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 (contd.)

0:00 I mean, it's true, or at least defundable, that you can make sense of this proof only with respect to the belief of someone else. That is, someone believing in true contradictions, then you are going to prove that he's wrong, or she's wrong. And this is what he calls, and he does not call this the argument anonym, but it just explains that the proof relates to the belief of someone else. And that's why it's called refutation. And why? It's just because there is no way that we can provide a demonstration, scientific demonstration, that is a deductive inference in order to prove the principle of non-contradiction because that begs the question. in order to prove the principle of non-contradiction independently of someone else's belief that you must already assume the principle of non-contradiction. Thank you. Again, an interesting presentation. We are not having very much time, so I will take two short questions. Yes? Sure? Actually, you can't run here analyzing the pressures of Aristotle with much stress in the fact that actually all this particular view given, and actually even in Lukasiewicz, you know, interpretation, are sort of always speaking of time, right? It's time every time. The country is quite wrong, so it's kind of confusion. And this, of course, spreads well, if you say that our stakeholders need a concrete speaker in mind. But I just wonder, how is this that important for your analysis? I mean, what would happen if we really try to sleep time out, but would correspond more to today's notion of principles not to do this problem, right?

2:30 So you might somehow use this reference to time as another way to say this principle but probably to interpret it differently than it's usually done. I mean, you're right to say that time is important because it's something, you know, always, I mean, a temporal location and also a special location. I mean, it's not something, a principle you can universalize. But I would say, I mean, if you want to translate, you know, this principle into a modern language, you could, you, we will call, I mean, this pragmatics. The fact that, you know, semantics is related to the use, of natural language, you know, is not only understood by semantics and syntax and semantics, but syntax semantics plus pragmatics, that is, how are we going to use this language and pragmatics require context. That is because a speaker uses a natural language with respect to a context, and the use is going to defer according to the particular context. Okay. Yes? I believe a lie sentence is both true and not true, at least so it seems to me. What would Aristotle have said about me? That means that you believe in everything. No, no. Are you saying that's what Aristotle is saying? Yeah, it's not to me or anything. I thought Aristotle doesn't believe in the explosion. yeah but that will mean that you're right but oh exactly right but you believe in everything now that does not mean that you are going to believe this in another context but in this context of this discussion in a way you

5:00 i could say that you believe in the opposite of what you are seeing well in a sense i do but i don't believe in everything i don't believe you know the lie sentence is six feet tall No, true. I mean, so, that's why, okay, you're right, it's not trivialism, but the fact that, in a way, I cannot know, you know, I cannot take your argument seriously. I mean, Aristotle will say this, because in a way, maybe you, I mean, if I follow your belief, that means that you believe in the opposite of what you are just saying. therefore I have to choose or maybe you believe you are both I mean you hold both opposite beliefs that's correct so in this case you know, I mean fair enough but I don't really understand what you mean because in a way you don't make a choice you know It's almost like, yeah, the skepticism, that is, you don't believe in both. I mean, in the sound that, you know, it's suspension. You suspend your belief. That is, you believe both that are, I mean, both beliefs are possible. Maybe we'll pursue this. Yeah, yeah, sure. Just one small question. Is there some explicit reference that I was talking to being uttered or evil? Here and now? We are going through a short coffee break and then we come back through the last session of this conference. Short coffee break, 10 minutes. The conference is going on. What? Yeah, hi. Oh, thank you. Yeah. Oh, that's fantastic. I really, really want to look at this. Well, no, no, as a matter of fact, I asked Alexander if he could get this to me, but I think he was able to get it before he went away. Yes, that's fantastic. That's really kind of... I was actually going to write to you on that. I was going to ask you if you have any money. That's very kind of you. And can I get a copy of that as well? Sure. Oh, I want to ask you a few minutes.

7:30 Marvellous. Thanks. This is obviously going to be a good one. Marvellous. Thanks so much. I'll email you as soon as I get back tomorrow, I'll send you all the details of that. I'm very tired but I will be there tomorrow. Are you leaving tonight? I'll be talking about the most, maybe I did breakfast on this, so we'll be around because we had the ceremony. How are you going back to North Chateau? I still want to be gone because I think this is the... Oh, you're staying here. No, I'm staying here. So I would be around, he is going back. Yeah, I know you're going back to North Chateau. Well, I just wanted to... when you have 100 people out there. What is that? I think my mind is 6, 7 years. I don't know if you're going to be a school. When you get around to see my face, I'm so close. I'm so close. I'm so close. I'm so close. I'm so close. I'm so close. I'm so close. I'm so close. I'm so close. I'll make sure I do that and leave it at the end of the day. Thank you. It's my pleasure to introduce Abe Stone,

10:00 who's at the University of California, Santa Cruz. He was introduced to me as a person who's read everything, and then proceeded to talk about Buffy the Vampire Slayer. He's trying to make me sexy, Abe, come on. I don't think he's actually in Buffy's. Exactly, exactly. But today we'll be talking about the disunity of beings and the disunity of beings. Thank you. I realized after having heard all the other talks that this was the long talk for this conference, and I really wish that I had said something about Hegelian logic and the foundations of mathematics instead. It seems to me like that's one piece, and I mean this in all seriousness, but that's one piece that's been missing so far, but instead I'm talking about this. I'm just going to read this, I'm sorry about that, but hopefully these pictures will help. Husserl's interpreters, and I might add also Kant's interpreters, have disagreed bitterly on almost everything, but especially on his realism about the external world. It is striking, therefore, that Husserl's most influential successors, Heidegger and Karnap, are in explicit agreement about the question of realism, that is, of the mind independence of external things. They both dismiss it. In fact, both do so using the exact same term, pseudo-problem, or Scheinthor-Went. Now, Husserl himself did regard this as a good question, if rightly understood. Still, it's no coincidence that his answer is difficult for us, that is, the intellectual errors of Heidegger and Carnap, to understand. Both were so concerned to rule out Husserl's answer that they not only tried to get the question itself dismissed, but also to undermine the key concepts in terms of which his answer is framed, the concept of a mode of being. That should sound like an odd thing to say if you're acquainted with Heidegger or Karnak, and I'll come back to that later. For who told the question whether objects of a certain kind exist, whether, for example, there are mind-independent external objects, is viciously ambiguous unless one first investigates the mode of being appropriate to such objects.

12:30 Now, the traditional roots of this concept lie on the one hand in the Aristotelian doctrine that being is not a genus, and on the other hand, in the Platonic distinction between the upper realm of true being and the lower realm of mere becoming. The two lines of thought were combined and developed, first of all, by Platinus, and then by various later Neoplatonists and Aristotelians, including eventually Thomas Aquinas and Dunstotus. And unlike Husserl himself, his teacher, Rentano, was something of an expert on the historical details. Leaving aside all such historical complexity, however, the basic ideas as follows. Generic or specific unity is a matter of different individual things having something essential in common. This is why the name of the genus or species signifies uniquely. The signified objects share not just a name but an essential feature. There is, however, no essential feature common to everything which is called a being. and hence, and you can see figure one here, there are highest genera of beings between which there is no community of essence. The term being applies significantly to the members of each highest genus that is equivocal as applied across their boundaries. Still, being is not absolutely equivocal. Unlike, say, the English word bank, it's based on some kind of sameness. And in fact, two different types of looser so-called analogic unity genera. And I'm going to call them analogy of relation and analogy of proportion. Now the first analogy of relation connects fundamental and derivative genera. In Husserl, the genera of individua and the genera of their, quote, logical modifications. And the traditional terms, of course, are substance and accident. Husserl's logical modifications include not only quality, quantity, relation, and so forth, but a huge variety of exotic categories, including and or set theoretic constructions. And importantly, too, his logical derivation can be iterated to yield further object types, sets of sets, qualities of relations, and so forth. Still, Husserl's system retains the key feature, a relation of priority by which all objects are tied to the fundamental ones. Thus, there can be a term which applies primarily in the fundamental case, but then also in the secondary sense in the derived cases. So Aristotle's example is health, you need to see figure 2 for this.

15:00 What is healthy in the primary sense is an animal, but other things are called healthy in secondary senses. A urine sample as a sign of health, a diet as a cause of health, and so forth. Similarly, given a single pious genus of individua and all the various genera of modifications built up from that base, we can say that they all, in some sense, have something in common. Now, if there were one gemus of all individua, then the something which it and all its derivative genera have in common would be being. That is, would be the meaning of the equivocal, but not absolutely equivocal, term being. Here, however, is where platonic doctrine traditionally comes in. If sensible substances are not substances in the truest sense, then the term substance, or for Husserl, the term individual, is itself a critical. It does not name a single genus. Hence, there are at least two, the typical Morse, systems of fundamental and derived genera, such as A and B in figure three. Now, these systems are what Husserl calls regions or spheres or realms of being. Because these separate systems are parallel, there arises an analogic unity of the second kind which involves a proportion, A is to C as B is to D. If, as in figure 4, A and B are highest genera of individua and B and D their respective systems of modifications, then A and B are in a way similar despite the absence of a central community. Each plays a proportionate role within its own region. Hence, the term individuum, although not uniquely applied to both, is not absolutely equivocal either. Now the cross-regional unities which thus arise are what Thomas calls logical genera. Husserl calls them formal categories. That's not quite right, close enough. The iterability of Husserlian logical derivation means, moreover, that in a way every type of object is analogous to every other. As you can see in Figure 5, B is to its own modifications, as A is to the whole system, C. Hence, Husserl's formal categories, unlike Plotinus' or Thomas', include the universal stoic category, something. The study of all beings from this point of view, as, quote, modifications of the empty something, is what Husserl calls formal ontology. So formal ontology studies all beings as such,

17:30 but it is no metaphysics that is no first philosophy because it exhibits beings as standing in similar relationships to one another but does not account for their being. It does not, in Aristotelian terminology, reveal their common causes and principles. That would require a single structure in which all the highest genera are explained as deriving from one fundamental type. Now such a structure is implicit in the original photonic motive for introducing different regions of being, which is a Heraclitian worry about the inconstancy of sensible things. According to Aristotle's report, Plato always believed to remain the Heraclitian about the sensible world. Aristotle goes on to explain that Plato therefore understood the Socratic quest for definitions, that is for essences, as concerned with the super-sensible. Here's a quote from the metaphysics, for there cannot be a common definition for any of the sensible things Now, under the influence of Aristotelian doctrines about being, this came to be understood as follows. The dispersion of sensible substances, that is, their constant flux, their spatial extension, and the distribution of their forms among numerically different individuals, is a sign that they are imperfectly existent. More importantly is a sign that their being is dependent. it. Since a dispersed thing is, so to speak, not quite self-identical, not quite what it is, its being anything at all depends on the presence of something unified and unchanging of which the dispersed pieces can be seen as imperfect manifestations. There's a picture of this in figure six. So actually, on this view, you can say that the gluon is an object in a higher sense than the things that it glues together. It's more of an object than the other things. But such a higher, super-sensible source of unity is not merely different in species or genus from the sensible beings dependent on it. Rather, it is what it is in a higher sense than they are. It differs from them in mode of being. So the disunity of beings in the sense of their dispersion is linked to the disunity of being in the sense of equivocality. Now, in this way, we can see figure seven for this, an analogy of relation connecting all fundamental genera gets built in in advance. Sensible objects depend on the super-sensible as the cause of such being and unity as they possess,

20:00 or more generally, relatively dispersed objects at lower levels depend on the more unified ones higher up. At the highest level is a genus of fully existent and unified substances, or individua, and all beings as such therefore have something in common namely their dependence on the members of that genus. The science that studies it, the science that studies that genus and all beings in general and their dependence on it is in metaphysics in the full sense that is the science of beings, quae beings with respect to their common causes and principles. Now Husserl however calls this science transcendental phenomenology. The new name represents a new view about what occupies the primary region, and you can see the comparison in figure eight. It now contains not forms or intellects or angels, but the contents of my own consciousness. Furthermore, the truest unity of all, which sits outside the whole system of beings, and on which even the highest ones depend, is no longer God or the one beyond being, but rather the transcendental ego. but this change is smaller than it seems traditionally the relation of metaphysical dependence is understood in two ways since dependence on a relatively unified upper being is supposed to supply the self-identity which the lower beings lack, the upper being must in some sense be just what the lower ones are, only more so one can see this as manifestation, so that's the way I showed it in figure six. The upper being brings the lower into existence as an imperfect image or representation of itself. And the medieval concept of eminent being, famously appealed to by Descartes' second meditation is a version of this approach. But one might also see the upper being as a consciousness or conscious state in which the lower being is represented and known, and that's just shown in figure nine. The existence of a lower being within the higher is then a form of intentional existence, or what they call objective being. So such knowledge by representation involves assuring inadequacy. Insofar as the thing known is very different, different in mode of being from its representation. But here the inadequacy is on the side of the known object, not of the known object. And as Thomas puts it, all material things pre-exist in the angels more simply and immaterially than in themselves.

22:30 Now, for Thomas, these two relationships are separate. Actually, I'm not sure about this now, but I'll say it anyway. For Thomas, these two relationships are separate. All angels know material things, but not every material thing is the assumed body of an angel. In Leibniz, however, as in Plotinus, the phenomenality of material things means that they are both appearances of intelligible substances and appearances to them. And Husserl, who refers to Leibnizian monotology as, quote, one of the greatest anticipations in history, that is, anticipations of Husserl, expresses this tight relationship as the law of correlation between noetic and normatic components of consciousness, between consciousness regarded as positing its objects and the intentional being of those same objects within consciousness. The correlation between the two means precisely that intentional being is eminent being. Hence, what we ordinarily describe as the fallibility of external cognition is reinterpreted as the inadequate being of dispersed external things. They may turn out not to be what they were, what they were posited as being, because they never are quite what they are. They never are quite self-identical. Now, such self-identity as they do possess derives from the process of perceptual synthesis by a continued harmonious positing within the same stream of consciousness. It derives, in other words, from a higher self-identity of the upper world and ultimately from the transcendental unity of our perception that is the absolute self-identity of the transcendental ego. Now, this picture clearly has its disturbing aspects. I will not go into detail here as to why Heidegger and Karnak both reject it, but we'll instead discuss their way of doing so. In particular, I want to focus not on the dismissal of transcendental phenomenological idealism along with all its apparent competing positions as pseudo-answers to a pseudo-question, but on the rejection of the concept mode of being in terms of which Husserl's position is stated. Now, it sounds odd to describe them that way, as I've already admitted. The early carnival actors are, after all, both much concerned to prevent confusion between different modes of being. As early as this habilitation script, Heidegger was already warning about the dangers of, quote, a fateful confusion of realms, if he's been angry, by which, for example, mathematical, logical, and real being might be confused.

25:00 He does so, moreover, by an explanation of the traditional structure we have been discussing in the shape given to it by Scotus. The concern continues into being and time, where one central theme is careful distinction between being as two-hand and height, four-hand and height, and dasein, along with the danger of leveling all down to one mode, namely that of the four-hand. Karnap, meanwhile, in an outbound, presents confusion of spheres there in Kammengung as a major source of philosophical error. And here the connection to Husserl is even clearer since Karnap's main spheres of being correspond precisely to Husserl's, his auto-psychological corresponding to Husserl's pure consciousness, physical, psychic, and geistish. Nevertheless, Heidegger and Carnap both also insist that being as such is unified in some strong sense not allowed by wisdom. Thus, the most important thesis of Carnap's constitution theory is that, quote, the objects do not disintegrate into different unconnected realms, but rather there is only one realm of objects, and therefore only one science. In contrast to the worry about Sperrin-Femming, which ceases to interest Carnap much once he adopts on allowable systems of logic, this latter thesis under the name Unity of Science was one Carnap would never give up. And in his autobiography he describes it as, quote, one of the main tenets of our general philosophical position. Sorry, conception. Here in the outbound he defends it in what looks superficially like orthodox Susurlian terms. All objects he tries to show belong to a single region. They are all logical derivations of a single fundamental object type. There is a single basis upon which all scientific objects can be constituted, that is, a single highest genus of what Husserl would call individua. The only obviously un-Husserlian element, and it is a profound one, is Carnap's insistence that we may choose, based on our purposes, which genus to take as basis. In Alfvau, Karnoff actually adopts as basis elements the fundamental erlebnisse, corresponding to the pure erlebnisse, which are the individual and individual vision of pure consciousness. But he makes it clear that other types of constitutional systems are equally possible. As for Heidegger, his fundamental ontological question, the question of the meaning of being,

27:30 is just the question that the traditional and Husserlian accounts would rule out. For Husserl, as for Aristotle, there is no meaning of the word being, which is equivocal, and also nothing like an essence of being, no higher possibility against which the actuality of being can be understood. What all beings share is first something merely formal logical, that is, the status of each as a something, and second, a dependence on the transcendental ego and its acts of positing pure consciousness. The first offers no kind of essential understanding any understandings that there are kinds of positive. What being comes to for a given object can be settled only by examining the sense of the appropriate positive acts. And even though Heidegger, in his habilitation, still sounds too surly on these issues, his choice to work on Scotus, who famously disagrees with Thomas about precisely the issue of whether being is predicated univocally, shows that he is starting to rethink it. And as in Karnap's sense, it is the concern with the unity of being which was to prove persistently central in Heidegger's later time. Now, it is true, however, that in their earliest mature periods, Heidegger and Carnap criticized their predecessors, and implicitly Husserl, both for over-unifying being, mixing different regions. There's kind of a table on the handout for you to follow this problem. They criticize their predecessors both for over-unifying being, that is mixing different regions, and for not unifying it enough. So let me pause to say a bit about that. Now the real target for both Heidegger and Carnac is neither the unity nor the disunity of being as such, but rather the relation of metaphysical dependence which supposedly stands in for a true generic unity. As they see it, we come to imagine such a relation when we officially declare to be mode of being is treated by us in a kind of double think or double speak, as if it were just one mode taken twice, as if, that is, the two sides were sitting next to each other in the same conceptual space. Hence, when Heidegger accuses his predecessors of leveling, he means that they actually, without realizing it, did give an answer to the fundamental ontological question, the answer that being is presence or . In particular, Dasein, the being for which something can be present, ultimately itself

30:00 gets interpreted as a special, absolute kind of presence, so that the asymmetry of presence for looks like the dependence of a relative contingent presence upon the absolute and necessary one. In Carnap, this connection between too little unity of being and too much is less explicit. He does suggest in section 180 of the Aufbau that Sparren-Premnemu is the root of all metaphysical but declines to give any details. But presumably the thought is that when we describe positing or leibnissat as prior to external things, we are really trying to propose a language, one with an autopsychological basis so that fundamental or leibnissat are objects and external things are quasi-objects. And given such a language, we could reduce suruchlurin external objects to autopsychological ones by invoking the relevant explicit definitions, hence eliminating all nearly quasi-proper names. But our existing customary language has no tag structure. So I can say Adel Neuwat weighs 150 pounds, and I can say this stone weighs 150 pounds, and I can say Adel Neuwat is thinking about Vienna, but I can't say meaningfully this stone is thinking about Vienna. That shows our customary language doesn't have a tag structure. On a plane of contradiction, therefore, it can contain no such object-quasi-object distinction. Hence, the relation of reducibility gets interpreted as phenomenological dependence of one kind of object on another, which in fact is precisely Husserl's use of the term Zerufu. Now, for both Heidegger and Carnap, in contrast, modes of being are not kinds of objects at all, and this is what allows beings to be unified even though or because its unity is not out of the genus. For Heidegger, and here you can see the left-hand side of the figure 10, the whole notion of a kind of object, that is, of a classification of objects according to their essential characteristics, is appropriate only in the realm of the poor high. In a sense, then, he recognized only one highest genus of individua, namely the genus of poor high individua. precisely the unity of being leaves no room for another such genus. But this one highest genus plays no fundamental role either ontologically or epistemically. What is true and immediate, to nuts and to maist, is the realm of the Suhaddonites,

32:30 while the ontologically primordial is Dasein in its temporality. Now for Carnap, similarly, there is only one highest genus of true objects, the constitutional basis. But the possibility of multiple constitutional systems means that while one need not recognize more than one most general kind of object, the choice of which to recognize is purely practical. So, again, the unity of the chosen basis is a consequence of the unity of being rather than its ground. Karnat, moreover, unlike Heidegger, does not speak in any system about beings lying outside the one highest genus. So for this you can see the right side of figure 10. Each thus accuses the other of extreme metaphysics. For Heidegger, Carnap is a final extreme leveler of being to forhandenite, because it's only forhandenite, whereas Heidegger for Carnap is an extreme doublespeaker who describes objects outside of all objects. In any case, whatever their other advantages or drawbacks, neither Heidegger nor Karnak's moves will be satisfactory if they cannot address the original motive for introducing modes of being. They must, in other words, encounter the possibility of synthetic unity, the unity within disunity which characterizes the sensible world. For it was the failure of the sensible Porhan to be fully Porhan, present all at once, which prompted the search for a realm of the absolutely present being lying behind it. Or, it was the recognition that self-identity within the sensible world, for example that of Heraclitus' river, is mere, quote, gen-identity, and is such arbitrary or conventional that originally prompted the search for a realm of absolute self-identity. Now, it's tempting to say that Heidegger and Carnap both replied that unity within diversity is the unified meaning of being itself, not a characteristic of sensible being as such. This way of putting things sounds too Hegelian, however, so that would have been the other talk I should give. As if the transcendental unity and diversity lay within being qua constant theoretical reason, being as the first and emptiest logical determination. Whereas in fact, both Heidegger and Karnapp, and therefore we too, are profoundly un-Hegelian and if anything closer to Schopenhauer.

35:00 They hold what one might say, although neither would approve this formulation, that all beings are characterized by merely relative unity, by unity within diversity, because the only absolute unity is the self-identical, that is, autonomous will. Hence, they reject both speculative metaphysics, phenomenology as a rigorous science, and logic as theoretical formal ontology, that is, as the study of an analogous structure to which we know all beings must conform. Both want to return rather to logos as speech or discourse with the fact that absolute practical unity, unity of practical reason, is the condition for ontologi in the sense of seriously, responsibly speaking and hearing what is. Hence, when Carnap repeats characterization of an object as that of which something can be predicated, he means that the very concept of a being, object or quasi-object, is relative to choice of language. More accurately, relative to choice of a logically perfect language, a system of symbols which can be seriously and responsibly chosen as a means of communication. Self-identity of objects in such a language means that seriously to say one thing about something else. So the principle of non-contradiction, in other words, is in its primary sense not a theoretical law but an imperative. In our, quote, difficult problem situation, this is a quote from the preface to the alpha, our duty is to avoid, quote, the worst thing that can happen to a scientific theory that it falls into contradictions. And similarly, for the empiricism side of logical empiricism, the principle of verifiability is in its primary sense a duty to speak in such a way that whenever a claim rules out its contradiction, we can defend it by appeal to public criteria. Logically perfect languages will contain theoretical echoes of the practical principles, so the law of non-contradiction will be a logical axiom or theorem, and the principle of verifiability but the practical principles which require us to choose such languages in the first place remain primary.

37:30 Similarly, although each such language acts as a definition of being insofar as every being must be constitutable, still the primary definition of being is practical. A being is that which we are duty bound to constitute. One might say then that for Karnap, the unified meaning of being, because of which all individual beings have merely relative unity, the problematicity of the problem situation, in the face of which the will is called to express its unity by choosing responsible speech. A being to kind at is that about which we must bind ourselves to say responsibly and unequivocally one thing rather than another. Heidegger's view is almost the same. Part two of Being in Time, which was never written, of course, which would have addressed the fundamental ontological question, oh, it was never written. But we know from the introduction that the reason fundamental ontology is phenomenology, the reason being is that which, quote, must be called phenomenon in a instinctive sense, is that individual beings are what always, at first and for the most part, show themselves. Being as such, the being of beings, is, quote, that which, at first and for the most part, does not show itself, which is concealed, and which yet is at the same time something, et vas, essentially belonging to the individual beings. Thus, being is that which distinctively demands phenomenology, a logos, a speech or discourse or veda, which shows forth what is to show itself, but for the most part does not. Here, too, then, the unified meaning of being is demand for speaking forth. And as we know from the later parts of the book, moreover, the fallenness of reda into gereda, the fallenness which is responsible for covering up, or pseudostai, that is, for covering the Zionist praga with shinefragen, with pseudo-questions, is characterized precisely by equivocation and irresponsibility. The meaning of being, then, is demand for a speaking forth which is unequivocal and responsible. And here, too, this unified meaning of being accounts for the disunity of all individual beings, for the meaning of being is essentially related to temporality. A demand for the uncovering of what is always already covered presupposes the structure of temporality as ecstasis, as standing outside oneself. All beings fail a complete self-identity because being as such, the being of beings, is a call

40:00 to reunify in speech what has always already failed to be itself. Now, to bring Heidegger and Kleiner together, I have not only employed terminology which neither would accept, but obviously also ignored some big differences between them. And two evident ones are, first, Heidegger's emphasis on Reda rather than Schraffer, and second, the fact that being is for him not only a call for disclosure, but also itself a something, or ethos, to be disclosed. The latter is tied to the difference mentioned earlier, Heidegger's willingness to speak about what lies outside the realm of Orhan, and also to the absence in Heidegger of the doubling, which in Carnap later on becomes the distinction between object language and A further and deeper difference is that while Carnac's emphasis is on the choice of a way to speak from now on, Heidegger's is on the interpretation of what has, so to speak, always already been said. And the logos of phenomenology, as he puts it, is a hermenele interpretation. All of these differences are closely related. Without going into them further, however, let me point instead to one final commonality between the two. At the early stage we have been discussing, each felt that the requirements of responsible speech mandate a reinterpreted form of Husserl's stepwise, egocentric system. Thus, Carnap calls for a tight theoretic language with auto-psychological basis, whereas Heidegger claims that fundamental ontology must proceed by an interrogation of Dawson. By the mid-thirties, however, Carnap came to advocate a physicalistic system in which the type structure, if any, no longer has any epistemic significance, and Heidegger, a version to fundamental ontology in which the primordial being to be interrogated is not Dasein, but Pusis. The structure of their common position soon moved both of them away from methodological solipsism and towards physicalism. Thank you. We have about ten minutes for discussion. Do you want to take the other questions or should I leave the answer? Sure. That's a question of clarification of being the highest genus. Yeah. I couldn't quite see why it seems. I mean, why is not the same shot to everything which exists to be a being rather than life?

42:30 Well, I don't want to answer that in general. I mean, there is actually an argument in Aristotle why being cannot be a genius. But it's a weird argument. But I think for the people I'm talking about, I guess I would say the strongest argument for it is that argument I was making about that the source of unity for the dispersed or dependent things has to be a meaning in a higher sense than they are. So the principle that Torkinus cites in this respect and those versions of this that go on is that they can't be prior and posterior in the same genus. Now, I mean, how can I make that, as opposed to saying this is a traditional principle which I don't have burdens, or how can I make it seem compelling? I think one of the ways they think about it maybe is that, and I try to kind of get into this in the way I explained in the paper, is that that source of unity is the reason for those lower beings having anything like an essence at all. So it can't share any part of their essence, because that would mean the essence is causing itself. This is the way Thomas Aquinas explains. I don't know if that, I don't know if it's better to try to get that actual explanation or just to say, look, they think, you know, an accident is not mean in the same sense as the substances, so they don't have anything in common, but is that helpful, though? It would be. I think you said we. Hi, please. Hi, right. Yeah, several times you said we, and I'm not sure if there is a we or what it means to do. And secondly, the impression I have that we want to be involved in the comment together to make both of them one comment or something like that.

45:00 And perhaps that's to be identified with the gene. So that's one of the issues. Okay, this is an important question. First of all, the reaction you're having, I did think that's the opposite reaction from people who are carnal enthusiasts. You're trying to make them both one-minded. So that makes me feel like I've reached out some of them in the right track. But I don't get to the question of who the we is here. But by saying we intellectual descendants of Heidegger and Carnap, I'm kind of implying that what we call analytic and continental philosophy, that Heidegger and Carnap are kind of foundational figures for those two sides. And I mean, I don't want to make them both the same thing at all. That's why at the end I talk a little bit about the differences between them. That's not the topic of this talk, really. The reason I want to bring them so close together is so I can actually say what the difference is. Before doing this, it looks like there's just no way to prepare them at all. You open up being in time and you just think, even though it's implausible in the face of it, because as Michael Friedman and other people have pointed out, they travel in the same circles, they read the same stuff, you can see they're citing the same words, but still you open it and you say these people are flying in totally different universes. I'm trying to show how that's not true, to bring them as close together as possible so I can say in this way, and of course the reason I want to do that is because I want to understand why we are not, why there's a difference between us. There is a mistake in putting it that way, though, which is of course that we're not, I mean, and after all, I'm well aware of this. My teacher, Stanley Covell, is my thesis advisor. He's neither a delightful descendant of Heidegger nor California. So I'm well aware that that read leaves someone out, including me perhaps. But still, I think there's something important about it. I mean, I do think that, because I think that someone like Stanley Covell exists in an institutional framework. I think this is true even for people who work in a continental philosophy in a certain type of annual American department.

47:30 They exist in an institutional framework which derives from Carnot, basically. Not just Carnot, obviously, I'm simplifying. Does that help? That's what I thought. You don't like it. Okay, since you mentioned Michael Friedman in this connection, on a point, if you like, about the general intellectual history rather than on the substantive issue, huge and certainly substantive metaphysical issue about the unity and disunity of being, it seems that the common origin to which both Carnap and Heidegger can plausibly be said to trace back the problematics of their respective positions is Kant, is their respective readings of Kant and that the bridging figure there is Kassara. Certainly this is a claim which Michael Friedman has made I think quite persuasively and that the point at which the Davos disputation was particularly crucial in this respect for the divergence of their readings of Kant. Would you have a view on that? Well, I mean, so first of all, obviously his treatment wasn't persuasive to me, because I disagree with it. But I'm not sure what to... I mean, first of all, I do think that Kant... I mean, Kant didn't come up in this paper, but I do think that Kant is very important to both Heidegger and Carnap. However, I think that Kant's theoretical philosophy is read by both of them very much to the original, not to the Sera. I don't think even Carnap was reading I actually have a paper about this as part of it. It deals with Nattrop, not Cassiro. It actually kind of mentioned Nattrop more than Cassiro. I think the Davos thing was kind of a red herring. It's true that they were all three together then, which is interesting. Anyway, I have this paper called Nattrop, Husserl, and Karnap on an Object as Infinitely Determinable acts and what i try to show them i think this is true the way to see the way to see i'm not to think about this is basically it's a it's it's one way of reading khan which goes to dishta and so forth and the way karnia stands that and therefore the way he reads misreads is based on a completely different branch of khan's interpretation which is kind of

50:00 an attempt to square Kant with Brintano by making him more like well that's obviously a very complicated dialectic so basically so you reject the idea that there ever was a point at which Carnap and Heidegger had a common point of departure no no I think they do but I think it was Husserl not Husserl and I do think that Kant's practical philosophy is important to both Heidegger and Carnap in basically the same way remember I skidded over like why no kind or kind of like this picture that this one ends up with and i think it's because they both feel like it blows out the possibility of human freedom um but that's a different side i just want to give a small comment on the problem of being not being the heroes it is related to the problem of the difference between existential and From Plato up to Kant you will find the difference. Being does not belong to, indeed, this hierarchy of being somethings. There is a difference and the difference is necessary between being, being something, because you want to be able to speak about something in countries without falling into contradiction. There's a logical formulation, but there is an ontological counterpart in the metaphysical system. That's why being itself does not belong to the hierarchy of properties, if you want to say it in a la-la-la-la-chronicity. But they need it for the consistency of their system. The way Heidegger treats it in the I'm, I think, to sign in sight is a bit tricky, because it's more like Thomas of Aquino than what Aristotle himself means with it, but that's basically the reason. Well, I mean, that's very close, I think, to Aristotle's actual argument for this. But as you say, it's not close to the mystics that have a numeric platonic reading of this, which, as you point out, is the one with actual influence on these people. I agree. So, therefore, it's a good answer, but not necessarily with much in this context, I think. But I'm not sure I know a lot about the students, you know.

52:30 Well, you ended the session on time. Thank you, Abe. No, I don't mind. Very interesting. I certainly see the point about the cells and the medicine of Scotland was obviously the guidance teacher. But I don't know. I can't more. Do you have this question? I have actually, I have actually attended... He also attended presently, of course. Well, yes. Well, yes. I was persuaded by the reading today of the importance of this. and I came up in the store, and I heard of electronics in the College of France, which was earlier this year. But then I know Heidegger, or a scholar, and my background is in the math, as you probably know, in the history of philosophy, and that. But still very interesting. One thing I was fascinated to learn is that Heidegger lit upon the history of Greek mathematics. So the final plenary speaker is Richard Arthur, who lives for many years as one of the main Newton's powers in the world, but published far more on white nets and the infinite, and I think also on space time, but I think it's not as familiar as your work here. And we have a great pleasure to introduce today's talk, X plus BX is X, white nets are convenient into decimals.

55:00 Well, thank you very much, and I would like to give a big thanks to the organizers, we may not have another opportunity to thank them, so I think they'd be good to thank them now. Yeah, yeah. So there's one for occasion. Like many of you, like many others here, I've had opportunities to think that maybe I'm not giving the right talk and should have given it on something else, and I've been many times thinking about what Leibniz has said about identity of bodies and his theory of substance, I thought I might well be talking on that. But I'm talking on this because I thought that this gets at a certain issue in the theory of identity. And I've been doing some work recently on this theory of the infinite and this theory of infinitesimals, and I'm basically going to be speaking about that. So, first of all, the problem. This is a paper by John Ehrman, and, well, first of all, there's the remark by Bishop Barclay, which is pretty famous. I admit that in the original paper, we should make this a bit darker, the room. Over there, over there. Yes, yes. That's better, yeah, much better. Yeah, this is a famous remark by Bishop Barclay on Newton's analytes. I admit that in the original notation, x plus o, o might have signified either an ultimate or nothing, and that was a lot of confusion about Newton's having used o and people thinking it was a zero. But then, which of these so ever you make it signify, you must argue consistently with such its signification and not proceed upon a double meaning. So in a classic paper, John Elman takes much the same line, he quotes this from Berkeley, and he says, for instance, to get the tangent of the curve of y equals x squared over a, Consider the point, he says this is how Leibniz proceeds, consider the point y plus dy, x plus dx, infinitesimally close to the point yx, thus arriving at the formula dy by dx is equal to 2x plus dx over a.

57:30 Here the ratio dy by dx is defined because neither dy nor dx is strictly zero, right? They're infinitesimals, they're not zero. Now Leibniz sets dx over a equal to zero on the grounds that relative to the finite quantity x, dx is incomparably small. Thus dy by x equals 2x plus a, which is the valid term in dx. But now this equation must be interpreted as an exact equality, not as an equality up to an infinitesimal quantity. To put the matter succinctly, a Leibnizian equation such as x plus dx equals x, taken as a literal equation, has only the solution dx equals 0. This would justify the move of ignoring the term in dx, but is inconsistent with the interpretation of the formula 2, in which neither dy nor dx was 0, so if you had dy and dx being a finite ratio. It's true that the error in equating 2 with 3 can be made smaller than any assignable, that's the equation from Leibniz, but I'm going to insist a contradiction is a contradiction, even if it is only a small one. I have a few things to say now about Leibniz's Law of Continuity. According to it, a parabola can be regarded as the limiting pace of an ellipse, as the second focus of the ellipse is removed from the first. In the same way, rest can be regarded as the limiting motion as it becomes gradually slower, or a point is the limit of a line as the line is reduced in magnitude, even though in each example the series of cases or species terminates in an opposing quasi-species, rather at rest or point. I'm just giving those quotations because there's some of Deleuze's criticisms which go on the same line. He gives a rather subtle interpretation, quoted about Ehrman insisting that a contradiction is a contradiction even if it's a small one, Guglielder does gives a more subtle interpretation, and he coined this term, vice-diction, I did not have it in the second, I was pointing the wrong thing, to describe such banishingly small contradictions. He says the neologism, vice-diction, is intended to capture this state of affairs,

1:00:00 although the parabola is essentially different from the ellipse, It includes it as a limit in case. It does not contain the other in essence, but only with respect to properties or in cases. Thus, in the infinitely small, the unequal vice-dix de-equal and vice-dix itself to the extent that it includes in the case what it excludes in essence. So this is Deleuze's somewhat enigmatic interpretation and as an attempt to get around this contradiction in my sense. But as he himself admits, it's not entirely successful. It doesn't really get rid of the problem. There's still a contradiction of the level of properties or cases. In reality, he says, the expression infinitely small difference does indeed indicate that the difference vanishes so far as intuition is concerned. it matters little whether the supposed negative of difference is understood as a vice-dicting limitation or as a contradicting limitation what is at issue, according to Deleuze is the alternative between infinite and finite representation so he says this is an extended quote that is why the metaphysical question was announced from the outset, why is it the differentials are negative and must disappear in the result. It's obvious that to invoke here the infinitely small, and the infinitely small magnitude of the error, if there is an error, is completely lacking in sense and prejudges infinite representation. Prejudges, then, in the sense of presupposes. The rigorous response, and this is, of course, a translation from the French, and I have to come back to check the original. The rigorous response was given by Carnot in his famous reflections on the methods of the infinitesimal calculus, but precisely from the point of view of a finite interpretation. So he's counterposing the infinite representation he sees as underpinning lightnesses viewpoint, one completely lacking in sense, with the finest interpretation of the calculus served by Cronaut, one that's now become the norm. On the finest interpretation, the fate of the calculus, as Deleuze's term,

1:02:30 is no longer tied to infinitesimals. Next slide. Another extended quote, and this is interesting for our purposes here. the numbering I've introduced here. We know in effect, says Deleuze, that the notion of limit has lost its poronomic character and involves only static considerations. That variability has ceased to represent a progression between all the values of an integral and come to mean only the disjunctive assumption of one value within that integral. That the derivative of the integral will become ordinal rather than quantitative concepts. And finally, that the differential designates left undetermined, so that it can be made smaller than any given number has required. And he goes on to say, and this is one of the reasons that I've introduced to Lewis into this, he goes on to say the birth of structuralism at this point coincides with the depth of any genetic or dynamic conditions of the calculus. So he's drawing quite an interesting and provocative connection between the birth of of the calculus. Now, as a statement of the received view of the status of Leibniz's infinitesimals, I think this appraisal cannot be faulted. This one up here. But I do not think it's a correct interpretation of Leibniz's own position on infinitesimals. And what is What's interesting about Deleuze's insight for remark about the connection of modern analysis with the birth of structuralism himself upheld for, and that's what I'm going to be arguing today, but he didn't uphold one, two, and three. And so if his position makes any sense, and if Deleuze's interpretation is correct, then Leibniz's position is quite interesting, because it gives an alternative to the point of view that, according to Deleuze, led to the, or at least it's connected with the rise of structuralism. Thus when Deleuze says there's a treasure buried within the

1:05:00 so-called barbaric, I don't know who's called it, or pre-scientific interpretations of the differential calculus, I agree, even if I do not believe it's correctly identified where the treasure lies. If I'm right about the foundation Leibniz gives its calculus, and Deleuze is correct in his analysis of the connection between the modern foundation and structuralism, then Leibniz's own interpretation of the calculus contains the seeds of an alternative to structuralism. Leibniz's commitment to the actual infinite has been much misunderstood, and that's I'm just going to say a little bit about this now. This is a quote from his new essays. It's perfectly correct to say that there is an infinity of things, i.e. that there are always more of them than can be specified, but it's easy to demonstrate that there is no infinite number, nor any infinite line or other infinite quantity, if these are taken to be genuine holes. For this reason, he's generally being accused of inconsistency by Cantorians, beginning with Cantor himself, and this is from Cantor's Grundlagen, it's my translation, even though I've quoted many places in Leibniz's works where he comes out against infinite numbers, and this would be one of them, I'm still, on the other hand, in a happy position of being able to cite pronouncements about the same thinker in which, to some extent in contradiction themselves unequivocally for the actual infinite, as distinct from the absolute. Actually, I'm not going to be saying anything about the absolute. Cantor and Leibniz agree completely on the absolute. Actually, they agree on many, many points, but they have a very subtle and very interesting disagreement about the actual infinite. And that's what I'm going to talk about. So, a lot of modern authors who agree with Cantor, people like Nicholas Resher and Gregory Brown, they've said, well look, he understands the need for the actual infinite, but because of his commitment to the Park Hall axiom, which is really old fashioned, he should have listened to Galileo, he didn't see this opportunity that Campbell had seen. So that's been a criticism of Leibniz. I think, however, this misses the sovereignty of Leibniz's position, and I've argued this in many papers over the years. According to the account that I've given, that I think is the correct one, the term infinite does not refer to an entity, even though it makes perfectly good sense in appropriate contexts.

1:07:30 It is, to use the terminology of the schools, a syn-categorimatic term, that's the term used for precisely the expressions that didn't refer but made perfectly good sense in specified contexts. Thus Leibniz holds that although there is an actual infinity of the parts into which matter can be divided, to use his leading example, this does not mean that there is a number infinity that can be assigned to those parts. Any piece of matter is actually, not merely potentially, divided into further parts, but there's no totality or correction of these parts. In fact, because if there were totality there would be an infinite number and vice versa to, as Cantor also recognized, do go hand to Cantor. His argument for his actual infinite revision is given in many places. This is, I think, one of the clearest. It's in an unpublished manuscript, in a volume that I published of English translations and Latin originals of his writings on the continuum. Created things are actually infinite. For anybody, whatever, is actually divided into several parts, whatever is acted upon by other bodies. And any part whatever of a body is a body by the very definition of body. So bodies are actually infinite, i.e. more bodies can be found than there are unities in any given number. So we can actually express this distinction, this follows what A.W. Moore in his book on the infinite laid out I couldn't get the quantifiers to come up, so forgive me, A being the wrong way up and E being Dr. Frank, right? For all x and for all y, so to say, to a certain infinity of things, synchetical amount is to say that for any finite number x that you choose to number the things, there is a number of things y greater than this, that is for all x there is a y such that if x is finite then y is greater than x. with, you know, with x and y's numbers. And that's to be counterposed with the categorical African term, which would be to assert that there exists some one number of things, y, which is greater than any finite number. And so simply the order of quantifiers is that it's different

1:10:00 to the next order of the difference in the world. That's the same example you often get when I'm talking about the quantifier, I can't remember what it's called, you know, the Inverting the Quantified Fallacy. I think it actually has various names. So, you know, for instance, for every person you take, that person has a mother, it doesn't mean to say that there is a mother who is the mother of every person. Alright, so that's the I gave a lecture on this in Berlin in 2001 and Hideo Ishiguro was in the audience and she said that that's completely in agreement with the interpretation I've given of Leibniz's infinitasimals and do you know of it? And I said no and she said here and she put in my hand a copy of a chapter of a second edition of the book on Leibniz and I think this is very much true. I'll read a few quotations from Hideo Ishiguro. Would you like me to say anything more about the syntagormatic infinite? Was that reasonably clear? I mean I can give a concrete example if that would make sense clearer. Would you like an example? Yes, yes. Let's go back to Euclid. And Euclid has proved that there are an infinity of primes. So that's normally taken to mean that there's an infinite number of primes, so that, you know, you have 0, you have power 0, that's the cardinality of the primes. But, let's look at the actual structure of Eugard's proof. He supposes that there is a greatest prime, so reductio argument, one of the first and most beautiful examples of a reductio. He supposes that there is a greatest prime, that is, that there exists some finite prime such that all primes are less than or equal to it. He then constructs a number as follows. Take the supposedly greatest prime and multiply it successively by each prime smaller than it. Then add 1 to the result. This is now a number which is greater than all the primes. It's greater than all the primes, but it's not divisible evenly by any of the primes,

1:12:30 because it's all the primes plus one. So it's therefore a prime itself, and therefore the one you started with was not the greatest prime number after all, so that negates the starting supposition. So in terms of the logic, you start out assuming that... Assume that f means that x is finite and x and y are ranging over prime numbers. You can assume that there is some great x prime. contradictions, so the contradictions, yeah, it's not the case that there is the greatest prime. It's not the case that there is the greatest prime. You can write them here, so and that's going to be equivalent to this. So if you look at the result of what you've got, you've got the same category of magic For every x, there exists a y, so for every finite number, for every finite prime number, there exists a prime n squared to x. That's the synchategorial argument, that's not the categorial argument. So you don't need to assume that there is an actual infinity in the Catholic sense of primes to make sense of Butherford's argument. Butherford's argument actually establishes the synchategorial argument. And so to assume that there is a cardinality, that there is an infinite number of them, in other words, would be an additional assumption. Of course, you know, Cato didn't make any such mistake. He was quite clear that it's an additional assumption to assume that there is a number after all the quiet encounters.

1:15:00 All right. I'll leave out a bit about contrasting this with the potential in the interest of time. So I was talking about Hideyashogoro's interpretation. She says, Lightness's position is analogous to Russell's regarding definite descriptions. This is a and have a rigorous language of infinity and infinite decimal while at the same time considering these expressions as being syn-categoromatic in the sense of the scholastics, i.e. regarding the words as not designating entities, but as being well defined in the proposition in which they occur. So this is contrary to our usual understanding faithfully recounted by Deleuze, where Leibniz is understood as committed to an infinite representation. Even Heng Boss, whose profound contribution to the understanding of Ivens's differential calculus we will depend on in this paper, takes Ivens to have provided two different approaches to interpreting infinitesimals. One is finest and intermediate, in which differentials are interpreted as finite differences that may be taken so small as to lead to an error less than any assignable, and the other is based on the law of continuity and accepts infinitely small quantities as true quantities of their own sort, but insists on interpreting them as fictions. But as Ishiguro has argued, these approaches are in fact two sides of the same coin. To say that dx is a fiction is not to say that there exist fixed entities with non-Archimidian magnitudes, of which Chauvin's proofs. The word infinitesimal does not designate a special kind of magnitude, it does not designate at all. This is what is meant by calling the interpretation syncategoromatic. Terms involving infinitesimals are ostensibly designating expressions which follow sui generis rules, and whose introduction Chauvin's proofs that they do not in fact designate real entities. The syncategoromatic interpretation explains how it's possible to treat infinitesimals as if they are infinitely small actuals and a certain long time conditions. As Ishikura puts it, when we make reference to infinitesimals in a proposition,

1:17:30 we're not designating a fixed magnitude in comparables smaller than our ordinary magnitudes. Gladness is saying that whenever, sorry, whatever small magnitude an opponent is present, one can certainly take a small magnitude. Okay, so in the remainder of the paper, what I want to do is show how, if you look at the examples of what Leibniz actually did in the Leibniz Foundation for the campus, you find that it beautifully exemplifies what she wrote and laid out. I think she may be based her interpretation just on quotations from the later Leibniz, and this is quite interesting that it does correspond extremely well with his mathematical practice. Excuse me. So, what is the Archimedean axiom? We also discussed something about that. In Euclid's elements, of course it was usual with the docks that's not Archimedes, but it's come to be called Archimedes axiom because he uses it so much. Magnitude is said to have a ratio of one another to be capable, when multiplied, if exceeding the number. That is, for any two geometric quantities x and y would be like a than x, a natural number n can be found such that nx is greater than y. So it will be a corollary, automatically, of that, that no matter how small a geometric quantity is given, a smaller one can be found. Now, Weibitz wrote this wonderful manuscript in 1676, which he left for the... He was trying to get entry to the Académie Francaise, and he left it with a trusted friend, who didn't do anything with it, and then he lost it. And he didn't have it for a long time, I'm not quite sure. He didn't know what he had, I think, and he sort of matched himself with stuff by making some sarcastic smile at the wrong point or something. Anyway, Eberhard Robloff in Berlin did a wonderful edition of this back in 1993, I think it was, and has given some expositions of it. And it really is quite a remarkable document. I'm going to give a simplified, his diagram for this is absolutely, well he thought of Spino's dissonance, and there's no reference to Spinoza there, it just means very, very thorny, and very difficult, and still flying through all over the place, so this is a simplified version of the heart of the thing, and this is proof of proposition 6, and also of 7 and 8, but this part is absolutely crucial,

1:20:00 As Knobloch has pointed out, what Leibniz does here is to give a perfectly rigorous demonstration of what's come to be called Riemannian integration. And Newton does too, so it's a bit of a mystery why it's called Riemannian integration, because both of you give a perfectly good justification of it. So, the basic idea is you're building up the area under the curve by lots of rectangles, but they can all be unequal. but it would be identical unlike the identical in width as well. And so he's looking at the mixed-linear figure, which is basically the area under the curve here, between these ordinances, and this is typical Widenst's terminology, the little subscript for the letter, 1L and 3D, and Q would be the sum of the elementary rectangles, now looking at the rectangles themselves. And what Leiden shows in this proof, that's the same figure there, is that the difference between A minus Q can be no greater than the area of a certain rectangle. It's really ingenious and very nice argument for this. A certain rectangle whose height is the maximum height of any of the elementary rectangles, and whose width is 1L through B. So A minus Q is less than equal to, again, I couldn't think to reproduce that symbol, less than or equal to, not having a pentamute symbol, that area. Now, this is like Mrs. Reason following this. But because the curve is assumed continuous, well this isn't actually this word yet, our medius axis implies, and this is what Lyons actually says. This greatest light, an abscissa, the abscissa is going vertically, I think they had the opposite notation to us, they were like, the x axis is going vertically and the y horizontal. Because the curve is assumed, this greatest height in the system can be chosen smaller than any given quantity because the curve is continuous. Thus the height, hm, which is the maximum height of all these little rectangles, even though it's greater than the heights of all the other elementary rectangles, can be assumed smaller than any assigned quantity for however small it's assumed, still smaller heights can be taken.

1:22:30 And it just appeals to this idea that no matter what quantity you're given, you can get a smaller quantity. So the area of the rectangle can also be made smaller than any given surface, therefore A minus Q can also be made smaller than any given quantity, UED. Now, there are some interesting things to say about this. it's worth recalling Deleuze's charge that it's obvious that to invoke here the infinitely small and the infinitely small magnitude of the error is completely lacking in sense and prejudges infinite representation. This would only be so if the infinitesimal areas in question were non-Archimedian or actual infinitesimals. In fact, however, what Leibniz has done is to invoke finite areas that can always be made smaller than any preset magnitude. He's justified perceiving in this case as if there were an infinity of infinitesimals precisely without assuming an infinite representation. It's also hard to see any difference in rigor between his justification and the finite, the finite justifications of Carmel and Cauchy. Thus like this appears justified in remarking about this theorem, it serves to lay the foundations of the whole method of indivisibles in the soundest way possible. so this was written at the end of his stay in Paris this was 1676 so this is just after he's actually finished the calculus this is one of the founding documents of the calculus so this isn't some late defense of the calculus against criticisms by Neely Moyes and Roth and people like that this is Linus' interpretation when has the calculus finished so the point is that he doesn't have was supposed, one committed to the existence of infinitesimals and the other Archimedean, nor is it the case that he simply uses the infinitesimal calculus and then earlier refers to the fact that one could instead have used an Archimedean method. It is that if examples like this demonstrate the Archimedean axiom justifies proceeding as if there are infinitesimals and at the same time demonstrates that what they really stand for, are finite quantities, which can be taken as small as desired. Well, thanks, I thought I was running out of time, just running out of water.

1:25:00 As Leibniz himself writes, nor is it necessary always to use inscribed or circumscribed figures and to infer by reductio absurdum and to show that the error is smaller than any assignable, although what we've said in Proposition 6, 7, and 8 establishes that it can easily be done by those means. So, the point about this proof is it's so general that it can always be done in any other quadrature at all. And so you can do a quadrature using dx and dy to stand for these arbitrarily small infinitesimals, which stand for arbitrarily small finite quantities, and proceed as if there are infinitely many of them without actually having an existence assumption that there are such things as quantities smaller than any given infinitely many of them in terms of number of them. Alright. How am I going for time, actually? We have another ten minutes if you want to wait in this session. Okay. So those, again, are examples of Leibniz's statement the law of continuity. So I just want to say something in brief. We've really only talked about quadratory, and it may be somewhat mystifying if he had this justification of the calculus early of why he didn't come out with it. Well, he did make the terms later, and I suppose he never satisfactorily completed them. There is rather a nice one that Ross analyzes very on justifying now finding tangents, so this gives a nice kind of counterpart to the quadrature one. This is, I'm just going to go through this relatively fast, so you'll have to excuse that. I'm sorry about the quality of this, I have to stick it in now. What we do, we can look at the characteristic triangle with dxdy and ds, and ds can be treated as the side of an inscribed finite polygon. So you're going around the curve with these little elements, ds. And Linus' procedure is to let this hook, I don't know how to refer to this, the d in brackets, this is another notation of this.

1:27:30 The d in brackets is going to designate a fixed line segment. So you can see it represented there on the diagonal. And dx and dy are not infinitely small. They are arbitrarily small finite quantities. And they are quantities that are variable so that they can be varied as opposed to the fixed ones which have the parentheses notation. Thank you.