Structuralism & the a priori

0:00 I've extended the break, and so we can all be off at the back, you see. But, well, it's nice to be able to introduce Dr. Eichmann, who will talk about structuralism as a priori. Thank you, Guido. It's a great pleasure to be here. I'd like to thank Michelle for coordinating, organizing, funding this illustrious event. And the CREA for supporting it. Many of you in the audience have my own thoughts about structuralism over the years, and particularly the young lady who has really inspired me to think about structuralism in a serious way. Happy birthday to her. So this is the agenda. I co-title here is Lessons from Hermann Weyl. I can make it a little larger. Now, click the minus, smaller steps here. How's that? Perfect. Well, James has set himself nicely on why we are now talking about structuralism. Structuralism is an argument, or it's a response to the No Miracles argument,

2:30 I would like to distinguish between three different varieties of structuralism. Transcendental is not structuralism. Transcendentalism is my own idealist version of structuralism. Instrumental, which I would associate with thoughts, and transcendent, that would be realism, 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 have from Voss and Hermann Weyl. And I want to say a little bit about Weyl's own program of structuralism, which I see as, a priori, constructive cognition in physics. How Lyle really discovered how to engage 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. Actually, the bulk of the talk is going to be on point four, against eliminated structuralism, since James has now surrendered that vice, that will be targeted on Stephen French, who is not here. 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. Yeah, I can try that again. Is that okay? Is that readable? Well, I would start with Kant, and the basic idea is that, as he put it, the order and regularity of appearances that we call nature, we introduce ourselves, and indeed we could not find them there had not we, or the nature of our mind, put them there originally.

5:00 I find similar echoes of that particular point of view in Henri Poincaré, writing in 1906, does the harmony between all the things he 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. 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's 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... It's modal ontological, modal structure. My hero, Hermann Weyl, writing in 1934, the structure of our scientific knowledge is conditioned by the circumstance that in the world, the goal of our scientific endeavors is not one that exists in itself but arises strongly 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 external world. It's just a world that we talk about in science. It's a world that arises from its own dynamic meaning and subject matter. And another hero of mine, a little more controversial, Arthur Eddington. Beautiful expression. In the end, what we comprehend about the university... It's precisely what we put into the universe to make it comprehensible, and that is just exactly, I think, what Kant said in his first quotation.

7:30 So, structuralism without the universe. I am taking some passages here from Voss's paper, Structured Perspective, that appeared in 1997. James has laid out what the semantical approach to scientific theories is, is the idea that a scientific theory is a family of models, as we'll see. For Baas, this view is now at once structuralist. By common consent, the semantic approach is the current general form, the current form of the general idea of structuralism. To present a family of models, that is, mathematical structures, offered for the representation of the various subject matter. Within mathematics, isomorphic objects are not relevantly different, so it is especially appropriate to review 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 structuralist position. Science's description of the subject matter is solely a structure and just a little bit more text. Properly understood, it is entirely correct to say that models represent nature only out of the 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 like for a non-realist in Voss's constructive empiricism or structural empiricism, and as Voss says to himself, the basic conception of the semantic approach was already stated in 1926 by Hermann Lyle in his wonderful book on philosophy and mathematics and the natural sciences. The science can determine its domain only up to an isomorphic mapping. In particular, it remains quite indifferent as to the essence, German word here is basing, of its object.

10:00 The idea of isomorphism demarcates the self-evident insurmountable boundary of cognition, the vial is self-evident. Another fundamental aspect of Biles' epistemology of exact science of mathematics and physics, really, is not emphasized by Biles, is Biles' transcendental idealism. Here is again a passage from the book on philosophy and mathematics and natural science. But posed as a problem, and that it cannot be constructed absolutely, but only in relation to an arbitrarily assumed coordinate system and their symbols. Now that particular locution is the language of Kant from the transcendental dialectic. This is essential in Kant's resolution of the so-called mathematical antinomies in the transcendental dialectic. In particular, Kant wants to say that totalities of which we think a condition part of given, the totalities themselves are not given as totalities, not essential to the resolution of the antinomies. So let's look at structuralism without realism now. According to Hermann Weyl, just some passports again. From a late paper of Weyl, a very interesting paper which unfortunately has never been translated into English, Science of 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 spatio-temporal material being, we are left only with a construction in pure symbols. Now that seems very mysterious, but we're just going to see if we can figure out what that might actually be saying. And there's one more thing about Biles' view of constructive cognition.

12:30 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 outside star, but in fact extending far beyond these closely founded. How far one should go here, including the nascent shell of Fussell's phenomenology, I prefer to leave in the dark. Understanding and expression, thinking of possible and destruction of symbols and formulas on the mathematical side. 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 Tirol Pusrol. It's language taken precisely from Pusrol. Pusrol does not use the term alveolar, meaning something like ostensible or evident. He uses other terms. That term is taken up by some papers of Kurt Riedemeister that Biles has been reading in 1954, and 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 active scene. And let's just take a look at what this Symbolic cognition is the key file. 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 destructive 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. This is a mind-independent reality. Even when the manipulations needed for the determination were not carried out, and by all things to be found, we're confronted with a fundamental limitation to this realist point of view.

15:00 And here's a thesis that, just to boil down these quotations from Weyl, I want to think of structure as the result of net possibilities into which the actual world, the experienced world, is embedded and posed as a problem within that net possibilities in order to describe it as the actual world. And it's built up by symbolic construction. In a 10-year period, one brief decade, Bile gives two astonishing examples of such a priori construction in physics that still yields 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. Which Weil himself carried over to Dirac's theory of the electron in 1928, where it now becomes a phase invariant. And the idea here is that Weil, in his theory of gravitation and electromagnetism, Weil 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 the 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, we have some formally unified theory of gravitation and electromagnetism, which is simply a theory of the space-time manifold. No additional fields are added into that manifold. Gravitation and electromagnetism are simply there, although he does not show that they can have a fundamental interaction.

17:30 The second attempt, about which I am beginning some work on right now, is Biles' attempt to derive fundamental relationships in quantum mechanics from group theoretical symmetry principles. And here I found the writings of Harvard mathematician George Mackey very helpful. He calls it Biles' 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 can be integrated for a filed form. On the basis of the Fios conjecture of the general correspondence between self and joint operators and unitary representations of a non-compact robotic lead group, the additive group is the real line. And crucial here is the step from finite to infinite groups that gives you the integrated form of the Heisenberg commutation relations. Again, this is 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 instruction or structuralism in physics. Well, I wanted to spend the bulk of the talk, the rest of the talk anyway, on what is the limited structuralist analysis. It has a certain expression in currency and 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.

20:00 In philosophy of physics, Eliminated structuralism is really the position of ontic structuralism, which I will now associate with Stephen French, as of this morning, and not James Ladyman. Reference to fundamental physical objects is to be understood as two objects reconceptualized entirely in terms of the mind-independent world structure. And this is basically the view that James laid out. Or was the view for James, anyway, that there are no such thing as individual properties, all you have, really, is structure without related, you have relations without related. And the classic example there is the metaphysics of non-relative duality of quantum objects. I want to look more at the problems with mathematical structure. Citing here Charles Parsons, in mathematics no one has shown how eliminated structures 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. And talk about the natural numbers is about any system of objects and relations that satisfy the definition. Here's the explicit definition, Dayton defines a simply infinite system as a system or a set end such that there's a distinguished element and a mapping Carrying n into n minus a term, 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. 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, n, o, s.

22:30 Now, 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 derivatives. So that we could write some statement of arithmetic or abbreviate it as A-N-O-S. And then understood it's about the natural numbers N-O-S of variables, so that that statement is elliptical for this one. So for any N-O-S, right, if the explicit definition of a simply infinite system, then the statement in its canonical form. But of course this doesn't do the job, because if there are no simply infinite systems, then it's factless and true. And if that's the case, also then if A is not a true canonical form, you would have an inconsistent arithmetic. The Vatican has to prove the existence of simply infinite systems. And how does he attempt to do this? He famously argues, in paragraph 66, 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. And this will itself be a new 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 fatality. 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 a limiting structure.

25:00 This is just one, it's a key example, but just one example of the failure of eliminated structuralism. We can turn to Parsons to discuss this. In detail, the shortcomings of other eliminated structural proposals, including structural conceptions and set theories. We could think of Hilbert's axiomatization of Euclidean geometry, which is often taken to be a kind of purely formal structuralist view. But it omits the fact that Hilbert begins his Foundations of Geometry with the Kantian statement that knowledge begins and intuition proceeds. From there to concepts, and then on to ideas, or as Hilbert called them, axioms. Another failure of eliminated structuralism, more familiar perhaps to certain philosophers, would be the failure in Carnet's outbell, where Carnet... This 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. Well, Parsons' own diagnosis, I think, is insightful of why the purely structured account doesn't seem appropriate. For the most elementary mathematical objects, like the natural numbers, these are, in some sense, pseudo-Guinier's abstract objects that are quasi-concrete in parts of the term, types determined by intrinsic relations to concrete objects, tokens. Concrete objects are, if you like, are intuition. Here we can think of Pilgrim's string of strokes or even just... Formal expressions in a kind of syntactical arithmetic. The concrete representation is something about the object that goes beyond the formal structure. For example, being an infinite sequence that they instantiate.

27:30 Moreover, the concrete representation is essential to the intuitability of the object. In Husserlian 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 paramoxy of an epistemological notion. How are the most basic abstract objects known to us? 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. You'll have to say that the natural numbers are not mathematical objects because they 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 termed 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 Hilbertian Scientism. But there are also those of Weyl's symbolic construction that begins with objects that are turned outside of us, that are, in some sense, livid to our intuition, that we see clearly and understand clearly.

30:00 I would say that Weyl's examples of such obfuscation physics are what Hilbert called a deepening of the foundations of human matter. Do the failures of eliminated structuralism in mathematics, one can ask, bear upon ontic structuralism in physics? And it might be argued that it doesn't. Rather, articulation of eliminated structuralism requires a revamped metaphysics, not mathematics. That is expressed by Stephen French and James Ladyman in their paper, Remodeling Structural Realism, that appeared 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. 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 a linear dictionary. I suppose you could call it recolage 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 for mathematics, because mathematics itself has this classical framework of individual objects and so forth and predications. We need to turn to a new kind of metaphysics. That will enable us to carry through completely the program of structuralism, eliminating structuralism of relations without relata.

32:30 And could such a metaphysics therefore provide the means of carrying out eliminated structuralism in physics? I think it's possible. I think anything is 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 systemologically and genealogically these structures are the bootstrap for more elementary domains of objects that are not pure structures. In other words, we can tell a plausible story about how we have... All of these 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 these non-standard structures of a revamped medical nation? Well, it couldn't be isomorphism because that would be having the same structure. And then just in conclusion, I would say that it's a transcendental dialectical diagnosis of eliminated structuralism. Structures are not given, they're not postcards from eternity, but are constructive or posed as a problem from the given. Nischke gave them, Zolder probably gave them. Thank you. We have time for some time for questions. One, two, three. My question is, in the beginning you mentioned you related structures to isomorphism, it's no way we think about structures something defined after an isomorphism, but I just wonder, I'm thinking about category theory, where you basically regard other morphisms as isomorphism, and I just wonder how it bears upon the notion of structure in particular to your account.

35:00 We just try to generalize it to the fact that not only isomorphism. And one thing that I'm thinking, saying, they have to mention that this quasi-concrete is not a structural property, but probably if we do this more generally. Understanding of what is structural, it becomes a structural property, because if you think about some representation theorems, like carry theorems for groups or store theorems for Boolean algebra or Yoneta lemma. I don't know in which set, but it seems to be that a sort of structural issue was involved. Yeah, well, with regard to the last question, that's a 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 natural numbers before you can prove any representation theorems about simple structures. I think that's just a matter of technological 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. 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 prior act. It is extremely empirical, because if things were to change, if we replaced them a few kilometers, we would never even consider proving such a theorem, right?

37:30 In a sense it's a question of how much with how little. Translation of symmetry is an obvious symmetry to impose, and then what can you get out of it? It's quite remarkable that that bio does give you that. Yes, just a little question. I would like to understand how you make a difference between... There is a vast dispersion of structurism that you call instrumental structurism and the transcendental version of structurism. In the things that you projected, I didn't see a very clear difference between the two. For instance, at the end, you say, well, at the end, structure is only constructed, not given. But I think in order to differentiate clearly between the two versions of search phrase, you have to add something, the condition to be added, I think, is the constitutive power of structure in transcendental ideas. That I thought was implicit in what I said. I agree with this 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. The column with Pompey-Figuelism about physics goes beyond the ones that you outlined for mathematics, because in mathematics the relational structures are always there and all of these are necessary in some sense, but, you know, just... If I can quote James, there's nothing more to the objects than the relational nexus in which they are located.

40:00 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, do these relational nexus have any contingent properties? Under what conditions does it exist or not exist? And in the case of mathematical structures, that would be inappropriate questions. Whereas here, it seems to me that trying to answer those questions you seem to get other structural questions. Let me see if I can grasp your point. So the question is, if 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 pneumological or something. Still, it's not necessary in mathematics. Well, here there's got to be some kind of account of physical law. And rather than... You get bogged down in laws and particular theories. I am much more wedded to thinking of structuralism really being about principles that are a framework within which we build, if you like, constructive theories. The structural elements that are a priori are not the laws themselves and they have no necessity, physical or otherwise, but the principles that we hold, the framework principles that we build. They 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 constituted by a concept. I think the neo-Kantians were more right about this, I think, here. And I think it's certainly the way, for example, in which Einstein thinks of general covariance as a condition on a unified field view. It's a condition that is not to be violated under any circumstance, and that's really where it gives susceptibility in its discussions about bell types, EPR types, if there is no background structure there without some dynamics.

42:30 So I don't know if that's an answer to your question or not, but I would put much more emphasis on principles than on law. So you understood that the main topic, of course, the main idea of structurism 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 it. The aim of this talk is twofold. The first aim is essentially historical. It is to show that from his so-called pre-critical period 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 essential features, 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

45:00 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-separability between states of subsystems, whose paradoxical flavor was conveyed in a well-known sentence of Amy Mermin. Correlations among differences subsystems have physical reality, but the correlators themselves do not. But following a boring 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 arrays. Some features of the quantum paradigm, especially holism, To thus find the 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, so... Incongruent counterparts are pairs of geometrical shapes that are perfect mirror images of one another yet can by no means occupy the regional 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...

47:30 A right hand will occupy exactly the same region of space delineated by the left hand. In other terms, there is no way of putting a right hand in a left slope. 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 will 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 of space, Kant appears to give a purely geometrical 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, nor 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. Only one hand is created in the universe. Actually, the universe consists only in one hand. And this hand, according to the relationist, should be indeterminate with respect to leftness or right. If the rest of the human body were later created, the primordial hand would fit either his left or his right arm. Kant's conclusion is that the difference between incongruent counterparts, I quote,

50:00 relates exclusively to absolute and original space. It relies on a relation, I quote again, to universal space as a unit. This is 1768 art. Secondly, 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 forward 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 externa already presupposes space. As a consequence, space can only be subjective and either, as he says. But which component of subjectivity is it for? To clarify this point, Kant insists that the difference between incongruent counterparts Cannot be described discursively or reduced to intellectual marks by any mental acumen, and so on. Therefore, there exist manifest perceptive differences that have no conceptual or level of equity. Kant then concludes that space is grounded in pure intuition. He construed as the fundamental form of our sensations rather than in understanding. Thirdly, in paragraph 13 of his Prolegomena to any Future of Metaphysics, and also in the first chapter of his Metaphysical Foundations of Natural Science, to which in 1783 and 1786, Kant inferred directly from the existence of incongruent quantum paths that spatial 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, especially incongruent counterparts could be determined by thinking alone.

52:30 They are no object of the understanding, and therefore they cannot pertain to the things in themselves. No wonder that the latter version of the argument of the incongruent counterparts is often taken to be the foundation of Kant's transcendental ideas. 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 Pure Reason, is determined by an answer to the question qui duris, namely by a backward inference from the fact of knowledge that its conditions are possible. But this strategy has already been used in 1768. It was already used. I quote Schumann. Absolute space, independently on the existence of all matter, and as it sets the ultimate foundation of the possibility of the compound character of matter, here is the condition, its lack of radical disconnectedness, a reality of its own. And of course, the fact is, the fact here, in its recipe, Is the compound character of math. And the conditional possibility is absolute space. So clearly there is the same type of reasoning as the later texts. True, the insistence on the intrinsic reality of absolute space sounds very pre-critical.

55:00 But actually even this claim of the reality of space has a reasonably close equivalent in the critique of heuristics. Here is a quote, space, says Kant, is real, it is, 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 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 the concept. So, even the word real is not to be banned in the critical context. 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. Possible outer sensations. This is very close to the critical theorism, except for a few data. The only momentary difference between this so-called critical statement 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. The continuity between the two stages of Kant's thought also arises in the opposite direction from critical to precritical text. The idea of the 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 1780.

57:30 The internal determination of every space is only possible by the determination of its external relation to the whole space. 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 precritical text, absolute space is referred to before the final remark that space is nothing else than an a priori formed fundamental concept. So there is a difference in priority, not a difference in complexity. And even more striking similarities 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 aptitude of conceiving. A larger space within which any given region of space is located. There is a sort of redefinition of that, a larger space with a greater location. These detailed similarities between the three versions of the argument of the current counterpart 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 relativity. The crucial concept of relation between objects that are dead cells 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.

1:00:00 This 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 count, followed by a self-conscious critical count, is not to be taken at face value. The reason why Kant was so easily awakened from his dogmatical slumber by Hume was likely that he was already receptive to every system from the outside. To realize this, one just has to pay more attention to the central section 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 body. In these central sections, Kant explains exactly how the spatial concept arises in relation to our body. To our body 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 can fill the entitled claim that, I quote, the completed determination of the corporal form depends on the reference of that physical form to universal absolute space. The brutal transition from the relation of forms to our human body to the relation of forms to absolute space is intriguing.

1:02:30 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 the then or even by the therefore. In order to understand transition between reference to our own body, to reference 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 in Husserl's German vocabulary, Körper and Life. In making a quick transition between the relation of things to our body Absolute space. Kant may well have elaborated 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 naturalist. These are all naturalized conditions of location. Accordingly, the lived body is the transcendental condition of a geometrical space which is too easily reified into one. This transition from an anthropological or natural attitude to a transcendental attitude was by no means foreign to Kant. He performed it repeatedly in his natural work. One celebrated example is once again the metaphorical use he made of Copernic's astronomy. In Copernic's 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.

1:05:00 Similarly, in Kant's Copernican revolution, appearances in general are ascribed not to things themselves, but to their relation with the human faculty as known. 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 the 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-eutonic. 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 matter. After all, the transcendental subject is bound to be present in the background of any one of its thought experiments, including the celebrated thought experiment of God, creator of the single hand. Here, there is already the transcendental subject, and it is this transcendental subject that imposes a functionally absolute space. 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-subjects.

1:07:30 Which is not grounded on any monadic property or internal relation of these states, counted for, holds because of their common relation to a cognitive background. The ungrounded character of the mutual relation is accounted for by their cognitive relativity. In the Prolegonena and the Critique of Pure Reason, Kant made these accounts as explicit as possible. Cantor thus presented a sort of converse of Leibniz's methodology. His reasoning, in three steps, develops from broadly as follows. 1. Relations between things themselves would necessarily be grounded on their monadic properties. 2. But there exist objects, such as quantum reality, whose mutual relations are not retrieved. 3. Therefore, these objects are not things in themselves. They are appearances to a transcendental subject and they are therefore related to it. In other terms, the irreducibility or ungroundedness of relations between objects is taken by Kant as a proof of the relational character of knowledge. The idea is all persuasive 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. In the transcendental aesthetic, the first direction of reasoning is briefly sketched, I quote, since through outer sense we are given nothing but mere relational presentations,

1:10:00 outer sense can, by the same token, contain in its presentation only the relation of an object to the subject. But later, in the anthropology of concepts of reflection and aesthetics of the critique of pure reason, the second direction of reasoning is also developed at Kant. There, Kant takes as a premise that though 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 determination of these things, I quote, Express mere relations without being based on anything in prison. 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 country's original deduction. Kant finally summarizes his ideas. By declaring that structuralism can only be intelligible in a transcendental version, his remark is as follows. It is startling to be sure to hear that the 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. 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 modality progress. 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. So I could develop on that, but I have skipped this point.

1:12:30 So let me examine the case of quantum non-supervillain relations in this period. The archetype of such relations Concerns a pair of EPR-correlated particles for which the mutual distance and the total momentum are well defined, but neither the individual positions nor the individual momentum are defined. This non-supervenience of relational determinations on monadic determinations can be interpreted in three ways. Third way is transcendent. The ontological interpretation of EPR correlations consists in ascribing global properties to the compound system itself. This is certainly the most popular misconception since it favors ground metaphysical speculations about the holistic nature of the universe. This kind of interpretation amounts to asserting, along with Einstein himself, that the lack of determination of individual positions and momenta are, in fact, due to an imperfect knowledge of them, and that, therefore, quantum mechanics is imperfect, as Einstein had defined. As for the transcendental or Boreal interpretation, 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 detail 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, is exclusive of maximal determination of local features, such as individual positions and momentum.

1:15:00 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 relations between incomparably countless. Now, this boolean transcendental understanding of non-severability is well known, and one must then wonder why it is so often found and something. My feeling is that it is only due to some widespread misunderstandings of both celebrated prefect formulations. Let me just give an example of such misunderstandings. Michael Eister, 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 the problem. What he rather meant was that the type of information to be drawn from certain experiments does divide. is by their very arrangement of a global nature. It's not the apparatus which is global or holistic, it's the type of information we wish to draw from them, which is very different. The holistic features do not pertain to the experimental devices who are material entities.

1:17:30 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 of us didn't really understand what Bohr 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, object square phenomena are positions in an abstract network of law-like relations provided in essence by the cognitive process. This doesn't favor any subjectivity. If one doesn't forget Kant's warning that representing something beyond phenomena is just nonsense. Thank you. I take a lot of what you say about similarities between critical and pre-critical concepts, so I just wondered what, what I'd like to clearly say, what did you, it seems to me there is something. 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 perceiving material thing as a complete whole.

1:20:00 And the idea of space as part of our institutions or our sensible position. And this is one of the points on which Kant exists very much in the current consultation. It's to say, oh well, I cannot communicate about the difference between these two counterparts. So there is nothing conceptual in this difference. Therefore, it is only part of my sense of the word. Therefore, space is only part of my sense of the word. I agree with that. It seems to me that it's very difficult for the presentation to come. He feels himself saying something revolutionary. It's wrong. He's such a racist. So there's something very straightforward about that, but that in turn suggests that Calvin didn't think the 68 was satisfactory. And I think in many ways he tried to explain where the difference lay. Yes, I'm very 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 their attention. They are not in this sort of absolute gem between the dogmatic and ontological camp and the critical and self-reflective and epistemological camp and so on. That's what I don't like. Now of course you're right, there are differences, maybe. 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 states a very large importance, and so on and so on.

1:22:30 But most of the ideas are already there, and even this splendid idea of going from the naturalized or anthropological version to transcendental position of our body is already there, and it's completely taken over in the later campus, later in the program, in continuity. In your discussion of non-separability, you say you have sort of your own interpretation of non-separability, and you refer, even if negatively only, to the material structures of these, say, these networks. But by referring, even if negatively, to the material structure of the instrument, you sort of seem to deny that the distortion is real. And then you may have the difficulty in explaining in what sense you... I don't understand the word matter or such things. So, isn't it, my question is, isn't it more satisfactory to think, they say, that separability is essentially negative? Explanations. We find that they are too naive and that they don't work. Even though, even if they look the same way, it's all such that the two ones that you mentioned, the ontological one and the Einstein one, well, they don't work.

1:25:00 I understand that, yes, I think you are right. The problem is that we have sometimes, in order to understand in a very clear 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 explain this is not to make explicit the material 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 thing with which this phenomenon is in relation, but we see that in fact we cannot give meaning to this phenomena without referring to some elements of our direct macroscopical surroundings, for instance the preparations we did. And the type of experiments we perform and so on and so on. And then say that these phenomena display the feature of non-separable reality. 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

1:27:30 which would completely eliminate Any reference to the apparatus and focus our attention only on the structure of the phenomena themselves, namely here the non-separability which is, as you say, a purely negative feature of our findings. 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 confusion. There is the position in the structural epistemology of the fifth tree, Gephthia, and distinguishes the fifth on the left. This was the last. Yes, the idea is that the reason why the... The relation between phenomena is ungrounded on any genetic property. It's that these phenomena are phenomena and that they are relative to our cognitive background. And the idea of Kant about this type of non-grounded relationship between phenomena... In contrast, counterpart was then transposed to the idea of non-supervenience and the non-supervenience relation in quantum mechanics, the whole idea. Well, yes, perhaps I don't remember now exactly the quantum mechanics, but something which has to do with clean objects, object exposition into the site. So you mean that what is an appearance phenomena can be seen as a position in a netball?

1:30:00 That's right. So this was my question. I can make a question. So what do you mean exactly with appearance? Is it something we need? Yes, it could be something we need. Of course, in Gantt it's phenomena in the process. ...very anthropological way, but we can transpose it, of course, from sensibility to artificial sensibility, say, in this space. So just, perhaps I don't understand what you are deducing very good for my question. For example, things both are identical, but which cannot be distinguished from one another. You don't observe it, but so what would be the position that if they have the same position they are? They have the same position. And that when you move, you either want to move or not, because this is what the phenomenon appears. So I just don't understand how you accommodate that. The situation is relative. I mean, you measure any of these, right? You don't measure identities, or two-parties, or other things. So, if this is the appearance, how can you see that there is sort of a kindness on the basis of a conditioning, or... So perhaps I could listen to you and give you some information on it. Yeah, well... I think you have the metric between local and global observables given here. You can have a global observable bearing on the pair of bottles, and local observables bearing on each one. And there is, and usually there is an incompatibility between the global and local observables. So if you know everything about global observables, about the pair of bottles... Then you don't know everything about the law of law.

1:32:30 So this is why, you know, the, the, the...