulk of his oracle.

47:30 one often thinks, because this is what Hilbert stresses in the introduction to the monograph, that independence really is a matter of showing the independence of the axioms. He wants to get a group of mutually independent axioms. And actually, I mean, getting the axioms independent was of some interest to Hilbert, but actually it was of subordinate interest. He was much more interested in looking at what I call the logical way of central propositions. By central propositions I mean things like Euclid's famous theorem that the base angles of the Finisosceles triangle are equal. Or another one, actually one that we'll look at a bit, is the Desartes theorem. What exactly is involved in proving this and why do you have to focus on it? That sort of question was much, much more important than the question of the So there are two centrally elements in this treatment of independence, this assessment of logical ways. One was the divorce of primitives from their fixed meanings, allowing reinterpretation. The second is what I'll call here the translation of theories from one framework, so translation from one language into another. general view that's behind the reinterpretation thesis, the divorce of meaning thesis, is summed up in lectures from 1894 and you find this repeated very, very often throughout Hilbert's statements on geometry, actually on text insistence generally from to the end of his working career. But it's summed up nicely

50:00 in this 1894 passage. In general, one would say our theory, in this case geometry, furnishes only the schema of concepts which are connected to one another through the unalterable laws of logic. It's left to the human understanding how it implies this to appearances, how it fills it with material. This can happen in a great many ways. Here's another statement of something somewhat similar from 1921 to 1922. According to this point of view, the method of the axiomatic construction of the theory presents itself as the procedure of the mapping of the domain of knowledge into a framework of concepts, which is carried out in such a way that the objects of the domain of knowledge and now correspond the concepts into statements about the objects that correspond the logical relations between the concepts. Through this mapping, the investigation is completely severed from concrete reality. The theory has now absolutely nothing more to do with the real subject matter or the intuitive content of knowledge. It's a pure object, a full construct of thought about one, which one cannot say that it's true or it's false. So much for that. mean that's something we can talk about. What do I mean by the notion of translation? Well, one finds this, this is not new with Hilbert, one finds it very, one finds it expressed in various places. He did some work in some other places, too, I would claim. Maybe I'll come back to that later. But one finds it very, very classically presented in a popular paper that Poincaré published in 1891 on non-intuitive geometries, an explanation of what is going on with Beltrami's work. And what one has here is a presentation of essentially what is now known as the Quankarae model of Lovacev's geometry. So Quankarae presents it, he doesn't present it in a schema like this, but he actually presents it as a translation, as if one is translating by use of a dictionary between two distinct languages.

52:30 So, on the left, in the left-hand column, you've got the non-Euclidean terms. In the right-hand column, you've got the Euclidean terms. So, I presented this as the terms in the language of non-Euclidean geometry. On the right, then, you've got the correctness of terms in the language of the Euclidean geometry. And you've got, then, this mapping. between them. So the space is the upper half space, U, determined by the fundamental plane. You just take a plane, any plane, or P, you call it the fundamental plane, and take the upper half space. Planes then become hemispheres in U, which intersect with P orthogonally, lines, semicircles, and so on and so forth. So there are translations for sphere, circle, angle, distance between two points as they give in a non-elementary way as a certain logarithm and so forth. So as I said, Poincaré presents it as if the words on the left are dictionary entries and the terms on the right for translations are the dictionary definitions given in all the language of the clitical geometry. The key assumption about this, which Quankery doesn't mention, but the key assumption is the TOR, this mapping, preserves the logical form. So in other words, that, you know, the TOR acting on it. So what is, of course, missing from this is how it goes on then to give translations of propositions in this language to propositions in the other languages. so the key thing about that is that Tor must preserve logical form so if you have a statement in this language of non-Euclidean geometry which is a conjunction then what you get is a conjunction of statements in Euclidean geometry and so on so a translation of a negation is the negation of a translation of what it negates and so on so if you once you've done that And then you'll see that a proper proof of psi from some principles phi 1 to phi n in the language of non-exciting geometry, under that translation what you get then is a proper proof from the premises tor 1, the translations of these premises of the proposition tor psi from those translations.

55:00 And if you add the further condition that these Tor, that whenever you have an axiom in the first language, in this case, non-ecclidian geometry, its translation is a theorem in the theory being appealed to on the right-hand side. once you've got that then it follows Torfei will be a theorem in the theory that you're appealing to over on the right now you can use that very quickly to show that if you assume the theory E so the theory given in this language over on the right here if you assume that that's consistent and you set this up in this way then it will follow automatically, they would never have a proof of contradiction in the first thing. So this is how Piper A. explains, uses this notion of translation. What one finds is a very, very similar thing in Hilbert. Here is an example of how Hilbert uses a similar scheme. And this is taken from the monograph very early on, after having set up the axioms. He then produces a consistency proof for the axioms. What this consistency proof amounts to is just the use of analytic geometry. Except that he constructs, he doesn't field of real numbers, what he constructs is this thing, which he calls Omega, capital Omega here, which is a minimal Pythagorean field. So it's a countable minimal Pythagorean field. Remember that he's only, the only continuity axiom he's got is the Archimedean axiom, so he doesn't, he doesn't use any principle which demands uncountability. He's only, he's

57:30 got to deal with Cantor-Glueben So what he presents is his field and then he shows that a standard analytic geometry based on pairs of elements taken from his field will give a model the axioms 1 to 5 The way he does it is by setting up this, essentially this translation where a point is given by a pair of numbers. In this field, a straight line is given by a linear equation. Over this field, a plane is given by a three-parameter linear equation. so the axioms in Euclidean geometry here translate to theorems in this analytic geometry so it follows that there can't be a contradiction in Euclidean geometry there are four things to note about this one is that this is fundamentally Poincare's scheme the second is that reductionist definitions are used But here the axioms are not reduced to analysis. They're just used for the purpose of this reinterpretation. They're just temporarily used for the purpose of reinterpretation. There's no requirement, unlike with Poincare, that the meanings for the tours are related to those of untranslated principles. So in practice, the language L-A and the corresponding theory A could be arbitrarily chosen. And as with Poincaré, the schema will change a bit if you change the language, you change the theory in such a way that under the translation just as in the case, independence of psi from sigma is the case where all the axioms, you choose a translation where all the axioms of

1:00:00 sigma are theorems of A but A actually yields not psi, so in other words you won't have, you can't have a rule of psi from sigma I think that Hilbert's procedure generalizes not just but I conjecture that many other geometrical investigations in the 19th century actually will be generalized