Individuality, identity, quantum mechanics / Maarten Van Dijk: New causal structure for nature / Jean-Louis Hudry: Paraconsistency, bivalence & identity / Abraham D Stone: Disunity Husserl Heidegger / Richard Arthur: Leibniz' Archimedean infinitesimals / Wim Christiaens: concluding remarks

0:00 To get into model interpretations later, and the questions of possible and actual entities to avoid certain problems which arise in quantum mechanics. So, the thing is, since already Aristotle conceived the concept of entity as a kind of solution to the problem of movement already proposed by the Presocratic leaders, it has been discussed this notion of identity in the conference quite a lot, I would like just to notice the fact that, of course, this notion of identity is conceived as a condition of possibility to determine what is to be an entity. And as Heidegger noticed, in relation to classical occidental philosophy, there is a kind of a critique that Heidegger made and also was present in Nietzsche, is the fact that within classical philosophy, within Occidental philosophy, there has been a forgotten conception of the being, always thought in terms of entity, which is entitizing the being. And in this sense, one could say that in classical classical physics, that has also happened in a direct correlation. Like classical physics can be conceived also as a theory of entities, very specific entities, like particles, waves, fields, and so on. Now, quantum mechanics is, I would say, completely engaged with the relation to this specific process of what was conceived in general terms at the end of metaphysics, already at the end of the 19th century. And it is, to some extent, a certain condition of possibility. All the criticisms involved in that period to bring quantum mechanics into stage. Now, there is, of course, a certain problem with the notions of entity in quantum mechanics,

2:30 specifically with identity, and this was recognized very much from the beginning of the theory. So, the problem is that quantum mechanics talks about elementary particles, of atomic process, which are thought in terms of elementary particles. But of course, we do not really know what an elementary particle is. I mean, there is no consistent picture whatsoever of what is an electron, for example. I would say that quantum particles are characterized by their intrinsic properties. that, of course, the notion of identity is sound, and as Schrodinger stated, he has a very direct quotation on this, she says, I mean this, that the elementary particle is not an individual, it cannot be identified, it lacks sameness. The fact is known to a professor that it is rarely given any prominence in surveys readable by a specialist. language, it is covered by saying that the particles obey a new-fangled statistics, either Einstein-Boswin or Fermi Dirac statistics. The implication, apart from obvious, is that the unsuspected episode of this is not quite properly applicable to, say, an electron, except with caution, in a restricted sense, and sometimes not at all. So, this of course brings into stage the fact that this is the traditional way of analyzing identity in quantum mechanics. It has a consequence that we count differently in quantum mechanics than we do in classical mechanics. So, we are interested in working out a different way of problematizing the notion of identity within the structure of quantum theory, and this is what we will do. There are two main points which I would like to bring to the stage, firstly a theorem which which is quite well known, the Cohen-Specker theorem, which states more or less that in a Hilbert space dimension greater than 2,

5:00 it is impossible to associate a definite numerical value of 1 or 0 with every projection operator, in such a way that if the commuting operators satisfy the formulae, the corresponding values of the operators, either 1 or 0, also satisfy this relation. Now, this means on the one side that if we have three observables, if we measure R together with B, or we measure A together with C, or we measure A alone, the valuation is different. is completely different. And this gets a quite big problem, which is the following, at least in physical terms. If I try to describe an entity in terms of the quantum structure, what I get is that, for example, if I measure the length of the table, I get one meter. And if I measure the color, I get it's brown. But if I would measure first the color, then I might have red. And if I measure then the length, I would get two meters. So there is a kind of incompatibility within the structure, which makes very difficult to retain the idea of entity. Of course, if we go to the structure of the theory, we can think back in terms of quantum logic the structure of it, we find out that there's a kind of nice way to see this. If you have the structure, the propositional structure of the lattice, which is the fortune-art-mobile lattice, and you can choose subsets of properties, which is normally called in physics the complete set of commuting or circles. We choose in a certain definite experiment. Then we can A subset can constitute a Boolean structure, and we can evaluate it into the 0-1 without any problem. But of course, if we try to do this with different substructures, then contextuality arises, which is well known. And we cannot consistently provide a single valuation to all this structure, which is a problem, of course.

7:30 Right? So, that's one point to the fact that within the quantum structure, the Cochrane spectrum theorem, as I am interpreting it, restricts very much the possibilities of thinking of objects with properties, with definite properties, of course. Now, there is another way which is quite not so much taken into account, especially because maybe the interpretation that is done to quantum mechanics is quite nostrilogical in some sense. So I'm more interested in trying to discuss the possibilities of deriving an ontology on quantum mechanics. Quantum superposition is a very deep principle within quantum mechanics and the act stated in this very known group that it was maybe one of the most important one. It says, the nature of the relationships which the superposition principle requires to exist between the states of any system is of a kind that cannot be explained in terms of familiar physical concepts. One cannot, in the classical sense, picture a system being partly in each of two states and see the equivalence of this to the system being completely in some other state. There is an entirely new idea involved to which one must get accustomed and in terms of which one must proceed to build up an exact mathematical theory without without having any detailed classical picture. Of course, yeah, you can choose not to go into the picture, but for me as a physicist, this is a kind of necessity. And a superposition is such a weird thing, which has, let's say, a property and can have at the same time its opposite. So there is a certain kind of violation of the principle of non-contradiction with this

10:00 very specific sense. Now, I would like now to go into what is called modern intuitions of quantum mechanics and the possibility of escaping all these problems through the idea of possibility. So let me introduce a little bit what is called the model interpretation. Model interpretations are quite related already to the Born rule, right? Because, of course, since the interpretation of Born of the quantum wave function in terms of probability, there is a model aspect engaged in the interpretation of one's mechanics. However, what is called a model in model interpretation is the development Van Frassen did in the 70s, which continued in the ages with several physicists and professors of science. was so an interpretation of quantum logic in terms of model logics. So we really formalized all the structure that was involved in describing certain processes. Then we have Koch and Dick's model interpretation, which relies hardly on the so-called bio-autogonal decomposition. We have Boobs-Hommean variant, which is a kind of hidden variable interpretation which has modalities involved, and we have the atomic model interpretation proposed by Dixon and Bacchaga-Lupi. Of course, there are a lot of problems with model interpretation, but One of them is the fact that one cannot consistently ascribe properties to systems in a consistent way without taking into account certain noble theories. I would try to give a general characterization of what is modern interpretation just to have in mind what we are

12:30 discussing. Well, so one of the most significant features of modern interpretations is that they stay close to the standard formulation of quantum mechanics. So we don't want to change anything in the structure of quantum theory. Like, I would say that there are two main lines of interpretations going on in the quantum domain, the quantum realm. One interpretation, certain interpretations try to go into the structure and try to find out certain hints in order to develop a certain ontology. This is the case, for example, of the model interpretation, but also I understand a little bit of quantum logic in this sense. But you have other attempts which try to do the opposite. They just fix the ontology, the ontological scheme, which they have in mind. And they try to see how they can touch a little bit the formulas in order to get what they want. For example, GRW. So model interpretations are also no-colapse interpretations. And the evolution is given always by the Schrodinger wave equation. And the collapse of the wave function is just the path from the possible to the actual. But it should not be conceived as a physical process. So, in this sense, model interpretations would talk about the possible state of affairs, not about the actual state of affairs, so to speak. Model interpretations describe possible properties to quantum systems, and the property description depends on the state of the system and applied regardless of whether we measure or not measure. In this sense, we try to escape the instrumentalistic viewpoint that you only can talk about measurement outcomes. There is also a distinction between the level of possibility and the level of actuality, which are related through an interpretation of rule, of course. And this is a point which we have been working out, is the fact that modality is not interpreted

15:00 in terms of ignorance. So this possibility should not be interpreted and cannot be interpreted in terms of ignorance. So there is no ignorant interpretation of the probability distribution assigned to the physical properties. The state of the system determines all there is to know. And in this sense, from all the interpretation, there are no such things as hidden variables. So, the question arises, because actually in the Koch-Inspector theorem, we talk about actual properties. But since quantum mechanics is talking about possibility, the question which arises is, so why can't we escape with the notion of possible entities? Couldn't we recover a notion of individuality, maybe not in terms of actuality, but rather in terms of possibility, and talk about a possible individual? Well, the answer is no. We cannot do that. we developed with another joint work in this case a Cochrane spectre theorem which talks about modalities so even in the case we enrich the structure with mobile operators so we extend the automobiles and we add mobile propositions to it So there is a kind of theorem which can be proved both for this enriched structure which shows that the contextual character of quantum mechanics cannot be avoided, even in the case of possibility, it's taken into account. So possibility, even possibility is a contextual, is contextual quantum mechanics, which is pretty strange, of course, in relation to an equalization condition. So, actually there is no possible individual in these terms. Now I'll give you the time. You have, so, five minutes left. Okay. So, the conclusions I would like to draw is the fact that we can analyze the identity, the form of structure of quantum mechanics, apart from identical particles.

17:30 This is the Cochrane-Specker and superposition principle. And the other conclusion is that model temptations do not allow to reconstruct the individual in terms of possibility. There are some further comments I would like to make. We are still working in several aspects of this problem. And in relation to what I said before regarding Heidegger, the history of philosophy, or something which Whitehead called in a phrase, I think, something like, after Plato there has been only footnotes. In this sense, we are trying to think, not in terms of entities, maybe an ontology is possible to develop, which does not talk about entities. Well, Bob gave a very nice talk in which I would say that the underlying ontology is processed. I'm very much in that line of thought. And in such case, the notion of definite particle number would not have a definite meaning. It would be just a wrong question. So it should not be completely applied to the structure, structure as it is done, I think, in quantum mechanics. Now this might give some questions. So, we are working with Fox space. I don't know if many people are acquainted with it, or physicists. This is quite a known type of space, which has the peculiarity, which is kind of very close to quantum Peel theory and does not have a definite particle number involved. So you can have superposition in the number of particles. So in this sense we are discussing the fact

20:00 that Fox-Space may be more adequate formalism for quantum mechanics to really grasp what the theory is talking about What I'm seeking for, sorry. Should we say it at the end? Okay, thank you very much for a clear, concise, and well-enantime speech. Any other questions? Yes, in one of your early slides, You stated that quantum mechanics was in violation with the principle of non-contradiction. Could you repeat that argument, please? Yeah, well, I restricted only to, in a very different sense, in the sense that a superposition state should be fought in terms of contradiction. You have something which is and is not. But the standard postulate in quantum mechanics for the ascription of properties is that we assign a property if and only if it is in an eigenstable. And the modal interpretation breaks with one half of that. But it retains the sufficient condition. if it is in a superposition then well the modal interpretation is silent unless some other condition is met the standard interpretation says simply there is no property full stop and there is no contradiction there so the problem is as I pointed out the modal interpretation makes a kind of layers of one layer of possibility and the other one of actuality. And, well, I've discussed this a lot, especially with the fact that the possibility should be integrated in terms of some kind of ontological ground, because that's the part which gives you the information of what's going on. Now, you have this interpretational rule, and you can take an osteological path, and say, okay, I have a superposition of states, and I just take it as a rule.

22:30 I take it as if I have, let's say, 50% spin up and 50% spin down, 50% of the times I will get up and 50% of the times I will get down. But of course, the interesting fact about quantum mechanics is that, Because, firstly, it does not allow an ignorant interpretation of the term, so I cannot apply this kind of statistical reading to it. And, on the other hand, if I only take it as a rule, I'm still without any grasping of what is that thing of which quantum mechanics is talking about, which is a superposition. So I really take quantum mechanics seriously and I want to talk about superpositions as existing entities. I am very much in this line. And actually I think this is not so crazy because all the experiments which are done nowadays are going in this direction. Like Schrodinger cancer are being made up in the laboratories. Of course, the problem with those experiments is that we don't really understand them. We do not really understand what's going on. But Christian, would you then mean in terms of contradictions in the sense that some property is and is not the case at a certain moment? Yeah. So just in that straightforward sense? Yeah. Okay. Somebody else? The intention at least of the Koch and Dix modal interpretation is exactly to restore an ignorance interpretation so this is simply in conflict with what you are asserting It depends about what it depends how you take it because of course Koch and Dix, what they would like to have first of all is something, about systems with properties. In this sense, they have to take into account open specotype contradictions, so they restrict the subset of properties which they discuss about. I grant you that. But now, and this is the problem, you have to deal with a lot

25:00 of other problems in order to go out with a consistent interpretation about some kind of a realistic interpretation of quantum mechanics. And this is what they have not been able to do. Yeah, but it has nothing to do with not being able to interpret probabilities in terms of ignorance. Because if the bi-orthogonal decomposition theorem requirement is met, then they simply do apply the ignorance interpretation. Yeah. I mean, this is, in principle, consistent. The problems they have is with the dynamic evolution of properties. But this is another issue. Yes, it is. Okay, so what they wanted to do, what Diggs wanted to do for the first time, is to get a joint probability distribution. So if you have systems, you have subsystems, you want to relate all these subsystems as existing, things which have definite properties. Yes. Now, if you could do that, then you could apply probability interpretation. Yes. But actually you cannot. And this has been proved by Bacchegallupi and by Vernaz. You know these results. And these results exactly talk about the fact that if you change the basis or you change the vectorization, you just cannot apply this this type of rule. So you are a little bit... But the Biorthogonal Decomposition Theorem selects a unique basis. Yes, exactly. So that is then the preferred basis. Okay, let me draw... Yeah, but because there is one order... So, you have two systems, let's say, alpha and beta. And you can make it be also another composition of this. Now, you can also subdivide it into another sub-system. And now, what Dietz wanted to do, at least in the start of the interpretation, was, his first started with this, and he said, OK, now I can have this probabilistic interpretation, and everything is OK, and I have a realistic interpretation, But I want to extend this to other subsystems. This would be nice. But for this I need a joint probability distribution.

27:30 And this is where all the type of noble theorems drop in. So you cannot do this. You cannot have a joint probability distribution for this. And then, if you take into account the fact that for every system, for every pure state, you can choose different sets of P or T on the composition, you can choose different factorizations, then you're in trouble, because there is an inconsistency underlying the definite-ness of the properties involved. Okay. I saw something that opened up. Yes. This is just a comment on what we were saying about the superimposed state being contradictory, which of course I'd very much like to believe. But you want to reject an epistemological, an ignorance analysis with superimposed state. That's fine. There's got to be something objective about the state. and then when you were discussing that you talked about potentiality when you were discussing that you talked about potentiality so you have to think of this thing as objectively being a case that is potentially spin up and potentially spin down now if that's what it comes to then that's not a contradictory state because possible idea and possible not idea isn't a contradiction of course I'm not thinking in terms of, there are two ways of thinking of potentiality. One is potentiality as becoming, so a seed can become a tree, or a baby can become a man. There is another way of discussing potentiality, which is the mode of being. So I have the faculty, for example, of raising my hand. And to some extent, when I talk about this kind of potentiality, there is, on the one hand, not the possibility that I will raise my hand It's instantaneously, at one single moment, the fact that I have the faculty. I'm not talking in terms of becoming now. No, no, the ability to raise your hand and the ability to raise your eye is not a contradiction.

30:00 No, okay, but I'm not thinking in terms of becoming. I'm thinking in terms of this process as existent here and now. But that's okay. I mean, the abilities are existent here and now, but they're just not contradictory abilities. I mean, they're abilities to realize Egypt's two contradictory states, but the possession of the ability is not itself a contradiction. Well, think about this. Thank you very much. We now have, and I suppose that the less formally versed or interested people will be very happy, we now have Martin Van Dijk. He's one of the really fine young historians and philosophers of science. He's from the University of Kent. He has done a lot of work on the 17th century natural philosophy and he's going to speak, I get to read it. a new causal structure for nature, the role of conservation principles in the scientific revolution. I think we have to wait a second for the team to start up. But as Karin already announced, indeed my talk will be more historically orientated than most of the talks we had at Cornell. But in a certain way one could see that the topic I'm going to talk about as some kind of the pre-history of, for instance, the previous talk we had about the problems on what is the system we're talking about when we're doing quantum mechanics. So I'm going to discuss the question, what are the physical systems that classical mechanics talks about, in a sense. And I'm going to discuss this question from a historical perspective, because the main involved in is more or less trying to understand the scientific revolution. Of course, there have been a lot of doubts about, well, should we think about it as a revolution, etc., etc. I think there are some good reasons why we can talk in a meaningful way about the scientific revolution. There are still some philosophical lessons to be learned there. So the project I am engaged in is trying to come up with some kind of philosophical perspective on what happens there.

32:30 So, for today, I will focus on these questions. What is the unity of a physical system? And also related, what are its conditions of identity through time, through change? When we have change in a system, why does it remain the same system from the perspective of physics? Well, as I said, I tried to come up with some kind of interesting philosophical perspective of what happens, and this philosophical perspective is more or less informed by some Kantian ideas mediated through Kassir for myself, but that's not terribly important at this moment. The main idea is something like this, change presupposes substance. If change is to be more than just a capricious succession of appearances, that is, if it is to be something that we can conceptually grasp, then we need to posit something permanent with respect to which change can then be determined. So this is a classical Kantian point. And furthermore, this determination of the change is then the function of something like causality. So it's causal reasoning that allows us to conceptually unify the substance, the thing that does not change, and the change with respect to this substance. And I believe that this is more or less something that remains unchanged throughout the scientific revolution. So this is something that is shared by both the kind of explanations that people try to give of change in the natural world before and after the scientific revolution. What does change, crucially, is the identification of what is this substance. And correlatively, of course, how change is then to be determined. And this identification, this will always be with respect to some regulative ideas about how causes operate. This is, of course, what I mentioned here with the function of causes. Ideas about the operation of causes which will allow you to pick out the substance and the change, how these relate to each other. And it is this operation which is, of course, which functions very differently in an Aristotelian and a Galilean program. I just called it Galilean program because Galileo is my own personal hero, but could have been a Eutonian program or whatever. Now, in both programs, we have the idea that there must be some kind of equivalence between what is a cause and what is an impact. But how this equivalence is to be interpreted, this is something that changes, of course.

35:00 we have like the equivalence between cause and effect will be qualitatively that's mine on the other hand it will be quantitatively that's on mine and this this comes together with the idea of the operation of the cause on the one hand there is a feeling an idea of cause operating is much more on the idea of generation something is hot well this can only be caused by another hot thing which generates hopelessness in the other thing whereas on the more We have more to do with the idea of conservation. So this is, of course, one of the main points I want to make today. So I'm going to try to spell out this much more clearly or give some more ideas about this. And on the one hand, the idea of generation is, of course, linked with power ontology, with idea of process, whereas conservation will be much more linked with magnitudes, with physical systems as states, instead of thinking about processes. This, of course, also hangs together with this other point that, in the Aristotelian view, you have diversity in matter. There are different kinds of matter in the world, whereas on the Galilean program, there's just one kind of matter, and this, of course, allows for the possibility of this kind of conservation across systems, whereas on the Aristotelian idea, where a hot thing always generates a hot thing, well, there you have to have some kind of different things which are responsible for the diversity that we can learn. in the world. And the explanations will be regulated on the one hand more through some kind of idea of a syllogistic logic as the instrument to figure out this causal structure of the world. On the other hand, we will have something like a geometrical logic. To sum up, there is like the Aristotelian idea. If you want to understand change in the world, well, you have to posit some substance in really the substantial sense that Aristotle On the other hand, we have more like, it starts on laws, some kind of functional dependencies. This is, of course, where Cassirer comes in for the ones who know the idea of substance. On the one hand, the metaphysics based on substance, on the other hand, one based on functions. But there is something important to point out with respect to these functions. That is that I believe that it's very important to be clear on the fact that these functional relationships

37:30 are, from the beginning, causally interpreted. So this goes counter to an idea that, for instance, Russell makes about causes in modern science. Well, we have functions, we don't need causes. Well, these functions in themselves are causal, or this is part of my claim. And it is the idea of conservation that gives you the logical foundation of causal explanation in the new Galilean program. So this is kind of the broad picture. So, but in what sense does the ideal of conservation make such a new causal logic possible? This is then, of course, a question we have to treat. And the clue, I believe, both historically and philosophically, lies in what we could call a Perpetuum mobile principle, which of course states that such a thing is not possible in the world. And in this respect, it's important to make the following analytical distinction. It's a principle that comes in two parts, I want to say. The first part is more or less some kind of constraining conservation principle, which states that there is not such a thing as spontaneous motion, spontaneous change. So you always need a force to put something in motion. To get something going, you need to put in something. The second part of the principle, which I call constructive conservation principles, for a reason that will become clear, is that there is no spontaneous rest. So once you have put something into motion, it will keep on moving out of itself. Once that, of course, has been exerted, it does not get lost anymore. And the second claim is, of course, a stronger claim, or it really has to be distinguished from the first claim. And if you just look at 16th and 17th century natural philosophers, you can see that there is quite a number of people who will accept the first claim and deny the second claim. And these are some more, if they give reasons for this, to be some kind of really empiricist program. And people I'm thinking about are Pedro Baldo del Monto in the 16th century and 17th century people like Robert Valls, They will hold on to the first part, but will deny the second part. For empiricist reasons, we see that force gets lost. We don't see this conservation, whereas we do see that you need force to put something into motion.

40:00 But this also implies that, from the same empiricist perspective, the first part will also be more or less like an empirical generalization. This is just what we see. It's not a matter of principle. In my opinion, it was the insight of Galileo that he realized that the ideal limit, which is expressed in the second part, that it is this ideal limit which allows for the unambiguous definition of fruitful mathematical concepts. so that you really need this idealization to come up with the right kind of concepts you want to use in your mathematical science. And I just very quickly want to illustrate this, what I mean with this idea, with two examples, which I both take from an early work by Galileo on the operation of mechanical machines. The first is just a proof of the law of the labor, which is more or less based on Archimedes' proof So Galileo starts by imagining that you have a solid which hangs, a uniform solid, hangs by its two endpoints and then the connecting line is supported in the middle, in the center. So this thing will be in equilibrium, for symmetry reasons, for empirical reasons, whatever. It is an equilibrium situation. This is something we can accept. Now he asks to imagine that you make two, so you divide it in two parts just at a random point and then you do away the support you had at the end and you make new supports exactly at the middle of these new two unequal bodies. And he says, well, it's clear that this will still be in equilibrium because these are both, like, again, symmetrical situations shows that for geometrical reasons, that in this situation, the conditions of the law of the lever will hold. So this is the way he tries to explain the validity of Archimedes' law of the lever. So, and he then comments on this. He comments on this. Yeah, thank he gives the following comments on this proof that he has given and from what it has been said

42:30 it seems to me clearly understood not only how the two unequal bodies CS and SD so these are the two separated bodies weigh equally when hanging from distances inversely proportional to their weight but moreover how in the nature of things this is the same effect as if equal weights were suspended at equal distances situation he started from. Since, in a certain sense, the heaviness of the weight CS virtually spreads out beyond the supported G, and that of the weight SD shrinks back from it, as any speculative mind can understand by examining closely what has been said about the present diagram. So, in this situation we see how in a striking piece of visual reasoning, Galileo readers to see, really, to grasp what makes for equilibrium in mechanical situations. So one can see how the relative positions of the respective centers of gravity are responsible for the fact that the effects of the separate body's weights are distributed over space in such a way that they are conceptually reducible to a situation where a single body is hanging from its two endpoints. and so in some sense one would say he tries to show how in a sense all equilibrium situations are the same and he then after having shown this introduces an abstract concept which expresses exactly the sameness that is he says well this means that both bodies have an equal memento a memento is something which is more or less in this kind of situation of static moments. And it is this concept which can then be mathematically characterized through the law of the levels, which gives you an invariant proportion. So, from this perspective, both the symmetrical and the asymmetrical situation are now characterized by the fact that momentum on both sides is equal. So there is an invariant mathematical proportions. Or to put it differently, we can introduce a mathematical concept with a coherent measure because we can see in what way these two situations, the one perfectly symmetrical, the other asymmetrical, have the same effect. So it's the fact that they have the same effect which allows you to introduce the mathematical measure for the abstract concept which Pelleo

45:00 introduces. And from this point on, in his treatise on all the mechanical instruments, he will characterize any instrument always by the conservation of momentum. Any equilibrium situation will always be characterized by the fact that this abstract magnitude is the same at both sides of the machine. Let me now quickly move on to a second example. Again, involving But now we're going to put the lever into motion. We're going to move this heavy body up with this light body there. And as the distance CE, so this distance is supposed to be five times that distance, whereas that body is, of course, five times as heavy as that body, this means that both bodies will have an equal momentum. And this will also mean, because this means that you have equilibrium, that just by adding an infinitesimal small weight at the body in the place D, you can move the other body up to the place G. But, Galileo then goes on to explain, considered from the perspective of conservation of momento, which characterizes all machines, this is exactly the same thing as saying that the body five times lighter placed at the place L. So, body five times lighter than the body at... No, sorry. If we here have a body which is five times lighter at place B, this body can also be moved by the same body which was at D but which we now put at L since if we do this, the proportionality that is expressed in the equality of moment remains invariant and if you repeat this, then five times then you can also move up the complete body which was originally in the point B so either you do five times this action or you do one time that action the effect will be the same the same effect. And here comes the importance of what I stressed about, thinking away all losses of forces that are being exerted, because of course you will have friction in the fulcrum, etc., as someone like Bido Baldo was quick to point out. But this identification of the two effects, either moving the whole body one time of their fight, by having the moving

47:30 body pass over that path, or repeating five times this one, this identification is only possible if you think away all friction, et cetera, et cetera. If you do not make this move, you will not be able to see how in the nature of things, as Galileo would put it, these are the same effects. So what does this tell us about the causal structure of the world? So, well, how should we think the causal relation between on one hand the moving force the load. Well, if we decide to think away friction, so if we really make this move, then we can see that something remains invariant throughout the interaction between both bodies. That is, the old idea that cause and effect are proportionate is given a precise mathematical interpretation. The causal relation now becomes really also a mathematical relation. And it It is in this way that the newly introduced concept of memento will reflect something crucial about the causal structure of the world. What does this invariant proportion characterise on one hand to moving weight and on the other hand to load in this kind of situation? That there is invariant proportion in the equilibrium situation and also in the movement situation. What does this actually tell you? Well, it is clear from Galileo's own way of talking about is that one of the important things about this proportion is that it gives you the limit of what can be achieved. This is the best you can do. This is, of course, related on one hand to the first part of the Perpetu-Mobile principle. You cannot create force out of nothing. But the second part also tells you something. Mainly, this is the best you can do. So if there were no friction, etc., or if you would go on reducing friction as friction, this would be the limit case of what is achievable. And it is in this sense, I think, that going also for accepting the second part of the Perpatian-Mobili principle gives natural philosophers such as Galileo something extra. Namely, this explicit introduction of these limits

50:00 can have a special objective function, in a sense. Now, there are a few things to be noticed about this claim that I make here. On one hand, we're talking about these limits. It is these limits which are, of course, expressed in this mathematical proportion. And it is also this that allows you to give a measure of the core that is actually exercised in this kind of interactions. Second, of what can be achieved. So this is, I think, the point where it's important to see that Galileo's treatment of this kind of situation is against some kind of very practical background. He just gives three ties on machines. And so he's really thinking what can we do. So this is why it becomes like something relevant or something interesting to introduce also these limits against this background. And I think this is, of course, not something that's explicit in Galileo, but this is something I can write. I think this is part of the philosophical movement that you can see going on there. Is that, in the end, these limits of what we can achieve, this is what constrains our actions in the world. So there is something out there about how the structure of the world is which we can exploit, which we have to take into account. part of the background of a more modern idea of scientific laws, if you want to put it like this. And it is these conservation principles, or these limits, which express these conservation principles, which now truly determine what is the systematic unity of a machine, how is the operation, the systematics behind the whole thing. And this is, of course, why I introduced the second part of the Perpetuum-Mobile principle, calling it a constructive part. This is about seeing what is the unity there, but also more especially allowing you to introduce these two concepts, like momento, which will express this conservation. And it is this conservation which now is responsible for the unity, Rather than from an Aristotelian perspective, if you look at the machine, what is the real unity of this machine will come from the intention of the maker or of the user. This has to be with the idea of generation as the main cause of concept. It is the maker who makes the machine who is responsible for its causal structure.

52:30 So this is much more tied up with the intention of the maker. Whereas here it's tied up with something mathematical in the end. Now, this can all sound very nice, but in the end, of course, many people will still be left a little bit unsecure and were also in this historical period. Well, this is quite a strong principle. So how are you going to justify this if it's not something that you can see empirically? And Galileo himself is rather vague on this. He just equates it with, well, this is the constitution of nature that things get conserved. You can see the cat trying to give a real strong metaphysical proof, starting from the nature of God. And you can also see Leibniz giving something like, well, more or less transcendental proof. Not only is he a little bit uncomfortable with how to think the stages of this kind of conservation principle, which, of course, Leibniz is also very, very important, but you can see some kind of transcendental proof, maybe, or could interpret it like this. like this in a discussion with Denis Papin on the measure of force, that is part of the vis-viva controversy, makes a very interesting observation. He says, well, the principle of the impossibility of perpetual motion, well, this is something like the principle of non-contradiction. It's just something you cannot deny. If you want to do physics, this is rock bottom, more or less. But then, of course, he tries to, at some point, he says, just robot, really, it's like, any rational mind must accept this point. He also tries to give some kind of argument through the principle of sufficient reason, which is not very successful, but he also gives another kind of argument. And this is, oh, I've changed my slides, okay. Okay. And it is this principle which makes meaningful measurement possible. If you do not accept this, you cannot introduce coherent measures for any physical property like its claims. Remember this kind of thinking by Galileo. You have to be able to identify these both effects. It's the same effect in the end. if you just do this five times or if you do this one time.

55:00 So it's really the fact that this is the same effect that gives you some kind of causal equivalence between different kinds of phenomena in the world, which allows you to introduce the coherent measure. These are identical changes in the world. And it is only because you have this kind of possibility that you can... Another way to put this is that the relation that holds between forces, if you want to have a measure for force, then you have to make sure that the relation that holds between forces must remain invariant. Whatever the particular effect you choose to measure with. Either it's moving a heavy body over this distance or a lighter body a few times over another distance. This must remain invariant. And it's of course what's not happening with the Cactus measurement. There is another interesting thing that Leibniz does in this respect. He introduces the idea of epipolence, which is a logical concept from medieval times already, which expresses something like true functional equivalence. If two propositions are epipolence, if you can interchange them, substitute them without changing the truth of the sentences where they appear. But the same concept of equipolens, or the same term, is also being used in the mechanical literature of 16th, 17th century to say that, well, they're equally strong both bodies on, for instance, on a balance. So Leibniz is really starting, is introducing this idea of equivalence is like, well, you have also some kind of caudal equivalence, which is more or less functioning like a true functional equivalence. And it's, of course, this which then provides the world with an in-principled mathematical syntax, which, of course, we now speak not like this. And the interesting thing, well, and this is why I call this more or less transcendental argument to start by saying, what is the possibility of meaningful measurement given the fact that we think that we can do something like measure meaningful in the world. We have some examples of this. So it is more or less, or I would claim that there is a claim to, or you can make a claim for this being some kind of synthetic prior principle already in my place, although we

57:30 would not of course have agreed or would not have put it like this. But to come to my final point, this syntax, this mathematical syntax of the world, well Fundamentally based on closed systems as its basic constituents, because closed systems, a system where there is conservation of some kind of magnitude. So the identity of a physical system throughout the change in the world will always be delineated by the conservation of a characteristic magnitude. And I think this is something very striking, just about what classical mechanics gives you. Just to give a very quick example, which will be the end of my talk. inertially. Well, there will be something conserved, just its amount of motion. So, here you have just, well, like, a state. Now, there is a change happening. So, this is a conserved system. This is a closed system. There is, like, one thing that is being conserved, the quantity of motion. Now, there is a change happening. There is some interference with this body. Well, how are you going to first notice that you're going to determine this change with respect to this conserved thing, which is playing the role of substance in the general idea that you need something substantial to determine change. But secondly, what you of course do in classical mechanics is, well, this is a new closed system. And this is what the three laws of Newton give you. you can start as this kind of layered picture where you always look like this is a physical system which in itself will again be characterized by the conservation of some magnitude. And this comes out most striking, if you look to classical mechanics, of course, by the peculiar role that the center of gravity plays. The center of gravity of our cosmological system or the center of gravity of two colliding bodies. The center of gravity will just move on inertially, as if there was no mistraction. It's here that you can see where the identity of the physical system lies from the classical perspective. These are the colliding problems. Thank you. Yes, this is such an interesting talk that I would like to ask many questions myself, which I'm not going to do.

1:00:00 Yes, Martin. In the case of the balance... The causes are identified with the forces. Take another conservation law from those times, the one of Huygens, of Conservation of Linear Momentum. There, well, in momentum, forces are not involved in the concept of momentum. So what is playing the role of the causes there? Well, this is very interesting about Hervens, in the sense that indeed explicitly does not enter in any dynamical considerations, tries to do it through relativity arguments as much as possible. But what you can, or how I understand this kind of reasoning by Huygens, is that he just says, well, there must be something conserved. And let's not assume anything about forces involved, about the causal processes, but by just, let's see what's invariant in this kind of situation. put in a minimal amount of empirical information on the symmetrical case of what is colliding and then through relativity arguments let's try to find out what's invariant and once you have found out what's invariant then you have learned something important and indeed Huygens at this point does not introduce any forces in the picture but this as you know is of course crucial for Newton's three laws of motion Newton starts from this result by Huygens in a collision. And this will be the backbone of his third law of motion, which of course crucially tells you something about forces. So this is a kind of exercise trying out what is invariant in this kind of situation. And once you've found out what is invariant, then this gives you a possibility to introduce the right causal concepts. Force, which is defined in the conceptually coherent way, such as that indeed causes and effects will And would you then say that for Huygens there is no conceptual unity of substance and change like you began? Well, it's a very good question. Because now I am indeed interpreting Huygens from the Newtonian perspective.

1:02:30 I am saying, well, what did Newton do with Huygens? How does it make sense? How does it make sense from Huygens' own perspective is a very good question and I am not going to try to answer it. I'm going to phrase this as a historical question. It's the thought that the no perpetual immobile principle is itself something of an achievement culturally, on both sides of it. But what's odd about the way you present the story is that it's as a word presupposed in Galileo Enlightenment, although I thought that historically it happens actually to reasons backwards from, I guess in your language, conservation principles, says, well, therefore, no perfection mobile. So, which may actually partly explain why, in response to Fred's question, you end up making Hawkins sound like Newton, because what the dynamics for Hawkins is going to be is going to be a different question altogether from this question. Well, at one point I introduced the importance of the practical background. I think this kind of no perpetual motion idea is something that is being present in practice or in more practice-orientated people for a long time. Nehra Dudahin, she talks about it. It's like an idea that crops up before people like Galileo really start using it in a way to introduce concepts, which is, in my point of view, the really crucial thing, that you're going to use it to introduce new concepts. But it was thought that it required proof, and now we have this... It's thought by some people that, like, for instance, someone like Galileo, this is part of the evolution, I think, from Galileo on, we have clear awareness that this is not something that we're going to prove, this is something that you must presuppose. in order to get our theory on the ground you can see people trying to do it also after Galileo

1:05:00 you can see Leibniz being quite uncomfortable with this then he says it's just something we rational people have to accept this is part of what he says very ambiguous, it's something that people indeed are uncomfortable with well I guess if you buy my line that this is some kind of synthetic Well, it's something to be uncomfortable about, unless you have some kind of very sophisticated philosophical position from which to do justice to principles like this. So I would say that the historical ambiguity surrounding the principle tells us something about the philosophically special role it plays in the practice. Yes. I want to say something about Newton, but I guess there was another question. Do you need much time to make your point? Well, it seems to me that what it is to be a closed system is precisely, and you know, I think, this is one of the central problems in trying to interpret the third law. So that what you just made, a new story, I want to say is question-begging there, because in order to be forces at all and to be able to assign closeness it's not obvious that Newton has the resources obvious resources to do that. I'd like to... Yes, sure. I should start this also question because it's in this ideal for physics of invariance to say Sympathetic Plastic or special relativity, so it's really a long-term idea, but could you probably just, at this point, to differ from the notion that there's still a new problem? Well, I think that one thing that's very important about the idea of invariance in physics is that this physics starts from geometry, or the mathematics that someone like Galileo is using is geometry. you're really reasoning on proportions, on constant proportions. So there is something about invariance that's really, that comes with the kind of mathematics with which you start looking at the world if you're a 16th, early 17th century mathematician. And then, of course, Galileo also starts giving relativity arguments in the Dialogo. That he gives them

1:07:30 then on basis of this inertial principle or some kind of inertial principle and this inertial principle itself is being introduced by Galileo in the context of this kind of mechanical systems also as well if you want to understand the invariant that characterizes this kind of systems you have to posit something like this and then you can start exploiting it to reason about space-time structures basically what you do with doing the relativity arguments but I don't know if it's really you're also thinking about invariants of things, invariants of properties that is, invariants will not be interpreted through Euclid through constant proportion expressed geometrically I think this is something or one of the important reasons Yes, and I think that will be the last question. You mentioned, you attribute the conservation principle approach to Galileo, but he doesn't actually use that terminology. And it's usually attributed to Descartes, and so there's a kind of implication in your talk that maybe Descartes guessed it from Galileo. But I think... Guess it from Beekman. Guess what I was going to say. Yes, that's what my research is. but I would say Beckman comes to this kind of principle and analogously with Galileus for somehow more or less the same reasons, less clearly thought out because Beckman is much more fragmentary I think that's a disservice to Beckman since he is from these parts I want to emphasize that, I think that he was a much more coherent thing it's really been underplayed in some different revolutions because much of the background this idea of conservation, for instance, was this dynamic? I agree. And it's really foregrounded with Beckman. Yeah, and it's in coherent, in a sense, it's really taught out well, but it's, of course, not like, it's not as accessible or as as as as. But I agree. Beckman is very important. We think Baldo and Beckman share the same, on your reading, you also share the same conceptual possibility space, right? Beckman, I think that it is. And I agree to Baldo and Beckman. So, uh, Becman is much closer to Galileo in the willing to make this idealizing move,

1:10:00 for instance, something that Becman does that Guilobaldo does not. So, but I would definitely agree with your point that this is, I tell it from the perspective of Galileo, but it's not just Galileo. Someone like Becman is, but he's more or less going for the same program, obviously. There are significant differences, but... Of course, it's kind of like me and Parma. Yeah, yeah, that's of course the right behind it. going to stop this one. Thank you very much. And we're now having the last of the contributed sessions. Ah, Jean-Pierre Brieg. He has something like an exemplification of the basic idea of this conference, because he starts from the French-speaking world and the cultural tradition there, and he's working in the Anglo-Saxon world in Edinburgh. Dublin now, okay, changed and he's going to give us a talk about bio-existency, violence, and identity. Thank you for giving me the opportunity to present my paper. Thank you. So, as you know, in classical logic, there is the notion of classical consistency which rejects the idea, which claims that contradictions are false and which is based on a law non-contradiction which is assumed to be true, which is true a priori. And often there is an historical justification for this law, is to refer to Aristotle, and And many claim that it is probably one of the rare defunct philosophical justifications of the law of non-contradiction.

1:12:30 But they also say that it is surprising to see how Aristotle's arguments are not very good. and either you want to defund the law of non-contradiction you are going to try to say what Aristotle says and to say more maybe than what he says, if it's possible or if you want to claim that there is, I mean the law of non-contradiction can be rejected, maybe, or at least neglected as an a priori principle, then you are going to use, you know, Aristotle just to say how the proofs are bad, and so maybe there is a reason for this. So, my point in this talk will be to show that actually Aristotle does not deal with the law of non-contradiction. So I'm not going to speak of the law of non-contradiction, but only of the principle of non-contradiction, and you are going to see why I make this distinction. In other words, when I am going to use the principle of non-contradiction, I mean what Aristotle says in the metaphysics, and I don't mean the law of non-contradiction as in classical logic, as implied by classical consistency. So, and, I mean, the reason for this historical tradition is partly due to Lukasiewicz, who wrote on the Principle of Contradiction in Aristotle in 1910. I mean, in a way he explains Aristotle's principle, he examines the different proofs and reputation and criticizes Aristotle and just claims that his proofs are not good. And so far, we have just repeated what Lucas Leavis said about Aristotle without, I guess, trying to study the text itself.

1:15:00 Sometimes he makes me think about what Aquinas said about the soul in Aristotle. And then when you read Aristotle's Denimap, you realize it has nothing to do with Aquinas' interpretation of the soul. In other words, Aquinas has an agenda in order to, you know, use Aristotle's argument, and Lukasiewicz again has an agenda as well when he tries to understand what Aristotle says. So, in short, it's not a logical principle for Aristotle because it is only a metaphysical belief. In other words, and maybe I'm going just to, yeah, before, no, let's deal with three. It's not completely short. I can see. It doesn't. There is no, stop. Ah, thank you. Sorry. So, I'm going to use, you know, three main claims. So, the first and the obvious definition is in the metaphysics. And Aristotle says, for the same thing, so this is what he called a principle, I mean we call this a principle of non-contradiction, even though he never says this, but anyway let's understand this as a principle of non-contradiction. For the same thing to all good and not to all good simultaneously on the same thing and in the same respect is impossible, given any further specifications which might be added against the dialectical difficulties. This, then, is the firmest of all principles, for it fits the specification stated, for it is impossible for anyone to believe that the same thing is and is not. Here, just a note, dialectical difficulties, I mean, dialectic in Aristotle has nothing to do with the German conception of dialectics, so it just that refers to the fact that a discussion between speakers and their conditions, and I'm going

1:17:30 to say more about it, but we can also read another claim. If it is not possible for contraries to hold good at the same thing simultaneously, given that the customary specifications are added to this proposition too, and the opinion to an opinion is that of the contradictory, then obviously it is impossible for the same person to believe simultaneously that the same thing is and is not. For anyone who made that error would be holding contrary opinion simultaneously. That is why others would demonstrate go back to this opinion. In the end, it is in the nature of things the principle of all the other axioms also. And the last claim, which is in his conclusion at the end of chapter 6 of Book Kama, and he says, it has now been fully enough stated that the opinion that opposite acceptions are not simultaneously true is the promise of all, and what the consequences of those who make the statement and why they make it. So each time the principle is associated to an opinion, to a belief, that is a doxa, which can be either true or false. Aristotle never speaks of knowledge. So, what is, in a way, this metaphysical belief? First, that it's always related to a speaker. In other words, a belief can be true or false, but depends on the speaker and depends on a given context. And I think this is very important to understand this context, that is, the context of a discussion, insofar as for Aristotle, what is true or false is only about assertions. In other words, there is not this notion, you know, of sentences as expressing a logical proposition. A sentence cannot have a true value unless it is uttered by someone at a given time, in a given context,

1:20:00 and so on. And you can apply this to writing as well. But the point is that an assertion, when it is a TERF is true because it says something about what it is, or it's false because it says something about what it is not. So, in other words, there is no, you know, if we need each time, in order to make sense of the principle of non-contradiction, if we need a speaker, and if we need a context, then we cannot make any generalization. Therefore, we cannot, you know, I mean, it's nothing to do with the law of non-contradiction. It's more like, if I want to use a modern term, it's more like pragmatics. It's pragmatics and not semantics. That is, Aristotle is interested in natural language in relation to a speaker. That is, the use of natural language. And in this, in that respect, then we are going to see that, I mean, the principle of non-contradiction that is understood like that is impossible to reject in those sounds that everybody in this room, of course, holds this principle and believes in this principle. And in the sounds of, by saying something, but I don't mean, you know, the opposite of what I am saying. So I assume, you know, the truth of my claims, and therefore I believe in the principle of my contradiction. But this belief only applies to now, to, I cannot generalize it, or only to this talk, because as such it's a pragmatic principle and not a purely semantic one. So therefore, we cannot say that Aristotle defends classical consistency. We cannot say that Aristotle would agree with the ex-falso-collibate, that is the fact that if you have contradictory premises, anything follows. So, this does not mean anything for us, but in the past there is, you know, no context. So there is only sentences, but sentences does not mean anything unless it is uttered or written.

1:22:30 But there is someone, you know, I mean, there is a speaker and there is someone else to give some meaning to the sentence which has just been uttered. So, in other words, there is, you know, there is no justification to say that Aristotle would disagree with an intuitive notion of paraconsistency. Contradictions can be meaningful and can be true in some particular context. And in the Deinterpretatione, he gives an example of a true contradiction. That is, it is true to say that a man is pale and a man is not pale. Just we need to understand the context of this true contradiction. And when I say context, it's a dialectical context. That is always, you know, what is about dialectical difficulties. That is a reference to a discussion between two participants. But that does not mean that Aristotle... Of course, Aristotle rejects trivialism, because again, it's a non-contextual principle, it's universalization, that all contradictions are false. He cannot agree with this. And with respect to dialectism, I don't think you would have agreed with this insofar as contradictions, even when they could be, I mean, when a contradiction is true, an exception, which confirms the rules that we showed as the belief in the principle of non-contradiction. In other words, to say that non-contradiction is true is not the same as to claim that a non-contradictory assertion is true.

1:25:00 contradictions for Aristotle must, you know, be avoided. Or, if they cannot be avoided, should be done very carefully and should be understood only as exception. So let's understand now Lukasiewicz's interpretation. And in 1910, he says, Aristotle formulates the law of contradiction in three ways, as an ontological, a logical, and a psychological law. He doesn't make explicit the differences between them. So, ontological formulation, it is impossible that the same thing should both belong and not belong to the same thing at the same time and in the same respect. Then you have the logical formulation, the most certain of all principles is that contradictory sentences are not true at the same time. And then you have the psychological formulation, which says that no one can believe that the same thing can at the same time be an ugly. If you compare with the original quotation, that Lucas Hedis is just tinkering with Aristotle the text. When he says autological formulation, he just adds this sentence and forgets the psychological justification, the fact that it is impossible for anyone to believe that the same thing is an in-note. And actually, of course, he does not want to see it as a justification because for him that there is another formulation. So, and he understands this as a psychological formulation. And again, with respect to the logical formulation, he just, that, deal with one sentence on its own and he completely, I don't know if it's unpurposed, but he interprets all as principles, whereas

1:27:30 here, all, firmest of all, refers to opinions. So in other words, here, the logical, I mean, if, by just following the text, we see that the so-called logical formulation only expressed belief, and there is no, I mean, the principle is not logical at all. So, to understand maybe Lukasiewicz's motivation, we can read what he says about and why he thinks that Aristotle is wrong. I mean, at least in his justification of the principle of non-contradiction. So he says, Aristotle thinks that the logical and ontological formulations are logically equivalent, for he treats sentences as with which it puts them in a one-one correlation. This one-to-one correlation between sentences and objectives entails the equivalence of the ontological and the logical laws of contradictions. So, here, you can see that Lukasiewicz understands his own formulations of the principle of non-contradiction are almost structures, at least on an intuitive level. That is, that there is, you know, the formulation dealing with sentences, that is, the sentences are elements. There is the ontological formulation dealing with objectives. And then, after that, it's going to deal with the psychological formulation dealing with mental facts. And then, from this, it tries to draw an equivalence

1:30:00 formulations. That is, the fact that there is a one-on-one correlation between sentences and objectives. But it is not, I mean, this is pure invention, because there is, first, there is no sentences as such. This, I mean, there is no, you know, propositional logic in Aristotle. So, assertances, again, is only, I mean, relies on analyzation, an utterance. And so, in other words, of course, there are objectives, and this is, in a way, maybe the only structure that we can find in Aristotle, the fact that there is a metaphysical foundation about things, you know, in the world, or things about, you know, genera, that is the essence of things. And this is part of knowledge, so it's something we can be sure about. But there is nothing corresponding to this. There is no sentences in a so-called logic which could correspond to the structure of genera in the metaphysical world. So in other words, I mean, he's right to say that it's equivalent, but I mean, he's wrong by saying that because there is nothing corresponding to sentences in Aristotle. There is only genera. So let's deal with the second quote. Aristotle attempts to prove the psychological law of contradiction on the basis of the logical law. If two beliefs answering to contradictory sentences could exist at the same time in a single consciousness, then contrary properties would hold of that consciousness at the same time. But by the logical law of contradiction, it is impossible for contrary properties to board of a single object at the same time, and to believe answering to contradictory sentences cannot exist in a single consciousness at the same time. He noticed that Lukács Yeditsch deals with belief, but only from a psychological point of view, that is with respect to some consciousness. And this, I mean, this kind of interpretation does not mean anything

1:32:30 for Aristotle, since consciousness is a Cartesian prejudice for him, and in other words, he thinks that mental states are physically reducible, so therefore he is not, obviously, I mean, he is not going to speak of a single consciousness. Yet, I mean, it is, Aristotle is interested in belief, but only on an epistemic to say that a belief can be either true or false, and it's not, you know, knowledge. So it's... But there's nothing to do with the fact that, to the question of how to understand the production of belief by, I mean, through consciousness. In a way, the psychological foundation does not exist. Even in the D'anima, Aristotle does not understand mental state as produced by consciousness, because there is no consciousness. I mean, what we call consciousness is explainable by something physical. So, in this case, it's better to speak of awareness. So, and then the last quote, Aristotle proves that the psychological law of contradiction is inadequate because he has not proved that belief answering to contradictory sentences are contrary. Aristotle falls into the common error of logicism in psychology. the converse of psychologism in logic. Instead of investigating mental fact, Aristotle considers the sentences answering to such acts and the logical relation holding between such sentences. So, here again, I mean, we have the fact that, and this is very interesting, logicism and psychology, Because in 1910, the conception of logic is derived from logicism, and also psychology in 1910 is very predominant in philosophy. So, in other words, he seems to use logicism and psychology in order to understand and to account for Aristotle's text.

1:35:00 Whereas the only argument we have in Aristotle is the fact that there is a metaphysical belief, which is dependent on the context and which is derived from a discussion. and, I mean, this is based even on intuition, the fact that by saying something by saying something about what it is that you this means that you are going to reject the opposite statements. So, to conclude that it's not a real conclusion, but I'm not going to have enough time. So I just would like to say with respect to the proof, because often it has been said that the proof for the principle of non-contradiction are extremely weak. They are seen as...