Einstein & Relativized A Priori

0:00 Well, it's a great pleasure to be here. I want to thank the organizers, Alexei, Michelle, and Jean, for arranging this incredible event in this amazing, beautiful place. I was just saying this morning, it is so relaxing and beautiful here, I actually have had no jet lag, which has never happened to me. Well, today, I suppose, is our day of philosophy, or more precisely, history of philosophy of science. So I actually have a paper in my hand, as philosophers are wont to do, not all, and I'm going to read it to you. While I show you various slides, which are mostly pictures, there'll be only one slide that has any equation on it, and it's just going to be the Lorentz contraction. So this is slightly different than we had yesterday. Kant's original version of transcendental philosophy took both Euclidean geometry and Newtonian laws of motion to be synthetic a priori constitutive principles, which from Kant's point of view function as necessary presuppositions for applying our fundamental concepts of space, time, motion, and matter to our sensible experience of the natural world. Although Kant had very good to view the principles in question, that is, Euclidean and Newtonian, as having such a constitutively a priori role, we now know, of course, that they are not, in fact, a priori in the stronger sense of being fixed necessary conditions for all human experience in general, eternally valid once and for all. And it is for precisely this reason that Kant's original version of transcendental philosophy must now be radically reconceived. It was Hans Reichenbach Here's the young Hans looking a little bit chubby in the 1920s in Berlin. In his 1920 book, Relativity, Theory, and a Priority Knowledge, who first proposed the idea of relativizing Kantian constitutively a priori principles of geometry and mechanics. Such principles still function, according to Reichenbach, throughout the development from Newton to Einstein, as necessary presuppositions for applying our changing conceptions of space, time, and motion but they are no longer eternally valid once and for all. Thus, for example, while Euclidean geometry and the Newtonian laws of motion

2:30 are indeed necessary conditions for giving empirical meaning to the Newtonian theory of universal gravitation, the situation in Einstein's general theory of relativity is quite different. The crucial mediating role between abstract mathematical theory and concrete sensible experience experience is now played by the light principle and the principle of equivalence, which together ensure that Einstein's revolutionary new description of gravitation by a four-dimensional geometry of variable curvature in fact says something about concrete empirical phenomena, namely the behavior of light and gravitationally interacting bodies. In my recent book, Dynamics of Reason, published in 2001, I have taken up and further developed Reichenbach's idea. You can see at the beginning of part two, there's a section called Relativized Apriori, and that's one of the main things discussed in the book. But my implementation of this idea of relativized constitutively apriori principles of geometry and mechanics essentially depends on a historical argument describing the developmental process by which the transition from Newton to Einstein actually took place, as mediated, in my view, in scientific philosophy, involving especially Hermann von Hemmholtz, Ernst Mach, and Henri Poincaré. However, since this argument depends on the concrete details of the actual historical process in question, it would therefore appear to be entirely contingent. How, then, can it possibly be comprehended within a properly transcendental Kantian philosophy? Indeed, once we have given up on Kant's original ambition to delineate in advance the a priori structure of all possible scientific theories, it might easily seem that a properly transcendental argument is impossible. We have no way of anticipating a priori, the specific constitutive principles of future theories, and so all we can do, it appears, is wait for the historical process to show us what emerges a posteriori as a matter of fact. So how more generally can we develop a philosophical understanding of the evolution of modern science that is at once generally historical and properly transcendental. Let us begin by asking how Kant's original transcendental method is supposed to explain the sense in which certain fundamental principles of geometry and mechanics are, in fact, both a priori and necessary. Kant's method appeals to the two rational faculties of sensibility and understanding.

5:00 The answer to the question, how is pure mathematics possible, appeals to the necessary structure of our pure sensibility. as articulated in the Transcendental Aesthetic of the Critique of Pure Reason. Here's the title page. Notice it's the 1787 second edition, which is important, and if anyone wants to know why it's important, I will talk about it later. The answer to the question, how is pure natural science possible, appeals to the necessary structure of our pure understanding, as articulated in the Transcendental Aesthetic. Yet there is an obvious objection to this procedure that has been made many times. transcendental explanations inherit the assumed a priori necessity of the sciences, namely mathematics and pure natural science, whose possibility they are purport to explain, unless we can also somehow establish that they are the unique such explanations. From our present point of view, for example, it does not appear that Kant's explanation of the possibility of pure mathematics is uniquely singled out in any way. On the contrary, our greatly expanded conception of purely logical or analytic truth suggests that an appeal to the faculty of pure sensibility may, after all, be entirely superfluous, a common argument in 20th century analytic philosophy. Indeed, from the point of view of the approach to such questions that has dominated much of 20th century analytic philosophy, again emphasizing logic, it appears that all consideration of our subjective cognitive faculties is similarly explanatarily superfluous and merely psychologistic. In Kant's own intellectual context, however, explanations of scientific knowledge in terms of our cognitive faculties were the norm for empiricism, rationalism, and, of course, for Aristotelians. Everyone agreed, in addition, that the relevant faculties to consider were the senses and the intellect. What was then controversial was the precise nature and relative importance of these two faculties. Empiricist views, which deny the existence of the pure intellect were for Kant simply out of the question, since they made a priori rational knowledge incomprehensible. Moreover, the conception of the pure intellect that was most salient for Kant himself was that of Leibniz, where the fundamental structure of this faculty, the intellect, is delineated in effect by the logical forms of traditional Aristotelian syllogistic. But this conception of the pure intellect, Kant rightly saw, is

7:30 entirely inadequate for representing the mathematics of space, time, geometry, and motion, which had recently proved itself to be both indispensable and astonishingly fruitful in Newtonian mathematical physics. Nevertheless, Newton's own conception of space, time, and so on, space as the sensorial of God, for example, was also entirely unacceptable on theological grounds and metaphysical grounds, and so the only live alternative left to Kant was the one he actually came up with. Space is a pure form of our sensibility as opposed to the divine sensorium wherein both Euclidean geometrical constructions and the perception of spatial objects in nature like the heavenly bodies thus then becomes first possible. It is of course entirely contingent that Kant operated against the background of precisely these intellectual resources which were determined in Kant's time, largely by Leibniz and Newton and the debate between them. Just that it's entirely contingent that Kant was born in 1724 and died in 1804. Given these resources, basically Leibnizian and Newtonian, however, and given the problems with which he was faced, the solution Kant came up with is not contingent. On the contrary, the intellectual situation in which he found himself had a definite inner logic, mathematical, logical, metaphysical, and theological, which allowed him to triangulate, as it were, on a practically unique, and in this sense necessary, solution. Beginning with this understanding of Kant's transcendental method and its associated rational necessity, we can then see a way forward for extending this method to post-Kantian developments in both mathematical exact science and scientific philosophy. We can trace out how the inner logic of the relevant intellectual situation evolves and changes after Kant in response to both new developments in the mathematical exact sciences and the manifold and intricate ways in which post-Kantian scientific philosophers attempted to reconfigure Kant's original version of transcendental philosophy in light of these developments. That each of these successive new intellectual situations has its own inner logic implies that the enterprise does not collapse into total contingency. That, in addition, they successively evolved out of and in light of Kant's original system

10:00 suggests that it may still count as transcendental philosophy, or so I shall argue. Hermann von Emels' Neo-Kantian scientific epistemology for example had deep roots in Kant's original conception and Himmels is very explicit and self-conscious about this in particular, Himmels developed a distinctive conception of space as what he called a subjective and quote, necessary form of our external intuition, end quote in the sense of Kant and Himmels' conception accordingly retained important elements, and Helmholtz himself uses this word, transcendental. More specifically, space is transcendental, Helmholtz says, insofar as the principle of free mobility, permitting arbitrary continuous motions of rigid bodies, is a necessary condition for the possibility of spatial measurement, and indeed for the very existence of space and spatial objects, which for Helmholtz are constructed out of our experience of free mobility, free motion. moreover the condition of free mobility itself represents a natural generalization of Kant's original Euclidean conception of geometrical construction in the sense that Euclidean constructions with straight edge and compass carried out within Kant's form of spatial intuition are generated by the group of specifically Euclidean rigid motions translations and rotations the essential point however is that free mobility also holds for the classical non-Euclidean geometries hyperbolic and elliptic. And so it is no longer a transcendental and necessary condition of our spatial intuition, for Hemmholtz, that the space constructed from our perception of bodily motion obeys the specific laws of Euclidean geometry. It could be any space of constant curvature. Nevertheless, Hemmholtz's generalization of the Kantian conception of spatial intuition is, in an important sense, the minimal, and in this sense unique, such generalization, consistent with the 19th century discovery of non-Euclidean geometries, namely the initial discovery of constant curvature spaces. And thus, in particular, it is only this more general structure, common to the three classical geometries of constant curvature, that is now seen as a necessary transcendental condition for both spatial measurement and all perception of our objects of our spatial experience. The great French mathematician Henri Poincaré

12:30 Here's a nice picture, a little bit dark, but at least he looks very Victorian here. Poincaré then transformed Helmholtz's conception in turn. In particular, Poincaré's use of the principle of free mobility, which plays a central role in his philosophy of geometry, is explicitly framed by a hierarchical conception of the mathematical sciences, beginning with arithmetic and proceeding through analysis, geometry, mechanics, and finally empirical physics, where, in particular, each level of this hierarchy, other than the first one, arithmetic, presupposes that all the previous levels are already in place. This hierarchical conception of the mathematical sciences, developed, of course, especially in Poincaré's great work, Science and Hypothesis, published in 1902, underlies Poincaré's fundamental disagreement with Hemmholtz. For Hemmholtz, as we have seen, the principle of free mobility expresses the necessary structure of our form of external intuition. And following Kant, Hemmholtz views all empirical investigation as necessarily taking place within this already given form. Hemmholtz's conception is Kantian, that is, insofar as space indeed has what he calls a necessary form, expressed in the condition of free mobility. But it is also empiricist, insofar as which of the three possible geometries of constant curvature, positive, negative, or zero, actually obtains inexperience. And Hamels thinks that determination is empirical, basically by making measurements with rods that you rigidly move around. For Poincaré, by contrast, although the principle of free mobility is still fundamental, our actual perceptual experience he calls bodily displacements, arising in accordance with this principle, is far too imprecise and indefinite, inexact, to yield the empirical determination of a specific precise geometry. Our only option at this point, Poincaré thinks, is to stipulate Euclidean geometry by convention as the simplest and most convenient idealization of our actual perceptual experience. In particular, experiments with putatively rigid bodies, for Poincaré, involve essentially physical processes. Rigid bodies, after all, are matter that's bound together by forces. And so it involves physical processes at the level of mechanics and experimental physics.

15:00 And these sciences, in turn, presuppose that the science of geometry is already firmly in place, according to his hierarchy. In the context of Poincaré's hierarchy of the sciences, therefore, expresses our necessary freedom to choose, by what he calls a convention or definition in disguise, which of the three classical geometries of constant curvature is the most suitable, convenient idealization of physical space. One of the most important applications of Poincaré's hierarchical conception involves his characteristic perspective on the problem of absolute space and the relativity of motion, which is articulated in his discussion of the next lower level of the hierarchy that is after geometry, mechanics. And by mechanics he means classical Newtonian mechanics. Poincaré's key idea is that what he calls the physical law of relativity rests squarely on what he calls the relativity and passivity of space where these latter conditions reflect the circumstance essential to free mobility and constant curvature that the space constructed from our experience of bodily displacements is both homogeneous and isotropic. All points in space and all directions through any given point are necessarily geometrically equivalent. Thus, Poincaré's conception of the relativity of motion depends entirely on his philosophy of geometry. It's because space is homogeneous in this way that motion is relative. And this is especially significant from our present point of view because Poincaré's ideas on the relativity of motion were also inextricably entangled with the deep problems then afflicting the electrodynamics of moving bodies, which were eventually solved, according to our current understanding, by Einstein's special theory of relativity. I shall return to Einstein in a moment, but I first want to emphasize that the connection Poincaré makes between his philosophy of geometry and the relativity of motion represents a continuation of a problematic, originally prominent Kant. In particular, whereas Helmholtz transformed Kant's philosophy of space and geometry, and Ernst Mach, among others, participated in a parallel transformation of Kant's approach to the relativity of motion, resulting in the 19th century concept of an inertial frame of reference, neither Helmholtz nor Mach established a connection between the two,

17:30 that is, geometry on the one hand, relativity of motion on the other. Yet it was an especially central feature of Kant's original approach to transcendental philosophy that these two problems were in fact closely connected. Whereas Kant's answer to the question, how is pure mathematics possible, essentially involved his distinctive perspective on Euclidean constructions. His answer to the question, how is pure natural science possible, appealed to an analogous constructive procedure by which Newton, from Kant's point of view, arrived at a successive approximation, a series of successive approximations, to absolute space. By means of a definite sequence of rule-governed operations, starting with our parochial perspective here on Earth, and then proceeding to the center of mass of the solar system, the center of mass of the Milky Way galaxy, the center of mass of the system of such galaxies, and so on, adding finita. And this conception is developed in Kansa, the Metaphysical Foundations of Natural Science, published in 1786, right before the second edition of the critique. Each of these successive approximations, from our modern point of view, is a closer approximation to a real inertial frame. That's the connection between these two. Indeed, the way in which Kant thereby established a connection between the problem of space and geometry and the problem of the relativity of motion was intimately connected in turn with overarching conception of the relationship between sensibility and understanding, and his characteristic perspective, more generally, on the relationship between what he calls constitutive and regulative transcendental principles. So absolute space for him is what he calls a regulative idea. You never get there because it's the limit of this infinite series of better and better inertial frames, as we would call them. Now it was Ernst Mach, as I suggested, between Kant's original solution to the problem of absolute space and the late 19th century solution based on the concept of an inertial frame of reference. And it is clear, moreover, in successive editions of the Science of Mechanics, Poincaré begins to learn more and more about the contemporary literature on the concept of inertial frame. That's discussed in the reference to R. DeSalle on the handle, if you're interested. and it is clear moreover that Poincaré himself was very familiar with this late 19th century

20:00 solution based on the concept of inertial frame however it is also clear that Poincaré's attempt to base his discussion of the relativity of motion on his philosophy of geometry runs into serious difficulties for Poincaré is here forced to distinguish what he calls the physical law of relativity from what he calls the principle of relative motion. This latter principle of relative motion applies only to inertial frames of reference, moving uniformly and rectilinear with respect to one another, while his latter principle, that is the principle of the law of relativity based on the homogeneity and isotropy of space, applies also to non-inertial frames of reference in states of uniform rotation. It follows from the relativity and passivity of space for Poincaré, since it's both homogeneous and isotropic, we change directions, doesn't make any difference, that uniform rotations of our frame of reference should be just as irrelevant to the motions of the physical system as uniform translations. There's no difference between those two from the point of view of space. Poincaré must also admit, however, that this more extended law of relativity that incorporates rotations as well as inertial motions does not appear to be in accordance with our experiments. For example, Newton's rotating bucket experiment. He has interesting and ingenious things and important things to say about this, but it's a problem with his conception of relativity. It is for this reason that Einstein appeals to what he calls the principle of relativity. His appeals, in his famous 1905 paper, here's Einstein sitting in the patent office, is entirely independent of Poincaré's law of relativity, and it is also independent, accordingly, of Poincaré's conventionalist philosophy of geometry. Einstein's principle is limited from the beginning to inertial frames of reference, of course, moving relative to one another with constant velocity and no rotation. And Einstein's concern is rather to apply this limited principle of relativity to both electromagnetic and mechanical phenomena. Thus, in particular, whereas Poincaré's law of relativity involves very strong a priori motivations based on the relativity and facility of space, which is a necessary feature of space, Einstein's principle of relativity rests only on the emerging experimental evidence

22:30 suggesting that electromagnetic and optical phenomena do not, in fact, distinguish one inertial frame from another. Einstein conjectures, as he puts it, and I'll show you this famous second paragraph from the 1905 paper, it's also on your handout. Einstein conjectures that this experimentally suggested principle holds rigorously and for all orders, and he proposes to elevate it, as he says, that's an important word I'll come back to, elevate this conjecture to a status of a presupposition or postulate upon which the consistent electrodynamics of moving bodies may then be erected. understanding of the principle of relativity is also entirely independent of Poincaré's carefully constructed hierarchy of the mathematical sciences. That is, it doesn't presuppose anything about geometry to begin with. And it is for precisely this reason, I suggest, that Poincaré himself could never accept Einstein's theory. Nevertheless, it appears overwhelmingly likely that, although Einstein did not embrace Poincaré's conventionalist philosophy of geometry, Einstein's use of the principle of relativity here was explicitly inspired by Poincaré's more general methodology described in Science and Hypothesis, according to which the fundamental principles of mechanics, as Poincaré puts it, are definitions or definitions in disguise, arising from experimental laws, as he says, that, quote, have been elevated into principles to which our mind attributes an absolute value. So I'm claiming that Einstein's use of this word elevation is based on what we're in. In Einstein's case, the experimental law in question comprises the recent results in electrodynamics and optics. And Einstein now proposes to elevate, as he says, of course using a German rather than French word, to elevate both the principle of relativity and the light principle, which together imply that the velocity of light is frames, to the status of a presupposition or posthumatic. So why should we think that Einstein is talking about Poincaré here? Well, here's an interesting thing that I don't think has been noticed before. If you look at Einstein's 1921 paper, Geometry and Experience, that I'll come back to later, here he is describing Poincaré's view, his view of geometry, which, as I'll talk about later,

25:00 he's about to reject for general relativity. But he uses is the same word that he used in 1905, now to describe Poincaré's conception. So this suggests to me that it's likely that he had Poincaré in 1905. Of course, it doesn't prove that. Perhaps he changed his mind, but it is rather striking. Okay, so in 1905, back to that, his two principles, which he just said he's going to elevate, then allow him to stipulate, a new definition of simultaneity based on the assumed invariance of the velocity of light, implying a radical revision of the classical kinematics of space-time and motion. In particular, whereas the fundamental kinematical structure of an inertial frame of reference is defined in classical mechanics by the three Newtonian laws of motion, that's basically what the 19th century tradition says, a revised version of this same structure, inertial frames in Einstein's theory, is rather defined by his two postulates. However, a central contention of Kant's original version of transcendental philosophy, as we know, is that the three Newtonian laws of motion are not mere empirical laws, but rather a priori constituted principles on the basis of which alone the Newtonian concepts of space, time, matter, and motion can then have empirical application and meaning. What we have just seen is that Einstein's two fundamental presuppositions or postulates play a precisely parallel role in the context of special relativity. But we have also seen significantly more. This is really the main thing I'm trying to add to that. For Poincaré's conception of how a near-empirical law can be elevated to the status of a convention or definition in disguise is a continuation in turn of Kant's original conception of the constitutive a priori. Whereas Hamilton's principle of free mobility generalized and extended Kant's original theory of geometrical destruction within what Emel's calls our subjective and necessary form of external intuition. Poincaré's idea that specifically Euclidean geometry is then imposed on this form by a convention or definition of disguise represents an extension or continuation of Emel's conception. In particular, specifically Euclidean geometry arises for Poincaré by precisely such a process of elevation,

27:30 by which the merely empirical fact that this geometry governs very roughly and approximately our actual perceptual experience of bodily displacements is transformed into a precise mathematical framework within which alone our properly physical theories can subsequently be formed with it. This same process of elevation in Einstein's hands then makes it clear how an extension or continuation of Kant's original conception can also accommodate new and surprising empirical facts. In this case, the very surprising empirical discovery to one or another degree of approximation that light has the same constant velocity in every inertial frame. It now turns out in particular that we cannot only impose, following Einstein, or as Poincaré did, already familiar and accepted mathematical frameworks, Euclidean geometry, on our rough and approximate perceptual experience, we can also, in appropriate circumstances, impose entirely unfamiliar ones, in Einstein's case, the kinematical framework of special relativity. Einstein's creation of special relativity from this point of view represents the very first instantiation of a relativized and dynamical conception of the a priori, which in virtue of precisely its historical origins through Poincaré, Hamilton, and eventually Kant, has a legitimate claim to be considered as generally constitutive in Kant's transcendental sense. Okay, so now we'll do a little general relativity. Yet Einstein's creation of the general theory of relativity, here's Einstein in 1914 and 15 as he's finishing up on the theory, involved an even more striking and explicit engagement with Poincaré's conventionalist methodology. which, I contend, makes the transcendentally constitutive role of this theory's fundamental postulates that of general relativity, the light principle and the principle of equivalence, even more evident. The first point to make in this connection is that the principle of equivalence together with the light principle plays the same role in the context of the general theory of relativity that Einstein's two fundamental presuppositions or postulates played in the context of special relativity. Namely, they define a new inertial kinematical structure for describing space, time, and motion. Because Newtonian gravitation theory involves an instantaneous action at a distance,

30:00 and therefore absolute simultaneity, it was necessary after special relativity to develop a new theory of gravitation where the interactions in question propagate with the velocity of light. Einstein solved this problem by means of the principle of equivalence kinematical structure wherein the freely falling trajectories in a gravitational field replaced the inertial trajectories described by free particles affected by no forces at all. The principle of equivalence in this sense replaces the classical law of inertia holding in both Newtonian mechanics of special relativity, but the principle of equivalence also rests on a well-known empirical fact, namely that gravitational and inertial mass are which is, of course, well-known since Galileo. In using the principle of equivalence to define a new inertial kinematical structure, therefore, Einstein has elevated this merely empirical fact to the status of a convention or definition in disguise, just as he had earlier undertaken a parallel elevation in the case of the new concept of simultaneity introduced in the special theory. Nevertheless, Einstein did not reach this understanding equivalence all at once, which is basically our modern understanding. He first operated instead within an essentially three-dimensional understanding of special relativity. He rejected Minkowski until about 1912, and he proceeded to develop relativistically acceptable models of the gravitational field by considering the inertial forces, like centrifugal and Coriolis forces, arising in non-inertial frames of reference, accelerating and rotating frames of reference within the framework of special relativity. It was in precisely this context in particular that Einstein came upon the decisive example of the uniformly rotating frame, the rotating disk. And it was at this point, and only at this point, that he then arrived at the conclusion in 1912 that the gravitational field may be represented by a non-Euclidean geometry. Here the idea, as you're probably familiar, is that these successive circumferences are Lorentz-contracted, while of course the radius, which is perpendicular to motion, is not contracted. So the ratio of the circumference to the diameter is different from pi, and the geometry is non-Euclidean. This is how non-Euclidean geometry got into Einstein's gravitational fields.

32:30 It was in precisely the context of this line of thought, the rotating disk, that Einstein found that he now had explicitly to oppose Einstein's conventional philosophy of geometry. Yet Einstein's argument, as described in his celebrated paper, Geometry and Experience, in 1921, which talks about the rotating disk, or at least refers to it, was far from a simple rejection of Poincaré's methodology in favor of a straightforward empiricism. For Einstein also explicitly says in the same paper that subspecie et ernie, as he puts it, Poincaré is actually correct. So that in particular, Einstein's reliance on a Hemholtzian conception of what he calls practically rigid bodies is here merely provisional, as Einstein says. I have suggested, therefore, that we can best understand Einstein's procedure as one of delicately situating himself between Hemholtz and Poincaré. Whereas Einstein had earlier followed Poincaré's general conventionalist methodology in elevating the principle of relativity together with the light principle to the status of a presupposition or postulate, he here follows Hamilton's empiricism in rejecting Poincaré's more specific philosophy of geometry in favor of practically rigid bodies. It does not follow, however, that Einstein is also rejecting his earlier embrace of Poincaré's general conventionalist methodology from science and hypothesis. Indeed, Einstein had already rejected Poincaré's specific philosophy of geometry in the case of special relativity, and for essentially the same reasons, he does so here. Poincaré's rigid hierarchy of the sciences, where geometry has to be there first, before either kinematics or physics. in both cases, special and general relativity, stands in the way of the radical new innovations Einstein himself proposes to introduce. But why was it necessary, after all, for Einstein to engage in this delicate dance between Humboldts and Poirier? The crucial point is that Einstein thereby arrived at a radically new conception of the relationship between the foundations of physical geometry on the one side and the relativity of space and motion. on the other. These two problems, as we have seen, were closely connected in Kant, but they then split apart and were accordingly pursued independently

35:00 in Helmholtz and Mach. In Poincaré, as we have also seen, these two problems, geometry, relativity, and motion, were perceptively reconnected once again, insofar as Poincaré's hierarchical conception of the mathematical sciences incorporated both the modification of Helmholtz's philosophy of geometry and a serious engagement with the late 19th century concept of an inertial frame of reference. Indeed, it is for precisely this reason, as we now see, that Poincaré's scientific epistemology was so important for Einstein. Einstein could not simply rest content with Helmholtz's empiricist conception of geometry because Einstein's most important problem that he was faced with was to incorporate the relativity of motion into his theory. But he could not rest content with Poincaré's conception either because Einstein's new models of gravitation arrived at by exploring the principle of equivalence in the years 1907 to 1912 culminating in the rotating disk had suggested that geometry has genuine physical, maybe gravitational, content. Einstein's radically new way of reconfiguring the relationship between the foundations of geometry and the relativity of motion therefore represents a natural but also entirely unexpected extension or continuation of the same conception of dynamical and relativized constitutive a priori principles he had first instantiated in the creation of special relativity. Just as Einstein had earlier shown how an extension or continuation of Kant's original conception, of course he doesn't explicitly say that, that's me saying that's what he was doing, could accommodate new and surprising empirical facts, here the discovery of the invariance of velocity of light. Einstein, in the case of general relativity, shows how a further extension of the same tradition can do something very similar in facilitating for the first time the application of a non-Euclidean geometry to nature. In this case, however, it is not the relevant empirical fact, namely the well-known equality of gravitational and inertial mass, that is surprising. unforeseen connection between this fact and the new geometry. That's what no one else saw. And what makes this connection itself possible for Einstein is precisely the principle of equivalence, which thereby constitutively frames the

37:30 resulting space-time geometry of general relativity in just the same sense that Einstein's two fundamental presuppositions or postulates had earlier constitutively framed his mathematical description of the electrodynamics of in special relativity, that is what we now call the geometry of Minkowski spacetime. Whereas the particular geometry in a given general relativistic spacetime is now determined entirely empirical by the distribution of mass and energy in accordance with Einstein's field equations, the principle of equivalence itself is not empirical in this sense. This principle is presupposed as a transcendentally constitutive condition for any such geometrical description of spacetime in GR have genuine empirical meaning in the first place. The historicized version of transcendental philosophy I am attempting to exemplify therefore sheds striking new light, I believe, on the truly remarkable depth and fruitfulness of Kant's original version. Finally, there he is. Kant's particular way of establishing a connection between the foundations of geometry and the relativity of motion, which, as we have seen, lies at the heart of his transcendental method, has not only led through the intervening philosophical and scientific work of Himmels, Marx, and Poincaré to a new conception of the relativized a priori, first instantiated in Einstein's theories, it has also led through this same tradition to a radically new reconfiguration of the connection between geometry and physics in the general theory of relativity itself. There can be no question, of course, of Kant's having anticipated general relativity in any way. The point, rather, is that Kant's own conception of the relationship between geometry and physics, which was limited of necessity to Euclidean geometry and Newtonian physics, then set in motion a remarkable series of successive reconceptualizations of this relationship in the light of profound discoveries in both pure mathematics and the empirical basis of mathematical physics, which finally eventuated Thank you very much for this very accurate study of the parallel. So I will ask Patricia.

40:00 Do you mean the quantum mechanics book, or do you mean the book, the Reichenbach? The relativity book. The relativity book. And Schley writes a paper to react to this deposition about the relativizing . And the argument of Schley. and stop to talk of the procedural principles that we don't hear them. How we can escape of Schlieb's critique of this position? Good, thank you. Yeah, one way to think of my paper is perhaps as a way of trying to escape what Schnick says. Yes, it's interesting because in that review, as you know, it's mostly about Kassir, but he also, at the very end of the review, mentions Reichenbach's book. And he and Reichenbach were actually corresponding at the time about this, because in Reichenbach's book, he criticizes Schnick's allgemeiner Tegeslera for not paying attention to the constituent role of principles. Schleg actually says to him, oh no, I accept this constituent role, but I don't want to call it synthetic a priori. I want to call it a convention in the sense of Poincaré. And actually, Reichenbach is convinced by that, sadly. And then he changes his view to his later philosophical degree that is familiar to us. What I'm basically saying here is if you look a bit more carefully at Poincaré, and how Poincaré's so-called conventionalism actually plays a role in leading to special relativity of what Einstein is doing there, it's actually going to support Reichenbach's view. One important element of what I'm saying is that supports Reichenbach's view. Supports Reichenbach's view because one important aspect

42:30 of my view is I want to adopt Poincaré's notion of elevation. but I don't want to adopt conventionalism in the sense that there is an arbitrary choice that could have been otherwise just as well and I don't even think that's what Poincare thinks about geometry Poincare thinks there's a unique best choice namely Euclidean geometry although there there's a very special situation because of the Helmholtz-Leaf theorem the possibility of three different geometries consistent with criminality That possibility no longer exists in the case of special relativity or the case of generality, even less. There's really no alternative to the principle of equivalence in Einstein's context for, as Reichenbach would say and as Schlick would say, coordinating their theory with experience. I still want to say, though, that the principle of equivalence follows Poincare's methodology of elevation, of something that is merely empirical to something that is constitutive, even though there's nothing like an arbitrary conventional choice at all. So in that sense, that's part of my answer to Schleck and defensive right. I think Einstein himself approved quite clearly Schleck. He did. And he then criticized very eagerly Cassero. He has a famous thing he says. He says, I have not read anything as perceptive in a long time. Exactly. And I think, if I could just comment on that, that what's going on there is that Einstein sees any kind of hint of Kantian and neo-Kantian idea as a conservative view that is going to keep him from making radical innovations. So what I'm trying to say is if we understand what he was really doing in terms of this elevation idea on Coray, can at the same time defend Reichenbach's relativized a priori and allow all the radical annotations that Einstein does. Just one other point about Kassir. Kassir, I think, actually doesn't have a relativized a priori. We can talk about that. I may not have understood, so let's just speak to the special relative I have a kind of worry about treating principles like principle of relativity and life postures

45:00 in special relativity as constitutive values, so taking them out of the game and making them part of the rules of the game. And the worry is that if you don't put any physical content in, you're not going to get any physical content out, so you can define things as you wish, you can specify rules which delimit the concepts you're going to employ and you And because you make the rules, they're not subject to reputation. They're true, but they're true by definition. And if you just stick to those things, you're never going to get the true by definition stuff. You're never going to get any empirical content out. So say Einstein could take the principle of the rules, could you make it a rule, take the life muster, make it a rule. But then he can't draw any empirical consequences from that. And what you need, in fact, of course, is that the actual bodies, the laws governing bodies, the imperative laws governing bodies, actually display the required symmetries. So, I mean, I struggle to see how you get any empirical content out if you make these things. Good, good. Well, that's this idea of elevation. So if you look at the second paragraph of the 1945 paper, we suggest, it starts by saying, this sort of, that's the magnetic conductor of this, suggests that as has already been shown to the first order of small quantities, empirical content, empirical content, the same laws of electrodynamics and optics will be valid for all frames of reference for which equations are valid, empirical content. Of course, he's going from some level of approximation. Actually, interestingly, he only says first order here, although he knows already that it's valid to second order. But in any case, he says for shorter. But that's the empirical facts. And he says, I'm going to elevate this conjecture, that's an empirical conjecture, to the status of a hosta, a presupposition. And also introduce another one and blah, blah, blah. So there is empirical content in these relative-based a priori principles. That's the idea, that they take something empirical. Are they true by definition or not? They're not true by definition in the sense you're thinking of. you're taking them to be something like analytic truths, which could never be revised. But of course, Freystein himself, it is revised, because in general relativity, the principle of relativity in this sense doesn't hold, there are no inertial frames. And instead you have the principle of equivalence,

47:30 you have local inertial frames, and what you have here is now a mere approximation. So what you here have constitutive, relativized principally principles, turn out to be mere empirical approximations to this new thing, with new constitutive principles, of course, might themselves have empirical content because principal equivalence rests on the equivalence of gravitational inertial mass. That may not hold for certain particles, and people speculate about this. That may be revised, too. So the whole idea is that these so-called a priori principles have empirical content and are revisable. Nevertheless, they are, in some sense, constitutive. And that's my argument. Just the way Kant, well, Reichenbach in 1920 distinguishes two meanings of a pre-work. One is necessarily true, fixed for all time, eternally valid, never changed. The other is constitutive of the concept of the object of scientific knowledge. And he tries to separate those two meanings. He says, look, that first meaning relativity shows is kaput. That's impossible. We cannot do that anymore. However, the second meaning of constitutively framing the more empirical parts of the field, so in special relativity, Maxwell's equations for the electromagnetic field. In general relativity, the field equations. That is the thing we can still maintain. So the whole idea of this story and of this idea of this notion is to have empirical content that is revisable but still leads to constitutive principles. I don't understand, you didn't talk about it as later, but if something could play that role of defining the concept of happening and that is the truth, but it would be much easier. It's more like a synthetic, if you're a truth. Yeah, Claude? You talk about these constitutive principles characterizing what sort of empirical experience is possible, but in the case, by the time we get to Einstein, take it that what is constrained are things like the behavior of highly idealized rigid rods and plots whereas when we go back to Kant you know empirical experiences much closer to you know what goes on in my immediate experience so there would appear to be a parallel or or

50:00 the concomitant development for this notion of empirical experience. So I'm just inviting you to comment. Good. Good. Now, the Khan case is very interesting, because one thing he has in mind as constituent principles are Newtonian laws of motion, or actually his slight alternative formulation of these three laws. So he clearly takes Euclidean geometry, taken as what we would call a very precise idealized theory. as a constitutive feature of experience in this sense. And he takes... So just a word about what his sense is there, the consens of empirical experience. Well, Ferkant includes at the same time our ordinary experience of space and the Newtonian-Euclidean description of space. And one interesting thing about that is there's no gap for Ferkant between the scientific and the manifest image, partly because the model is Newtonian gravitation theory, which does not proceed by postulating some kind of micro world. It basically proceeds by taking what we experience, the sun, moon, planets, the earth, falling bodies, pendula, and describing them more exactly and precisely than we do in everyday experience. So it is a persistification, an idealization of everyday experience, But it's really quite continuous with it, right? So the two ideas, ordinary experience, ordinary perception, and highly idealized scientific description for Kant are very close because of the . Now, of course, when you get to the electron that enables the moving bodies of the theory of the electrons, and you are talking, and for Poincaré very much, you're talking about the microwave. Einstein, of course, wants to say, well, my principle of relativity but he's actually dependent on it. And you don't have to worry about that. That's part of the principle versus constituency. That's really just a footprint. So by the time we get to relativity, in any case, things are further from ordinary experience than they were for Kant. But in some sense, it is a continuous extension of the same kind of thing that Newton did. But take it further. And reading, involving very unexpected things which doesn't happen every day in the study.

52:30 Thank you very much, Paul. I am almost completely in agreement with you. It's a beautiful presentation. And in particular, I am in agreement with the last one when you said that the validation is not the validation of something similar to an analytic principles, and to a synthetic principle, so we are speaking about the synthetic principles relative to the field. On this I completely agree and I agree to respect the people of the movement. I have only a question of technological clarification. You were reminded, say, Reichenbach and Schlinger, Reichenbach is distinguished between the two meanings of the And this was considered by Schlich as a negation of the original Kantian scientific a priori. Because for Schlich, the two aspects of the scientific a priori are to better become, you know? And so they found an agreement that there is the negation, is the thesis, the scientific a priori, there was a disagreement about compression and not and after right and back changes his mind but there is a point of when right and back in 1922 he gave a survey of the several interpretations of the And let's think about this position, the position of Schlinger says surely our position is that it is impossible as a transcendental method. A method for proving, for establishing some in absolute sense. So my question is please excuse me for the same problem. when you use the idea of the why you are also the objective you also about the principles of you speak about the transcendental and what is the meaning of this

55:00 transcendental surely when she arises this idea is the real come Crascendental is the knowledge of the explanation why some truths are absolutely valid, not this is original meaning. There are other meanings in Canada. This is original. And they refuse this meaning of the word Crascendental. So, why do you continue to do very well, because it's okay, transcendental, why? Right, good, thank you. That's really another kind of good point of the paper. Now, transcendental, well, one meaning of transcendental, a kind of famous neoconics in a way, is explaining the fact of science. the fact of exact science, and you look for the necessary conditions or necessary presuppositions of exact scientific theory. So let's take it in that sense for a moment. Now for Kant, his application of that, say explaining the Newtonian science that he found himself faced with, took that science to be necessary, unrevisable, eternal. his explanation of that fact appealed to the fixed, changeless, timelessly valid faculties of the human mind of all human beings at all times and at all places. So his transcendental method, with its appeal to the cognitive faculties of, as it were, the idealized transcendental subject, the transcendental human being, if you like, took that to be fixed, reflecting his idea that geometry and physics is also fixed, right? So I'm not doing that, obviously. So what I'm doing is I'm trying to replace, so then at the very beginning of the paper, you may remember, what I did is say, well, how did Kant's transcendental arguments actually work? Let's just take his faculty explanations. What makes his explanations, as they have to be, not only something that explained the possibility, but uniquely explained it? This is a common objection. Why is your explanation about the faculty of pure intuition the only possible explanation? And the move I made there was to say, it's unique given the resources that Kant was faced with

57:30 in his time, in his context. So that's where I already relativized the transcendental method to. And so, and then what I did is say, these resources, what the state of pure mathematics is at the time, what the view of logic is at the time, various things that are happening in physics and empirical science at the time, various philosophical and metaphysical views like Leibniz and Newton's collision in the Leibniz-Clark correspondence, all of that frames Kant's transcendental arguments. I mean, you look explicitly, Kant says, look, there's the Leibnizian alternative, there's the Newtonian alternative, mine is the that explains geometry and isn't subject to the problems that libraries and museums have. So what I'm trying to do is say, well, let's now reinterpret what Kahn is doing and historicize that, too. He's not talking about the necessary faculties of the human mind. He's talking about what is the uniquely best understanding of the foundations of the exact sciences at a certain time in light of all of these resources that he's dealing with. So then I historicize the transcendental method, too, so let's look how that context changes to who still uses the word transcendental as I pointed out meaning to say look Kant had it right except he was stuck with Euclid that was his mistake otherwise he's right so I'm relativizing historicizing transcendental at the same time I'm sorry we have a very short time So your transcendental has nothing to do with the in 1922. Nothing to do with the last logical invariant of thought. No, that's not a relative . That's an absolute universal . I am very good. Thank you very much. OK, two last . I'll try to shorten my question into sort of complementary to Chris's question. So what is happening with the elevation process is that we're looking at some experimental laws where we're making with suggesting conjectures on the basis of those and elevating them to the status of the principle. Now what I'm worried about is not what Chris is worried about. I'm worried about the status of these synthetic principles

1:00:00 now, because these are no more principles of the physical theory. Because indeed, we can have different and even contradictory experimental laws, experimental data that we will be tempted to elevate to the status of principles. So now we're going to have a whole landscape of different physical theories, starting from some principles. You know, this theory starts with this principle, and that theory starts with that principle. Right. So will you accept that now this whole move of relativizing a priori takes us from doing the physical theory into actually having a variety of possible physical theories and this way, you know, then trying to see which of those is more adequate to us? Exactly. Let's take what actually was going on at the time, which is you have Einstein's special relativity, and then you have Lorenz Fitzgerald Poincaré trying to have mathematically and empirically the same theory, the same Lorenz group, but keeping classical mechanics as the constitutive framework. So there actually were two different proposals at the time to accommodate the surprising discoveries in electrodynamics. One, Poincaré is very clear about this. Mechanics for him means Newtonian mechanics. And that has to be presupposed before you get to electrodynamics. So what you do in electrodynamics is something about the electromagnetic field and the micro world of the electron. But you're not doing anything at the level of mechanics. For Einstein, you are doing it at the level of we're conquering the quantum mechanics. But the theory is good. The theories are very good. Yeah, so there are an equal sort of justification. In fact, I would argue that at the time, actually, until you get to general relativity, there actually isn't a decisive reason to go in Einstein. Thank you.