Hilbert Foundations, Einstein's GR

0:00 And I will try and read because we don't want to get to coffee on time. Okay, so I'm going to tell a historical story about the foundations of physics. And the reason for telling this story is because I want to display Hilbert's approach to the foundations of physics by showing it at work in a particular example and talking about the results that he arrived at. This is work that is joint work with Tom Reitman, so any mistake that I make, I'll just explain. Okay, so the period of interest begins in 1915. And as you all know, in the autumn of 1915, Einstein and Hilbert are working on related projects, and famously, at the end of November 1915, Einstein arrives at his field equations of general relativity. as we know them. Now, that leaves us with the question of what was Hilbert doing and what did he achieve? That's what I want to talk about. So let me start by just introducing the materials that I'm going to focus on and then I'll talk a little bit about what I'm going to say. So in 1915, at the end of 1915, Hilbert made two presentations to the Royal Goething Academy of Sciences, each under the title Foundations of Physics, the first communication, and the second communication. The first communication is more well known than the second one, and what it contains is Hilbert's use of his axiomatic more about what that is in a minute, to the foundations of physics and its demonstration of various results that follow from this application. And a particular two sets of results that I'm going to talk about today. The first concerns causality, and the second concerns the relationship between the electromagnetic field equations and the gravitational field equations. There's Second communication, and that's less well known, and what I want to talk about is how the first communication

2:30 and the second are related to one another. In this second communication, Hilbert goes back to his discussion of generally covariant theories. And again, the axiomatic method is at work, and what Hilbert relates to, carries on the theme of the previous talk. What happens in the second communication is that Hilbert makes explicit what modification he thinks is necessary to Cantina Pystomology in the face of different variant physics. Okay, so this is the basic outline of what I want to do. I'll say something about Hilbert's axiomatic method, just say in general terms what that is, and then look at this specific case of the application of the method in the end of the year, and ask what results did he obtain, general covariance is one of his axioms, and he shows how this leads to two sets of consequences, one concerning the problem of causality in a theory that's generally covariant, and then in the second communication we have the solution to this problem which he presents with the impact of the technology, and then a second group of results that concern electromagnetism in the covariant theory. Okay, so first of all, then, something about Hilbert's axiomatic method. So this is a method of logical analysis of a field of study. So what you're supposed to do is you start with our experience of the world, you start with intuition, and then you organize these experiences under different concepts. You look at the logical relations between these concepts and arrive at sort of overarching of the theory, from which various logical consequences follow. And for our purposes, there are kind of three features of this method that are really important. So once you've got these axioms in place, what you can then do is investigate the logical consequences of these axioms without any further appeal to intuition. So this method is most familiar, of course, from Hilbert's application of it to geometry. But the same thing is in play here. We're trying to separate out, when we're doing our proofs within mathematics or within physics, we're trying to separate out what we can derive through logic and what

5:00 input intuition has to have. So we're trying to make sure that we don't use, sort of sneak things in from intuition when we're doing our own proofs. But nevertheless, for Hilbert, the relationship to intuition is very important. So once we've done our logical analysis, we want to relate our results back to intuition. This is what gives this content to the results. So it's not just an empty kind of formal game. We have to relate what we're doing back to intuition. And more than that, we have to make sure, this is going to be really important, that any appearance of conflict between what we know through intuition and what we've gained by doing this conceptual work has to be removed. The whole thing has to hang together with So that's kind of very rough and ready hand-waving, but what we want to do is to look at this method in the context of a specific example, see it at work, and this will be somewhat clearer. Okay, but before I get into that, I just want to add some more detail now to the text that I'm going to be using for the purpose of this talk and just put a little bit more historical context into place so that we know what materials are that I'm going to be talking about. So, as I said, I'm going to be talking about these two communications that Hilbert made to the Goethe-North Academy. on the 20th of November, 1950, and was published on the 1st of March. In 1996, Corrie discovered some proofs of this first communication that barely printed on the 6th of December, 1915. And these proofs have important differences between the proofs and the published version, which of course tells us that there are important changes that Hilder made between the presentation that he gave on the 20th of November, and the eventual published version in March, and there are going to be some differences that we need to pay attention to there. The second communication was, well, he made two presentations under the heading of the second communication on the Foundations of Physics, one in December, on the 4th of December, one at the end of February, so about three months later, in 1916. but we don't have any record of what he said at those presentations.

7:30 All we have is the published version, and as you can see, that comes much later. It was not submitted until the end of 1960, and then published only the following year. So that indicates, again, that substantial changes, it seems likely that substantial changes were made between the presentations that were made to the Academy. and the published version. So let me put that in the context of what was going on with Einstein at the time, and then we can get to the meat of what I want to say about Hilbert's own work. So, as you all know, in the summer of 1915, Einstein visited Goettingham and lectured on his approach to finding new theory of gravitation. And at this time, Einstein's still convinced that no generally covariant theory of gravitation is to be had. The theory that he's going to get has to be restricted in some way. But he's delighted with the response that he gets, particularly from Hilbert in Goethe, and he thinks he's found the last people who understand what he's trying to do. But then this ensues in this apparent race at the end of the year with Hilbert. So on the 20th of November, as he said, Hilbert presents his first communication. And then on the 25th of November, Einstein now presents his field equations, we get what we now call the final form of the field equations, because they're published in 2nd December 1915. Now, in the proofs version, that have the 6th December date of Delbert's first communication, the Einstein field equations don't appear in their explicit form, but by the time we get to the published version, they do appear in their explicit form, and this is going to be important for the story that I want to tell. And in particular, it's going to be important because it means that the results that survive from the proofs version into the published version, therefore don't depend on the specific form of the Einstein field equations, because he doesn't have those, he doesn't know what they are at the time that he makes his original, originally proves his results. So it shows us, it makes very clear that the results that Hilbert arrives at are more general, they don't depend on the specific form of the Einstein field equation. All right, so that's some date and some context.

10:00 Okay. So now we get to the kind of main body of what I want to say, or what I was kind of by way of background. So what Hilbert does in 1915 is to apply his axiomatic method to the as-yet-unknown theory of gravitation, combined with Hilbert's generally covariant version of Mies theory. theory of electromagnetism, this being one of the candidate theories of electromagnetism that's available at the time. So he's taking what he thinks is kind of the best available physics at the time, Mee's theory, and what he's sort of extracting what he thinks is important about what Einstein is doing, what he's learned in the summer of 1915, and trying to apply his axiomatic method to that to see what he can learn about what the completed physics is going to look like. And he starts with these two axioms. And he doesn't argue for these axioms in a sense of trying to give physical justification for them. It's, as I've said, just taking, sort of extracting what he thinks is at the core of what's going on in the best physics of the time. So the first axiom is this one, that the world function H, which is a hydrogen, depends on the metric and its first and second derivatives, and the keys here are the electromagnetic potential, so it depends on the electromagnetic potential and its first derivative. And then Hilbert says, you know, we're just going to use standard variational techniques in order to arrive at the field equations associated with gravitation and with electromagnetism. The second axiom is the axiom of general invariance, and this calls it, but what he's referring to is what we call general covariance now, so I'm going to be horribly anachronistic and call it general covariance from now on, just to avoid confusion. And given these axioms, here's the first result that Robert arrives at his theorem 1. This theorem says that if these axioms are satisfied, then there will be an undetermination problem in the theorem. So we're going to have only 10 independent equations for the 14 potentials postulated in axiom 1. So this theorem says that regardless of the details of the field equations, we don't know what they are yet, you're going to have this problem, you're going to have an underdetermination problem.

12:30 And I won't say a lot more about this, but first I just want to go on and set before you the two kind of main classes of results. So there's this first one concerning underdetermination, And then the second set of results concerns electromagnetism. And then I'll come back and say a little bit more about each set of results. So theorem 1, the end of the determination problem. And then Hilbert proves a bunch of results concerning the relationship between electromagnetism and gravitation. So let's see, he says that there are various consequences for electromagnetism, And in particular, we arrive at strong restrictions on the form of the electromagnetic part of the theory as a consequence of the structure of the gravitational part of the theory. And then Hilbert says, in the sense indicated, and we'll come to that in a minute, his claim is that electrodynamic phenomena are effects of gravitation. So he's showing that, again, we don't know what the form of the field equations is, but whatever they are, there's going to be this very tight relationship between gravitation and the electrodynamic phenomena. So what exactly does he mean by this? Two kind of main results which are, I think, for our purposes. So first of all, he shows that axioms 1 and 2 lead to the gauge structure of electromagnetism. So all I mean by that is basically this anti-syncrasy condition. So it shows that given the two axioms of the requirement of general covariance and the postulation of which potentials we're going to have in the theory, which fields we're going to have in the theory, general gate variance, you immediately get out this result that the electromagnetic part is going to have to satisfy this condition. So K and L are separated with a randine algorithm for these two part sizes, familiar. So that's the first part, the gauge structure of electromagnetism. I reckon the bank was to the previous slide. Great job. Okay. And then the second thing is that given satisfaction of the gravitational field equations,

15:00 so it's assumed that they hold whatever they are, then you can arrive at four mutually independent linear combinations of the electromagnetic equations, or rather with the left-hand side of the Euler-Lebron equations for the electromagnetic fields and their first derivatives. And this is just a special case of what is later proved as Neurton's second theorem. So this, he says, is the exact mathematical expression of the above generally stated assertion that we had earlier concerning the character of electrodynamics as an accompanying phenomenon of gravitation. Just take these two axioms, and from that, you can demonstrate that given, if you just assume that the gravitational field equations are satisfied, you get this restriction on the structure of the electromagnetic equations. Okay, so those are the results concerning electromagnetism. And these, along with the under-determination problem, appear both in the proofs and in the published versions. So they appear in the version where he doesn't have the explicit form of the Einstein field equations and also in the published versions to survive the changes between those two versions of the paper. there is however one very important difference between the proofs and the published version regarding the underdetermination problem that I want to say more about and this is how we should resolve the underdetermination problem the solution that Hilbert offers to getting rid of the underdetermination is that the original office is completely cut from the proofs version And in fact, no solution is offered in the published version of the first communication. It's not until we get the published version of the second communication that we eventually get Gilbert's new solution to the underdetermination problem. So for the rest of what I want to say, I want to talk about what exactly this underdetermination problem is, say something about his original solution in the crux, and then spend the rest of the time talking about his more settled view.

17:30 communication about how to resolve this under-determination problem. And it's there that he makes explicit how things that generally covarian physics forces us to revise counting epistemology. So, back to the underdetermination problem that's stated in the first communication in his theorem 1. so what Hilbert does in the proofs version of the first communication is to state this problem to say that there's going to be this underdetermination problem and then he appeals to energy conservation as a way of solving this problem. What he's done in this first communication is to use the axiomatic method to diagnose an apparent problem, an apparent conflict between general covariance and causality. And then he characterizes this problem in terms of whether any theory satisfying axiom 1 and axiom 2 permits a well-posed quotient problem. And what theorem 1 does is to suggest that it doesn't. And this in turn then marks out some kind of strategy for trying to resolve this underdetermination problem. What you need to do then is to find four conditions that you can add to your field equations in order to restore the evolution of your potentials. And this is exactly what Hilbert does in the unpublished version of the first communication. And this is what gets discarded between the proofs version and the published version. So in some sense it's of historical interest only, but I want to say just a little bit about it because it provides some context for the final solution that Hilbert offers in the published version of the first communication. Both this problem, the problem of causality, and the solution regarding energy conservation, the seeds of this are in what Hilbert would have heard from Einstein in 1915. When Einstein was there lecturing in Göttingen, As I already mentioned, he still believed that no generally covariant field equations were going to be physically acceptable. One reason for this is the whole argument, which is related to problems of determinism and causality.

20:00 And another reason, another argument that he had about this concerned, energy conservation. he thinks that in the summer of 1915 Einstein thinks that requiring the matter energy momentum tensor plus the gravitational pseudo tensor energy momentum requiring that they jointly satisfy a conservation law with an ordinary derivative he thinks that you have that requirement plus the field equations put these together, and somehow you put these together, and you arrive at some identities, which are non-covariant identities that Einstein thinks you need to use to restrict the covariance properties of your theory, so that it's kind of all a bit odd, but you arrive at these four conditions, and then given a solution of the Entworth-Build equations, these identities restrict the transformations that you're allowed to make on that solution, So they were strictly allowed coordinate transformations. And it doesn't take Einstein very long to realize that this is not the right way to proceed. But at the time that he's lecturing and ingesting, he thinks that the requirement of conservation of energy results in four non-generally covariant conditions that have to be satisfied. And he's using those to restrict the covariance properties of his theory. So these kind of two key features of the possible problems of causality and the role of energy conservation get picked up by Hilbert, but understood and used in a very, very different way. So, for example, as we've seen, Hilbert has general covariance as an axiom. So whereas Einstein says, oh, these problems with the whole argument mean that we can't have a generally covariant theory, Hilbert says, no, no, axiom, general covariance. It gives priority to general covariance over things relatively like, whereas Einstein has it the other way around. Also, with the energy conservation thing, so Hilbert, so there's a similarity, Hilbert follows Einstein's lead in believing that energy conservation holds only for a restricted class of coordinate systems. but what he does is to use energy conservation

22:30 what Hilbert does is to use energy conservation to arrive at four non-generally covariant conditions and then instead of using those to restrict the covariance properties of your theory what he says is we'll have a generally covariant theory but we'll use these conditions to kind of extract out of our theory a Cauchy determinate structure so let's kind of summarize those differences So for Einstein, these four non-genomic covariant conditions ensure any conservation and restrict the covariance class of the field equations, whereas for Gilbert, these four conditions, you take your genomic covariant theory, and then within that, you find for yourself a creature to determine the structure using these four conditions. And so we get in the proofs that this, so all this discussion is only in the proofs, we get the third axiom in which it says that the space-time coordinates are such world parameters for which the energy conservation law is valid. So, as a kind of a thread between the proofs version and the published version, just to So notice here that the space-time coordinates then is picking out just some of the, what we would generally refer to as coordinates, a subclass of those. So we have the world parameters, which are what we would generally talk about in terms of coordinates. And then there's just a subclass of these that we refer to as the space-time coordinates. So the spatio-temporal structure of our theory is associated with energy conservation being satisfied, this structure within our generally ovarian picture um okay so and then just that final point on that slide is just to say that he means that you need these four conditions to complete the system of physics that it enables us to relate what we're doing with physics to our experience of the world in which energy conservation is satisfied So what I want to do now, having this kind of explain how he tries to go about solving mental determination problem in the first communication, is to now look at the second communication where, so this gets discarded and look at what he offers us instead. So, as we said, between the proofs version that we just looked at and the published version

25:00 of the first communication, Einstein publishes his field equations in which, of course, energy conservation no longer places any restriction on the theory. So, Einstein's published version is generally covariance and energy conservation holds them as no problem. So, this means that Hilbert has to jump his original solution to the under-determination problem. It's not, we can't use that anymore, so it all gets scrapped and it doesn't appear in the published version. But what we get in the second communication then is a new attempt to solve this problem of causality. And we get more than that because Hilbert goes into more detail about what exactly this problem is. So in addition to the under-determination problem, he inserts an additional problem that is a prior problem that comes before the under-determination problem. We call that the problem before the order. So I'll say a little bit about what each of these problems is, as stated in the second communication, and then something about how Hilbert solves it. Again, very explicitly in the second communication, Hilbert's claim is that he's diagnosing this problem by means of his axiomatic method, and he's using the axiomatic method to help him in coming up with a resolution of the problem. So I'm going to start then by kind of skipping ahead a bit, and just trying to put in front of you a basic sketch of the resources that he's going to use for modification to Kantian epistemology that he thinks is necessary, details of the problems and exactly how this gets applied to the specific papers. So this is kind of an outline of what Hilbert's going to conclude about how we have to modify the Kant's interstimology in the face of this new generally covariate physics. So our physical meaningfulness for Hilbert, as for Kant, is tied to our understanding. In the Kant's infirmary objectivity is tied up with conditions of possible experience, to be a possible object is just to be a possible object of experience. What Hilbert's going to do is perhaps pull those two things apart so that he's going to have different conditions and what it is to be a possible object of experience versus being a possible object of physics. In particular, general covariance is going to be the criterion of physical objectivity

27:30 in the sense of the criterion of being a possible object of physics. Whereas being a possible object of experience still relies on spatio-temporal notions, but our experience is spatio-temporal, our experience is causal, objects as we experience them are in space and time and causally related to one another. So that's kind of abstract how it works in the context of the two problems So here's the first thing that Hilbert says, and this is the problem of court and order, which he now places before his discussion of the problem of underdetermination. So Hilbert says, a conflict with the court and order would arise in two world points lying along the same time I could, and standing in the relation of court and effect can become a order that becomes simultaneous. So the new physics is to be compatible with the experience court and ordering events over the fact that our experiences of the poorly ordered world need to restrict the allowed ordinary systems, such as under a form of transformation, any time might go, may be a time might go. And he says that's called such transformation proper. Conclusion, the concepts of cause and effect lying at the basis of the principle of causality never leads in and contradiction as soon as we restrict ourselves to use proper place-time for ordinary. So, we've got this generally covariant physics, but if we're going to make this consistent with our experience of the causally audit world, then we have to restrict the coordinate systems that we're allowed to use such that we keep the, such that the events are causally audit in a consistent way. But just emphasize at the end, this has to do with our experience of the world, and not to do with possible objects of physics in themselves. It's just between possible objects of experience, which are to be in space and time, in order to be ordered, versus possible objects of physics. That's the first problem that he diagnoses and then attempts to solve. And now back to the problem of underdetermination. So this is the one that appears in the first communication where we get a solution,

30:00 and then the solution gets junked, and now we get a new solution. So the strategy that Hilbert adopted in the proofs has to be dropped, of course, following the publication of Einstein's field equations. And the main point of the second communication is to provide this new solution to the problem under the termination. Now, before I kind of go any further in this, I guess I should say that I'm trying to do exposition, and this doesn't mean that I endorse the view, necessarily. I'm just putting to one side a critical analysis of what he was trying to do, and just trying to put on the table what it is that he's trying to say in this second communication. So that's kind of one disclaimer. And the other thing, perhaps, just to put some context, contemporary context here, is that, And so following the discovery of the proofs, a few people have gone back to the second communication, have a look at it, and in particular, the major piece of contemporary scholarship on this is by Jürgen Renan and John Statschall. And they say, well, here's the first communication, that's, you know, there's a lot of good things about that. Here's the second communication. It's completely disconnected from the first one. It's nothing to do with it. And it shows Hilbert, what's the phrase, in agony in the collapse of his research program, and just tackling a string of different problems within the context of Einstein's theory of general relativity. So part of the background to the story that I'm telling here is just to say that that's not the right way of reading the second communication. Another part of the contemporary background is that even the people who are more sympathetic to the idea that Hilbert is continuing with his program here say that Hilbert was a victim of Einstein's whole argument, that Hilbert's problem of causality is the same as the problem that Einstein had with his whole argument. And part of what we're trying to do here is to say that it's not the case. I mean, there's a construction in Hilbert's work that's very similar to Einstein's whole argument, but the lesson that Hilbert takes is very different. His problem of causality is very different, and the solution that he offers is very different. And Einstein's point-coincidence argument wouldn't even touch the problem that Hilbert has, to be different problems, because the point coincident argument is supposed to extract us from the whole argument, but it has no bearing on the problem that all of us. That's kind of brackets to one side of the context of what's going on here, but all for today

32:30 what I want to do is just put in front of you what Hilbert says and see what you think about it. So here's what Hilbert does in the face of his underdetermination problem. He revisits what's meant by physical meaningfulness, the physical meaningfulness of propositions in physics. And he gives us two conditions. He says that for a proposition to be physically meaningful, it must, first of all, have a generally covariant formulation. And that, of course, is very familiar. I'd be completely happy with that. But this is not sufficient. And why is it not sufficient? It's not sufficient because any physically meaningful proposition through measurement. Physical meaningfulness has to do with our understanding, and so it's to do with the world as we experience it. So it's ineliminably, physical meaningfulness is, I can't say it, but ineliminably, spatial, temporal, and causal. And because of that, we have to add a second condition. And this second condition is this, that when expressed with respect to, when this proposition is expressed with respect to coordinate system, so when we give a description that presupposes space and time and temporal term ordering, the truth of that description has to be uniquely determined by an appropriate space-like past hyperservice. In other words, when this proposition, when we express the propositions of physics in terms of possible objects of experience, so spatially, atomically embedded and causally related, those statements are physically meaningful only if they're employment in this sense here. And according to Hilbert, these two conditions are individually necessarily jointly sufficient for physical meaningfulness. So just to go back to the basic strategy that I outlined above, while Hilbert separates physical objectivity from meaningfulness or possible objects of physics or possible objects of experience, he makes general covariance necessary and sufficient for objects of physics, but he separates this from the conditions of possible objects of experience. He says that causality has to do with, not with the objects of physics, but with our experience. He said, hence we need for the second condition. But from Helmut's point of view, the requirement of causality

35:00 is anthropomorphic has to do with our experience of the world and not the objective world. Finish up to try and leave a bit of time for discussion before coffee on time. So conclusions then. Hilbert's 1915 work on the foundations of physics as I was saying earlier, has generally been judged by saying, okay let's have a look at what Einstein did and managed to do and where did Hilbert go wrong? And what Tom and I have been trying to say is that actually it's much more, you get a lot more out of Hilbert and understand better Hilbert's project if you just take it on its own terms. So don't say, you know, how did he succeed or fail relative to Einstein's project? Just look at Hilbert and say, what is he trying to do and what did he achieve? And once you look at the project in its own terms, then there's quite a lot of philosophical richness and meat here that's relevant to the foundations of physics, both in terms of methodology and in terms of the specific results that we can arrive at. What we want to say about those results is a kind of further critical step. So what can we achieve then? So these strong results can be concerning electromagnetism and then this diagnosis and resolution of the problem of causality to think about the modification of the content of the same. Thank you. Thank you very much. I think that's really, really important is the stuff about the difference between the conditions or experiencing the conditions of physics. So, yes. I have a short question. If you remove the condition 2, does Hilbert think that what you lose, which is basically causality, is just linked to this anthropomorphic aspect that, you know, it's us doing the theory? Or do we lose something which is in the theory as such? So if we remove 2, do we only lose the anthropomorphic part of it? My question might be related to that.

37:30 I guess what I want to ask is, first of all, I thought that was absolutely great and unbelievably illuminating. So it's not the same as Einstein's whole argument and Einstein's response to determinism. It's a different kind of response. And I guess Hilbert is paying attention to two things. One is this Kantian medieval of the crop is the Cauchy problem, which I guess Einstein is not particularly talking about when he's talking about determinism. So could you say maybe a little bit more about the relationship between the point of coincidence resolution of the so-called determinism problem and what additional goodies do you get, and I'm convinced you do, by what you said, by moving to this Kantian-Kauchy problem way of thinking? And I guess the last question would be, could you incorporate the goodies of the Kauci problem solution without the Kantian business? Or are those closely connected? Okay, so the first bit and then the Kantian bit. So the first bit about the point coincidence argument, one way to see that they're different is that Hilbert would accept before he's even started point coincidence argument and say, okay, we've got all these events, if you like, but now we have a further problem. We need to tie them together. So we need to order them pausally. Our experience comes up. We need to tie these together, order them pausally, and if you like, join them up uniquely so that you have to turn the pictures between these different point events. So that's the further thing that you're getting. So we can start. So he's working with Göttingen at a time when people have worked, and he himself is working with re-cramatization. He's not going to fall into the kind of issues that Time Bill Einstein had. So he's going to kind of take that as a given and then say, but we need something more, we need to sew these events together. So that's the more that he's trying to get. Whether you think you need that or not is another matter. Whether you think that he's right that our experience tells us that you need more, is a kind of a separate issue for discussion. And then, do you need the counting stuff? I don't know. I haven't. Again, this is kind of critical, further critical reflection.

40:00 Clearly, you don't need, you know, clearly you don't need, you can just take the issue, okay, we need to solve the Koshy problem in order to be able to evolve data forwards. We need to be able to do that. We can just do that in terms of mathematics and not worry about the Koshy issue. It involves, I guess, a space-like hypersurface and a certain, you know, foliation. Yeah. So it involves saying, you know, the manifold splits up into space and time, you certainly guess. Yeah. Which Einstein apparently, I mean, Einstein just didn't get that the Cauchy problem was how to think about it. No, Hilbert is the first to understand this is what's going on here and to make use of out some coordinate systems in order to be able to do basic calculations and do just yeah evolve data for so i i don't think and maybe someone here can tell me this i'm not sure at what point there's recognition of the sort of the first part of the problem that you need to be able to find a surface on which you can write consistent data i don't find that in hilbert so it's concentrating on what you do to how to evolve things forward. And I don't know where realization was. I think, you know, the initial value problem in general is the realization that, you know, you're having a hyperbolic differential equation. And all that came much later with the work of Chiquet, Fouillard, and others in the 50s. The 50s. Yeah, yeah. That's at least that's what the thesis is. So, yeah, fantastic, that was really great. My question was really the same as Michael's. I was puzzled by saying that you were saying the whole problem is really different from his own, his own determination, but I was thinking anachronistically the whole problem is just being a problem about being able to apply time and gauge transformation and getting different evolution. But what was the role of the, in his statement of the problem, the worries about causality of having the two points on the worldwide particle and worrying about, you know, transforming them into a simultaneous, because a different morphism is not going to do that, is there? Well, he was worried that you can take two points and then have them, so they could be time-right related and then form a transformation and the space-like related to another transformation you reverse the different one, you reverse the causal order. So this is his worry that if you just take arbitrary coordinate transformations, you can do all these things.

42:30 That's not how I understood what happened in that. To apply it to the metric as well as to the matter fields, the metrical relations between events is effectively the same. So you can't transfer events outside it. Yeah, so if you're learning about something you shouldn't have done, right? But is he saying you have to have a specific foliation? If you don't have foliation, you could do it. Well, I'm not sure about that. I mean, he's talking about the particulars, I mean, he's talking about the ordinance which has stated and tools and instruments in some sense. Yeah. So you can start with phobiation and suddenly have the corner which is, you know, constant on that phobiation and you can have a global transformation which sort of fits the order of events with respect to that corner, but that corner, I mean, that's the longer a specific corner, so I think it's sort of a requirement that the processes have to some degree of space. Yes, I want to return to the fact that for Gilbert to be represented in this time is a condition for being the possible object of the experience, but no longer for being the possible object of physics. I think that to generalize Kant, we have to return to the basis, the basis of transcendental studies. Physics is generally a categorial legalization and a mathematical reconstruction of phenomena, only phenomena, observable phenomena. And phenomena are relational entities, and they have to be accessed. So the conditions of accessibility phenomena are constitutive of the objectivity of the objects of physics. Now, for sensible phenomena, microscopy phenomena of our sensible world, through perception, space-time, guarantee, accessibility,

45:00 But for other types of physical phenomena, spectral phenomena for instance, or cosmological phenomena, etc., you need another type of accessibility. You need apparatus, etc. And so you have to drastically generalize the perception of perception. in fact you have to transform the concept of the phenomena in the two-day concept of observance the phenomena is an observance it's something which is accessible there are different ways to respond to this but one, I'm not sure exactly what the worry is but everything that is part of our experience the experimental apparatus. Nevertheless, what we experience, we don't experience the spectral phenomena directly, what we experience is through the apparatus. No, no, no, no, no, no, no, no. When you measure something, you use your perception. But the fact of the accessibility we include in the axiops of the theory, for instance the quantum mechanics, is not the same. The prospect of a race, Heisenberg, we don't have, at the beginning of the answer, we don't have the same. So I'm not sure whether you're saying we have something similar going on here, are you saying we have something similar going on here, where we have to separate out the possible objects of physics and the possible objects of experience? Or are you saying that our possible objects of experience are things that are non-spatial. It's a prolongation of my question. Phenomenal must be accessible. So, of course, at the end of the accessibility process, that is, you may have felt. But there is a part of the conditions of accessibility which must be constitutive of the feasibility.

47:30 And there is absolutely no obligation to include exceptions at that level. for instance in quantum mechanics to include the measurability of measurability by some apparatus in the axioms of the theory that you don't include perceptual condition of the results of the measure of the apparatus that's what's beside It sounds like you're accepting for those decisions. Absolutely. So it sounds like it is a divergence between experience and physics. And to reconnect experience and physics in a certain way, for instance, is a great challenge. This was a comment after the previous discussion concerning this proper transformation. I would interpret them as the proper orthogonal transformation which are in a large Lorentz group. And all transformations of the Lorentz group preserve the causal structure, but only the proper one preserve the other, because the unproper would change fast into a different the future and so on. And it's true that all physics is invariant, locally, I mean, under the big Rens group, including Unproper, but what you would not accept is that the future has changed in the past, more or less, but this is related more to the observer than to the objective physical relative. But this has nothing to do with time, in fact, because because the causal structure and all this transformation are perfectly well-defined without any time function defined. You don't need the metric, in fact. You need only the conformal part of the metric

50:00 to define the causal structure. That's a different sense of causal structure. Any other questions? Any other questions? OK, so I think it's time to have a question. The difference between Hermann Weil and Eric Carton approaches to differential geometry in the 20s and early 30s, and so there's lingering consequences. Carton's method in differential geometry is called L'Aventure de Pierre-Mobile, the method of moving frames. We heard a little bit about that yesterday from Mark. you'll see that there's some difference in the 20s and 30s as to what the method actually is and how it's just today the reason for this talk is broadly very broadly speaking the reason for most of the work that I try to do with philosophy is to try to understand the role of mathematics in physical theory I particularly like this quote by Richard Curran which has only recently come to my attention rather than a more well-known one of Eugene Wigner who speaks about the unreasonable effectiveness of mathematics and physical science Wigner, as you may remember, was on to say that it's a gift that we've neither earned nor deserved well that's just St. Thomas Aquinas talking about the gift of revelation so I don't like that I also like the fact that Quran very directly points out that mathematics is an emulation of human mind that's got to be right this has rightly attracted the concern of the philosophers so I take that as license to proceed here are my presuppositions in this talk you don't have to endorse them but that's at least where I'm coming from thinking of the role the relation between mathematics and physics as one of application or coordination I think is misleading of course there is something to that but I like to think that the central role of mathematics in physical theory is formative a term that Gaston Bachelard famously used it's a role of creative invention

52:30 and to quote Roberto Peretti it's the plan for the intellection of nature with Jean-Petitot I believe that Israel taught us that intellection implies intelligibility I think that Kant has told us that mathematical knowledge involves construction of concepts Levi Chivita Bayer and Elikartan taught us that intelligibility within a continuum based or I often use the term constituted and a geometry of connections primitively involving only relations of comparison and structures between points that are infinitesimally adjacent the issues in this talk are primarily two for me I was puzzled by the fact that it was Vile who discovered as we all know gauge theory in 1918, gauge theory in a slightly different sense, where we have really just a scale invariance, whereas the contemporary geometrization programs in physics, as well as the fiber bundle formalism of differential geometry, stem not from Bile, but from Kata. And I wanted to find out why this is. So did you put quotation marks to this color? Well, yes. I mean, you could say he invented gauge theory just as well without the quotation marks, perhaps. I'm looking to particularly the nature of Weyl's. The reason that Weyl didn't actually comes down to this world geometric objection to Catan's method of moving frames. This is a very brief slide. I could spend a lot of time on Vial's Pure Infinitesimal Geometry Program in field physics. But just to summarize here in only one slide, this Pure Infinitesimal Geometry-based

55:00 So basically, you've got the objects of Maxwell and Einstein theory constituted in a differential human construction built up from elements and operations possessing. And this is crucial. A directly evident, visualizably evident is another way that you can think of its meaning. And a canonical example of this would be the congruence of two figures by the superposition of one figure over another figure. this is a systemological program in Husserl's sense as well as a speculative hypothesis and there are further details how does this relate to physics you ask by what Eddington and Dirac called a method of identification Bob doesn't use this term both Eddington and Dirac refer to it so objects constructed in of pure infinitesimal geometry. These differential geometric structures are shown to be mathematically identical to known vector and tensor structures of Einstein-Maxwell. Well, you asked the question to what mathematical space corresponds the space of intuition in which the primitive elements and operations of pure infinitesimal geometry have their evident meaning? And I was quite clear about this, it's just the tangent space at any point, and you remember that. And here's a little bit of text to support that claim. It comes from a later paper, the Rouse Ball Lecture at Oxford in 1930. Only the spatiotemporal coinciding and its immediate spatiotemporal neighborhood have a clear meaning directly exhibited in intuition. to the space of intuition belongs the ego center, the hich centrum, whereas the coincidences, the relation of the space of intuition to battle physics, becomes vega the further the distance from the ego center. This is mere than theoretical construction in the relation between the curve's surface and its tangent plane at the point p. what about Cartan we can think of Cartan geometry

57:30 resulting from a generalization of Klein geometry which is itself a generalization of Euclidean geometry and a generalization of Riemannian geometry which of course is itself a generalization of Euclidean geometry Let's start with Klein, a famous quote from the Germans say the Erlanger program, in English we always say the Erlanger program, I don't know which is proper, but the point of that program is right here, given a manifold and a group of transformations on it to investigate those properties of the figures belonging to the manifolds, unchanged by the transformations of the group. And so the principal idea here is the how group of, the principal group or the fundamental group of the geometries. just a little bit of technical stuff here to make it a bit more precise a Klein geometry is a connected manifold a lead group of motions acting transitively Klein geometries are fully homogeneous transitivity, the group action is on 2, but not 1, 1. So you have something here, a closed subgroup of G, which stabilizes X, which basically means that X is left invariably fixed. Klein geometry traditionally, these days, is described by the pair GH, where H is the stabilized group, and you've got a projection, which induces It's a bijection, actually, between the and them. This is from Cartan's famous notice that he wrote in 1931 to become a member of the Academy of Sciences. It's quite remarkable to me that Cartan was not made a member of the Academy of Sciences until 1931.

1:00:00 But it turns out that it was Cartan's work on BIO's program in differential geometry, really, that brought him to the attention of the rest of the, at least the people who matter, who controlled nominations to the Academy of Sciences in France. Cartan writes, in the movement of ideas, this is my translation following the general theory of relativity I was led to introduce the notion of new spaces more general than Riemannian spaces and playing the same role to the different spaces of the fundamental group that Riemannian spaces play in relation to Euclidean space that vast synthesis famous quote vast synthesis often you see that I sketch no doubt is connected to Klein's ideas as formulated in the famous Elanen program Though it goes much further. Indeed, it includes Riemann's geometry, which have constituted an entirely isolated branch of geometry in the framework of a very general schema in which the notion of group again plays a fundamental role. So there's the synthesis of Klein and Riemann. Well, you might think, do the inhomogeneous spacetimes of general relativity exclude basing the space of physics on the notion of a group? Cartan says, not at all, because the fundamental hypothesis of Einstein's theory is not, as many people have believed, that it is possible to formulate the laws of physics in any arbitrary coordinate system, which is a simple pathology, but that in any sufficiently small region of space-time the laws of classical physics, as expressed in the special theory of relativity, are true in first approximation. And that's a quote that's going to come back to us. That certainly is not Plyle's view. A Cartan geometry, therefore, based on a generalized notion of a connection, carries Klein's view of geometry over to in-homogeneous spaces and spacetimes. Plyle disagrees. In a paper that he wrote as late as 1931, After the death, the cure is fine. The general theory of relativity says mark .