Struralism and the a priori / Kant's ungrounded relations & quantum non-supervenient relations

0:00 I can make it a little larger. How's that? Perfect. Well, James has set us up nicely on why we're now talking about structuralism. Structuralism is an argument, or it's a response to the numericals argument that has been advanced by certain scientific realists. Transcendental is not structuralism. Transcendental is my own idealist version of structuralism. Instrumental, which I would associate with thoughts, and transcendent, that would be structural realism in the Kantian sense of transcendent realism. The next thing I will talk about after that will be Structuralism Without Realism, which is two versions that I will have from Voss and Hermann Weil.

2:30 And I want to say a little bit about Weil's own program of structuralism, which I see as a theory of constructive cognition in physics. I'm going to say a little bit about, not very much, about how Lyle really discovered how to gauge structures in field theory, local symmetries in field theory, in an a priori way, and just a few words about Lyle's program in quantum mechanics. And actually, the most bulk of the talk is going to be on point four, against eliminating structuralism. Since James has now surrendered that advice, that will be targeted on Stephen French, not you. So, what about, I'm not going to say anything in particular about transcendent structuralism until the end. I will talk for the moment about what is this transcendental structuralism and why we would want to talk about transcendental structuralism. Could you make it bigger again? Yeah, I can try that again. Let's see if I can make it bigger. Is that okay? Is that readable? Just fractionally bigger, if it can be made slightly bigger, yeah, it needs to be a bit bigger. Well, I would start with Kant, and the basic idea is that, as he put it, the order and regularity and appearances that we call nature, we introduce ourselves, and indeed we could not find them there had not we, or the nature of our minds, been there originally. I find similar echoes of that particular point of view in Henri Poudret, writing in 1906. Does the harmony that human intelligence thinks it discovers in nature exist outside this intelligence? No, beyond doubt. A reality completely independent of the mind that sees or feels it is an impossibility.

5:00 A world as exterior as that, even if it existed, would be for us forever inaccessible. But what we call objective reality is, in the last analysis, what is common to many thinking beings and could be common to all. This common part can only be the harmony expressed by mathematical laws. It is this harmony, then, which is the sole objective reality. And perhaps that's going to tie up a little bit with what James said about modal ontological structure. My hero, Hermann Weyl, writing in 1934, the structure of our scientific knowledge is conditioned by the circumstance that the world, the goal of our scientific endeavors, is not one existing in itself but arises from and exists only by means of the meaning of subject and object. That is not an expression of subjective idealism. That's not to say that it's not an experimental world. It says that the world that we talk about in science is a world that arises from and exists only by means of a regime of subject and object. And another hero of mine, although more controversially, Arthur Eddington, beautiful expression, These are exactly, I think, what concepts in the first quotation. So, structuralism without realism. I'm taking some passages here from a Boston paper, Structured Perspectives, that appeared in 1997. James has laid out what the semantical approach to scientific theories is. This view is now at once structured by common consent, the semantic approach to the current general idea of structuralism. And he goes further, according to the semantic approach, to present the scientific theory is, in the first instance, to present a family of models, that is, mathematical structures, offered for the representation of the theory's subject matter.

7:30 Within mathematics, isomorphic objects are not relevantly different, so it is especially appropriate to refer to mathematical objects as structures. Given that the models used in science are mathematical objects, therefore scientific theoretical descriptions are structural. They do not cut through isomorphism. So the semantic approach implies a structural disposition. Science's description of the subject matter is solely a structure. Properly understood, it is entirely correct to say that models represent nature only up to isomorphism. They only represent structure. Therefore, it is really a consequence of the semantic view that science describes only structure. Now, I find there's a lot to lie for a non-realist in Baas's Constructive Empiricism or Structural Empiricism. The basic conception of the semantic approach was already stated in 1926 by Hermann Weil in his wonderful book on philosophy of mathematics and the natural sciences. A science can determine its domain only up to an isomorphic mapping. In particular, it remains quite indifferent as to the essence of its object. The self-evident insurmountable boundary of cognition, for Weyl this is self-evident. Well, another fundamental aspect of Weyl's epistemology of exact science, of mathematics and physics really, is not emphasized by Weyl, is Weyl's transcendental idealism. Here is again a passage from the book on philosophy of mathematics and natural science. Science concedes to idealism that it should be its objective reality is not given but posed as a problem and that it cannot be constructed absolutely but only in relation to an arbitrarily assumed coordinate system and mere symbols.

10:00 Now that particular locution is the language of Kant from the Transcendental Dialectic. In fact, it's essential in Kant's resolution of the so-called mathematical antinomy in the Transcendental Dialectic. Kant wants to say that totalities of which we think are conditioned, part are given, the totalities themselves are not given as totalities, and that's essential to the resolution of the antinomies. So let's look at structuralism according to Hermann Weyl, and just some passages again. From a late paper of Weyl, a very interesting paper which unfortunately has never been translated into English, Science is symbolic construction. The development of physics itself has modified the view of the ultimate elements, the bricks of the objective world's construction. Instead of a real spatial temporal material being, we are left only with construction in pure symbols. Now that seems very mysterious. We're going to see if we can figure out what that might actually be saying. And there's one more thing about Weyl's view of constructive cognition. He actually outlines it in four steps, this most clearly in an address at Columbia University in 1954. At the basis of all knowledge there lies intuition, mind's originary act of seeing what is given to it, limited in science to the aufweisbare, but in fact extending far beyond these boundaries. How far one should go here, including the Wiesenschau of Husserl's phenomenology, I prefer to leave in the dark. Understanding an expression, thinking the possible, and construction of symbols and formulas on the mathematical side, the construction of measuring devices on the empirical side. Now there are a couple of things about this passage. One is that this talk of the mind's originary act of seeing, what is given to it, is tyrannical. It's language taken precisely from Husserl. Husserl does not use the term meaning something like ostensible or evident.

12:30 Other terms. That term is taken up from some papers of Kurt Riedemeister that Biow has been reading in 1954. He finds a particularly apt German expression for what it is that we start with when we begin our symbolic constructions. We start with things that are evident to the mind, present to the mind in the mind's originary act of seeing. Symbolic cognition is impenetrable. In physics, he says, we apply an a priori construction of the possible into which the actual is embedded on the basis of values and attributes indirectly determined by reactions. And then he says in his book on theory of groups and quantum mechanics, natural science is of a constructive character. The concepts with which it deals are not quantities or attributes that can be obtained from the objective world by direct cognition. But scientists, he goes on to say, have long held the opinion that such constructive concepts were nevertheless intrinsic attributes of the thing in itself, of a mind independent reality, even when the manipulations needed for the determination were not carried out, and by all things that in quantum theory were confronted with a fundamental limitation to this realist point of view. And here's a thesis that, just to boil down these quotations from Weyl, I want to think of structure as the resultant net of possibilities into which the actual world, the experienced world, is embedded and posed as a problem within that net of possibilities in order to describe it as the actual world. And it's built up by symbolic destruction. Well, in a ten-year period, one brief decade, Weyl gives two astonishing examples of such a priori construction in physics that still yield fruitful developments. One is the part that I've actually done a good deal of work on, is the idea of local gauge invariance, originating in his theory of gravitation and electromagnetism.

15:00 Which Biles himself carried over to the director of physics theory at the electron in 1928, where it now becomes a phase invariant. And the idea here is that Weyl, in this theory of gravitational electromagnetism, Weyl starts with a more general geometry than the Riemannian geometry of general relativity. In particular, it doesn't allow comparisons of magnitude at a distance as a Riemannian metric does. Distance comparisons can only be made locally. And then if you carry out the mathematical construction of the manifold that does this, out pops the electromagnetic field. So he has a unified, in some formal sense, not in a physical sense, he has some formally unified theory of gravitation and electromagnetism, which is simply a theory of a space-time manifold. No additional fields are added into that manifold. Gravitation and electromagnetism are simply there, although he does not show that they have a fundamental interaction. The second attempt, about which I am beginning some work on right now, is Weyl's attempt to derive fundamental relationships in quantum mechanics from group theoretical symmetry principles. And here I found the writings of the Harvard mathematician George Mackey very helpful. He calls it Weyl's program, that is, the program to derive fundamental relationships in quantum mechanics from group theoretical symmetry principles. And, among other things, this is the program to find some a priori justification for the fact that the self-adjoint operators of position and momentum components should satisfy the Heisenberg commutation relations in that form with all other pairs including And the resultant statement of these relations in integrated or vile form on the basis of vile conjecture of a general correspondence between self and joint operators and unitary representations of a non-compact lead group, the additive group of the real line. And crucial here is the step from finite to independent groups that gives you the integrated form of the Heisenberg commutation relations.

17:30 Taking the existing quantum mechanics and reconstructing it from the point of view of the theory of groups. Just two examples there of Weyl's own approach to symbolic destruction or structuralism. Well I wanted to spend the bulk of the talk, the rest of the talk anyway, on what is eliminating structuralism now. It has a certain expression in concurrency in philosophy and mathematics where it means something like the fact that the reference to mathematical objects is always in the context of some background structure. Mathematical objects have no more to them than can be expressed in terms of the basic relations of the structure. A classic example, cited many times, is Dedeckin's supposedly structuralist treatment of the natural numbers. James pointed to Russell's criticism of Dedeckin's structuralist treatment. Eliminated structuralism is really the position of ontic structuralism, which I will now associate with Stephen French, as of this morning, and not James Ladiman. Reference to fundamental physical objects is to be understood as to objects re-conceptualized entirely in terms of a mind-independent world structure. And this is basically the view, as James laid out. Or was the view for James, anyway, that there are no such thing as individuals with intrinsic properties. All you have, really, is structure without related, you have relations without related. And the classic example there is the metaphysics of non-visuality of quantum objects. I want to look more at the problems with mathematical structuralism. Citing here Charles Parsons, in mathematics no one has shown how eliminative structuralism can be carried through. Let's just look at the example of Dedeckin. For Dedeckin, the natural numbers are but an instance of a simply infinite system.

20:00 This is in paragraph 71 of Here's the explicit definition. They can define the simply infinite system as a system or a set n such that it's a distinguished element and a mapping carrying n into And minus the terms so you can think of if you start the natural numbers with zero carrying that zero into a system beginning with one and one into two and so forth and so on which is one one and on two such that mathematical induction holds. And that's just what that symbolism says. This is an explicit definition of a simply infinite system. Instanced by the natural numbers, which I'll abbreviate, following Charles Parsons, as just omega NOS. Now, statements about the natural numbers are then implicitly general. Any statement about the natural numbers is about this particular explicit definition of a simply infinite system. This would mean that any statement of arithmetic can be rewritten in terms of the primitives. So that we could write some statement of arithmetic or abbreviate it as A-N-O-S. And then understood as about the natural numbers, N-O and S are variables, so that that statement is elliptical for this one, so for any N-O and S, right, if the explicit definition of a simply infinite system, then the statement in its canonical form.

22:30 But of course this doesn't do the job, because if there are no simply infinite systems, then it's vacuously true. And if that's the case, also then if A and not A have true canonical forms, you would have an inconsistent arithmetic. So Dadekin has to prove the existence of simply infinite systems. And how does he attempt to do this? He famously argues, in paragraph 606, that the totality of things that can be the objects of my thought is infinite. For given any such object S, object of my thought, we can let S of S be the thought that S can be an object of my thought. But then, S is a one-one mapping of the potential objects of my thought into themselves, and by virtue of that fact, it's an infinite totality. Well, this is to, as Parsons rightly said, this is to prove the existence of simply infinite systems by a kind of transcendental psychology, which certainly transgresses eliminated structuralism. This is just one, it's a key example, but just one example of the failure of eliminating structuralism. We could turn to Parsons to discuss in detail the shortcomings of other eliminating structuralist proposals, including structural conceptions of set theory. We could think of Hilbert's axiomatization of Euclidean geometry, which is often taken to be a kind of purely formal structuralism. But it omits the fact that Hilbert begins his foundations of geometry with the Kantian statement that knowledge begins in intuition and proceeds from there to concepts and then on to ideas, or as Hilbert calls them, axioms. Another failure of eliminated structuralism, more familiar perhaps to certain philosophers, would be the failure in Carnap's Aufbau. Runs aground on trying to formalize the one empirical relation of the system, recollection of similarity among elementary experiences, and gets into problems with how that might be singled out in a univocal way using only formal means.

25:00 Well, Parsons' own diagnosis I think is insightful of why the purely structuralist account doesn't seem appropriate. For the most elementary mathematical objects, like the natural numbers, right, these are, in some sense, pseudogenerous abstract objects that are quasi-concrete, in Parsons' terms, types determined by intrinsic relations to concrete objects. Concrete objects are, if you like, are intuition. Here we can think of Hilbert's strings of strokes or even just formal expressions in a kind of syntactical arithmetic. Their concrete representation is something about the objects that goes beyond the formal structure, for example, being an infinite sequence that they instantiate. Moreover, the concrete representation is essential to the intuitability of the objects. This again is an intuition founded in Heserlian terms in ordinary perception and imagination. I should say that Parsons also recognizes an intuition that, like Poincare, that would apply to general propositions about objects, for example mathematical induction. But here we have the paramountcy of an epistemological motive. And we have to be able to answer this question for the more concrete domains of mathematical objects since these still play an illimitable role in the explanation and motivation of other mathematical concepts and theories that is of mathematical structures generally. This is making a claim really about the epistemological basicness of these. Quasi-concrete objects like the natural number and how we come to know such objects. In sum, I would say that eliminating structuralism does not go through without legislating what counts as a mathematical object.

27:30 You can't say that the natural numbers are not mathematical objects because we cannot give a complete structural count of the natural numbers. In particular, it necessarily omits the explanatory and justificatory role of the most elementary mathematical objects that Parsons terms quasi-concrete. And why? It's because this relation of quasi-concrete representation is additional to that of interstructural relations. These, of course, are the epistemological motivations of puberty and finitivism, but they're also those of Weyl's symbolic construction that begins with objects that are turned ausweissbar, that are, in some sense, exhibited through our intuition, that we see clearly and understand clearly. I would say that Weyl's examples of such obdurate conditions... These are what Hilbert called the deepening of the combinations of certain matters. Do the failures of eliminating structuralism in mathematics, one can ask, bear upon ontic structuralism in physics? And it might be argued that it doesn't. Rather, articulation of eliminating structuralism requires a revamped metaphysics. Not mathematics. And this is the position that I take is expressed in this quotation from by Steven French and James Ladyman in their paper, Remodeling Structural Realism, that appeared in CINDES in 2003. The structuralist finds herself hamstrung by the descriptive inadequacies of modern logic and set theory, which retains the classical framework of individual objects represented by variables, and which are the subject of predication or membership, respectively.

30:00 In lieu of a more appropriate framework for structuralist metaphysics, one has to resort to a kind of spatchcock approach. That's not a word you'll find in an English dictionary. I suppose you could call it bricolage or something in French. Treating the logical variables and constants as mere placeholders, which allow us to define and describe the relevant relations which bear all the ontological weight. So the idea here is, well, we're not going to find what we need for an eliminated structuralist metaphysics from mathematics because mathematics itself has this classical framework of individual objects and so forth and predication. We need to turn to a new kind of metaphysics that will enable us to carry through completely the program of the structuralism, eliminating structuralism of relations without relata. And could such a metaphysics therefore provide the means of carrying out eliminated structuralism in physics? I think it's possible, but I'm skeptical for these reasons. Physical theory requires physical interpretation of particular mathematical structures that indeed appear as pure structures of various kinds, but epistemologically and genealogically these structures are bootstrapped from more elementary domains of objects that are not pure structures. In other words, we can tell a plausible story about how we have epistemic access to the pure structures of mathematics. Whereas on the physical side, we're forced to reconceptualize structure altogether. Another problem with this ontic structuralism is what kind of realist correspondence could there be between mathematical structures and the non-standard structures of the dramatic metaphysics of nature? Having the same structure. And then just in conclusion, I would say that there's a transcendental dialectical diagnosis of eliminating structuralism.

32:30 Structures are not given, they're not postcards from eternity, but are constructed or posed as a problem from the given. Nishke Geben, Sondre Avke Geben. Thank you. In the beginning you mentioned you related structures to isomorphism, it's not the way we think about structures, something defined up to an isomorphism, but I just wonder, I'm thinking about categories where you basically regard other morphisms than only isomorphism, and I just wonder... How it bears upon the notion of structure in your account. We just try to generalize it to the fact that not only isomorphism. And one thing which I'm thinking, say, In the end, we mentioned that this quasi-concrete is not a structural property, but probably if you look at this more general understanding of what is structural, it becomes a structural property, because if you think about some representation theorems, like Kelly theorems for groups, or Storn theorems for Hooghoud and Algebras, or Jona D'Alema... I don't know in which sense, but it seems to me that it's a sort of structural issue. Yeah, well, with regard to the last question, that's the kind of chicken and egg question. I'd say the egg comes before the chicken. I think that the important thing is to have the national numbers in order to include any representation about simple stuff, simple causes, I think that's just a mathematical, psychological priority. As for the former question about morphisms and category theory, that's a very good question, and I actually do not have a good answer to it. I would have to think about that. Yes, I have a slight problem with the way you presented this on one of your first slides.

35:00 You know, a priori symmetry considerations and then deriving equations from it because this this theorem that you mentioned between how to derive say algebraic structure between observables from the additivity along the real line, But the real empirical input there is of course translation symmetry in space, right, so it is sort of an extremely obvious fact, almost too obvious to notice, that if you replace things a few kilometers that really nothing happens, so it is not a priori, it is extremely empirical. Well, because if things were to change, if we replace them a few kilometers, we would never even consider proving such a theorem. Yeah, okay, but in a sense it's a question of how much you can do with how little. And translation of symmetry is an obvious symmetry to impose as a constraint. And then what can you get out of it? It's quite remarkable that Weyl does get this out of it. I would like to understand how you make a difference between Basse's version of structuralism that you call instrumental structuralism and the transcendental version of structuralism. In the things that you projected, I didn't think of a very clear difference. Structure is only constructed, not given. But I think in order to differentiate clearly between the two versions of structuralism, you have to add something to these conditions. The condition to be added, I think, is the constitutive power of structure in transcendental idealism. That I thought was implicit in what I said. I agree with you completely. And it's a constituent structure that is in some sense working out some kind of description of the actual world by fitting it into some wider system of possibilities and showing why this one particular realization is what we deserve.

37:30 The problem with ontic realism about physics goes beyond the ones that you outlined for mathematics because in mathematics the relational structures are always there and all their properties are necessary in some sense. But, you know, just see if I can quote James, there's nothing more to the objects than the relational nexus in which they are located. Something like that, right? This is the new improved version. Yeah, the new improved version, but still presumably only a slightly weaker version than the radical one. But I want to ask, well, does this relational nexus have any contingent properties? Under what conditions does it exist or not exist? And in the case of mathematical structures, those would be inappropriate questions. Whereas here, it seems to me that if you're trying to answer those questions, you seem to get out of structuralism. So the question is, is what's going to differentiate structuralism and physics is going to be something about the contingent character of, you call it contingent, James might call it nomological or something. Yes, but still, it could be different. Still, it's not necessary in the sense of mathematics. Well here there's got to be some kind of account of physical law and rather than You can bog down in laws and particular theories. I am much more wedded to thinking of structuralism as really being about principles that are a framework within which we build, if you like, constructive theories in physics. The structural elements that are a priori are not the laws themselves, and they have no necessity, physical or otherwise.

40:00 The framework principles in which we built it have a certain kind of necessity that is fallible, if you like. They're regulated by deals and consents, and this is violating strict Kantian doctrine. Regulating principles cannot be constitutive for Kant, but I think the neo-Kantians were more right about this, like Cassir, and I think it's certainly the way, for example, in which Einstein thinks of general covariance. As a condition on unified field theory, it's a condition that is not to be violated under any circumstance, and that's really where he gets separability in his discussions about bell type, EPR type theory. There is no background structure there without some dynamics. So I don't know if that's an answer to your question or not, but I would put much more emphasis on principle than on law. Thank you. We're glad to talk with you. You understood that the main topic, of course, the main idea of structuralism was relations without any property, intrinsic property to be related, which is really one of the paradoxes of structuralism, but my aim here is to find the historical root of this idea. So, the aim of this talk is twofold. The first aim is essentially historical. It is to show that from his so-called pre-critical periods to his mature critical philosophy, Kant consistently pursued a single line of argument about what he called the ungrounded relations between objects, not grounded on any intrinsic property.

42:30 Kant's permanent claim was that the reason why the relations between objects are not grounded on any monadic properties or essential features is that these objects are pure phenomena, that in other terms the related objects are themselves relative to a certain cognitive process. This crucial idea of the relativity of relations between objects brings me to my second aim. In quantum mechanics, one finds very striking examples of ungrounded or, as we call them, non-supervenient relations. These are the relations of non-supervenientity between states of subsistence. whose paradoxical flavor was conveyed in a well-known sentence of David Mermin. Correlations among differences subsystems have physical reality, but the correlata themselves do not. But following Borean trend of thought, it proves quite easy to account for such an extreme case of ungroundedness of relations. This is precisely due to the relativity of mutually related values of observables to well-defined experimental arrangements. This would be the second part of my book. Some features of the quantum paradigm, especially holism, thus find a natural Kantian interpretation. So, let me first develop Kant's idea about ungrounded relations at length. By strongly insisting on their historical continuity, since they were mainly derived from his reflection about the so-called incongruent counterparts, I'll undertake a short reconstruction of this reflection. But what are these incongruent counterparts? I have to say something about that, I know you know everything, Simon.

45:00 Incongruent counterparts are pairs of geometrical shapes that are perfect mirror images of one another, yet can by no means occupy the original space delineated by the other. Kant's favorite sample of incongruent counterparts were right and left hands. Right and left hands are mirror images of one another. Yet, no global geometrical transformation in three-dimensional space, such as translation or rotation, can bring a right hand to occupy exactly the same region of space delineated by a left hand. In other terms, there is no way of putting a right hand in a left globe. It is currently accepted that incongruent counterparts were used by Kant in three successive arguments whose conclusions are usually said to be very different, but I try to show that they are not so different. Firstly, in a short text of 1768 concerning the ultimate ground of the differentiation of the directions in space, Kant appears to give a purely geometrical The argument in favor of the thesis of absolute space, as opposed to the Leibnizian doctrine of the reducibility of space to a system of relations between moments, the core of the argument can be summarized thus. The relations and positions of the parts of the left hand to each other are exactly identical to the relations and positions of the parts of the right hand to each other. The difference between the left and the right hand, therefore, cannot be reduced to any difference in the internal relations between their parts, or more generally between particular places in space. This irreducibility is illustrated by a dramatic thought experiment, made by Kant, in which God creates only one hand.

47:30 And this hand, according to the Relationists, should be indeterminate with respect to leftness or right. But, says Kant, this is impossible, since if the rest of the human body were later created, the primordial hand would fit either its left or its right arm. Kant's conclusion is that the difference between incongruent counterparts I quote, relates exclusively to absolute and original space. It relies on a relation, I quote again, to universal space as a unity. This is 1768 argument. Secondly, in paragraph 15 of the inaugural dissertation of 1770, Kant denies from the outset that space is something real out there. According to him, so apparently it's contradictory to his former statement about the reality of absolute space, apparently, according to him, the concept of space cannot be abstracted from external sensations since the very possibility of sensations qua external already presupposes space. As a consequence, space can only be subjective and ideal. But which component of subjectivity clarifies this point can insist that the difference between incongruent counterparts cannot be described discursively or reduced to intellectual marks by any mental acuteness, and of course. Therefore, there exist manifest perceptive differences that have no conceptual or verbal equivalence. Kant then concludes that space is grounded in pure intuition and construed as the fundamental form of our sensations rather than in understanding. Thirdly, in paragraph 13 of his Prolegomena to any Future Metaphysics, and also in the first chapter of his Metaphysical Foundations of Natural Science, to which in 1783 and 1786,

50:00 Kant inferred directly from the existence of incongruent quantum parts that special relations do not concern the things in themselves. Indeed, the things in themselves are things, I quote, as the pure understanding would know them. But none of the differences between the partially incongruent quantum parts could be determined by thinking alone. They are dual objects of the understanding and therefore they cannot pertain to the thinking themselves. No wonder that the latter version of the argument of the congruent counterparts is often taken to be the foundation of Kant's transcendental idealism. Besides the striking differences between the three versions of the argument on incongruent counterparts, however, there are very deep similarities that reveal a lot about the roots of transcendental idealism and especially on its insistence on ungrounded relations. To begin with, the structure and latent content If not, the explicit formulation of Kant's later Copernican revolution already appears in the so-called pre-critical text of 1768. As we all know, the structure of the Copernican revolution Which is described in the second preface of the Critique of the Purism, is determined by an answer to the question qui duris, namely by a backward inference from the path of knowledge and its condition of possibility. But this strategy has already been used in 1768. It was already there. I quote Kahn. Absolute space, independently on the existence of all matter, and as itself the ultimate foundation of the possibility of the compound character of matter, here is the conditional possibility, its lack of radical disconnectedness has a reality of its own.

52:30 Is the compound character of math. And the condition of possibility is absolute space. So clearly there is the same type of reasoning as in the later texts of Kant. True, the insistence on the intrinsic reality of absolute space sounds very pre-critical. But actually, even this claim of the reality of space has a reasonably close equivalent in the critique of heuristics. Here is the proof. Space, says Kant, is real. It is. It is objectively valid in regard to everything that we can encounter externally as objects. But space is also ideal in regard to things when reason considers them in themselves. So from this, we can gather that the reality of space or anything else is a word that can still be used in critical context, provided one makes clear that it applies to the immanent domain of objectified phenomena, not to the transcendental domain of things themselves. So even the word real is not to be found in a critical system. If we push the inquiry a little further, we find that even the contents of the critical conception of space was already suggested in the text of 1768. I quote, absolute space is not an object of outer sensation. It is rather a fundamental concept which first of all makes possible all such outer sensations. A fundamental concept that makes possible outer sensations. This is very close to computerism, except for a few data. The only momentary difference between these so-called critical statements and the critical conception is that space is no longer construed as an a priori fundamental concept, but rather as an a priori form of sensibility. This is the main difference.

55:00 The continuity between the two stages of Kant's thought also arises in the opposite direction from pre-criticals. Sorry, from critical to pre-critical texts. The idea of a necessary relation of each special object with universal space as a unity, first formulated in 1768, is still taken for granted in the Prolegomena in 1782. I quote the Prolegomena The internal determination of every space is only possible by the determination of its external relation to the whole space. So here again, Kant refers to the relation of the parts of space to the whole space. Is it so different from the text of the New Testament? The only difference is that, in the critical text, this reference to the whole space is preceded by a statement according to which this space is nothing else than the form of the external intuition. Whereas, conversely, in the pre-critical text, absolute space is referred to before the final remark that space is nothing else than an a priori formed fundamental term. So there is a difference in priority, not a difference in content. And even more striking similarity is found in the metaphysical foundations of natural science of 1786. There Kant is still willing to speak of an absolute space, provided it is made clear that absolute space is nothing by itself. But only our unbounded attitude of conceiving a larger space within which any given region of space is located. So there is a sort of redefinition of that sort of space, of conceiving a larger space within which any given region can be located. These detailed similarities between the three versions of arguments of the Greenbrand counterthought allow us, I think, to consider them as stages of a single reasoning and to use the later versions to illuminate retrospectively the first and most detailed version of 76.

57:30 The crucial concept of relation between objects that are densed as relative to a cognitive background will be clarified by this study of Kant's thought in the long term. Using the later versions of the arguments of incongruent counterparts to illuminate retrospectively the original version of 1768 means understanding that Epistemological or reflective considerations were already crucial from the beginning, even in the so-called critical texts. The usual picture of a strictly dogmatic and ontological Kant, followed by a self-conscious critical Kant, is not to be taken at face value. The reason why Kant was so easily awakened from his dogmatical slumber by Hume It was likely that he was already receptive to Neapolitan space from the outside. To realize this, one just had to pay more attention to the central sections of the text of 1768 on the directions of space, and not only as usual. To the initial sections that assert the reality of absolute space, and the final sections in which the thought experiment of the hand isolated in the universe arises in relation to our own bodies. In these central sections, Kant explains exactly how the spatial concepts arise in relation to our bodies. So our body is taken as a reference. According to him, the knowledge of the position of places, I quote, would be of no use to us unless we could also orientate the things thus ordered by referring them to the sides of our own body. It is only after a long discussion about the crucial role played by our body and its structure in any process of location that Kant feels entitled to claim that, I quote,

1:00:00 the complete determination of the corporeal form depends on the reference of that physical form to universal absolute space. The relation of forms to absolute space is intriguing. It is all the more puzzling that Kant uses a strong logical connective to make the transition. It is also in German that can be translated by then or even by therefore. In order to understand transition between reference to our own body to reference between Between the bodies and absolute space, I propose to use a distinction made by Husserl and then by Merleau-Ponty between our body construed as an object and our body as lived from within. Object body and lived body, or to use Husserl's German vocabulary, Körper and life. In making a quick transition between the relation of things to our body and their relation to universal absolute space, Kant may well have adumbrated another transition from the role of the object body as a basis for reference frames to the constitutive role of the linked body. The object body, with its planes of symmetry, its distinction between right and left, and also its coordinated movements, is the anthropological or naturalized condition of location. Accordingly, the lived body is the transcendental condition of a geometrical space which is too easily reified into absolute space. This transition from an anthropological or natural attitude to a transcendental attitude was by no means foreign to Kearns, of course. He performed it repeatedly in his material work. One celebrated example is once again the metaphorical use he made of Copernic's Astronomy.

1:02:30 In Copernicus' astronomy, apparent motions of planets were ascribed not to themselves in the absolute, but to their displacement relative to human bodies located on the orbiting Earth. Similarly, in Kant's Copernican revolution, appearances in general are ascribed not to things in themselves. But to their relation with the human faculty of knowledge, Copernic's revolution is based on an anthropological relativization, whereas Kant's Copernican revolution consists in a transcendental relativization. The mere fact that Kant explicitly used a strategy of conversion between an anthropological or naturalized epistemology on the one side And the transcendental epistemology on the other side gives ground to my assumption according to which he was undertaking exactly the same kind of conversion, although less self-consciously, in his text on the direction of space of 1768. If I am right, Kant's reference to an absolute space is already in 1768. Very critical and non-Newtonian, the ontological absolute space has in fact been replaced by Kant with a transcendentally absolute space already. The reason why an isolated hand must be left or right according to Kant is not its relation to a substantially absolutized space. It is rather its relation to a functionally absolutized space. Which a transcendental subject posits as a conditional possibility of certain features of experience, such as the compound character of math. After all, the transcendental subject is bound to be present in the background of any one of its thought experiments, including the separated thought experiment of God, creator of a single hand. Here there is already the transcendental subject, and it is this transcendental subject that imposes a functionally absolute state.

1:05:00 To sum up, the effective distinction between two incongruent counterparts, which are not differentiated by any intrinsic property or any intrinsic relational feature, is not explained by their common relation to an ontological absolute space, but by their common relation to a functionally absolute space, presented in advance by the transcendental sub-substances. Which is not grounded on any monadic property or internal relation of these counterparts, holds because of their common relation to a cognitive background. The ungrounded character of their mutual relation is accounted for by their cognitive relativity. In the prolegomena and the critique of purism, Kant made these accounts as explicit as possible. Kant thus presented a sort of converse of Magnet's monadology. His reasoning, in three steps, develops roughly as follows. 1. Relations between things in themselves would necessarily be grounded on their monadic properties. 2. But there exist objects, such as incongruent counter-values, whose neutral relations are not retreated. They are irreducible to anything intrinsic to them. Therefore, these objects are not things in themselves. They are appearances to a transcendental subject and they are therefore relative to it. In other terms, the irreducibility or ungroundedness of relations between objects is taken by Kant as the proof of the relational character of knowledge. The idea is all pervasive in the critique of purism, where it is generalized and formulated in two opposite directions, from the ungroundedness of the relation of an object to the relativity of knowledge, and from the relativity of knowledge to the ungroundedness of mutual relations of objects.

1:07:30 In the transcendental aesthetic, the first direction of reasoning is briefly sketched. I quote, since through autosense we are given nothing but mere relational presentations, autosense can, by the token, contain in its presentation only the relation of an object to the such. But later, in the amphibology of concepts of reflection and appendix of the critique of pure reason, the second direction of reasoning is also developed by Kant. There, Kant takes as a premise that those things that are objects of our knowledge, I quote, are not things in themselves, but are merely appearances, end of quote. And he then infers that the determinations of these things, I quote, Express mere relations without being based on anything intrinsic. In other terms, Kant's deduction runs thus. The things related are relative to a cognitive act. Therefore, their mutual relations are not grounded on any intrinsic monadic property. This is a real deduction. Kant finally summarizes his ideas by declaring that structuralism can only be intelligible in a transcendental version. This is very polemical. His remark is as follows. It is startling to be sure to hear that a thing is supposed to consist altogether of relations. Such a thing, however, also is mere appearance. As I mentioned in the introduction, it seems to me that Kant's idea can easily be applied to quantum non-separability, provided two alterations are made to Kant's theory. The first alteration only concerns the vocabulary. Instead of ungrounded relations, I will refer, as it is now unusual, to non-supervenient relations. to which relations that do not supervene on the basis of monadic properties. The second alteration is more important, however. It consists in replacing Kant's components of the human faculty of knowledge, such as sensibility, with experimental devices.

1:10:00 So I could develop on that, but I'll skip this point. The archetype, so let me examine the case of quantum non-supervenence relations in this period. The archetype of such relations concerns a pair of EPR-related particles for which the mutual distance and the total momentum are well defined, but neither the individual positions nor the individual momenta are defined. This non-supervenience of relational determinations on monadic determinations can be interpreted in three ways. One way is ontological, second way is subjectivist, third way is transonomic. The ontological interpretation of linear correlations consists in ascribing global properties to the compound system itself. This is certainly the most popular conception since it favors grand metaphysical speculations about the holistic nature of the universe. The subjectivist interpretation amounts to asserting, along with Einstein himself, that the lack of determination of individual positions and momenta are in fact due to our imperfect knowledge of them, and that therefore quantum mechanics is incomplete. This is what Einstein had in mind. It ascribes non-supervenient or non-supervenient relatedness neither to nature as such nor to our missing knowledge of nature, rather non-supervenient relatedness is ascribed to the fact that the related features are pure phenomena in both sense, that they are only defined relative to a certain experimental arrangement whose detailed Functioning is pushed in the back. The idea here is that in many cases local and global observables are mutually exclusive. Therefore, maximal determination of global features, such as distance or total momentum,

1:12:30 is exclusive of maximal determination of local features, such as individual positions and momentum. Hence, non-supervenience. Non-supervenience or relational features or monadic features is due to the fact that relational features are defined relative to experimental contexts that exclude the experimental contexts relative to which monadic features are defined. This is non-ontological supervenience, very similar to the non-ontological ungroundedness of Kant's relation between incongruent counterparts. Now, this Borean transcendental understanding of non-superveniency is well known, and one must then wonder why it is so often found unconvincing. My feeling is that it is only due to some widespread misunderstandings of both celebrated cryptic formulations. Let me just give an example of such misunderstandings. Michael Esper, an advocate of a form of ontological holism, rejects both solutions. Because he thinks that it is tantamount to ascribing the holistic features of quantum systems to the holistic features of experimental devices. If this were true, that would just mean pushing the problem of holism under the carpet of experimental apparatuses. All triggering some infinite regress. There is holism of objects because there is holism of apparatuses, and there is a holism of apparatuses because there is holism of other apparatuses, and so on and so forth. But Bohr could not have meant such a simplistic denial of a problem. What he rather meant was that the type of information to be drawn from certain experimental devices is by their very arrangement of a global nature. It's not the apparatus which is global or holistic, it's a type of information we wish to draw from them, which is very different.

1:15:00 The holistic features do not pertain to the experimental devices who are material entities. but to the experimental devices who are conditions for generating relevant pieces of information. In other terms, they are not connected to the naturalized aspects of the apparatuses, but to their transcendental aspects. But this is much easier to understand with due reference to Kant, of course, than in isolation. And this is probably the reason why most people didn't really understand what Paul had in mind. In the context of this workshop on structuralism and physics, my aim was just to illustrate that it is by far more natural to hold a transcendental structuralism than a transcendental structuralist position. The idea that structure is all there is, or that things should be replaced by positions in an abstract network of relations, is to be rephrased thus, here is my rephrase, structure is all that appears, one, and two, objects qua phenomena are positions in an abstract network of law-like relations provided in advance by the cognitive process. If this doesn't favour any subjectivist view, if one doesn't forget Kant's warning that representing something beyond phenomena is just pointless. Thank you. It seems to me there is something to be said about that. One sees it already in the Nobel visitation. That's my thought. What we do say is that it did change. Well, I think I pointed out some of the changes. The first one, for instance, was the change between the idea of space as a fundamental concept, which is a fundamental probability for perceiving the material being as old, and the idea of space as

1:17:30 Part of our intuition, or our sensible intuition. This is the difference. And this is one of the points on which Kant exists very much in the editorial dissertation. It's to say, oh well, I cannot communicate about the differences between these counterparts. So there is nothing conceptual in this difference. Therefore, space is only part of my sensible conclusion. I actually agree with that. It seems to me that it's very clear in the organization that Trump feels himself to be saying something revolutionary, that we must grant the censor basis to previously thought to be pure understanding, and we must grant it because there are cases where we simply can't explain where in my eyes it is taking place. But we see it nonetheless. So there's something very straightforward about that, because then that in turn suggests that Kant didn't have the 68, satisfactorily, and I think, in many ways, he tried to explain where it lay. So, yeah, and then I'm puzzled, because, you know, I agree that, of course, there are differences, it's very clear there are differences, but these differences are not exactly where people place them. They are not in sort of an active sense into the dogmatic and ontological camps and the critical, self-reflective and epistemological camps of the world. That's what I insist about. Now, of course, you're right, there are differences, many differences. And also there are differences on the order of arguments, there are differences on the points he insists on. He doesn't insist very much on this idea of space as a condition of possibility, whereas afterwards it's something which takes a very large importance, and so on and so on.

1:20:00 But most of the ideas are already there. And even this splendid idea of going from the naturalized or anthropological version to a transcendental position of our body is already there and it's completely taken over in the later chapters about the continuity and discontinuity in the history of science. In your discussion of non-separability, well, you say you have sort of your own interpretation of non-separability, and you infer, even if negatively only, to the material structures of the field structure. You say it's not the field structure. The material structure of the instrument is not involved, but by referring, even if negatively, on the material structure of the instrument, you sort of seem to imply that this notion is painful, and then you may have difficulty in explaining in what sense you I understand the word matter and so isn't it, my question is, isn't it more satisfactory to think, they say, non-separability is essentially negative. We find that they are too naive and that they do not work. Even though, even if they look extremely natural, such as the two ones that you mentioned, the ontological one and the Einstein one, well, they don't work. So, that's negative.

1:22:30 Interpretation of non-savoir-habilité is, I mean, a safe position. I understand that, yes. Yes, I think you are right. The problem is that we have sometimes, in order to understand in a very clear and... And very easy to represent way, the idea of relativity of attributes or phenomena to a certain something, experimental device or context or anything, we have to make this a little bit concrete and say, oh, this is a material structure and so on, but in fact, the proper way to be able to explain these things would not be to... I don't want to make explicit the natural structure of the apparatus, but just say that there are phenomena which are contextual, not relative, but contextual. What does it mean? It means that we don't really represent the things with which phenomena are in relation, but we see that in fact we cannot These give meaning to these phenomena without referring to some elements of our direct macroscopical surroundings, for instance the preparations we did and the type of experiment with which we perform and so on and so on. And then say that these phenomena display the feature of non-separability. The problem is that at the same time we want to give a sort of naturalized picture of contextuality and at the same time we also wish to give a completely transcendental description which would completely eliminate Any reference to the apparatus and focus our attention only on the structure of the phenomena themselves, namely, here, their non-separate rapidity, which is, as you say, a purely negative feature of our fair findings.

1:25:00 So I know that there is always a sort of tension here, but it's a tension which is useful for pedagogical... And I'm sorry if there are some confusions about that. Yes, I would like to come back to your conclusion. That is, you're replacing the second, replacing the science. I don't think that's mathematical because I don't remember exactly what it is. Confusion, I think, yeah. So that's not the reason. That is the position, the structure of the phenomena. Just appearance. Just appearance. And to distinguish it, it's pink on itself. This was the left. Yes, the idea is that the reason why the relation between phenomena is ungrounded on any non-analytic property is that these phenomena are phenomena and that they are relative to our background. The idea is that... The grounded relations between the two incongruent counterparts was then transposed to the idea of non-separability and to the non-supervenience of relations in quantum mechanics, so it's the whole idea. To pose my question, there's something which has to do with seeing objects as position in the sides, in a network, so you mean that what is an appearance phenomena can be seen as a position in a network. That's right. So this was my question. So what do you mean exactly with appearance? Is that something we need to know?

1:27:30 Yes, it could be something in the future. Of course, you can't say it's phenomena in the first place in a very anthropological way, but we can transpose it, of course, from sensibility to artificial sensibility, say, and it's going through. So just, perhaps I don't understand you right. This is the reason for my question. For example, take bosons, identical parts, but which cannot be distinguished as simultaneously external. And you don't observe a difference, right? But, so, what could be the position that if they have the same position, they are? They have the same position. But when you measure one integral particle, this is what the phenomena does here. So I just don't understand very well how you accommodate this situation. Oh, the situation of the particles? I mean, you measure an integral, you don't measure identical two particles which are not distinguished. So if this is your theorem, how can you see that it's defined on the basis of the position in it? So perhaps I have misled you and this is the reason why I'm asking. Yeah, well, I think you have to make the difference between local and global observables even here. You can have global observables bearing on the pair of bosons and local observables bearing on each one. And there is, usually there is an incompatibility between the global and local observables. So if you know everything about global observables, about the pair of protons, then you don't know everything about the local observables. So this is why, you know, so the... The phenomenon of relation is incompatible with the phenomenon of individual properties.

1:30:00 This is why I say that phenomena can have this type of ungrounded relation. Ungrounded relation is when you have a global phenomenon for the two. Well, we can discuss it in detail, but I think it's... Yes, probably I had misunderstood what you were saying. Yeah, because here you were referring to the local observable. Of course, when you have local observables, you can say, well, this one has this position and this one has another position. But now they are not in this relation of non-separability. They are in the relation of non-separability when there is an observable. A global observable, which takes on the two together. For instance, here in the EPR case, it's very clear, because you have a global observable that gives the distance or the total momentum. In this case, you have relation without relation. Because to characterize the real attack, you would need to make additional individual observations which are not available in that sort of complementarity. Let's thank Pete for the day. The afternoon session starts. So the speakers, of course, are invited to the lunch, and some of those are also invited. I guess I won't be here, I went to nature, so... Yeah, okay. So, where do you go? If I don't get a chance to see you, I'll see you. What do you do for lunch? No, no, no, I don't know. I haven't, I don't know, basically. Would you like to come with me?

1:32:30 Or have you got the invite? I have got the invite to go with them. I'm assuming. Oh, thank you. There's an excellent place just across the road. Which may well be where they're going anyway. Okay, just give me two minutes. It's in the building. It's a really good little place just over there. Thank you.