Synthesis and TimeWe are returning to Kant. May this be an occasion for you to skim, read or re-read The Critique of Pure Reason. There is no doubt that a tremendous event in philosophy happens with this idea of critique. In going into it, ourselves, or in going back into it, I had stopped reading it a very long time ago and I read it again for you, it must be said that it is a completely stifling philosophy. It's an excessive atmosphere, but if one holds up, and the important thing above all is not to understand, the important thing is to take on the rhythm of a given man, a given writer, a given philosopher, if one holds up, all this northern fog which lands on top of us starts to dissipate, and underneath there is an amazing architecture. When I said to you that a great philosopher is nevertheless someone who invents concepts, in Kant's case, in this fog, there functions a sort of thinking machine, a sort of creation of concepts that is absolutely frightening. We can try to say that all of the creations and novelties that Kantianism will bring to philosophy turn on a certain problem of time and an entirely new conception of time, a conception of which we can say that its elaboration by Kant will be decisive for all that happened afterwards, which is to say we will try to determine a sort of modern consciousness of time in opposition to a classical or ancient consciousness of time. Why it is that it was Kant who created the philosophical concepts of this new consciousness of time, making his philosophical expression possible, does not concern us or in any case does not interest me, but what I would like to say is that it is indeed this sort of consciousness of time which takes on a philosophical status in Kant, and which is completely new. I will proceed by numbered points because I'm always working with the idea that to each point corresponds a type of concept, and once again, I will be happy if you grant me at the end of these lessons that philosophers are precisely this, that they are no less creative than painters or musicians, simply that they create in a determinable domain that is the creation of concepts. Firstly, what does Kant understand by the a priori which he opposes to the a posteriori? These are common terms. In some cases new words must be invented, and this happens with Kant when he creates the notion of the transcendental, which is a very strange notion, transcendental subject... no doubt you will tell me that the word existed before, but it was rarely used and it marked no difference from the ordinary word transcendent, whereas Kant gives it a very special sense: the transcendental subject, he almost created a word... in the case of the a priori and the a posteriori he borrows a word, but he completely renews its sense. A priori, in the first place, means: independent of experience, that which does not depend on experience. In opposition to a posteriori which means: given or givable in experience. What things are a priori? Note that I don't ask myself: does the a priori exist, which is to say, are there things independent of experience? The question of existence is secondary, we must first know what a thing is in order to be able to say and reply to the question of existence: does it exist or not? I'm saying that if it exists, what is something that would be independent of experience? Thus not givable in experience. Nothing complicated so far, Kant takes this up very quickly, the a priori in this sense is the universal and the necessary. Everything that is necessary and universal is said to be a priori. Why? It certainly fulfills the first condition of the a priori: not given in experience, because, by definition, experience only gives me the particular and the contingent. With expressions of universality and necessity it is always so necessarily, as also with certain uses of the future tense, or expressions of the type "each time": each time I bring water to 100 degrees it will boil. Philosophers have said this for a very long time: there is something in this which is not given in experience. What is it? It's the expressions: "always", "necessarily", or even the future tense. What experience has given me is, strictly speaking, that each time I have effectively brought water to 100 degrees, it has boiled, but in the formula "water necessarily boils at 100 degrees", the necessarily is not an object of experience. Similarly if I say "all objects of experience" - do I have the right to say this? We don't even know if "all objects of experience" is not nonsensical. Supposing that it is not nonsensical, "all objects of experience" are not given in experience, for the simple reason that experience is ???? Thus you can always make a summation, a sum of the objects you have experienced, but this sum is indefinite.Thus the universal and the necessary by definition are not givable in an experience since an experience is always particular and contingent. So that gives us a second determination of the a priori. The a priori was first of all what is independent of experience, in the second place it is what is universal and necessary.Third point: how can this universal and necessary be defined? There is already something extremely delicate here. To say that something is independent of experience doesn't prevent this something perhaps being applied to experience and only to it. The question of application is entirely different. When I say "water will always come to a boil at 100 degrees", I don't know where this idea of "always" comes from, since it is not given to me in experience, I don't know where this idea of necessity comes from, since it is not given to me in experience, this doesn't prevent the fact that "always" is applied to water, boiling, 100 degrees, all things which are given in experience. Let's suppose then that the a priori is itself independent of experience but applies to objects given in experience. In other words the universal and the necessary are said of objects of experience; perhaps they are said of other things as well, but they are said of objects of experience. What is universal and necessary? What would these universals and necessaries be which can be said of objects of experience? Here is introduced a notion which is famous in philosophy, that of the category. A certain number of philosophers have even made or proposed what are called tables of categories. There is a famous table of categories in Aristotle. With Kant, who did not escape a strong influence from Aristotle, there will be another table of categories. What is a category? A category is not just anything in philosophy, it's as rigorous as a scientific notion in another domain. What is called a category is a universal predicate, or universal attribute if you want. Which is to say a predicate which is attributed to, or predicated of, or said of any object. This notion of "any object" is bizarre. I say "the rose is red". What is that? "The rose is red" is not complicated, it's a relation between two concepts, the rose and red, and if I say "what is universal or necessary in that?" I can reply: nothing. Not all objects are roses, not all roses are red. Not all reds are the colour of roses. I would say that there is an experience of the red rose and that this experience is particular, contingent, a posteriori like all experience.Compare this judgement: "the rose is red" to this other judgement: "the object has a cause" or even "the rose has a cause".I see a difference straight away, which is that the concept of rose defines what will be called a class in so far as it is an a posteriori concept, the concept of rose defines a class or set. Red is a property of a subset of this set, the subset formed by red roses. I can define a set according to what it excludes and in relation to what it excludes: all that is not a rose. The set of roses is carved out of a broader set which is that formed by flowers, and the set of roses can be distinguished from the rest, which is to say all the flowers which are not roses. When I say "all objects have a cause", am I not in another domain completely? Evidently I am, I am completely in a different domain because to have a cause is a universal predicate which is applied to all objects of possible experience, to the point that I don't even need to - or I believe that - but that makes no difference because "I believe" will become an act that we will have to analyse - I believe that if an unknown object emerged in experience before my eyes, this object would not be an object if it didn't have a cause. To have a cause or to be caused is a predicate of a wholly other type than the predicate "red". Why? Because the predicate "to be caused" - to the point where we can wonder, after reflection, is that really a predicate or is it something else? - the predicate "to be caused" is predicable of any object of possible experience, to the point where it is not going to define a set or a subset within experience because it is strictly coextensive with the totality of possible experience.Moreover, we must go back. When I said that the totality of possible experience has perhaps no sense, now we have the response: the totality of possible experience makes no sense in itself, but it is precisely to the extent that there are predicates which are attributed to all possible objects, which are thus more than predicates, and this is what Kant will call conditions, they are the conditions of possible experience, it is thus via the notion of conditions of experience that the idea of a whole of possible experience will take on a sense. There is a whole of possible experience because there are predicates or pseudo-predicates which are attributed to all possible objects and these predicates are precisely what are called categories. I'll cite some examples of categories according to Kant: unity, plurality, totality (with Kant they come in threes).Reality, negation, limitation.Substance, cause, reciprocity.I'll stop there. In what sense are these categories and not predicates of the type red, green, etc...? They are categories or conditions of possible experience for the simple reason that any object is only an object to the extent that it is conceived as one, but also as multiple, having the unit parts of a multiplicity, and in this forming a totality, any object whatever has a reality. On the other hand, it excludes what it is not: negation, and by virtue of this it has limits: limitation. Any object whatever is substance, any object whatever has a cause and is itself cause of other things.That's enough to be able to say that my notion of object is made in such a manner that if I encountered a something which did not allow the categories be attributed to it, I would say that it is not an object.So there we have as a last determination of the a priori, they are the conditions of possible experience, which is to say universal predicates as opposed to empirical predicates or a posteriori predicates.I could define the categories in the simplest way as being the predicates of any object whatever. Thus you can yourselves make your list of categories according to your mood, according to your character... what would be good would be to see if everybody came up with the same list of categories. In any case you do not have the right to cheat with the word. To make your list of categories is for you to ask yourselves what is for me predicable of any object whatever. I have already given a certain list of them, with nine categories. In fact, for Kant, there are twelve of them, but I left three aside for later; you see: unity, plurality, totality, affirmation, negation and limitation, substance, cause, reciprocity or community.To finish with this first point, I am saying that the categories, qua predicates of any object whatever, are a priori, and they are conditions of possible experience; understand that it is through them that the notion of possible experience takes on a sense.To the question: does the whole of possible experience mean something? No meaning [sens] at all if we remain in an a posteriori approach, because in an a posteriori approach I am led to make an addition: the roses, the flowers other than roses, the plants which are not flowers, the animals, etc.... I could go to infinity like that and nothing tells me that I have a whole of possible experience. On the contrary, experience is fundamentally fragmented, it is opposed to a totalisation. If Kant launches this very very new notion of a totality of possible experience it is because he is in a position to define, to say: yes, there is a level where the whole of possible experience takes on a sense, it is precisely because there are universal predicates which are attributed to all things, which is to say are attributed to any object whatever. Thus it is a priori that the notion of the totality of possible experience will be founded.Is there anything else besides the categories that can be a priori, which is to say, universal and necessary? The reply is yes, and this other thing is space and time. Because every object is in space and in time, or at least in time. But you will say to me straight away, very well then, why not make a category of them, why not add space and time as two categories? Because space and time are also, it seems, predicates. Obviously, Kant has the most serious reasons to not want to and he will go to great pains to distinguish the categories on the one hand, and on the other hand space and time. There will thus be two sorts of a priori elements: the categories and space and time. Why doesn't he want space and time to be among the categories? I will give a reason very quickly which will become clear afterwards: it is that the categories qua predicates of possible experience are concepts, whereas Kant fundamentally holds that, these are a priori representations, a priori representations or concepts, while space and time are presentations. There you also have something very new in philosophy, it will be Kant's work to distinguish presentation and representation. So there will be two sorts of elements in the a priori.My second point is Kant's importance at another level, which is the notion of phenomenon, and that also is very important. There Kant operates a kind of essential transformation of a word which was frequently employed previously in philosophy. Previously philosophers spoke of phenomenon to distinguish what? Very broadly we can say that phenomenon was something like appearance. An appearance. The sensible, the a posteriori, what was given in experience had the status of phenomenon or appearance, and the sensible appearance was opposed to the intelligible essence. The intelligible essence was also the thing such as it is in itself, it was the thing in itself, the thing itself or the thing as thought; the thing as thought, as phenomenon, is a Greek word which precisely designates the appearance or something we don't know yet, the thing as thought in Greek was the noumenon, which means the "thought". I can thus say that the whole of classical philosophy from Plato onwards seemed to develop itself within the frame of a duality between sensible appearances and intelligible essences. You can see clearly that this already implies a certain status of the subject. If I say that there are appearances and that there are essences, which are basically like the sensible and the intelligible, this implies a certain position of the knowing subject, namely: the very notion of appearance refers to a fundamental defect in the subject. A fundamental defect, namely: appearance is in the end the thing such as it appears to me by virtue of my subjective constitution which deforms it. The famous example of appearance: the stick in water appears broken to me. It's what is called the rich domain of sensory illusions. So much so that in order to reach the thing in itself the subject must in fact overcome this sort of constitutive infirmity which makes it live amongst appearances. It's Plato's theme: leave appearances to find essences.With Kant it's like a bolt of lightning, afterwards we can always play clever, and even must play clever, with Kant a radically new understanding of the notion of phenomenon emerges. Namely that the phenomenon will no longer at all be appearance. The difference is fundamental, this idea alone was enough for philosophy to enter into a new element, which is to say I think that if there is a founder of phenomenology it is Kant. There is phenomenology from the moment that the phenomenon is no longer defined as appearance but as apparition. The difference is enormous because when I say the word apparition I am no longer saying appearance at all, I am no longer at all opposing it to essence. The apparition is what appears in so far as it appears. Full stop. I don't ask myself if there is something behind, I don't ask myself if it is false or not false. The apparition is not at all captured in the oppositional couple, in the binary distinction where we find appearance, distinct from essence.Phenomenology claims to be a rigorous science of the apparition as such, which is to say asks itself the question: what can we say about the fact of appearing? It's the opposite of a discipline of appearances. What does an apparition refer to? The appearance is something that refers to essence in a relation of disjunction, in a disjunctive relation, which is to say either it's appearance or it's essence. The apparition is very different, it's something that refers to the conditions of what appears. The conceptual landscape has literally changed completely, the problem is absolutely no longer the same, the problem has become phenomenological. For the disjunctive couple appearance/essence, Kant will substitute the conjunctive couple, what appears/conditions of apparition. Everything is new in this.To make things a little more modern, I would just as well say: to the disjunctive couple appearance/essence, Kant is the first who substitutes the conjunctive couple apparition/sense, sense of the apparition, signification of the apparition. There is no longer the essence behind the appearance, there is the sense or non-sense of what appears. Grant me at least that even if what I say remains just a matter of words, it's a radically new atmosphere of thought, to the point where I can say that in this respect we are all Kantians.It's obvious that thought, at that time, was changing elements. People had for a long time thought in terms which didn't come from Christianity but which fit in very well with Christianity, in the appearance/essence distinction, and towards the end of the eighteenth century, prepared no doubt by all sorts of movements, a radical change takes place: for the whole appearance/essence duality which in a sense implies a degraded sensible world, which even implies if need be original sin, is substituted a radically new type of thought: something appears, tell me what it signifies or, and this amounts to the same thing, tell me what its condition is.When Freud comes up and says that there are certain phenomena which appear in the field of consciousness, what do these phenomena refer to, Freud is Kantian. How so? In a way that is at the same time very general but also very rigorous, namely that, like all those of his era and since Kant we spontaneously think in terms of the relation apparition/conditions of the apparition, or apparition/sense of what appears, and no longer in the terms of essence/appearance.If you don't see the enormity of the reversal, admire the fact that the subject, in my second couple, the subject is not at all in the same situation. In the disjunctive couple appearance/essence, the subject is immediately condemned to grasp appearances by virtue of a fragility which is consubstantial with it, and the subject requires a whole method, it needs to make a whole effort to get out of appearances and reach the essence. In the other case, what makes the subject take on an entirely different value? It's when I say that every apparition refers to the conditions of the appearing of the apparition, in this very statement I am saying that these conditions belong to the being to whom the apparition appears, in other words the subject is constitutive - and understand this well, otherwise it's a radical misinterpretation - the subject is constitutive not of the apparition, it is not constitutive of what appears, but it is constitutive of the conditions under what appears to it appears to it.I mean that the substitution of the conjunctive couple phenomena-conditions, or apparitions-conditions ensures a promotion of the subject in so far as the subject constitutes the very conditions of the apparition, instead of constituting and being responsible for the limitations of appearance, or the illusions of appearance. There is indeed a subject, Kant will say, which is subordinated to appearances and which falls into sensory illusions; it will be called the empirical subject, but there is another subject which is evidently neither you nor me, which above all is not reducible to any empirical subject, which will be from that point on named the transcendental subject for it is the unity of all the conditions under which something appears, appears to whom? Appears to each empirical subject. It's already beautiful as a system of ideas. I hope you can feel its extent, it's a tremendous machine.To finish this second point, I'll make two corrections: Kant is at the turning-point of something, so it's more complicated than I'm making it out to be because he keeps something of the old essence-appearance difference, and effectively he will say all the time: do not confuse the phenomenon with the thing in itself, the thing in itself is the pure noumenon, which is to say it is what can only be thought, while the phenomenon is what is given in sensible experience. So he maintains the disjunctive duality phenomenon/thing in itself, noumenon. It's the duality of the couple appearance/essence. But he gets out of it and he is already in another type of thought for a very simple reason for he says that the thing in itself, it is so by nature or the noumenon - the thing in itself can be thought, it is thus noumenon, but it cannot be known. So if it can be determined, it is a completely different point of view than that of knowledge; so we don't bother with it or at least we will bother about it in very special conditions.What counts from the point of view of knowledge and of all possible knowledge is the other couple, apparition-conditions of appearing, conditions of the fact of appearing.Once again if I sum up this reversal it's the one which consists in substituting for appearance-essence, apparition-conditions or apparition-sense of the apparition.If you ask me what these conditions of appearing are, fortunately we have got somewhere because our first point gave the answer, the conditions of appearing, which is to say the conditions of the phenomenon, in so far as the phenomenon is what appears, we will not look for an essence behind the phenomenon, we will seek the conditions of its apparition, and in fact the conditions of its apparition are, the categories on one hand and on the other space and time.Everything which appears appears under the conditions of space and time, and under the conditions of the categories. By this fact space and time on the one hand and on the other the categories are the forms of all possible experience and they belong not to things as they are in themselves, but as forms of all phenomena, as forms of all apparition, space and time on the one hand, the categories on the other hand are the dimensions of the transcendental subject. Time is already completely involved here. Are there any questions?Richard: How is the difference between the transcendental subject and the empirical subject distributed? How is it very different from the domain of being?Gilles: Obviously he needs another notion. We start from the idea: phenomenon equals apparition. The phenomenon is not the appearance behind which there would be an essence, it's what appears in so far as it appears. I can add that it appears to someone, all experience is given to someone. All experience is related to a subject, a subject which can be determined in space and time. It's here and now that I put my little saucepan on to boil and light the fire. I would say that all apparition appears to an empirical subject or to an empirical self. But all apparition refers not to an essence behind it but to conditions which condition its very appearing. The conditions of the apparition - these are thus forms since apparitions appear in these forms, or under these forms - the conditions of the apparition are space and time and the categories. In other words space and time are the forms of representation of what appears.Given this if the apparition presupposes conditions which are not like objective essences behind it, but are like the conditions of its apparition to a given empirical self, we already have no more choice: the formal conditions of all apparition must be determined as the dimensions of a subject which conditions the appearing of the apparition to an empirical self, this subject cannot itself be an empirical self, it will be a universal and necessary self. It's for this subject that Kant feels the need to forge or to extend a word which only had a very restrained theological use till then, thus the need to invent the notion of the transcendental, the transcendental subject being the instance which the conditions of all apparition are related to, while the apparition itself appears to empirical subjects. That doesn't tell you yet very well what the transcendental subject is, you'll have to wait because it will be so involved with the problem of time.We just need for one little thing to suddenly become concrete, we mustn't demand continuous concreteness. There is the concrete and the opposite of the concrete, the true opposite of the concrete is not the abstract, it's the discrete. Discretion is the moment of thought. My aim is to arrive at a fabulous conception of time.Comptesse: inaudible commentGilles: The synthetic a priori was my third point. We have to begin somewhere. If I had begun there I would have needed a completely different organisation. Quite simply it seems to me that in all I have said I have not needed to assume synthetic judgements. Third point: what is a synthesis for Kant?It is common to distinguish two types of judgements. Judgements which are called analytic and judgements which are called synthetic. By definition, a judgement is called analytic if it expresses a predicate which is already contained in the subject, i.e. there will be an analytic relationship between two concepts when one of these concepts is contained in the other. An example of an analytic judgement: A is A, it's the principle of identity. When I say "A is A" I don't go outside of concept A. I predicate A of itself, I attribute A to itself, I'm in no danger of making a mistake. "Blue is blue", you will say to me that that doesn't go very far, it's obvious... because when I say "Bodies are extended" what is that? We want to reply that it's an analytic judgement. Why? Because I couldn't have thought the concept "body" - we're not saying "thing" - without having already included the concept of extension, thus when I say "Bodies are extended" I am formulating an analytic judgement. I think Kant would say something very malicious like: OK all bodies are extended is an analytic judgement, but on the other hand "all phenomena appear in space or in extension" is a synthetic judgement because if it is true that the concept "extended" is in the concept "body", on the other hand the concept "extended" is not in the concept "phenomenon" nor the concept "body" in the concept "phenomenon". Well, let's suppose that "all bodies are extended" is an analytic judgement. At least we can be sure of one thing which is that an analytic judgement is perhaps useless but it's true. "A is A" is true, no one has ever denied "A is A". In Hegelian-style dialectical contradiction no one says "A is not A", they say "A is not non-A", but just that a thing includes in its being this non-being that it is not. So they take seriously the formula "A is not non-A" in saying that the being of the thing is inseparable from the negation of the negation (is not...not), but they don't deny at all the principle of identity.In experience we have synthetic judgements, it's even in this way that we know things. When I say "Oh look, the rose is red", it's an encounter. "Red", at first glance is not contained in the concept of rose, the proof is that there are roses which aren't red. You will say that this is stupid because isn't "red" contained in the concept of this rose here? It gets complicated because is there a concept of this rose here, is there a concept of the singular? We'll leave that aside. We will say very broadly that, apparently, "the rose is red" is a synthetic judgement. You can see how this sorts itself out. All analytic judgements are a priori, it's independently of any experience that I can say that a thing is what it is. "A is A" is an a priori judgement. Still at first glance, the synthetic judgement seems by nature to be the combination of two heterogeneous concepts, the rose and the red, it establishes a link or a synthesis between two heterogeneous concepts and is by virtue of this a posteriori. The form of this judgement is "A is B". In a certain way, I'll just say very quickly, classical philosophy before Kant, just as I was saying a moment ago, is caught in the dualist couple, in the disjunctive duality essence/appearance, classical philosophy was caught, at least in appearance, in a certain duality: either a judgement is a priori and it is analytic, or it is synthetic and it is empirical or a posteriori.It became very complicated to know in what conditions an empirical judgement could be true. There is a famous and very prodigious attempt, Leibniz' attempt, before Kant. In order to found the notion of truth, he is led to try and show that all judgements are analytic, we just don't know it, we believe in the existence of synthetic judgements because we never take the analysis far enough, which is to say to infinity, it's because of this that we believe that there are synthetic judgements. But if we could take the analysis far enough, when we truthfully affirm one concept of another, the affirmed concept is always interior and contained in the one we affirm it of, to the point that - this gives Leibniz' famous theses - Caesar crossed the Rubicon, this proposition which seems eminently to be a synthetic proposition, implies the link between two representations: Caesar crosses the Rubicon on such and such a date, at such a point in space, here-and-now, which seems to be the very signature of the a posteriori, Leibniz says that if in the concept of Caesar there was the concept "crossing the Rubicon"... is it any accident that it's the same man who is one of the creators of differential calculus, which is to say a mathematical form of infinite analysis? Evidently not, it's not an accident. What does he mean when he manages to treat "crossing the Rubicon" as a predicate which is contained in the concept Caesar exactly as "extended" is contained in the concept body? Obviously he too will have to engage in a quite astonishing sort of gymnastics of concept-creation, because afterwards he will have to save freedom, he holds to this for his own reasons, so how can Caesar be free when from the beginning of time "he crossed the Rubicon here and now" is included in his concept? And what does such a proposition of Leibniz's imply, namely: there are only analytic judgements? That necessarily implies that space and time, the here-and-now be reducible and reduced to the order of concepts. Spatio-temporal position will be treated as a predicate, which is to say as an attributable concept.Why does Kant hold so fiercely to the heterogeneity of space and time on the one hand, and on the other hand the categories, i.e. a priori concepts. Precisely because he needs there to be something which is irreducible to the order of the concept.Classical philosophy is a long discussion between the respective proportion of a posteriori synthetic judgements and a priori analytic judgements. The possibility of reducing one to the other, or else the impossibility of reducing...Richard: How is it that we don't manage to derive the principle of identity from experience? In the example "A is A".Gilles: Because it's a pure empty form, A is A. A is not at all given as a generality, it's pure thought, it's generic thought. Moreover, as soon as there is an identity in experience, it's a temporal identity, which is to say that it's not a necessary identity. So "A is A" is said to be a priori precisely because it is strictly without content, it will be a rule for all possible content.So now Kant comes along and everything happens as if he discovered a new type, a third type of judgement, and he will have to invent the concept to designate this third type of judgement, namely synthetic a priori judgement. In doing so he effects an amazing forced takeover [coup de force]. For a classical thinker, still very broadly, analytic a priori judgement, that meant something, synthetic a priori judgement, that meant something, but synthetic a priori judgement - that's truly a monster. So a philosopher cannot but create monsters as new concepts. It's a prodigious monster. What on earth can it mean? Here I will use some examples which aren't even in Kant, in order to be more faithful, to try and be clearer than he is, because he has other things to do.The triangle is white. If I blithely ask you what that is you will reply it's a synthetic a posteriori judgement. I'll reply: very good, you've passed the course. If I say "we call triangle a figure formed by three straight lines enclosing a space", three straight lines enclosing a space, what is that? I can say that it is an analytic judgement. Why? Because I'm not saying anything but "A is A". The concept of triangle is precisely three straight lines enclosing a space. This was broadly the distribution in the world of classical philosophy, the terminological coordinates of classical philosophy. Kant comes along and says: if I say that the three angles of a triangle are equal to two right-angles - elementary geometrical proposition - what is that? Is it an a priori analytic judgement or an a posteriori synthetic judgement? Stunned silence! And yet this was something everybody had known for a long time, but nobody had used this case to explode the insufficiency of certain philosophical categories, the a priori analytic judgement and the a posteriori synthetic judgement. Here he is in the process of finding something which really appeals to the taste of philosophy qua philosophy, namely the simplest thing in the world which bursts a conceptual frame. In effect this story is very curious: the three angles of the triangle are equal to two right angles. It is the very example of what is called a geometrical necessity. It's universal and necessary, and yet is it analytic?As for Leibniz, he would have laughed at Kant's observation, this is why philosophy is so good. Leibniz's simple reply is: yes of course the concept of the triangle, if you take the analysis far enough, it's obvious that its angles being equal to two right angles is contained in the concept. But again, under what condition can Leibniz say that? Because he has also invented a mathematical discipline which he has determined as already being a topology, and which allows a sort of reduction of spatial determinations to conceptual ones. But under what condition?Kant began by noting the impossibility according to him of reducing spatio-temporal determinations to conceptual ones. In other words, there is an order of space and time which is irreducible to the order of the concept. So Kant: I say that [the equation of] the three angles of the triangle is so little contained in the concept that to demonstrate it you have to extend a side of the triangle, raise a parallel on the opposite side... already Leibniz would say that he doesn't agree, and he would be right because if he accepts something here he would be screwed, but we'll let it go, we'll go along with this attempt of Kant's. So here is my concept: three straight lines enclosing a space. To demonstrate the equality of three angles to two right-angles, I take for example the base of the triangle and I extend it; at point C I raise the parallel to AB and I show that the three angles of the triangle are equal to two right-angles. Kant tells us we mustn't get carried away, the side didn't grow all by itself, the triangle is not a flower, it doesn't raise a parallel to one of its sides all alone, parallel to a side of the triangle isn't part of the concept of the triangle thus it's a synthetic judgement. But it's a very curious type of synthetic judgement, not at all of the "the rose is red" type, since it's a universal and necessary synthetic judgement. How are you going to explain such a judgement?I'll take another example. "The straight line is black". Everyone understands, no problem: synthetic a posteriori judgement; I encounter it in experience, which is to say I come across a straight line which has been drawn in black. I take Euclid's definition: "The straight line is the line which is ex aequo in all its points", it doesn't matter if you use another definition. In any case, I would say that it's an analytic judgement, it's already contained in the concept of the straight line, it's even the statement of the concept of straight line. And then comes the monster, I say: "the straight line is the shortest path between two points." Is it analytic, can I say that the shortest path is contained in the concept "straight line"?Once again, Leibniz would say: yes. Kant says no. Why? For several reasons. I'll give a vulgar reason and a scholarly reason. The vulgar reason: if one looks very closely at "the shortest", is it a predicate or an attribute? It's a question of diagnostics. Is it something else? When I say "the straight line is the shortest path", it's bizarre, is "the shortest" an attribute? If you managed to demonstrate that it's an attribute, it would be via a very complex route. It wouldn't be an attribute because "the shortest"... I'll try putting it another way: if you want to find the straight line, take the shortest, what does that mean? The shortest appears to be a predicate, but it's not a predicate. In fact, it's a rule of construction. It's the rule according to which I produce in experience a line as a straight line. You will say to me; we still have to know what "the shortest" is... the shortest is not a predicate that I attribute to the straight line, it's a rule of construction for constructing straight lines in experience in order to determine a line as straight. We find this example in one of his disciples, Salomon Maïmon, a great, great philosopher. So the shortest is the rule of construction of the line as straight, it's the means of producing in experience a line as a straight line. What does that mean?It's obvious that a concept does not give the rule of construction for its object. In other words, the rule of construction is outside the concept. Once again Leibniz would say "not at all"; if he admitted that his whole system is screwed. At first glance the rules of construction are something very different from concepts because the rule of construction is the rule according to which one produces in experience an object which conforms to the concept. It's thus obligatory that it's not in the concept, by definition. You say: "the circle is where points are situated at an equal distance from a common point named centre", that is the concept of circle, that doesn't give you any means of producing a circle. We are already at the heart of the problem of time. When you say that a straight line is a line ex aequo in all its points, you have no means of producing a straight line in experience, you still need a rule to produce a line that is ex aequo in all its points, you still need a rule of construction to produce a figure such that it presents points situated at an equal distance from a common point named centre. And when you have said that the triangle is three straight lines enclosing a space, you have no means of producing a triangle in experience. The rule of construction of a triangle will be something else completely which will go via the circle, by the way. To produce a triangle you have to go via the circle. It's bizarre.What does Kant mean when he says it's a judgement of a synthetic kind? In effect you will define the rule of construction of a triangle by saying that if you give me a segment of a straight line - it assumes the straight line, that goes without saying, and the means of producing the straight line -, if you give me a segment of straight line, if the two end-points are taken as the centre, whether of the same radius or varying radii, if the two circles cross, if you link the two ends of the straight line to the point where the circles cross, if the circles are of equal radius, this triangle will be called equilateral. (correction: if the radius is equal to the circle). There, I have a rule of construction.You see that there is something amazing in the a priori synthetic judgement, it's that instead of operating a synthesis between two heterogeneous concepts, it operates a synthesis between the concept, between a conceptual determination, the triangle or the circle, and a group of spatio-temporal determinations. In effect, a rule of construction is a spatio-temporal determination. Why is it a synthesis? We have seen it, the rule of construction fundamentally relates heterogeneous concepts. Where does this power of necessarily relating heterogeneous concepts come from, since the only way we thought that heterogeneous concepts could be linked was through the contingency of experience: ah yes, this rose is red. But when I say that the straight line is the shortest path, I claim to be saying something necessary, in this sense a priori, it's geometrical necessity; it doesn't depend on experience. It is said of experience, I can check on any straight line that it is in fact the shortest path, but I don't need to. I know it from the first time, I know it at the same time that I understand the judgement. I know that it is necessarily and universally valid for all straight lines.... namely what underlies the necessary relation between the concepts is a group of spatio-temporal determinations by which one of the concepts is put into a necessary relation with the other.At this point my scholarly reason comes in. When I say "the straight line is the shortest path between two points", at first glance I don't see how that gives me the means to construct a straight line, but in fact, those who were here other years will remember that I had tried to show something quite obvious in geometry. Namely that "the straight line is the shortest path between two points" is not a Euclidean-style proposition, it's an Archimedean-style proposition because it implies a fundamental comparison between two heterogeneous concepts, that of the straight line and that of the curve. In effect, "the straight line is the shortest path between two points" only has a meaning in the very precise situation of the arc of a circle and the chord. In other words, it implies the method "the straight line is the shortest path between two points", it's what would be called an already pre-differential proposition referring to a pre-differential calculus which is the famous calculus of Archimedes, the calculus of exhaustion by which one stretches a broken line towards a curved line, to infinity, it implies the passage to the limit. That is why the straight line is the shortest path between two points even though the curve is not stated explicitly, the concept of the curve is not named. This judgement is devoid of sense if we don't see that it effects a synthesis between two concepts, the straight line and the curve, that it's uniquely in the comparison between the straight line and the curve in the very precise Archimedean situation that this judgement is expressed, with the passage to the limit and exhaustion, and that Kant's response on this level is: you can clearly see that it's not an analytic judgement because two heterogeneous concepts are... just as in the example of triangles, once again in order to demonstrate the equality of three angles to two right-angles, you have to erect a parallel, but the parallel is a concept exterior to the triangle. What welds these heterogeneous concepts together in the synthetic a priori judgement? Solely an operation which consists this: being a determination of space and time.It's the determination of space and time, for example in the figure of the circle's arc and the chord, in the elevation of the parallel to one side of the triangle, it's this spatio-temporal determination which will make possible the necessary link between these concepts which are nevertheless not contained in each other, i.e. you will have at that moment a synthetic a priori judgement.What are Kant's reasons for telling us that space and time are not reducible to categories, that is, that there are two sorts of a priori forms: space and time on the one hand, the categories on the other hand, or if you like space and time are irreducible to the order of concepts. He gives lots of reasons, but he invites us to engage in at least one thought-experiment, as it's the simplest it's the one I'll give you. He says, you see two hands, it's the paradox of non-superimposable symmetrical objects. You see two hands, not only do you see two hands but you think two hands. Let's suppose that, in reality, there are never two hands, there are always little differences, prints, traits, from the point of view of thought that is of no interest, you can always say that there are no two things alike. But you can still think, you can still represent to yourself two absolutely identical hands. Note that if I make Leibniz speak from off-stage, he would say: not at all, you believe you think it, but you can't think it, you've just stopped the concept. But we will accept this sort of dare of Kant's.So you can think two hands which are strictly identical in their concept. And however far you go in the concept, in the characteristics of the concept and you can even think that such a line is on each. And yet... Leibniz would say: OK maybe, but if you do that you will see that there remains only one hand. Kant says that there is something irreducible in them. Kant says that he can think two strictly identical hands and that there are nevertheless two of them. They are strictly identical in their concept, each characteristic of the one has its identical correlate in the other. And yet there are two of them. And why are there two? One is the right hand, the other is the left. Or else one is before and the other is after or behind. How can that be thought, in the two strictly identical hands, that one is on the right and the other on the left? You know that however well they can be thought as identical in each of their characteristics, they are not superimposable. They are absolutely symmetrical in their smallest details and yet they are not superimposable. Kant will say that that's what finitude is.That's what the irreducibility of space and time is. The right, the left. Here-now. Before, after. You can conceive of two objects whose concept is strictly the same, there are still two objects, for this very reason that the one is here and the other there. One is on the right, the other on the left, one is before, the other is after. There is a spatio-temporal order irreducible to the conceptual order.But Kant doesn't invoke that reason. He also gives this famous example: two like trihedrons, opposed at their vertex, you cannot make them coincide. Why is it that you can't make them coincide? Because superimposing two figures or making them coincide implies a rotation, a rotation in a dimension that is supplementary to the figure's number of dimensions. When you have two triangles opposed at the vertex, you can make them coincide, which is to say put one on the other by making one of the triangles undergo a rotation in the third dimension. You have in that case a supplementary dimension to the dimensions of the figure. When you come to volumes, i.e. three-dimensional figures, like the two hands or the two trihedrons opposed at the vertex, you can easily make the two hands superimpose on each other if you have a fourth dimension of space. You would effect the rotation in the fourth dimension. Finitude is the fact that space irreducibly has three dimensions and not n dimensions, or that time has one dimension. We could always be told that there are theories or spaces with n dimensions, or else that time has several dimensions. I think that there's little interest in such a thing because the idea of a space with n dimensions already implies a system of problems and concepts which have nothing to do with Kant's system of concepts and problems.Why are space and time irreducible to the order of the concept?It's because spatio-temporal determinations don't allow themselves to be reduced to conceptual determinations, to the extent that however far you take the identity of two concepts, the corresponding thing or things will always be able to be distinguished not only by contingent a posteriori characters, but by their situation in space and time. By their position in space and time. Spatio-temporal position is not a conceptual property.In which case we are assured of the following principle that the a priori synthesis happens less between two concepts, it doesn't happen between two concepts because in the first place, because it happens between the general concept on the one hand, and the spatio-temporal determination on the other hand. The true a priori synthesis is not between concepts like the empirical synthesis, the true a priori synthesis goes from the concept to the spatio-temporal determination, and vice-versa. That is why there can be a priori syntheses between two concepts, because space and time have woven a network of determinations which can make two concepts, however different they are, from the moment that there are rules of production, form necessary relations with each other. Thus space and time will acquire a constitutive power [pouvoir] which will be the constitutive power of all possible experience.To better mark the difference between the order of the concept and the spatio-temporal order, I'll return to terms that I used just before. Space and time are the forms of appearing, or the forms of presentation of what appears. In effect, we can understand this because space and time are indeed a form of appearing, but they contain no specific unity. What appears is always diverse, an apparition is always an apparition of diversity: the red rose, a smell, a colour etc. So what appears is, by nature, diverse. Space and time are forms of perception, but you can see that space and time themselves have a diversity, namely the diversity of "heres" in space, any point in space being a possible "here", and the diversity of moments for time, any point in time being a possible moment.We have thus to distinguish the diversity of what appears in space and in time and the diversity of space and time themselves. The first diversity will be said to be empirical diversity, the second diversity, the diversity of space itself or of time itself will be a priori diversity. Diversity of space. Diversity of time. The a priori diversity of space and of time constitute the forms of presentation. By contrast, empirical diversity belongs to what appears. The categories or concepts, which we have just seen are of another order than space-time determination, have a unity, it's even the function of the concept to unify a diversity. To the extent that you can in fact sense that the concept will have to bear, in a certain way, on space and time. Space and time as the forms of appearing of what appears are what Kant calls Forms of Intuition. Intuition is precisely the presentation, intuition is the immediate. Phenomena are immediately in space and in time, which is to say immediately appearing in space and in time. Space and time are the forms of immediacy. The concept is always what we call a mediation. The concept refers to the concept and it effects a unification. It is in this sense that it is not simply a form of presentation of what appears, it will be a form of the representation of what appears. The prefix re- indicates here the activity of the concept in opposition to the immediate or passive character of space and time which are given or which are the form of what is given.Space and time are, Kant says, the form of our receptivity, while the concept is the form of our spontaneity or our activity.What incredibly new thing does Kant bring to the history of time? Once it is said that determinations of space and time are irreducible to conceptual determinations, there would be no possible knowledge unless nevertheless and despite everything we were able to establish a correspondence between spatio-temporal determinations and conceptual determinations, and that's the sort of miracle of knowledge. And Kant constructed his whole system of new concepts to get to that point.He's an austere philosopher, a severe philosopher, he uses all sorts of complicated words but they're never just for effect, he's not a lyrical type. I refer you to his secretaries who wrote things about his life, he has a very calm life, very ordered? Thomas de Quincey has translated and somewhat arranged, embellished the accounts of Kant's secretaries, in "The Last Days of Immanuel Kant". It's a splendid text.There is an formula, a first formula about time which seems to me to be one of the most beautiful things said about time, it's Hamlet who says it. The formula suits is so well: "the time is out of joint". It's beautiful! It's a very beautiful formula if we understand it. What is the joint? The joint is, literally, the hinge [pivot]. The hinge is what the door pivots around. But the door? we have to imagine a revolving door, and the revolving door is the universal door. The door of the world is a revolving door. The door of the world swings and passes through privileged moments which are well known: they're what we call cardinal points. North, South, East, West. The joint is what makes the door swing in such a way that it passes and re-passes through the privileged co-ordinates named cardinal points. Cardinal comes from cardo; cardo is precisely the hinge, the hinge around which the sphere of celestial bodies turns, and which makes them pass time and again through the so-called cardinal points, and we note their return: ah, there's the star again, it's time to move my sheep!"The time is out of joint", time is no longer coiled up in such a way that it is subordinated to the measure of something other than itself, such as, for example, astronomical movement. Time has ceased to be the number of nature, time has ceased to be the number of periodical movement. Everything happens as if, having been coiled up so as to measure the passage of celestial bodies, time unrolls itself like a sort of serpent, it shakes off all subordination to a movement or a nature, it becomes time in itself for itself, it becomes pure and empty time. It measures nothing anymore. Time has taken on its own excessiveness. It is out of its joints, which is to say its subordination to nature; it's now nature which will be subordinated to it. I can say, going quickly, that the whole of ancient philosophy maintained a subordination of time to nature, even in its most complex forms; that classical philosophy, however complicated its conceptions of time were, never put into question this very very general principle. The famous definition: "time is the number of movement".With Kant there is an indescribable novelty. It's the first time that time is liberated, stretches itself, ceases to be a cosmological or psychological time, whether it's the world or the soul makes no difference, to become a formal time, a pure deployed form, and this will be a phenomenon of extreme importance for modern thought. This is the first great Kantian reversal in the theory of time.So I take Hamlet's formula literally to apply it to Kant: "the time is out of joint". It's with Kant, from the point of view of the concept of time, that we can effectively say that time is out of joint, which is to say has ceased to be subordinated to the measure of movement, and on the contrary movement will be completely subordinated to it. And time will be this sort of form which is also pure, and this kind of act by which the world empties itself, becomes a desert. This is why one of Kant's best disciples - it won't be a philosopher, we never find those who understand philosophers among philosophers - is Hölderlin, and Hölderlin who, drawing on Kant against the Kantians, understood by developing a theory of time which is precisely the pure and empty form in which Oedipus wanders.Next time I would like to see what the formula "the time is out of joint" means, applied to Kant. It really means something quite literal.The second formula that I want to develop truly belongs only to Kant and it is part of his last, most obscure texts. Kant, at the end of his life, compiles a book which will appear after his death. He begins a sketch of something which will be called the Opus Postumum. And the Opus Postumum is very strange because it's a mix of everything. There are laundry lists, there are little impressions of everyday life, and then there is a wonderful page. In these texts near the end the idea that time is like the form of auto-affection appears more and more. It's the form under which the subject affects itself. If anything is mysterious, that is. It would be clear for space, but he also says it of time. See how he divides things up: space is the form under which something exterior affects me and time is the form under which I affect myself. It's even more mysterious than "the time is out of joint".They're Kant's three oracles: firstly disguised as Hamlet, time is out of joint, secondly disguised as himself he says time is the form of auto-affection, the form under which I affect myself. But why does he say that? He couldn't do otherwise. If you followed the first point, time is out of joint, it no longer measures a movement, it is no longer subordinated to nature. Already, on the most basic level it's very new. What is new with someone must already be grasped on the most basic level. Before him, what did they say, very broadly. With Leibniz no problem, time is the order of possible successions, space is the order of possible coexistences. Kant wants nothing of this and can no longer accept it. The whole way in which he has posed the problem means that he cannot: it's obvious that to define time by the order of possible successions implies, at first glance, a subordination of time to a content which measures it, a content to which it is subordinated. It must be the case that time is subordinated to succession. So once he has conceived of formal time, the pure form of time detached from a movement to measure, once he has straightened time, once he has let it go like a spring, he can no longer define it by an order of succession. It's all the more significant given that to define time as succession means nothing but - of course succession is temporal, but it's only a mode of time, as coexistence or simultaneity by which we claim to define space, is another mode of time, it's not space. It's a very bad distribution. Space cannot be defined by the order of coexistence since coexistence is an idea which can only be understood in relation to time, it means at the same time. Time cannot be defined by succession because succession is only a mode of time, coexistence is itself another mode of time. You can see that he arranged things to make the simple distribution: space-coexistence, and time-succession. Time, he tells us, has three modes: duration or permanence, coexistence and succession. But time cannot be defined by any of the three because you cannot define a thing through its modes. Moreover space cannot be defined as the order of coexistence since coexistence is a mode of time. He is very very good on this point.He will say - and I want you to admire the simplicity - you will define space as simply the form - and above all not the order since order still refers to a measure of something to measure in time - as the pure form, of what? Space is the form of exteriority. That doesn't mean that it comes from outside, but it means that everything which appears in space appears as exterior to whoever grasps it, and exterior from one thing to another. It is not exteriority which ???? space, it's space which constitutes the form of exteriority or which constitutes exteriority as form, as pure form. As he has just defined space as the form of exteriority, it must be the case that time is the form of interiority. It's the form under which we affect ourselves, it's the form of auto-affection. Time is the affection of self by self.I ask you to consider that this second point follows from the first.So, the first paradox is what does it mean that time is out of joint; the second is what does it mean that time is the form of interiority.Why wouldn't there also be a synthesised or electronic way of handling philosophy?Last time I tried to determine a certain number of very precise Kantian notions: a priori, synthesis, etc... but very much as a function of what seemed the essential thing to me, namely a radical reversal in the position of the problem of time in relation to philosophy. It's a critical reversal, like a critical point.I proposed last time that we take as three arbitrary formulae, but it's very dangerous, but never mind there are three key formulae that aren't Kant's but under which, it seems to me, the three great novelties or the three great reversals that Kant operates on the notion of time group themselves.So if we can manage to eliminate everything that is facile in this evocation of literary formulae in relation to a conceptual study of philosophy, the first formula to which Kant would give a powerful content is that of Hamlet: the time is out of joint. The second formula is anonymous, and would be something like this: till now the task we have given ourselves was to represent space, the moment has come to think time. Third famous formula, given by an author who had nothing to do with Kant: "I is an other". I believe that if we separate these expressions from their contexts, they suit Kant admirably, if you take them as abstract declarations. Maybe that will allow me to understand in itself the formula "I is an other", as well as to understand in itself the formula "the time is out of joint".I have asked Gilles Châtelet to bring a contribution to the commentary of this first formula. So I'm taking us back to the level of the first formula "the time is out of joint", how is it that Kant's philosophy posits a time which is in the process of getting out of joint. The joint was this sort of pivot around which time turned, in other words, in a certain tradition of antiquity, time is fundamentally subordinated to something which happens in it, and this something can be determined as being change, the subordination of time to change, time will thus measure the changing of what changes, or else, which amounts to the same thing on another level, it will be subordinated to movement, the subordination of time to movement, I say that that amounts to the same thing on another level because movement qua local movement is the purest form of change, which is to say the perfect form of change; that goes back to things in Aristotle and which cover the whole of Greek philosophy. Or else, which again amounts to the same thing on another level, subordination of time to the course of the world, and it's in this sense that the classical definition of the Greeks appears: time is the number of movement. What does that imply?That implies a subordination of time to change, to movement, to the course of the world. That implies that time is as if bent, it becomes circular. It is a time, independent or not of questions of the eternal return which are posed in a completely different manner, time is cyclical. And indeed, to the extent that it is the number of movement, it will measure the movement of the planets - see all of Plato's prose writings on the eight movements of the eight planets - and the great circle that will measure the time it takes for the eight planets to come back to the same respective position, the eight circles of the world, you would have thus a great circle of circles whose point would be assigned by the planet's return to the same respective position, you would have the world's year. But this time become circular is but one with time subordinated to change, to movement, and to the course of the world, and it's the great idea which runs through all of ancient philosophy: time as the image of eternity. The circle of time, in so far as it measures planetary movement, and the return of the same, it's precisely this time become circular. In the Timaeus there were some very beautiful pages on the arc of the Demiurge which makes circles, this bending activity.However, this time as an image of eternity, the cyclical figure of time subordinated to movement and whose secret will be the periodic return of planets to the same position relative to each other, is indeed a time which is the image of eternity. I would say that all of the time of antiquity is marked by a modal character, and in effect the word appears all the time: time is a mode and not a being. No more than number is a being, it's a mode in relation to what it quantifies, in the same way time is a mode in relation to what it measures.Obviously, it's not a matter of just taking Kant like that, it doesn't happen only in his head, there's a very long scientific evolution which find its philosophical expression there, but it had already found, no doubt with Newton, a scientific expression. Everything happens as if time "deployed itself" [se déployait], but we must take "deployed itself" in its strict sense, which is to say unrolled itself, which is to say lost its cyclical form. What does that mean that time becomes a pure straight line. It's exactly as if you were holding a coiled spring and you let it go.Time becomes a pure straight line. It reminds me of Borges, the true labyrinth is the straight line. When time becomes a straight line, what does that mean and what change does that imply?Still speaking musically, I would say that with Kant time acquires a tonal character, it ceases to be modal. We can find no other images to indicate the violence of such an operation in relation to the thought that, truly, the circle snaps, like a spring that uncoils itself, which becomes a pure straight line. You can see that the cyclical line, when time is cyclical, is a line which limits [borne] the world and just saying that time becomes a straight line means that it no longer limits the world, it will traverse it. In the first case, cyclical time is a time which limits and which thus carries out - which has always been the supreme act for the Greeks - carries out the act of limitation. When time becomes a straight line, it no longer limits the world, it traverses it, it is no longer a limit in the sense of limitation, it is limit in the sense: it's at the extremity [bout], it never ceases to be at the extremity, it's the sense of a passage to the limit. The same word "limit" radically changes in sense, it's no longer the operation which limits something, it's on the contrary the term towards which something tends, and at the same time the tendency and that towards which it tends, that's time. How can we explain that. It's precisely a matter of assigning the importance of this time become straight line. It's not a simplification, it changes everything in the very atmosphere of time and in the operation of time.The simplest way is to refer ourselves to a poet who claims to be inspired by Kant. That's Hölderlin. For the moment our problem is solely to say what is the importance of the change when time ceases to be circular, ceases to be a circle in order to become a straight line. We must keep in mind both that Hölderlin claims to be inspired by Kant and that he has many things to say on what happens when time becomes a straight line.Hölderlin poses the problem at the level of Greek tragedy, and in particular he opposes Greek tragedy such as it appeared in Aeschylus and Greek tragedy such as it appeared in Sophocles, and above all in Oedipus and in Antigone. You will see straight away that the schema that Hölderlin develops, and that other commentators of Sophocles took up afterwards, concerns the very heart of our problem. It amounts to telling us that there is a certain sense of the tragic for the Greeks which is the tragic element of cyclical time. We find it very easily in Aeschylus. What is the tragic cycle of time? The tragic cycle of time is, broadly, like three unequal arcs of a circle; there is the moment of limitation; limitation is nothing other than justice, it's the lot assigned to each. And then there is the transgression of the limitation, the act which transgresses.The moment of the limit is the great Agamemnon, it's the beauty of royal limitation. Then there is the transgression of the limit, which is to say the excessive act [l'acte de la démesure]: it's Clytemnestra assassinating Agamemnon. Then there is the long atonement, and the tragic cycle of time is the cycle of limitation, of transgression and of atonement. The atonement is Orestes who will avenge Agamemnon. There will be the re-establishment of the equilibrium of the limit which for a moment was overstepped. Notice that this tragic time is modeled on astronomical time since in astronomical time you have the sphere of fixed points which is precisely the sphere of perfect limitation, you have the planets and the movements of the planets which, in a certain way, break through the limit, then you have the atonement, which is to say the re-establishment of justice since the planets find themselves in the same position again.And in this formula of the famous tragic destiny, as they say, it's settled from the beginning, and when the tragedy begins it's already done: as Aeschylus' text itself says, at the moment when Agamemnon goes into his palace and is about to be assassinated by Clytemnestra, it's already done. But at the moment when Clytemnestra assassinates him, an act of excess and injustice, of violation of the limit, the atonement is already there. It's this sort of cyclical destiny. Time is a curve.Whereas in some splendid pages, Hölderlin says: what is the novelty of Sophocles? In what respect does Sophocles found in the end the modern sense of the tragic? He is the first to un-curve [décourber] time. It's the time of Oedipus. He says that before Sophocles, in the Greek sense of the tragic, it's man who eludes the limit. You can see, in the limitation-limit, man transgresses the limit and in so doing eludes the limit; but with Oedipus one can no longer say that it has the atmosphere of someone who transgresses the limit, who eludes the limit. In the case of Oedipus, it's the limit which is elusive. Where is it? It's the limit which becomes passage to the limit. Splendid expression of Hölderlin's: in Oedipus, the beginning and the end no longer rhyme. And the rhyme is precisely the arc of the time bending in such a way that beginning and end rhyme with each other. There was atonement for the injustice. In Oedipus time has become a straight line which will be the line on which Oedipus wanders. The long wandering of Oedipus. There will no longer be any atonement, even if only in the form of a brutal death. Oedipus is in perpetual suspension, he will travel his straight line of time. In other words, he is traversed by a straight line which drags him along. Towards what? Nothing. Heidegger will be able to say later that it's towards death. Heidegger for his part will draw from the straight line the idea, which is not wholly un-Kantian, the idea of a sort of being-towards-death.We can see well indeed that in the case of Oedipus, in Sophocles' tragedy, the beginning and the end do not rhyme, and moreover there is a zero-instant. Hölderlin adds that this un-curved time, such that the beginning and the end no longer rhyme together, and it's precisely because there is a caesura in this time, thus a pure present, that there will be - and it's this caesura that will distribute it - a before and an after, and it's this before and this after which do not rhyme. For the schema of cyclical time is substituted a time as straight line, marked by a caesura, a caesura which distributes a non-symmetrical before and after. It's very important for us for time as a straight line contains the possibility of distributing a non-symmetrical before and after, of producing a non-symmetrical before and after using a caesura. We can call this caesura the pure present. Hölderlin's analysis is admirable however because he tries to show that this form of time, the caesura which distributes a before and an after, thus the linear form of this time marked by a pure present according to which a past and a future are produced in time, well this time is that of the modern consciousness in opposition to the consciousness of antiquity.Since I borrowed the formula from Hamlet, what strikes me, independently of dates, is the extent to which the whole schema that Hölderlin constructs for us to understand what he considers to be the novelty of Oedipus, the extent to which that applies to Hamlet. For those who remember Hamlet, it's curious the extent to which it's a linear time where something is always elusive, it is no longer Hamlet who eludes the limit, it's the limit which eludes Hamlet, as if it was spinning the straight line. And there is a caesura. For Oedipus, Hölderlin assigns the moment of the caesura to the intervention of Tiresias, the intervention of the seer. It will constitute the pure instant, the pure present from which a past and a future will be produced on the straight line, which is to say a before and an after which no longer rhyme together. And in Hamlet there is a moment which seems extraordinary to me: Hamlet hesitates a great deal in his task of avenging his father: the limit is literally elusive. When he hesitates a great deal to avenge his father it's the same story as Oedipus. For a long time it's as if it's the time before, but we can't yet say "before" since the before and after are only distributed by the caesura which is to say the moment of the pure present; and then his step-father, who wants to get rid of him, sends him on a sea trip. Well the sea trip is so fundamental that Hamlet returns from it saying: "there is something dangerous in me", which he would never have said before, as if the sea trip had made him capable of something which he was not capable of before. The sea trip has played the function of the caesura and has distributed on the straight line of time a before and an after which are non-coincident, non-symmetrical.We will see all that in this quite beautiful, obscure but beautiful text of Hölderlin's: "At the extreme limit of the rift nothing in fact remains any more except the conditions of time or of space [here Hölderlin is speaking like a Kantian]. At this limit man forgets himself because he is wholly inside the moment. God forgets because he is nothing but time. And there is infidelity on both sides, etc." The categorical turning-away [détournement], what is it? It's that in so far as time is cyclical, there is a sort of God-man relationship which is one with destiny in Greek tragedy. When time becomes a straight line, there is also something which separates. In Hölderlin's very beautiful commentary it's the double deviation in the same course of linear time which will separate man and God, God turns away from man who turns away from God. Which is why Oedipus is said by Sophocles to be "atheos", which does not mean atheist, but he who is separated from God. So much so that God is no longer the master of time, the one who curves time, and man is longer himself ???? encircled in a sort of harmony with God, in this sort of relationship with God, man is no longer anything but the caesura which prevents the before and after from rhyming together, which distributes a before and an after which do not rhyme together.I would simply like you to begin to feel the importance of this time which becomes a straight line. It doesn't mean simplification of the figure of time at all, on the contrary I would like you to feel an intense complication of the figure of time. Time is no longer subordinated to something which happens in it, on the contrary it's everything else which is subordinated to time. God himself is no longer anything but empty time. Man is no longer anything but a caesura in time. In The Critique of Pure Reason, there is a very famous passage, also very very beautiful, which is called "Anticipations of Perception". I would just like to show that, at a completely different level, Kant tells us a story which is the same one that Hölderlin told afterwards. But it's not in relation to Greek tragedy. Oddly enough it happens to be in relation to scientific physics. So there are twelve extraordinary pages entitled "Anticipations of Perception". Kant tells us that space and time are what are called extensive magnitudes. What does extensive magnitude mean? It's not complicated, in Latin an extensive magnitude is one which accepts the formula "partes extra partes", the exteriority of parts, which is to say an extensive magnitude is one whose parts are apprehended successively so that, all quantity being at the same time multiplicity and unity - when you say, for example, this is twenty metres long, it's the unity of a multiplicity - extensible magnitude or extensive magnitude will be defined in the following way: the multiplicity refers to a gathering of parts into a whole. That's an extensive quantity. But time is like that: a minute, another minute, and then you say that's it, that an hour has passed. You can see the succession of parts in their apprehension, the gathering into a whole: an hour.Space and time are extensive quantities, no difficulty there. Kant adds: but there you have it, the real in space and time - you recall that the real in space and time is what appears in space and time, it's the phenomenon since with Kant the phenomenon is no longer an appearance, it's the fact of appearing - the real in so far as it appears in space and time, no doubt it also has an extensive quantity, there is the space of the table. There's no more to go over on this point; it's precisely what Kant calls a synthesis. But the real in space and in time doesn't only have an extensive quantity, it also has an intensive quantity. What is an intensive quantity? It's what fills space and time to such or such a degree.We can see straight away the difference between extensive quantity and intensive quantity since the same extensive space can be filled to varying degrees. An example: the same space can be filled by a more or less intense red, the same room can be filled with a more or less intense heat, the same volume can be filled with a more or less dense matter. Kant will even distinguish the two questions fundamentally: can emptiness in space and time be conceived, and another question, namely that space and time can be filled without there being any void in them, can be filled varying degrees.So what is the intensive quantity of the real in so far as it fills space and time? Moreover, there is not just a real which fills space and time, there is a real of space and time, it's intensive quantity. In opposition to what we have just said about extensive quantity, the two fundamental characteristics of intensive quantity according to Kant - and this will be very important for all subsequent theories of intensity - first characteristic: the apprehension of an intensive quantity is instantaneous, which is to say that its unity no longer comes from the sum of its successive parts, the unity of a given intensive quantity is apprehended in an instant. Which amounts to saying that when I say "it's 30 degrees", the 30-degree heat is not the sum of three times ten degrees, it's at the level of extensive quantities that thirty is 10+10+10, but thirty degrees is not three 10-degree heats. In other words, the rules of addition and subtraction are not valid for intensive quantities. The apprehension of the unity of an intensive quantity happens in an instant. Second characteristic: the multiplicity contained in an intensive quantity is no longer referred to a succession of parts exterior to each other, but refers to a variable proximity to degree zero. I can say that each time there is something which fills space and time, I would say or rather Kant would say that he has before him an empirical intuition. Intuition, you will recall, is the faculty of receiving what is given, but the given is given in space and time, so intuition is not at all a magical faculty, it's the faculty of receptivity. I receive something which is given, and in this sense I have an empirical intuition. But to the extent that what is given has an intensive quantity, which is to say a degree, I grasp it in a relation to its production starting from zero, or its extinction... or the real which fills space and time from the point of view of its intensive quantity is grasped as produced starting from degree zero or as extinguishing itself, i.e., rejoining degree zero.At that point the question is not at all one of knowing if there is an empty space and time, the question is of knowing that in any case there is an empty consciousness of space and time. And there is an empty consciousness of space and of time as consciousness determined by and as a function of degree zero as the principle of production of all reality in space and time - production starting from zero or the principle of extinction.I don't want to make associations that are too forced, but at the physical level of intensity in Kant, you can do what Hölderlin ?????, namely the straight line of time marked by a caesura which is intuition = 0; what he will call the empty formal intuition, from which the real which fills space and time will be produced, and it's this intuition = 0, this empty intuition which constitutes the caesura. It's according to this caesura, this degree zero implied by all intensive quantity, which is naturally correlated with time as empty form, as pure line. So on time as a pure line the caesura of degree zero is marked, which will mean that before and after will no longer rhyme together. Again the question is not: is there an empty time and space, the question is whether there is an empty consciousness of time, by virtue of the nature of time itself. In other words God has become time, at the same time that man became caesura. It's hard, we understand nothing, but it's beautiful. That's all I wanted to say on time that's out of joint.Intensive quantity effects a synthesis between the degree zero that it implies, from which it is produced, and time as pure line or empty form. Intensive quantity as degree of the real which fills a space and a time effects the synthesis between a degree zero from which this real is produced or in which it extinguishes itself, and on the other hand time as empty form or pure line. So much so that there will be a complementarity between the function of the caesura which intensive consciousness plays in time and the empty linear form that time takes on. Hence, as Hölderlin will say: man (the consciousness of time) is no more than a caesura, God is no more than empty time. It's the double turning-away [détournement]. Kant didn't go as far as that, for a simple reason that I will explain: in effect Kant subtracted God and the soul from knowledge. He gave them a function in the field of knowledge, but God and the soul were not known as such since we only know phenomena, we only know what appears. But he didn't suppress either God or the soul since he was to give them a quite different function, a moral, practical function. But from the point of view of knowledge, Gods passes into empty time just as the soul passes into the caesura.Is that any better? True lived experience [le vécu] is an absolutely abstract thing. The abstract is lived experience. I would almost say that once you have reached lived experience, you reach the most fully living core of the abstract. In other words, lived experience represents nothing. And you can live nothing but the abstract and nobody has ever lived anything else but the abstract. I don't live representation in my heart, I live a temporal line which is completely abstract. What is more abstract than a rhythm?For the Stoics, they are at once so new in relation to antiquity, and at the same time they have nothing to do with it, they employ "limit" in a wholly different sense. The limit for them is no longer the limit assumed by philosophers of the Platonic type, neither is it the other limit... Kant's ?Anticipations of Perception? means something very simple, which is that you can't say anything about perception, a priori, if there is a colour that is called red and another that is called green, that's to do with the given, you cannot say it independently of experience, it's given in experience. There are two things that you can say a priori, which are: whatever there is that is given in space and time, what is given in space and time is an extensive quantity, but also has a degree, which is to say an intensive quantity. That is an a priori judgement. Which is to say nothing would come and fill space and time as extensive quantities if what comes to fill them did not also have a degree. So I anticipate perception since in this I have a determination, it's the only a priori thing I can say. So there is anticipation. With Epicurus it's not at all in this sense. The Epicurean definition of time will not even be the novelty of a Stoic form of time, it's typically modal time. Here I would very much like Gilles Châtelet to come in and say, from his rather mathematical point of view, precisely how this conception of time as straight line is fundamental.Gilles Châtelet (summarised because the taped recording is inaudible): With Plato there is a time which is created, which is to say there is a transcendence somewhere which is above time and which has, in correlation with this, a higher dimension. This time of Plato's measures periods, it's a set of periods and it assures the repetition of identities in the stars, the calendar. The fundamental thing to retain is that time is a number. This time above the market measures order. Time in Plato describes order, chaos has no time for example. Time is a sort of calendar that expresses the order of the world: it's a system of coordinates of order, it is in the world, it's a worldly being.In Aristotle everything is set out through movement and time is in movement, it is interior to mass. Time is attached to the body. Time will be purely astrological, but we owe to Aristotle the notion of an eternal, infinite and uniform time. But with Plato and Aristotle we have a cyclical representation.In Plotinus there is an abstract operator which is called the One, which is without any qualification and something degrades once we leave the One. Certainly time measures degradation in relation to eternity. Plotinus says that time is the irreparable addition of being to itself. Time is a fall, i.e. a degradation, and Plotinus speaks of aspiring towards God. The mathematical figure which would go with what Plotinus says is called a projective straight line, time is a straight line, but a straight line which has been curved. It's not a circle either. It's a circle minus one point (the One). Time in Plotinus would be a sort of projective time, there is already the idea of irreversibility. In Plotinus time flows from the One and the One is transcendent to time. Time is not exactly a cosmic being, it's the soul which appreciates time in so far... Time is already an equivalent of eternity, it has neither beginning nor end and the point outside the circle is not in time, the One is above, we never begin. It's rather paradoxical. In Kant time becomes a condition of possibility of phenomena. The succession of phenomena implies time, so it is time which is transcendent. Time is what is called a multiplicity, it's clearly said, it is uni-dimensional and above all it is ordered. In the end he says that it tends towards a straight line. But what is a straight line?... Time as a parameter gives the trajectory... The real straight line is a function, time becomes the condition of a function; it's not the image of representation, it's the function itself. There is the possibility of having a function of time. In what sense is Kant completely modern? Because temporality is defining a topology... a straight one... But Kant's essential idea is that his abstract space is pure parameter.There are two things in Kant: firstly a technological revolution in the sense that it is clearly affirmed that time is a real straight line, but there is also a notion of function.Gilles Deleuze: You're saying something very important, namely that with Kant time ceases to be a number or measure and becomes parameter. I would like you to explain the difference between a number or measure and a parameter?G. Châtelet: The parameter is not a result. A number, for the Greeks, is simply a measure, here the measure of time is possible because... In mathematics parameter has no definition, it's simply a notion. Time become parameter is no longer a result, it becomes an initial given. A parameter is what is given, what varies.Deleuze: I think that it amounts to exactly the same thing: to say that time ceases to be a number or that time ceases to measure something and thus is subordinated to what it measures, and that time becomes a parameter, time is related to a problem of constitution. When I said that time un-curves itself, becomes a straight line... There is something equivalent in this modern conception of time where it is at the same time that an empty form of parametric time appears and a complementarity with something which makes a function, whether it is the caesura in the tragedy, or else the cut in mathematical instrumentation. I am just a bit bothered by the key role that Gilles Châtelet gives to Plotinus. In antiquity it is much more complicated than has been said till now. There were in fact two directions and the two directions had at least something in common: in the two directions time only has a modal character and never a ???? character. However the two directions are time as number of movement, thus subordinated to the physical cosmos, subordinated to physis, and then Plotinus breaks away there, but he is not the first to break away, and he makes a conception of time which is subordinated not to physis but to the soul. I wouldn't completely agree with Gilles Châtelet on the importance of this point, of Plotinus, and on the one hand the two attempts: time subordinated to the soul, time subordinated to physis maintain or at least have in common the affirmation of a purely and uniquely modal character of time, thus time as the image of eternity, a secondary and derived character of time, and the two have a point of convergence in the Antique theory of the soul of the world. I would not make of Plotinus a...Comptesse: [inaudible comment]Gilles: Transcendent in relation to Kant. Once again there are two notions. The Kantian notion is transcendental, time is transcendental, but the whole Kantian notion of the transcendental is created in order to refute the classical notion of the transcendent. The transcendental is above all not transcendent.I would like to move very quickly to the second point. I'm going very quickly. I would say that the second formula that I would like to apply to Kant is... but thinking time is really the most difficult thing - it's the phase of philosophy as critical philosophy, as modern philosophy defined by Kant under the form of a critical philosophy. In classical philosophy, what is the other of thought. The other of thought is above all space. It's space. Space is conceived as limitation. It was conceived as an obstacle and a resistance, it is also limitation. Why? Because it happens that my thought is referred to a thinking substance that is itself unextended, thought is the attribute of a thinking substance that is itself unextended, but this thinking substance is finite in body. It is finite in body: it's the famous problem which will poison classical philosophy, namely the union of the soul as thinking substance and the body as extended substance. And the fact that the soul is finite in body, even though the soul is in itself unextended (you can see that it's an inextricable problem: how is it that something unextended can be finite in something extended, it will produce all sorts of paradoxes), this in fact introduces a fundamental limitation of thought since it will be the source of all the errors, of all the illusions which not only create an obstacle to thought, but limit thought. Third characteristic: if space is the other of thought, I'm saying that it's an other of, literally, alterity. Extended substance is other than thinking substance even though it is uni-substantially opposed, hence the well-known position of Descartes in which there were three substances: thinking substance, extended substance and the union of thinking substance and extended substance. With the Kantian transformation the aspect of everything changes. Why? We remember time become straight line, and I can no longer say that what is important is space as obstacle or resistance to thought, or as limitation of thought. Here it's time which ceases to be subordinated to space, it takes on an independence at the same time that it acquires this form that we have seen, this pure form, and it's not time which takes the place of space, it is not an obstacle to thought, it is the limit which works thought from the inside. For the notion of external limitation is substituted the notion of internal limit. Time is the limit which works thought over, which traverses thought through and through, it is the inherent limit, a limit interior to thought, whereas in classical philosophy it's space which is determined as the exterior limitation of thought.So everything happens as if the "enemy" of thought was within. It does not receive it from outside. There we have a sort of fundamental change. To think time means to substitute for the classical schema of an exterior limitation of thought by the extended, the very very strange idea of an interior limit to thought which works it from the inside, which doesn't at all come from outside, which doesn't at all come from the opacity of a substance. As if there was in thought something impossible to think. As if thought was worked over from the inside by something that it cannot think. From this point the problem, in Kant, will no longer be that of the union of the soul and the body, which is to say the union of two substances one of which is extended and the other unextended. The problem will no longer be the union of two distinct substances, it will be the coexistence and the synthesis of two forms (they're completely different, two forms and two substances) of one and the same subject. Instead of the union of two substances, the synthesis of two forms of the same subject, which implies that the subject is not substance.What are these two forms which will have to unite - I can no longer even say in the same subject since substance will not be inherent in the subject - they are two forms for the same subject. Now this subject will be traversed by this line of time; the subject is as if traversed by two forms and is himself nothing other than the synthesis, namely the most mysterious point, the synthesis of these two forms. What are these two forms? They're on the one hand the form of thought, and on the other hand the form of the internal limit of thought. What does that mean in concrete terms? The form of thought is in the first place the act of "I think", the "I think" as act or as determination. To say "I think" is to determine something. What? We will see later.The form of equal thought, in the most universal sense "I think" which is to say that it's thought in so far as it is related to a subject; but I don't have the right to say that it's a substance. Second determination of the form of thought: as Kant says, "I think" is the slightest [la plus pauvre] of representations, it's the slightest of thoughts which accompanies all thoughts. Self = self, it's the "I" of "I think". The "I think" is the universal form of determination, but in a sense I determine nothing and in "I think" the determination is at its emptiest.Concretely acts of thought are concepts. We have seen that a priori acts of thought are particular concepts called categories. So the form of thought is the "I think" and the categories taken together, the "I think" together with what it is that "I think", namely the categories or the predicates of any given object. These are what the forms of thought are. Kant will also use the term ?forms of spontaneity?, when "I think" is the act of determination and that implies an activity which is the activity of thought. Kant will reserve the word ?spontaneity? to qualify the form of thought in these two cases. But what else is there besides these two forms of thought? We have seen the form of receptivity or the form of intuition. In the form of intuition we also have two things, just as a moment ago we saw that the form of thought is the self, the "I" of "I think" and it's also the concept as act of thought, the a priori concepts, which is to say the categories, the forms of receptivity are space and time.There are two forms twice. Last time I said that space is the form of exteriority, time is the form of interiority, this doesn't prevent these two forms from having in common the fact of being two forms of intuition or two forms of receptivity. The form of receptivity is double: form of exteriority = space, form of interiority = time, but the two together are the form of receptivity. On the other hand there is the form of spontaneity which is the "I think" and the categories. You can see, and this is very important, how it unfolds: you have a first great duality: form of intuition and form of spontaneity, form of receptivity and form of spontaneity, and each one of these two great forms has two aspects. The form of receptivity has two aspects: exteriority-space, interiority-time, the form of spontaneity has two aspects: the self of the "I think", the I = I, and the concepts that I think, the a priori concepts.Kant's problem is howthe same subject, or self, can have two forms which are irreducible to each other (irreducibility of space and time on the one hand, and of the concept on the other hand), how = the same subject can have two forms, principally the form of time and the form of thought, and 