On Leibniz

We are going to be involved for a short while in a series on Leibniz. My goal is very simple: for those who don't know him at all, I want to present this author and to have you love him, to incite in you a sort of desire to read his works.
To begin reading Leibniz, there is an incomparable working instrument. It is the life work, a very modest work, but a very profound one. It is by a lady, Madame Prenant, who had long ago published selected excerpts by Leibniz. Usually a collection of excerpts is of doubtful value, but this one is a work of art, for a very simple reason: Leibniz had writing techniques which no doubt were rather frequent during his era (beginning 18th century), but that he pushed to an extraordinary extent. Of course, like all philosophers, he wrote huge books. But one might almost be tempted to say that these huge books did not constitute the essential part of his works, since what was essential was in the correspondence and in quite tiny memoirs. Leibniz's great texts often ran 4 or 5, 10 pages, or were in letters. He wrote to some extent in all languages and in some ways was the first great German philosopher. He constitutes the arrival in Europe of German philosophy. His influence was immediate on the German Romantic philosophers in the 19th century, then continues particularly with Nietzsche. Leibniz is a philosopher who best helps us understand a possible answer to this question: what is philosophy? What does a philosopher do? What does philosophy grapple with?
If you think that definitions like search for the true or search for wisdom are not adequate, is there a philosophical activity? I want to say very quickly how I recognize a philosopher in his activity. One can only confront these activities as a function of what they create and of their mode of creation. One must ask, what does a woodworker create? What does a musician create? For me, a philosopher is someone who creates concepts. This implies many things: that the concept is something to be created, that the concept is the product of a creation.
I see no possibility of defining science if one does not indicate something that is created by and in science. And, it happens that what is created by and in science, I'm not completely sure what it is, but not concepts properly speaking. The concept of creation has been much more linked to art than to science or to philosophy. What does a painter create? He creates lines and colors. That suggests that lines and colors are not givens, but are the product of a creation. What is given, quite possibly, one could always call a flow. It's flows that are given, and creation consists in dividing , organizing, connecting flows in such a way that a creation is drawn or made around certain singularities extracted from flows.
A concept is not at all something that is a given. Moreover, a concept is not the same thing as thought: one can very well think without concepts, and everyone who does not do philosophy still thinks, I believe, but does not think through concepts. If you accept the idea of a concept as the product of an activity or an original creation.
I would say that the concept is a system of singularities appropriated from a thought flow. A philosopher is someone who invents concepts. Is he an intellectual? No, in my opinion. For a concept as system of singularities appropriated from a thought flow... Imagine the universal thought flow as a kind of interior monologue, the interior monologue of everyone who thinks. Philosophy arises with the action that consists of creating concepts. For me, there are as many creations in the invention of a concept as in the creation by a great painter or musician. One can also conceive of a continuous acoustic flow (perhaps that is only an idea, but it matters little if this idea is justified) that traverses the world and that even encompasses silence. A musician is someone who appropriates something from this flow: notes? Aggregates of notes? No? What will we call the new sound from a musician? You sense then that it is not simply a question of the system of notes. It's the same thing for a philosopher, it is simply a question of creating concepts rather than sounds. It is not a question of defining philosophy by some sort of search for the truth, for a very simple reason: this is that truth is always subordinate to the system of concepts at one's disposal. What is the importance of philosophers for non-philosophers? It is that although non-philosophers don't know it, or pretend not to be interested, whether they like it or not they think through concepts which have proper names.
I recognize the name of Kant not in his life, but in a certain type of concept signed Kant. Henceforth, one can very well conceive of being the disciple of a philosopher. If you are situated so that you say that such and such a philosopher signed the concepts for which you feel a need, then you become Kantian, Leibnizian, etc.
It is quite necessary that two great philosophers not agree with each other to the extent that each creates a system of concepts that serves as his point of reference. Thus that is not all to be judged. One can very well only be a disciple locally, only on one point or another, philosophy is detachable. You can be a disciple of a philosopher to the extent that you consider that you personally need this type of [concept]. Concepts are spiritual signatures, but that does not mean it's in one's head because concepts are also ways of living. And this is not through choice or reflections, the philosopher reflects no more than does the painter or musician. Activities are defined by a creative activity and not by a reflexive dimension. Henceforth, what does it mean to say: to need this or that concept? In some ways, I tell myself that concepts are such living things, that they really are things with four paws, that move, really. It's like a color, like a sound. Concepts really are so living that they are not unrelated to something that would, however, appear the furthest from the concept, notably the scream .
In some ways, the philosopher is not someone who sings, but someone who screams. Each time that you need to scream, I think that you are not far from a kind of call of philosophy. What would it mean for the concept to be a kind of scream or a kind of form of scream? That's what it means to need a concept, to have something to scream! We must find the concept of that scream. One can scream thousands of things. Imagine something that screams: "Well really, all that must have some kind of reason to be." It's a very simple scream. In my definition, the concept is the form of the scream, we immediately see a series of philosophers who would say, "yes, yes"! These are philosophers of passion, of pathos, distinct from philosophers of logos. For example, Kierkegaard based his entire philosophy on fundamental screams.
But Leibniz is from the great rationalist tradition. Imagine Leibniz, there is something frightening there. He is the philosopher of order, even more, of order and policing, in every sense of the word "policing." In the first sense of the word especially, that is, regulated organization of the city. He only thinks in terms of order. But very oddly in this taste for order and to establish this order, he yields to the most insane concept creation that we have ever witnessed in philosophy. Disheveled concepts, the most exuberant concepts, the most disordered, the most complex in order to justify what is. Each thing must have a reason. In fact, there are two kinds of philosopher, if you accept the definition by which philosophy is the activity consisting of creating concepts. But there are perhaps two poles: there are those who engage in a very sober creation of concepts; they create concepts on the level of a particular singularity well distinguished from another, and I dream finally of a kind of quantification of philosophers in which they would be quantified according to the number of concepts they have signed or invented. If I say: Descartes! That's the type of philosopher with a very sober concept creation. The history of the cogito, historically one can always find an entire tradition, precursors, but there is nonetheless something signed Descartes in the cogito concept, notably (a proposition can express a concept) the proposition: "I think therefore I am," a veritable new concept. It's the discovery of subjectivity, of thinking subjectivity. It's signed Descartes.
Of course, we could always look in St. Augustine's works, to see if it wasn't already in preparation. There is certainly a history of concepts, but it's signed Descartes. Haven't we made rather quick work of Descartes though? We could assign to him five or six concepts, an enormous feat to have invented six concepts, but it's a very sober creation. And then there are exasperated philosophers. For them, each concept covers an aggregate of singularities, and then they always need to have other, always other concepts. One witnesses a mad creation of concepts. The typical example is Leibniz. He never finished creating something new.
That's all I wanted to explain.
He is the first philosopher to reflect on the power of the German language as a concept, as German being an eminently conceptual language, and it's not by chance that it can also be a great language of the scream. Multiple activities, he attends to all, a very great mathematician, great physics scholar, very good jurist, many political activities, always in the service of order. He does not stop, he is very shady . There is a Leibniz-Spinoza visit (he who was the anti-Leibniz): Leibniz has him read manuscripts, and one imagines Spinoza very exasperated, wondering what this guy wants. Following that when Spinoza was attacked, Leibniz said that he never went to see him, he said it was to monitor him... Abominable, Leibniz is abominable. His dates: 1646-1716. It's a long life, straddling plenty of things.
Finally he had a kind of diabolical humor. I'd say that his system is rather like a pyramid. Leibniz's great system has several levels. None of these levels is false, these levels symbolize each other, and Leibniz is the first great philosopher to conceive of activity and thought as a vast symbolization.
Thus, all these levels symbolize, but they are all more or less close to what we could provisionally call the absolute. And that belongs to his very body of work. Depending on Leibniz's correspondent or on the public to which he addressed himself, he presented his whole system at a particular level. Imagine that his system is made of levels more or less contracted or more or less relaxed; in order to explain something to someone, he goes to situate himself on a particular level of his system. Let us assume that the someone in question was suspected by Leibniz of having a mediocre intelligence: very well, he is delighted, he situates himself on one of the lowest levels of his system, and if he addresses someone of higher intelligence, he jumps to a higher level. As these levels belong implicitly to Leibniz's own texts, that creates a great problem of commentary. It's complicated because, in my opinion, one can never rely on a Leibniz text if one has not first discerned the system level to which this text corresponds.
For example, there are texts in which Leibniz explains what, according to him, is the union of soul and body, right, and it's to one particular correspondent or another; to another correspondent, he will explain that there is no problem in the union of soul and body since the real problem is that of the relation of souls to one another. The two things are not at all contradictory, it's two levels of the system. The result is that if one does not evaluate the level of a Leibniz text, then one will get the impression that he constantly contradicts himself, when in fact, he does not contradict himself at all.
Leibniz is a very difficult philosopher. I would like to give titles to each part of what I have to propose to you. The principal #1 I would call "a funny thought" . Why do I call it "a funny thought"?, Well, because among Leibniz's texts, there is a small one that Leibniz himself calls "funny thought." Thus I am authorized by the author himself. Leibniz dreamed a lot, he has a whole science-fiction side that is absolutely amazing, all the time he imagined institutions. In this little "funny thought" text, he imagined a very disturbing institution that would be as follows: an academy of games would be necessary. In that era, as well with Pascal, certain other mathematicians, and Leibniz himself, there developed a great theory of games and probabilities. Leibniz is one of the great founders of game theory. He was impassioned by mathematical game problems, he must have been quite a games player himself. He imagined this academy of games as necessarily being at once - why at once? Because depending on the point of view in which one is situated to see this institution, or to participate in it - this would be at once a section of the academy of sciences, a zoological and botanical garden, a universal exposition, a casino where one gambled, and an enterprise of police control. That's not bad. He called that "a funny thought."
Assume that I'm telling you a story. This story consists in taking up one of the central points of Leibniz's philosophy, and I tell it to you as if it were the description of another world, and there I also number the principal propositions that go into forming a funny thought.

a) The thought flow, eternally, brings with it a famous principle that has a very special characteristic because it is one of the only principles about which one can be certain, and at the same time one can not see at all what it offers to us. It is certain, but it is empty. This famous principle is the principle of identity. The principle of identity has a classical formula, A is A. That is certain. If I say blue is blue or God [is] God, I am not saying with this that God exists, in a sense I am in certainty. Only there it is, do I think something when I say A is A, or am I not thinking? Let us nonetheless try to say what results from this principle of identity. It is presented in the form of a reciprocal proposition. A is A means: subject A, verb to be, A attribute or predicate. There is a reciprocity of subject and predicate. Blue is blue, a triangle is a triangle, these are empty and certain propositions. Is that all? An identical proposition is a proposition such that the attribute or the predicate is the same as the subject and reciprocates with the subject. There is a second case just a bit more complex, notably that the principle of identity can determine propositions which are not simply reciprocal propositions. There is no longer simply reciprocity of the predicate with the subject and subject with the predicate. Suppose that I say: "The triangle has tree sides," this is not the same thing as saying, "The triangle has three angles." "The triangle has three angles" is an identical proposition because it is reciprocal. "The triangle has three sides" is a little different, it is not reciprocal. There is no identity of subject and predicate. In fact, "three sides" is not the same thing as "three angles". And nonetheless, there is a supposed logical necessity. This logical necessity is that you cannot conceptualize three angles composing a single figure without this figure also having three sides. There is no reciprocity, but there is inclusion. Three sides are included in the triangle. Inherence or inclusion.
Likewise, if I say that matter is matter, matter and matter is an identical proposition in the form of a reciprocal proposition. The subject is identical to the predicate. If I say that matter is in extension <étendue>, this is again an identical proposition because I cannot think of the concept matter without already introducing extension. Extension is in matter. This is all the more a reciprocal proposition since, inversely, perhaps I really can think of extension without anything filling it in, that is, without matter. This is therefore not a reciprocal proposition, but it is a proposition of inclusion; when I say "matter is in extension," this is an identical proposition by inclusion.
I would say therefore that there are two kinds of identical propositions: there are reciprocal propositions in which the subject and predicate are one and the same, and propositions of inherence or inclusion in which the predicate is contained in the concept of the subject. If I say "this page has a front side and a back side," OK, let's leave that, I withdraw my example. If I am looking for a more interesting statement of the identity principle, I would say in Leibnizian fashion that the identity principle is stated as follows: every analytical proposition is true.
What does analytical mean? According to the example we have just seen, an analytical proposition is one in which either the predicate or the attribute is identical with the subject, for example, "the triangle is triangular," reciprocal proposition, or proposition of inclusion such as "the triangle has three sides." The predicate is contained in the subject to the point that when you have conceived of the subject, the predicate was already there. It suffices therefore to have an analysis in order to find the predicate in the subject. Up to this point, Leibniz as original thinker has yet to emerge.

b) Leibniz emerges. He arises in the form of this very bizarre scream . I am going to give it a more complex expression than I did earlier. Everything that we're saying is not philosophy, but pre-philosophy. This is the terrain on which a very prodigious philosophy will be built. Leibniz arrives and says: OK, the identity principle gives us a certain model. Why a certain model? In its very statement <énoncé>, an analytical proposition is true, if you attribute to a subject something that constitutes a unity with the subject itself, or that is mixed up with or is already contained in the subject. You risk nothing in being wrong. Thus, every analytical proposition is true.
Leibniz's stroke of pre-philosophical genius is to say: Let's consider reciprocity! Something absolutely new and nonetheless very simple starts there, since this had to be thought through. And what does it mean to say, "it had to be thought through"? It means that one had to have need of that, that had to relate to something quite urgent for him. What is the reciprocity of the identity principle in its complex statement, "every analytical proposition is true"? Reciprocity poses many more problems. Leibniz emerges and says: every true proposition is analytical.
If it is true that the identity principle gives us a model of truth, why are we stumped by the following difficulty, notably: it is true, it doesn't make us think anything. The identity principle will force us to think something; it is going to be reversed, turned around. You will tell me that turning A is A around yields A is A. Yes and no. That yields A is A in the formal formulation which prevents the reversal of the principle. But in the philosophical formulation, which still amounts to exactly the same thing, "every analytical proposition is a true proposition", if you reverse the principle: "every true proposition is necessarily analytical," what does that mean? Each time that you formulate a true proposition, it must be analytical (and this is where there is the scream!), whether you want it or not, that is, it is reducible to a proposition of attribution or of predication, and not only is it reducible to a judgment of predication or attribution (the sky is blue), but it is analytical, that is the predicate is either reciprocal with the subject or contained in the concept of the subject? Does that go without saying? He throws himself into a strange undertaking , and it is not from preference that he says that, rather he needs it. But he undertakes an impossible task, in fact he needs some entirely crazy concepts in order to reach this task that he is in the process of giving himself. If every analytical proposition is true, every true proposition certainly must be analytical. It does not go without saying at all that every judgment is reducible to a judgment of attribution. It's not going to be easy to show. He throws himself into a combinatory analysis, as he himself says, that is fantastic.
Why doesn't it go without saying? "The box of matches is on the table," I'd say that this is a judgment, you know? "On the table" is a spatial determination. I could say that the matchbox is "here." "Here," what's that? I'd say that it's a judgment of localization. Again I repeat very simple things, but they always have been fundamental problems of logic. It's only to suggest that in appearance, all judgments do not have as form predication or attribution. When I say, "the sky is blue," I have a subject, sky, and an attribute, blue. When I say "the sky is up there" or "I am here," is "here" - spatial localization - assimilable to a predicate? Can I formally link the judgment "I am here" to a judgment of the kind "I am blond"? It's not certain that spatial localization is a quality. And "2+2=4" is a judgment that we ordinarily call a relational judgment. Or if I say, "Pierre is smaller than Paul," this is a relation between two terms, Pierre and Paul. No doubt I orient this relation upon Pierre: if I say "Pierre is smaller than Paul," I can say "Paul is larger than Pierre." Where is the subject, where is the predicate? That is exactly the problem that has disturbed philosophy since its beginnings; ever since there was logic they have wondered to what extent the judgment of attribution could be considered as the universal form of any possible judgment, or rather one case of judgment among others. Can I treat "smaller than Paul" like an attribute of Pierre? It's not certain, not at all obvious. Perhaps we have to distinguish very different types of judgment from each other, notably: relational judgment, judgment of spatio-temporal localization, judgment of attribution, and still many more: judgment of existence. If I say "God exists," can I formally translate it into the form of "God is existent," existent being an attribute? Can I say that "God exists" is a judgment of the same form as "God is all-powerful"? Undoubtedly not, since I can only say "God is all-powerful" by adding "yes, if he exists". Does God exist? Is existence an attribute? Not certain.
So you see that by proposing the idea that every true proposition must be in one way or another an analytical proposition, that is identical, Leibniz already gives himself a very hard task; he commits himself to showing in what way all propositions can be linked to the judgment of attribution, notably propositions that state relations, that state existences, that state localizations, and that, at the outside, exist, are in relation with, can be translated as the equivalent of the attribute of the subject.
In your mind there must arise the idea of an infinite task.
Let us assume that Leibniz reached it: what world is going to emerge from it? What very bizarre world? What kind of world is it in which I can say "every true proposition is analytical"? You recall certainly that ANALYTICAL is a proposition in which the predicate is identical to the subject or else is included in the subject. That kind of world is going to be pretty strange.
What is the reciprocity of the identity principle? The identity principle is thus any true proposition is analytical; not the reverse, any analytical proposition is true. Leibniz said that another principle is necessary, reciprocity: every true proposition is necessarily analytical. He will give to it a very beautiful name: the principle of sufficient reason. Why sufficient reason? Why does he believe himself fully immersed in his very own scream? EVERYTHING MUST SURELY HAVE A REASON. The principle of sufficient reason can be expressed as follows: whatever happens to a subject, be it determinations of space and time, of relation, event, whatever happens to a subject, what happens, that is what one says of it with truth, everything that is said of a subject must be contained in the notion of the subject.
Everything that happens to a subject must already be contained in the notion of the subject. The notion of "notion" is going to be essential. It is necessary for "blue" to be contained in the notion of sky. Why is this the principle of sufficient reason ? Because if it is this way, each thing with a reason, reason is precisely the notion itself in so far as it contains all that happens to the corresponding subject . Henceforth everything has a reason.
Reason = the notion of the subject in so far as this notion contains everything said with truth about this subject. That is the principle of sufficient reason which is therefore justly the reciprocal of the identity principle. Rather than looking for abstract justifications I wonder what bizarre world is going to be born from all that? A world with very strange colors if I return to my metaphor of painting. A painting signed Leibniz. Every true proposition must be analytical or still more, everything that you say with truth about a subject must be contained in the notion of the subject. You sense that this is getting crazy, he's got a lifetime of work ahead of him.
What does "notion" mean? It's signed Leibniz. Just as there is a Hegelian conception of the concept, there is a Leibnizian conception of the concept.

c) Again, my problem is what world is going to emerge, and in this sub-category c), I would like to begin to show that, from this point, Leibniz is going to create truly hallucinatory concepts. It's truly a hallucinatory world. If you want to think about relations between philosophy and madness, for example, there are some very weak pages by Freud on the intimate relation of metaphysics with delirium . One can only grasp the positivity of these relations through a theory of the concept, and the direction that I would like to take would be the relationship of the concept with the scream. I would like to make you feel this presence of a kind of conceptual madness in Leibniz's universe as we are going to see it be born. It is a gentle violence, let yourself go. It is not a question of arguing. Understand the stupidity of objections.
I will add a parenthesis to complicate things. You know that there is a philosopher following Leibniz who said that truth is one of synthetic judgments. He is opposed to Leibniz. OK! How does that concern us? It's Kant. This is not to say that they do not agree with each other. When I say that, I credit Kant with a new concept which is synthetic judgment. This concept had to be invented, and it was Kant who did so. To say that philosophers contradict one another is a feeble formula, it's like saying that Velasquez did not agree with Giotto, right! It's not even true, it's nonsensical.
Every true proposition must be analytical, that is such that it attributes something to a subject and that the attribute must be contained in the notion of the subject. Let us consider an example. I do not wonder if it is true, I wonder what it means. Let us take an example of a true proposition. A true proposition can be an elementary one concerning an event that took place. Let's take Leibniz's own example: "CAESAR CROSSED THE RUBICON". It's a proposition. It is true or we have strong reasons to assume it's true. Another proposition: "ADAM SINNED".
There is a highly true proposition. What do you mean by that? You see that all these propositions chosen by Leibniz as fundamental examples are event-ual propositions , so he does not give himself an easy task. He is going to tell us this: since this proposition is true, it is necessary, whether you want it or not, that the predicate "crossed the Rubicon," if the proposition is true, but it is true, this predicate must be contained in the notion of Caesar. Not in Caesar himself, but in the notion of Caesar. The notion of the subject contains everything that happens to a subject, that is, everything that is said about the subject with truth. In "Adam sinned," sin at a particular moment belongs to the notion of Adam. Crossing the Rubicon belongs to the notion of Caesar. I would say that here, Leibniz proposes one of his greatest concepts, the concept of inherence. Everything that is said with truth about something is inherent in the notion of this something.
This is the first aspect or development of sufficient reason.

d) When we say that, we can no longer stop. When one has started into the domain of the concept, one cannot stop. In the domain of screams, there is a famous scream from Aristotle. The great Aristotle -- who, let us note, exerted an extremely strong influence on Leibniz -- at one point proposed in the Metaphysics a very beautiful formula: it is indeed necessary to stop (anankstenai). This is a great scream. This is the philosopher in front of the chasm of the interconnection of concepts. Leibniz could care less, he does not stop. Why? If you refer to proposition c): everything that you attribute to a subject must be contained in the notion of this subject. But what you attribute with truth to any subject whatsoever in the world, if were it Caesar, it is sufficient for you to attribute to it a single thing with truth in order for you to notice with fright that, from that moment on, you are forced to cram into the notion of the subject not only the thing that you attribute to it with truth, but the totality of the world. Why? By virtue of a well-known principle that is not at all the same as that of sufficient reason. This is the simple principle of causality. For in the end, the causality principle stretches to infinity, that's it's very characteristic. And this is a very special infinite since, in fact, it stretches to the indefinite . Specifically, the causality principle states that everything has a cause, which is very different from every thing has a reason. But the cause is a thing, and in its turn, it has a cause, etc. etc. I can do the same thing, notably that every cause has an effect and this effect is in its turn the cause of effects. This is therefore an indefinite series of causes and effects.
What difference is there between sufficient reason and cause? We understand very well. Cause is never sufficient. One must say that the causality principle poses a necessary cause, but never a sufficient one. We must distinguish between necessary cause and sufficient reason. What distinguishes them evidently is that the cause of a thing is always something else. The cause of A is B, the cause of B is C, etc..... An indefinite series of causes. Sufficient reason is not at all something other than the thing. The sufficient reason of a thing is the notion of the thing. Thus, sufficient reason expresses the relation of the thing with its own notion whereas cause expresses the relations of the thing with something else. It's limpid.

e) If you say that a particular event is encompassed in the notion of Caesar, "crossing the Rubicon" is encompassed in the notion of Caesar . You can't stop yourself in which sense? From cause to cause and effect to effect, it's at that moment the totality of the world that must be encompassed in the notion of a particular subject. That becomes very odd, there's the world passing by inside each subject, or each notion of subject. In fact, crossing the Rubicon has a cause, this cause itself has multiple causes, from cause to cause, into cause from cause and into cause from cause of cause. It's the whole series of the world that passes there, at least the antecedent series. And moreover, crossing the Rubicon has effects. If I limit myself to the largest ones: commencement of a Roman empire. The Roman empire in its turn has effects, we follow directly from the Roman empire. From cause to cause and effect to effect, you cannot say a particular event is encompassed in the notion of a particular subject without saying that, henceforth, the entire world is encompassed in the notion of a particular subject.
There is indeed a trans-historical characteristic of philosophy. What does it mean to be Leibnizian in 1980? They exist, or at least it's possible that they exist.
If you said, conforming to the principle of sufficient reason, that what happens to a particular subject, and which personally concerns it, then what you attribute it with truth, having blue eyes, crossing the Rubicon, etc. ... belongs to the notion of the subject, that is encompassed in this notion of the subject; you cannot stop, one must say that this subject contains the whole world. It is no longer the concept of inherence or inclusion, it's the concept of expression which, in Leibniz's work, is a fantastic concept. Leibniz expresses himself in this form: the notion of the subject expresses the totality of the world.
His own "crossing the Rubicon" stretches to infinity backward and forward by the double play of causes and effects. But then, it is time to speak for ourselves, little matter what happens to us and the importance of what happens to us. We must say that it is each notion of subject that contains or expresses the totality of the world. That is, each of you, me, expresses or contains the totality of the world. Just like Caesar, no more, no less. That gets complicated, why? A great danger: if each individual notion, if each notion of the subject expresses the totality of the world, that means that there is only a single subject, a universal subject, and the you, me, Caesar, would only be appearances of this universal subject. It would be quite possible to say: there would be a single subject that would express the world.
Why couldn't Leibniz say that? He had no choice. It would mean repudiating himself. All that he had done before that with the principle of sufficient reason would then make what sense? In my opinion, this was the first great reconciliation of the concept and the individual. Leibniz was in the process of constructing a concept of the concept such that the concept and the individual were finally becoming adequate to one another. Why?
That the concept might extend into the individual, why is this new? Never had anyone dared that. The concept, what is it? It is defined by the order of generality. There is a concept when there is a representation which is applied to several things. But identifying the concept and the individual with each other, never had that been done. Never had a voice reverberated in the domain of thought to say that the concept and the individual were the same thing.
What had always been distinguished was an order of the concept that referred to a generality and an order of the individual that referred to a singularity. Even more, it was always considered as going without saying that the individual as such was not comprehensible via the concept.
It was always understood that the proper name was not a concept. Indeed, "dog" is certainly a concept, but "Fido" is not a concept. There is certainly a dogness about all dogs, as certain logicians say in a splendid language, but there is no Fido-ness about all Fidos. Leibniz is the first to say that concepts are proper names, that is, that concepts are individual notions.
There is a concept of the individual as such. Thus you see that Leibniz cannot fall back on the proposition since every true proposition is analytical, the world is thus contained in a single and same subject which would be a universal subject. He cannot since his principle of sufficient reason implied that what was contained in a subject -- thus what was true, what was attributable to a subject -- was contained in a subject as an individual subject. Thus he cannot give himself a kind of universal mind. He has to remain fixed on the singularity, on the individual as such. And in fact, this will be one of the truly original points for Leibniz, the perpetual formula in his works: substance (no difference between substance and subject for him) is individual.
It's the substance Caesar, it's the substance you, the substance me, etc. ... The urgent question in my sub-category d) since he forbids himself from invoking a universal mind in which the world will be included ... other philosophers will invoke a universal mind. There is even a very short text by Leibniz entitled "Considerations on universal mind," in which he goes on to show in what way there is indeed a universal mind, God, but that does not prevent substance from being individual. Thus irreducibility of individual substances.
Since each substance expresses the world, or rather each substantial notion, each notion of a subject, since each one expresses the world, you express the world, for all times. We notice that, in fact, he has a lifetime of work because he faces the objection that's made to him immediately: but then, what about freedom? If everything that happens to Caesar is encompassed in the individual notion of Caesar, if the entire world is encompassed in the universal notion of Caesar, then Caesar crossing the Rubicon only acts to unroll --odd word, devolvere, which comes up all the time in Leibniz's works -- or explicate (the same thing), that is to say, literally to unfold , like you unfold a rug. It's the same thing: explicate, unfold, unroll. Thus crossing the Rubicon as event only acts to unroll something that was encompassed for all times in the notion of Caesar. You see that it's quite a real problem.
Caesar crossed the Rubicon in a particular year, but even were he crossing the Rubicon in a particular year, it was encompassed for all time in his individual notion. Thus, where is this individual notion? It is eternal. There is an eternal truth of dated events. But then, how about freedom? Everyone jumps on him. Freedom is very dangerous under a Christian regime. So Leibniz will write a little work, "On freedom," in which he explains what freedom is. Freedom is going to be a pretty funny thing for him.
But leave that aside for the moment.
What distinguishes one subject from another? That, we can't leave aside for the moment, unless our flow were to be cut off. What is going to distinguish you from Caesar since just like him, you express the totality of the world, present, past, and future? It's odd, this concept of Expression. That's where he proposes a very rich notion.

f) What distinguishes an individual substance from another is not very difficult. In some way, it has to be irreducible.
Each one, each subject, for each individual notion, each notion of subject has to encompass this totality of the world, express this total world, but from a certain point of view. And there begins a perspectivist philosophy. And it's not inconsiderable. You will tell me: what is more banal than the expression "a point of view"? If philosophy means creating concepts, what does create concepts mean? Generally speaking, these are banal formulae. Great philosophers each have banal formulae that they wink at. A wink from a philosopher is, at the outside limit, taking a banal formula and having a ball , you have no idea what I'm going to put inside it. To create a theory of point of view, what does that imply? Could that be done at any time at all? Is it by chance that it's Leibniz who created the first great theory at a particular moment? At the moment in which the same Leibniz created a particularly fruitful chapter in geometry, called projective geometry. Is it by chance that it's out of an era in which are elaborated, in architecture as in painting, all sorts of techniques of perspective? We retain simply these two domains that symbolize that: architecture-painting and perspective in painting on one hand, and on the other hand, projective geometry. Understand what Leibniz wants to develop from them. He is going to say that each individual notion expresses the totality of the world, yes, but from a certain point of view.
What does that mean? Of so little import is it, banally, pre-philosophically, that it is henceforth as equally impossible for him to stop. That commits him to showing that what constitutes the individual notion as individual is point of view. And that therefore point of view is deeper that whosoever places himself there.
At the basis of each individual notion, it will indeed be necessary for there to be a point of view that defines the individual notion. If you prefer, the subject is second in relation to the point of view. And after all, to say that is not a piece of cake, it's not inconsiderable. He established a philosophy that will find its name in the works of another philosopher who stretches out his hand to Leibniz across the centuries, to wit Nietzsche. Nietzsche will say: my philosophy is a perspectivism. You understand that it becomes idiotic or banal to whine about whether perspectivism consists in saying that everything is relative to the subject; or simply that everything is relative. Everyone says it, it belongs to propositions that hurt no one since it is meaningless. So long as I take the formula as signifying everything depends on the subject, that means nothing, I caused, as one says ...

. . . What makes me = me is a point of view on the world. Leibniz cannot stop. He has to go all the way to a theory of point of view such that the subject is constituted by the point of view and not the point of view constituted by the subject. Fully into the nineteenth century, when Henry James renews the techniques of the novel through a perspectivism, through a mobilization of points of view, there too in James's works, it's not points of view that are explained by the subjects, it's the opposite, subjects that are explained through points of view. An analysis of points of view as sufficient reason of subjects, that's the sufficient reason of the subject. The individual notion is the point of view under which the individual expresses the world. It's beautiful and it's even poetic. James has sufficient techniques in order for there to be no subject; what becomes one subject or another is the one who is determined to be in a particular point of view. It's the point of view that explains the subject and not the opposite. For Leibniz, every individual substance is like an entire world and like a mirror of God or of the whole universe that each substance expresses in its own way: kind of like an entire city is diversely represented depending on the different situations of the one who looks at it. Thus, the universe is seemingly multiplied as many times as there are substances, and the glory of God is redoubled equally by as many completely different representations of his/her/its . He speaks like a cardinal. One can even say that every substance bears in some ways the characteristic of infinite wisdom and of all of God's power, and limits as much as it is able to.
In all this, I maintain that the new concept of point of view is deeper than the concept of individual and individual substance. It is point of view which will define essence. Individual essence. One must believe that to each individual notion corresponds a point of view. But that gets complicated because this point of view would be in effect from birth to death for an individual. What would define us is a certain point of view on the world. I said that Nietzsche will rediscover this idea. He didn't like him , but that's what he took from him. The theory of point of view is an idea from the Renaissance. The Cardinal de Cuse , a very great Renaissance philosopher, referred to portraiture changing according to point of view. From the era of Italian fascism, one notices a very odd portrait almost everywhere: face on, it represented Mussolini, from the right side it represented his son-in-law, and if one stood to the left, it represented the king.
The analysis of points of view in mathematics -- and it's again Leibniz who caused this chapter of mathematics to make considerable progress under the name of analysis situs --, and it is evident that it is connected to projective geometry. There is a kind of essentiality, of objectity of the subject, and the objectity is the point of view. Concretely were everyone to express the world in his own point of view, what does that mean? Leibniz did not retreat from the strangest concepts. I can no longer say "from his own point of view." If I said "from his own point of view," I would make the point of view depend on a preceding subject , but it's the opposite. But what determines this point of view? Leibniz : understand, each of us expresses the totality of the world, only he expresses it in an obscure and confused way. Obscurely and confused means what in Leibniz's vocabulary? That means that the totality of the world is really in the individual, but in the form of minute perception. Minute perceptions. Is it by chance that Leibniz is one of the inventors of differential calculus? These are infinitely tiny perceptions, in other words, unconscious perceptions. I express everyone, but obscurely, confusedly, like a clamor.
Later we will see why this is linked to differential calculus, but notice that the minute perceptions of the unconscious are like differentials of consciousness, it's minute perceptions without consciousness. For conscious perceptions, Leibniz uses another word: apperception. Apperception, to perceive , is conscious perception, and minute perception is the differential of consciousness which is not given in consciousness. All individuals express the totality of the world obscurely and confusedly. So what distinguishes a point of view from another point of view? On the other hand, there is a small portion of the world that I express clearly and distinctly, and each subject, each individual has his/her own portion, but in what sense? In this very precise sense that this portion of the world that I express clearly and distinctly, all other subjects express it as well, but confusedly and obscurely.
What defines my point of view is like a kind of projector that, in the buzz of the obscure and confused world, keeps a limited zone of clear and distinct expression. However stupid you may be, however insignificant we all may be, we have our own little thing, even the pure vermin has its little world: it does not express much clearly and distinctly, but it has its little portion. Beckett's characters are individuals: everything is confused, an uproar , they understand nothing, they are in tatters ; there is the great uproar of the world. However pathetic they may be in their garbage can, they have their very own little zone. What the great Molloy calls "my properties." He no longer moves, he has his little hook and, in a strip of one meter, with his hook, he grabs things, his properties. It's a clear and distinct zone that he expresses. We are all the same. But our zone is more or less sizable, and even then it's not certain, but it is never the same. What is it that determines the point of view? It's the proportion of the region of the world expressed clearly and distinctly by an individual in relation to the totality of the world expressed obscurely and confusedly. That's what point of view is.
Leibniz has a metaphor that he likes: you are near the sea and you listen to waves. You listen to the sea and you hear the sound of a wave. I hear the sound of a wave, that is, I have an apperception: I distinguish a wave. And Leibniz says: you would not hear the wave if you did not have a minute unconscious perception of the sound of each drop of water that slides over and through another, and that makes up the object of minute perceptions. There is the roaring of all the drops of water, and you have your little zone of clarity, you clearly and distinctly grasp one partial result from this infinity of drops, from this infinity of roaring, and from it, you make your own little world, your own property.
Each individual notion has its point of view, that is from this point of view, it extracts from the aggregate of the world that it expresses a determined portion of clear and distinct expression. Given two individuals, you have two cases: either their zones do not communicate in the least, and create no symbols with one another -- there aren't merely direct communications, one can conceive of there being analogies -- and in that moment, they have nothing to say to each other; or it's like two circles that overlap: there is a little common zone, there we can do something together. Leibniz thus can say quite forcefully that no two individual substances have the same point of view or exactly the same clear and distinct zone of expression. And finally, Leibniz's stroke of genius: what will define the clear and distinct zone of expression that I have? I express the totality of the world, but I only express clearly and distinctly a reduced portion of it, a finite portion. What I express clearly and distinctly, Leibniz tells us, is what relates to my body. We will see what this body means, but what I express clearly and distinctly is that which affects my body.
Thus I obviously do not express clearly and distinctly the passage of the Rubicon, since that concerned Caesar's body. There is something that concerns my body and that only I express clearly and distinctly, in relation to this buzz that covers the entire universe.
f) In this story of the city, there is a problem. OK, there are different points of view. These points of view preexist the subject who is placed there, good. In this event, the secret of point of view is mathematical, geometrical, and not psychological. It's at the least psycho-geometrical. Leibniz is a man of notions, not a man of psychology. But everything urges me to say that the city exists outside points of view. But in my story of expressed world, in the way we started off, the world has no existence outside the point of view that expresses it; the world does not exist in itself. The world is uniquely the common expressed of all individual substances, but the expressed does not exist outside that which expresses it. The world does not exist in itself, the world is uniquely the expressed. The entire world is contained in each individual notion, but it exists only in this inclusion. It has no existence outside. It's in this sense that Leibniz will be, and not incorrectly, on the side of the idealists: there is no world in itself, the world exists only in the individual substances that express it. It's the common expressed of all individual substances. It's the expressed of all individual substances, but the expressed does not exist outside the substances that express it. It's a real problem!
What distinguishes these substances is that they all express the same world, but they don't express the same clear and distinct portion. It's like chess. The world does not exist. It's the complication of the concept of expression. Which is going to provide this final difficulty. Still it is necessary that all individual notions express the same world. So it's curious -- it's curious because by virtue of the principle of identity that permits us to determine what is contradictory, that is, what is impossible, it's A is not A. It's contradictory: example: the squared circle. A squared circle is a circle that is not a circle. So starting from the principle of identity, I can have a criterion of contradiction. According to Leibniz, I can demonstrate that 2 + 2 cannot make 5, I can demonstrate that a circle cannot be squared. Whereas, on the level of sufficient reason, it's much more complicated, why? Because Adam the non-sinner, Caesar not crossing the Rubicon, is not like the squared circle. Adam the non-sinner is not contradictory. Understand how he's going to try to save freedom, once he has placed himself in a bad situation for saving it. This is not at all impossible: Caesar could have not crossed the Rubicon, whereas a circle cannot be squared; here, there is no freedom.
So, again he's stuck, again Leibniz has to find another concept and, of all his crazy concepts, this will undoubtedly be the craziest. Adam could have not sinned, so in other words, the truths governed by the principle of sufficient reason are not the same type as the truths governed by the principle of identity, why? Because the truths governed by the principle of identity are such that their contradictory status is impossible, whereas the truths governed by the principle of sufficient reason have a contradictory status that is possible: Adam the non-sinner is possible.
It's even all that distinguishes, according to Leibniz, the truths called truths of essence and those called truths of existence. The truths of existence are such that their contradictory status is possible. How is Leibniz going to get out of this final difficulty? How is he going to be able to maintain at once that all that Adam did is contained forever in his individual notion, and nonetheless Adam the non-sinner was possible. He seems stuck, it's delicious because from this perspective, philosophers are somewhat like cats, it's when they are stuck that they get loose, or they're like fish, the concept becoming fish. He is going to tell us the following: that Adam the non-sinner is perfectly possible, like Caesar not having crossed the Rubicon, all that is possible, but it did not happen because, if it is possible in itself, it's incompossible. That's when he created the very strange logical concept of incompossibility. On the level of existences, it is not enough for a thing to be possible in order to exist, one must also know with what it is compossible. So Adam the non-sinner, though possible in himself, is incompossible with the world that exists. Adam could have not sinned, yes, but provided that there were another world. You see that the inclusion of the world in the individual notion, and the fact that something else is possible, he suddenly reconciles the notion of compossibility, Adam the non-sinner belongs to another world. Adam the non-sinner could have been possible, but this world was not chosen. It is incompossible with the existing world. It is only compossible with other possible worlds that have not passed into existence.
Why is it that world which passed into existence? Leibniz explains what is, for him, the creation of worlds by God, and we see well how this is a theory of games: God, in his understanding , conceives an infinity of possible worlds, only these possible worlds are not compossible with each other, and necessarily so since it's God who chooses the best. He chooses the best of possible worlds. And it happens that the best of possible worlds implies Adam as sinner. Why? That's going to be awful . What is interesting logically is the creation of a proper concept of compossiblity to designate a more limited logical sphere than that of logical possibility. In order to exist, it is not enough for something to be possible, this thing must also be compossible with others that constitute the real world. In a famous formula from the Monadology, Leibniz says that individual notions have neither doors nor windows. That arrives to correct the metaphor of the city. No doors or windows means that there is no opening. Why? Because there is no exterior. The world that individual notions express is interior, it is included in individual notions. Individual notions have no doors or windows, everything is included in each one, and yet there is a world common to all individual notions: for what each individual notion includes, to wit the totality of the world, the notion includes it necessarily as a form in which what it expresses is compossible with what the others express. It's a marvel. It's a world in which there is no direct communication between subjects. Between Caesar and you, between you and me, there is no direct communication, and as we'd say today, each individual notion is programmed in such a way that what it expresses forms a common world with what the other expresses. It's one of the last concepts from Leibniz: pre-established harmony. Pre-established, it's absolutely a programmed harmony. It's the idea of the spiritual automaton, and at the same time, it's the grand age of automatons at this end of the seventeenth century.
Each individual notion is like a spiritual automaton, that is what it expresses is interior to it, it's without doors or windows; it is programmed in such a way that what it expresses is in compossibility with what the other expresses.
What I have done today was solely a description of the world of Leibniz, and even so, only one part of this world. Thus, the following notions have been successively laid out: sufficient reason, inherence and inclusion, expression or point of view, incompossibility.




The last time, as we agreed, we had begun a series of studies on Leibniz that should be conceived as an introduction to a reading, yours, of Leibniz.
To introduce a numerical clarification, I relied on numbering the paragraphs so that everything did not get mixed up.
The last time, our first paragraph was a kind of presentation of Leibniz's principal concepts. As background to all this, there was a corresponding problem for Leibniz, but obviously much more general, to wit: what precisely does it mean to do philosophy. Starting from a very simple notion: to do philosophy is to create concepts, just as doing painting is to create lines and colors. Doing philosophy is creating concepts because concepts are not something that pre-exists, not something that is given ready made. In this sense, we must define philosophy through an activity of creation: creation of concepts. This definition seemed perfectly suitable for Leibniz who, precisely, in an apparently fundamentally rationalist philosophy, is engaged in a kind of exuberant creation of unusual concepts of which there are few such examples in the history of philosophy.
If concepts are the object of a creation, then one must say that these concepts are signed. There is a signature, not that the signature establishes a link between the concept and the philosopher who created it. Rather the concepts themselves are signatures. The entire first paragraph caused a certain number of properly Leibnizian concepts to emerge. The two principal ones that we discerned were inclusion and compossibility. There are all kinds of things that are included in certain things, or enveloped in certain things. Inclusion, envelopment. Then, the completely different, very bizarre concept of compossibility: there are things which are possible in themselves, but that are not compossible with another.
Today, I would like to give a title to this second paragraph, this second inquiry on Leibniz: Substance, World, and Continuity.
The purpose of this second paragraph is to analyze more precisely these two major concepts of Leibniz: Inclusion and Compossibility.
At the point where we ended the last time, we found ourselves faced with two problems: the first is that of inclusion. In what sense? We saw that if a proposition were true, it was necessary in one way or another that the predicate or attribute be contained or included -- not in the subject --, but in the notion of the subject. If a proposition is true, the predicate must be included in the notion of the subject. Let’s allow ourselves the freedom to accept that and, as Leibniz says, if Adam sinned, the sin had to be contained or included in the individual notion of Adam. Everything that happens, everything that can be attributed, everything that is predicated about a subject must be contained in the notion of the subject. This is a philosophy of predication. Faced with such a strange proposition, if one accepts this kind of Leibnizian gamble, one finds oneself immediately faced with problems. Specifically if any given event that concerns a specific individual notion, for example, Adam, or Caesar -- Caesar crossed the Rubicon, it is necessary that crossing the Rubicon be included in the individual notion of Caesar -- great, O.K., we are quite ready to support Leibniz. But if we say that, we cannot stop: if a single thing is contained in the individual notion of Caesar, like "crossing the Rubicon," then it is quite necessary that, from effect to cause and from cause to effect, the totality of the world be contained in this individual notion. Indeed, crossing the Rubicon itself has a cause that must also be contained in the individual notion, etc. etc. to infinity, both ascending and descending. At that point, the entire Roman empire -- which, grosso modo, results from the crossing of the Rubicon as well as all the consequences of the Roman empire -- in one way or another, all of this must be included in the individual notion of Caesar such that every individual notion will be inflated by the totality of the world that it expresses. It expresses the totality of the world. There we see the proposition becoming stranger and stranger.
There are always delicious moments in the history of philosophy, and one of the most delicious of these came at the far extreme of reason -- that is, when rationalism, pushed all the way to the end of its consequences, engendered and coincided with a kind of delirium that was a delirium of madness. At that moment, we witness this kind of procession, a parade, in which the same thing that is rational pushed to the far end of reason is also delirium, but delirium of the purest madness.
Thus, if it is true that the predicate is included in the notion of the subject, each individual notion must express the totality of the world, and the totality of the world must be included in each notion.
We saw that this led Leibniz to an extraordinary theory that is the first great theory in philosophy of perspective or point of view since each individual notion will be said to express and contain the world. Yes, but from a certain point of view which is deeper, notably it is subjectivity that refers to the notion of point of view and not the notion of point of view that refers to subjectivity. This is going to have many consequences in philosophy, starting with the echo that this would have for Nietzsche in the creation of a perspectivist philosophy.
The first problem is this: in saying that the predicate is contained in the subject, we assume that this brought up all sorts of difficulties, specifically: can relations be reduced to predicates, can events be considered as predicates? But let us accept that. We can find Leibniz wrong only starting from an aggregate of conceptual coordinates from Leibniz's. A true proposition is one for which the attribute is contained in the subject; we see quite well what that can mean on the level of truths of essences. Truths of essences, be they metaphysical truths (concerning God), or else mathematical truths. If I say 2+2=4, there is quite a bit to discuss about that, but I immediately understand what Leibniz meant, always independently of the question of whether he is right or wrong; we already have enough trouble knowing what someone is saying that if, on top of that, we wonder if he is right, then there is no end to it. 2+2=4 is an analytical proposition. I remind you that an analytical proposition is a proposition for which the predicate is contained in the subject or in the notion of the subject, specifically it is an identical proposition or is reducible to the identical. Identity of the predicate with the subject. Indeed, Leibniz tells us: I can demonstrate through a series of finite procedures, a finite number of operational procedures, I can demonstrate that 4, by virtue of its definition, and 2+2, by virtue of their definition, are identical. Can I really demonstrate it, and in what way? Obviously I do not pose the problem of how. We understand generally what that means: the predicate is encompassed in the subject, that means that, through a group of operations, I can demonstrate the identity of one and the other. Leibniz selects an example in a little text called "On Freedom." He proceeds to demonstrate that every number divisible by twelve is by this fact divisible by six. Every duodecimal number is sextuple .
Notice that in the logistics of the nineteenth and twentieth centuries, you will again find proofs of this type that, notably, made Russell famous. Leibniz's proof is very convincing: he first demonstrates that every number divisible by twelve is identical to those divisible by two, multiplied by two, multiplied by three. It's not difficult. On the other hand, he proves that the number divisible by six is equal to that divisible by two multiplied by three.
By that, what did he reveal?
He revealed an inclusion since two multiplied by three is contained in two multiplied by two multiplied by three.
It's an example that helps us understand on the level of mathematical truths that we can say that the corresponding proposition is analytical or identical. That is, the predicate is contained in the subject. That means, strictly speaking, that I can make into an aggregate, into a series of determinate operations, a finite series of determinate operations -- I insist on that -- I can demonstrate the identity of the predicate with the subject, or I can cause an inclusion of the predicate in the subject to emerge. And that boils down to the same thing. I can display this inclusion, I can show it. Either I can demonstrate the identity or I can show the inclusion.
He showed the inclusion when he showed, for example... -- a pure identity would have been: any number divisible by twelve is divisible by twelve -- but with that, we reach another case of truth of essence: any number divisible by twelve is divisible by six, this time he does not stop at showing an identity, he shows an inclusion resulting from finite operations, quite determinate.
That's what truths of essence are. I can say that inclusion of the predicate in the subject is proven by analysis and that this analysis responds to the condition of being finite, that is, it only includes a limited number of quite determinate operations.
But when I say that Adam sinned, or that Caesar crossed the Rubicon, what is that? That no longer refers to a truth of essence, it's specifically dated, Caesar crossed the Rubicon here and now, with reference to existence, since Caesar crossed the Rubicon only if it existed. 2+2=4 occurs in all time and in all places. Thus, there are grounds to distinguish truths of essence from truths of existence.
The truth of the proposition "Caesar crossed the Rubicon" is not the same type as 2+2=4. And yet, by virtue of the principles we saw the last time, no less for truths of existence than for truths of essence, the predicate must be in the subject and included in the notion of the subject; included therefore for all eternity in the notion of the subject, including for all eternity that Adam will sin in a particular place at a particular time. This is a truth of existence.
No less than for truths of essence, for truths of existence, the predicate must be contained in the subject. Granted, but no less, that does not mean in the same way. And in fact, and this is our problem, what initial great difference is there between truth of essence and truth of existence? We sense it immediately. For the truths of existence, Leibniz tells us that even there, the predicate is contained in the subject. The "sinner" must be contained in the individual notion of Adam, just look: if the sinner is contained in the individual notion of Adam, it's the entire world that is contained in the individual notion of Adam, if we follow the causes back and if we track down the effects, as it's the entire world, you understand, that the proposition "Adam sinned" must be an analytical proposition, only in that case, the analysis is infinite. The analysis extends to infinity.
What could that even mean? It seems to mean this: in order to demonstrate the identity of "sinner" and "Adam," or the identity of "who crossed the Rubicon" and "Caesar," this time an infinite series of operations is required. It goes without saying that we aren't capable of that, or it appears that we aren't. Are we capable of making an analysis to infinity? Leibniz is quite formal: [no], you, us, men, are not able to do so. Thus, in order to situate ourselves in the domain of truths of existence, we have to wait for the experience. So why does he present this whole story about analytical truths? He adds: yes, but infinite analysis, on the other hand, not only is possible, but created in the understanding of God.
Does it suit us that God, he who is without limits, he who is infinite, can undertake infinite analysis? We're happy, we're happy for him, but at first glance, we wonder what Leibniz is talking about. I emphasize only that our initial difficulty is: what is infinite analysis? Any proposition is analytical, only there is an entire domain of our propositions that refers to an infinite analysis. We are hopeful: if Leibniz is one of the great creators of differential calculus or of infinitesimal analysis, undoubtedly this is in mathematics, and he always distinguished philosophical truths and mathematical truths, and so it's not a question for us of mixing up everything. But it's impossible to think that, when he discovers a certain idea of infinite analysis in metaphysics, that there aren't certain echoes in relation to a certain type of calculus that he himself invented, notably the calculus of infinitesimal analysis.
So there is my initial difficulty: when analysis extends to infinity, what type or what is the mode of inclusion of the predicate in the subject? In what way is "sinner" contained in the notion of Adam, once it is stated that the identity of sinner and Adam can appear only in an infinite analysis?
What does infinite analysis mean, then, when it seems that there is analysis only under conditions of a well-determined finitude?
That's a tough problem.
Second problem. I just exposed already a first difference between truths of essence and truths of existence. In truths of essence, the analysis is finite, in truths of existence, the analysis is infinite. That is not the only one, for there is a second difference: according to Leibniz, a truth of essence is such that its contradictory is impossible, that is, it is impossible for 2 and 2 not to make 4. Why? For the simple reason that I can prove the identity of 4 and of 2+2 through a series of finite procedures. Thus 2+2=5 can be proven to be contradictory and impossible. Adam non sinner, Adam who might not have sinned, I therefore seize the contradictory of sinner. It's possible. The proof is that, following the great criterion of classical logic -- and from this perspective Leibniz remains within classical logic -- I can think nothing when I say 2+2=5, I cannot think the impossible, no more than I think whatever it might be according to this logic when I say squared circle. But I can very well think of an Adam who might not have sinned.Truths of existence are called contingent truths.
Caesar could have not crossed the Rubicon. Leibniz's answer is admirable: certainly, Adam could have not sinned, Caesar could have not crossed the Rubicon. Only here it is: this was not compossible with the existing world. An Adam non sinner enveloped another world. This world was possible in itself, a world in which the first man might not have sinned is a logically possible world, only it is not compossible with our world. That is, God chose a world such that Adam sinned. Adam non sinner implied another world, this world was possible, but it was not compossible with ours.
Why did God choose this world? Leibniz goes on to explain it. Understand that at this level, the notion of compossibility becomes very strange: what is going to make me say that two things are compossible and that two other things are incompossible? Adam non sinner belongs to another world than ours, but suddenly Caesar might not have crossed the Rubicon either, that would have been another possible world. What is this very unusual relation of compossibility? Understand that perhaps this is the same question as what is infinite analysis, but it does not have the same outline. So we can draw a dream out of it, we can have this dream on several levels. You dream, and a kind of wizard is there who makes you enter a palace; this palace... it's the dream of Apollodorus told by Leibniz. Apollodorus is going to see a goddess, and this goddess leads him into the palace, and this palace is composed of several palaces. Leibniz loved that, boxes containing boxes. He explained, in a text that we will examine, he explained that in the water, there are many fish and that in the fish, there is water, and in the water of these fish, there are fish of fish. It's infinite analysis. The image of the labyrinth hounds him. He never stops talking about the labyrinth of continuity. This palace is in the form of a pyramid. Then, I look closer and, in the highest section of my pyramid, closest to the point, I see a character who is doing something. Right underneath, I see the same character who is doing something else in another location. Again underneath the same character is there in another situation, as if all sorts of theatrical productions were playing simultaneously, completely different, in each of the palaces, with characters that have common segments. It's a huge book by Leibniz called Theodicy , specifically divine justice.
You understand, what he means is that at each level is a possible world. God chose to bring into existence the extreme world closest to the point of the pyramid. How was he guided in making that choice? We shall see, we must not hurry since this will be a tough problem, what the criteria are for God's choice. But once we've said that he chose a particular world, this world implicated Adam sinner; in another world, obviously all that is simultaneous, these are variants, one can conceive of something else, and each time, it's a world. Each of them is possible. They are incompossible with one another, only one can pass into existence. And all of them attempt with all their strength to pass into existence. The vision that Leibniz proposes of the creation of the world by God becomes very stimulating. There are all these worlds that are in God's understanding, and each of which on its own presses forward pretending to pass from the possible into the existent. They have a weight of reality, as a function of their essences. As a function of the essences they contain, they tend to pass into existence. And this is not possible for they are not compossible with each other: existence is like a dam. A single combination will pass through. Which one? You already sense Leibniz's splendid response: it will be the best one!
And not the best one by virtue of a moral theory, but by virtue of a theory of games. And it's not by chance that Leibniz is one of the founders of statistics and of the calculus of games. And all that will get more complicated...
What is this relation of compossibility? I just want to point out that a famous author today is Leibnizian. What does it mean to be Leibnizian today? I think that means two things, one not very interesting and one very interesting. The last time, I said that the concept is in a special relationship with the scream. There is an uninteresting way to be Leibnizian or to be Spinozist today, by job necessity, people working on an author, but there is another way to make use of a philosopher, one that is non-professional. These are people who are able not to be philosophers. What I find amazing in philosophy is when a non-philosopher discovers a kind of familiarity that I can no longer call conceptual, but immediately seizes upon a familiarity between his very own screams and the concepts of the philosopher. I think of Nietzsche, he had read Spinoza early on and, in this letter, he had just re-read him, and he exclaims: I can't get over it! I can't get over it! I have never had a relation with a philosopher like the one I have had with Spinoza. And that interests me all the more when it's from non-philosophers. When the British novelist Lawrence expresses in a few words the way Spinoza upset him completely. Thank God he did not become a philosopher over that. What did he grasp, what does that mean? When Kleist stumbles across Kant, he literally can't get over it. What is this kind of communication? Spinoza shook up many uncultivated readers ... Borges and Leibniz. Borges is an extremely knowledgeable author who read widely. He is always talking about two things: the book that does not exist...

...he really likes detective stories, Borges. In Ficciones, there is a short story, "The Garden of Forking Paths." As I summarize the story, keep in mind the famous dream of the Theodicy.
"The Garden of Forking Paths," what is it? It's the infinite book, the world of compossibilities. The idea of the Chinese philosopher being involved with the labyrinth is an idea of Leibniz's contemporaries, appearing in mid-17th century. There is a famous text by Malebranche that is a discussion with the Chinese philosopher, with some very odd things in it. Leibniz is fascinated by the Orient, and he often cites Confucius. Borges made a kind of copy that conformed to Leibniz's thought with an essential difference: for Leibniz, all the different worlds that might encompass an Adam sinning in a particular way, an Adam sinning in some other way, or an Adam not sinning at all, he excludes all this infinity of worlds from each other, they are incompossible with each other, such that he conserves a very classical principle of disjunction: it's either this world or some other one. Whereas Borges places all these incompossible series in the same world, allowing a multiplication of effects. Leibniz would never have allowed incompossibles to belong to a single world. Why? I only state our two difficulties: the first is, what is an infinite analysis? and second, what is this relationship of incompossibility? The labyrinth of infinite analysis and the labyrinth of compossibility.
Most commentators on Leibniz, to my knowledge, try in the long run to situate compossibility in a simple principle of contradiction. They conclude that there would be a contradiction between Adam non sinner and our world. But, Leibniz's letter already appears to us such that this would not be possible.
It's not possible since Adam non sinner is not contradictory in itself and the relation of compossibility is absolutely irreducible to the simple relation of logical possibility.
So trying to discover a simple logical contradiction would be once again to situate truths of existence within truths of essence. Henceforth it's going to be very difficult to define compossibility.
Still remaining within this paragraph on substance, the world, and continuity, I would like to ask the question, what is infinite analysis? I ask you to remain extremely patient. We have to be wary of Leibniz's texts because they are always adapted to the correspondents within given audiences, and if I again take up his dream, I must change it, and a variant of the dream, even within the same world, would result in levels of clarity or obscurity such that the world might be presented from one point of view or another. So that for Leibniz's texts, we have to know to whom he addresses them in order to be able to judge them.
Here is a first kind of text by Leibniz in which he tells us that, in any proposition, the predicate is contained in the subject. Only it is contained either in act -- actually -- or virtually. The predicate is contained in the subject, but this inclusion, this inherence is either actual or virtual. We would like to say that that seems fine. Let us agree that in a proposition of existence of the type Caesar crossed the Rubicon, the inclusion is only virtual, specifically crossing the Rubicon is contained in the notion of Caesar, but is only virtually contained. Second kind of text: the infinite analysis in which sinner is contained in the notion of Adam is an indefinite analysis, that is, I can move back from sinner to another term, then to another term, etc... Exactly as if sinner = 1/2+1/4+1/8, etc., to infinity. This would result in a certain status: I would say that infinite analysis is virtual analysis, an analysis that goes toward the indefinite. There are texts by Leibniz saying that, notably in "The discourse on metaphysics," but in "The discourse on metaphysics," Leibniz presents and proposes the totality of his system for use by people with little philosophical background. I choose another text that seems to contradict the first; in a more scholarly text, "On Freedom," Leibniz uses the word "virtual," but quite strangely he does not use this word with reference to truths of existence, but to truths of essence.
This text suffices already for me to say that it is not possible for the distinction truths of essence/truths of existence be reduced to saying that in truths of existence, inclusion would only be virtual, since virtual inclusion is a case of truths of essence. In fact, you recall that truths of essence refer to two cases: the pure and simple identity in which we demonstrate the identity of the predicate and the subject, and the discovery of an inclusion of the type ‘every number divisible by 12 is divisible by 6,’ (I demonstrate the inclusion following a finite operation), and it is for the latter case that Leibniz says: I have discovered a virtual identity. Thus it is not enough to say that infinite analysis is virtual.
Can we say that this is an indefinite analysis? No, because an indefinite analysis would be the same as saying that it's an analysis that is infinite only through my lack of knowledge, that is, I cannot reach the end of it. Henceforth, God with his understanding would reach the end. Is that it? No, it's not possible for Leibniz to mean that because the indefinite never existed in his thinking. We have here notions that are incompatible, anachronistic. Indefinite is not one of Leibniz's gimmicks . What is the indefinite, rigorously defined? What differences are there between indefinite and infinite?
The indefinite is the fact that I must always pass from one term to another term, always, without stopping, but without the following term at which I arrive pre-existing. It is my own procedure that consists in causing to exist. If I say 1=1/4+1/8, etc...., we must not believe that this "etc." pre-exists, it's my procedure that makes it appear each time, that is, the indefinite exists in a procedure through which I never stop pushing back the limit that I confront. Nothing pre-exists. It's Kant who will be the first philosopher to give a status to the indefinite, and this status will be precisely that the indefinite refers to an aggregate that is not separable from the successive synthesis that runs through it. That is, the terms of the indefinite series do not pre-exist the synthesis that goes from one term to another.
Leibniz does not know that. Moreover, the indefinite appears to him to be purely conventional or symbolic; why? There is an author who said quite well what creates the family resemblance of philosophers of the 17th century, it was Merleau-Ponty. He wrote a small text on so-called classical philosophers of the seventeenth century, and he tried to characterize them in a lively way, and said that what is so incredible in these philosophers is an innocent way of thinking starting from and as a function of the infinite. That's what the classical century is. This is much more intelligent than to tell us that it's an era in which philosophy is still confused with theology. That's stupid. One must say that if philosophy is still confused with theology in the 17th century, it's precisely because philosophy is not separable at that time from an innocent way of thinking as a function of infinity.
What differences are there between the infinite and the indefinite? It's this: the indefinite is virtual; in fact, the following term does not exist prior to my procedure having constituted it. What does that mean? The infinite is actual, there is no infinite except in act . So there can be all sorts of infinites. Think of Pascal. It's a century that will not stop distinguishing orders of infinities, and the thought of orders of infinity is fundamental throughout the 17th century. It will fall back on our heads, this thought, at the end of the 19th and 20th centuries precisely with the theory of so-called infinite aggregates. With infinite aggregates, we rediscover something that worked at the basis of classical philosophy, notably the distinction of orders of infinities: this obviously includes Pascal, Spinoza with the famous letter on infinity, and Leibniz who would subordinate an entire mathematical apparatus to the analysis of the infinite and orders of infinities. Specifically, in what sense can we say that an order of infinities is greater than another, what is an infinite that is greater than another infinite, etc...? An innocent way of thinking starting from the infinite, but not at all in a confused way since all sorts of distinctions are introduced.
In the case of truths of existence, Leibniz's analysis is obviously infinite. It is not indefinite. Thus, when he uses the words virtual, etc..., there is a formal text that supports this interpretation that I am trying to sketch, it's a text taken from "On Freedom" in which Leibniz says exactly this: "When it is a matter of analyzing the inclusion of the predicate sinner in the individual notion Adam, God certainly sees, not the end of the resolution, but the end that does not take place." Thus, in other words, even for God there is no end to this analysis. So, you will tell me that it's indefinite even for God? No, it's not indefinite since all the terms of the analysis are given. If it were indefinite, all the terms would not be given, they would be given little by little. They would not be given in a pre-existing manner. In other words, in an infinite analysis, we reach what result: you have a passage of infinitely small elements one to another, the infinity of infinitely small elements being given. Of such an infinity, we will say that it is actual since the totality of infinitely small elements is given. You will say to me that we can then reach the end! No, by its nature, you cannot reach the end since it's an infinite aggregate. The totality of elements is given, and you pass from one element to another, and thus you have an infinite aggregate of infinitely small elements. You pass from one element to another: you perform an infinite analysis, that is, an analysis without end, neither for you nor for God.
What do you see if you perform this analysis? Let us assume that there is only God that can do it, you make yourself the indefinite because your understanding is limited, but as for God, he makes infinity. He does not see the end of the analysis since there is no end of the analysis, but he performs the analysis. Furthermore, all the elements of the analysis are given to him in an actual infinity. So that means that sinner is connected to Adam. Sinner is an element, it is connected to the individual notion of Adam by an infinity of other elements actually given. Fine, it's the entire existing world, specifically all this whole compossible world that has passed into existence. We are getting at something quite profound here. When I perform the analysis, I pass from what to what? I pass from Adam sinner to Eve temptress, from Eve temptress to the evil Serpent, to the apple. It's an infinite analysis, and it's this infinite analysis that shows the inclusion of sinner in the individual notion Adam. What does that mean, the infinitely small element? Why is sin an infinitely small element? Why is the apple an infinitely small element? Why is crossing the Rubicon an infinitely small element? You understand what that means? There are no infinitely small elements, so an infinitely small elements means obviously, we don't need to say it, it means an infinitely small relation between two elements. It is a question of relations, not a question of elements. In other words, an infinitely small relation between elements, what can that be? What have we achieved in saying that it is not a question of infinitely small elements, but of infinitely small relations between two elements? And you understand that if I speak to someone who has no idea of differential calculus, you can tell him it's infinitely small elements. Leibniz was right. If it's someone who has a very vague knowledge, he has to understand that these are infinitely small relations between finite elements. If it's someone who is very knowledgeable in differential calculus, I can perhaps tell him something else.
Infinite analysis that goes on to demonstrate the inclusion of the predicate in the subject at the level of truths of existence, does not proceed by the demonstration of an identity, even a virtual one. It's not that. But Leibniz, in another drawer, has another formula to give you: identity governs truths of essence, but not truths of existence; all the time he says the opposite, but that has no importance. Ask yourself to whom he says it. So what is it? What interests him at the level of truths of existence is not identity of the predicate and the subject, it's rather that one passes from one predicate to another, from one to another, and again on from one to another, etc.... from the point of view of an infinite analysis, that is, from the maximum of continuity.
In other words, it's identity that governs truths of essence, but it's continuity that governs truths of existence. And what is a world? A world is defined by its continuity. What separates two incompossible worlds? It's the fact that there is discontinuity between the two worlds. What defines a compossible world? It's the compossibility of which it is capable. What defines the best of worlds? It's the most continuous world. The criterion of God's choice will be continuity. Of all the worlds incompossible with each other and possible in themselves, God will cause to pass into existence the one that realizes the maximum of continuity.
Why is Adam's sin included in the world that has the maximum of continuity? We have to believe that Adam's sin is a formidable connection, that it's a connection that assures continuities of series. There is a direct connection between Adam's sin and the Incarnation and the Redemption by Christ. There is continuity. There are something like series that are going to begin to fit into each other across the differences of time and space. In other words, in the case of truths of essence, I demonstrated an identity in which I revealed an inclusion; in the case of truths of existence, I am going to witness a continuity assured by the infinitely small relations between two elements. Two elements will be in continuity when I will be able to assign an infinitely small relation between these two elements.
I have passed from the idea of infinitely small element to the infinitely small relation between two elements, that's not enough. A greater effort is required. Since there are two elements, there is a difference between the two elements: between Adam's sin and the temptation of Eve, there is a difference, only what is the formula of the continuity? We will be able to define continuity as the act of a difference in so far as it tends to disappear. Continuity is an evanescent difference.
What does it mean that there is continuity between the seduction of Even and Adam's sin? It's that the difference between the two is a difference that tends to disappear. I would say therefore that truths of essence are governed by the principle of identity, truths are governed by the law of continuity, or evanescent differences, and that comes down to the same.
Thus between sinner and Adam you will never be able to demonstrate a logical identity, but you will be able to demonstrate -- and the word demonstration will change meaning --, you will be able to demonstrate a continuity, that is, one or several evanescent differences.
An infinite analysis is an analysis of the continuous operating through evanescent differences.
That refers to a certain symbolic, a symbolic of differential calculus or of infinitesimal analysis. But it's at the same time that Newton and Leibniz develop differential calculus. And the interpretation of differential calculus by the evanescent categories is Leibniz's very own. In Newton's works, whereas both of them really invent it at the same time, the logical and theoretical armature is very different in Leibniz's works and Newton's, and the theme of the differential conceived as evanescent difference is proper to Leibniz. Moreover, he relies on it greatly, and there is a great polemic between Newtonians and Leibniz. Our story becomes more precise: what is this evanescent difference? . Differential equations today are fundamental. There is no physics without a differential equation. Mathematically, today, differential calculus has purged itself of any consideration of the infinite; the kind of axiomatic status of differential calculus in which it is absolutely no longer a question of the infinite dates from the end of the 19th century. But if we place ourselves at the time of Leibniz, put yourself in the place of a mathematician: what is he going to do when he finds himself faced with the magnitude and quantities of different powers, equations whose variables are to different powers, equations of the ax2+y type? You have a quantity to the second power and a quantity to the first power. How does one compare? You all know the story of non-commensurable quantities. Then, in the 17th century, the quantities of different powers received a neighboring term, incomparable quantities. The whole theory of equations collides in the 17th century with this problem that is a fundamental one, even in the simplest algebra; what is differential calculus for? Differential calculus allows you to proceed directly to compare quantities raised to different powers. Moreover, it is used only for that.
Differential calculus finds its level of application when you are faced with incomparables, that is, faced with quantities raised to different powers. Why? In ax2+y, let us assume that by various means, you extract dx and dy. What is that? We will define it verbally, conventionally, we will say that dx or dy is the infinitely small quantity assumed to be added or subtracted from x or from y. Now there is an invention! The infinitely small quantity... that is, it's the smallest variation of the quantity considered. It is unassignable by convention. Thus dx=0 in x, is the smallest quantity by which x can vary, so it equals zero. dy = 0 in relation to y. The notion of evanescent difference is beginning to take shape. It's a variation or a difference, dx or dy; it is smaller than any given or givable quantity. It's a mathematical symbol. In a sense, it's crazy, in a sense it's operational. For what? Here is what is formidable in the symbolism of differential calculus: dx=0 in relation to x, the smallest different, the smallest increase of which the quantity x or the unassignable quantity y might be capable, it's infinitely small. The miracle dy/dx is not equal to zero, and furthermore: dy/dx has a perfectly expressible finite quantity.
These are relative , uniquely relative. dx is nothing in relation to y, dy is nothing in relation to y, but then dy/dx is something.
A stupefying, admirable, and great mathematical discovery.
It's something because in an example such as ax2-by+c, you have two powers in which you have incomparable quantities: y2 and x. If you consider the differential relation, it is not zero, it is determined, it is determinable.
The relation dy/dx gives you the means to compare two incomparable quantities that were raised to different powers since it operates a depotentialization of quantities. So it gives you a direct means to confront incomparable quantities raised to different powers. From that moment on, all mathematics, all algebra, all physics will be inscribed in the symbolism of differential calculus... It's the relation between dx and dy that made possible this kind of co-penetration of physical reality and mathematical calculus.
There is a small note of three pages called "Justification of the calculus of infinitesimals by the calculus of ordinary algebra." With that you will understand everything. Leibniz tries to explain that in a certain way, differential calculus already functioned before being discovered, and that it couldn't occur otherwise, even at the level of the most ordinary algebra.
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x is not equal to y, neither in one case, nor another, since it would be contrary to the very givens of the construction of the problem. To the extent that, for this case, you can write x/y = c/e, c and e are zeros.
Like he says in his language, these are nothings, but they are not absolute nothings, that are nothings respectively.
Specifically, these are nothings but ones which conserve the relational difference. Thus c does not become equal to e since it remains proportional to x and x is not equal to y.
This is a justification of the old differential calculus, and the interest of this text is that it's a justification through the easiest or most ordinary algebra. This justification puts nothing into question about the specificity of differential calculus.
I read this very beautiful text:
"Thus, in the present case, there will be x-c=x. Let us assume that this case is included under the general rule, and nonetheless c and e will not at all be absolute nothings since they together maintain the reason of CX to XY, or that which is between the entire sine or radius and between the tangent that corresponds to the angle in c. We have assumed this angle always to remain the same. For if c, C and e were absolutely nothings in this calculus reduced to the case of coincidence of points c, e and a, as one nothing has the same value as the other, then c and e would be equal and the equations or analogy x/y = c/e would make x/y = 0/0 = 1. That is, we would have x=y which would be totally absurd."
"So we find in algebraic calculus the traces of the transcendent calculus of differences (i.e. differential calculus), and its same singularities that some scholars have fretted about, and even algebraic calculus could not do without it if it must conserve its advantages of which one of the most considerable is the generality that it must maintain so that it can encompass all cases."
It's exactly in this way that I can consider that rest is an infinitely small movement, or that the circle is the limit of an infinite series of polygons the sides of which increase to infinity. What is there to compare in all these examples? We have to consider the case in which there is a single triangle as the extreme case of two similar triangles opposed at the vertex. What Leibniz demonstrated in this text is how and in what circumstances a triangle can be considered as the extreme case of two similar triangles opposed at the vertex. There you sense that we are perhaps in the process of giving to "virtual" the sense that we were looking for. I could say that in the case of my second figure in which there is only one triangle, the other triangle is there, but it is only there virtually. It's there virtually since a contains virtually e and c distinct from a. Why do e and c remain distinct from a when they no longer exist? e and c remain distinct from a when they no longer exist because they intervene in a relation with it, continue to exist when the terms have vanished. It's in this way that rest will be considered as a special case of movement, specifically an infinitely small movement. In my second figure, xy, I would say it's not at all the triangle CEA, it's not at all the case that the triangle has disappeared in the common sense of the word, but we have to say both that it has become unassignable, and however that it is perfectly determined since in this case, c=0, e=0, but c/e is not equal to zero.
c/e is a perfectly determined relation equal to x/y.
Thus it is determinable and determined, but it is unassignable. Likewise, rest is a perfectly determined movement, but it's an unassignable movement. Likewise, the circle is an unassignable polygon, yet perfectly determined.
You see what virtual means. Virtual no longer means at all the indefinite, and there all Leibniz's texts can be revived. He undertook a diabolical operation: he took the word virtual, without saying anything -- it's his right -- he gave it a new meaning, completely rigorous, but without saying anything. He will only say it in other texts: that no longer meant going toward the indefinite; rather, that meant unassignable, yet also determined.
It's a conception of the virtual that is both quite new and very rigorous. Yet the technique and concepts were required so that this rather mysterious expression might acquire a meaning at the beginning: unassignable, yet determined. It's unassignable since c became equal to zero, and since e became equal to zero. And yet it's completely determined since c/e, specifically 0/0 is not equal to zero, nor to 1, it's equal to x/y.
Moreover, he really had a professor-like genius. He succeeded in explaining to someone who never did anything but elementary algebra what differential calculus is. He assumed no a priori notion of differential calculus.
The idea that there is a continuity in the world -- it seems that there are too many commentators on Leibniz who make more theological pronouncements than Leibniz requires: they are content to say that infinite analysis is in God's understanding, and it is true according to the letter of his texts. But with differential calculus, it happens that we have the artifice not to make ourselves equal to God's understanding, that's impossible of course, but differential calculus gives us an artifice so that we can operate a well-founded approximation of what happens in God's understanding so that we can approach it thanks to this symbolism of differential calculus, since after all, God also operates by the symbolic, not the same way, certainly. Thus this approximation of continuity is such that the maximum of continuity is assured when a case is given, the extreme case or contrary can be considered from a certain point of view as included in the case first defined.
You define the movement, it matters little, you define the polygon, it matters little, you consider the extreme case or the contrary: rest, the circle is stripped of any angle. Continuity is the institution of the path following which the extrinsic case -- rest contrary to movement, the circle contrary to the polygon -- can be considered as included in the notion of the intrinsic case.
There is continuity when the extrinsic case can be considered as included in the notion of the intrinsic case.
Leibniz just showed why. You find the formula of predication: the predicate is included in the subject.
Understand well. I call general, intrinsic case the concept of movement that encompasses all movements. In relation to this first case, I call extrinsic case rest or the circle in relation to all the polygons, or the unique triangle in relation to all the triangles combined. I undertake to construct a concept that implies all the differential symbolism, a concept that both corresponds to the general intrinsic case and which still includes the extrinsic case. If I succeed in that, I can say that in all truth, rest is an infinitely small movement, just as I say that my unique triangle is the opposition of two similar triangles opposed at the vertex, simply, by which one of the two triangles has become unassignable. At that moment, there is continuity from the polygon to the circle, there is continuity from rest to movement, there is continuity from two similar triangles opposed at the vertex to a single triangle.
In the mid-19th century, a very great mathematician named Poncelet will produce projective geometry in its most modern sense, it is completely Leibnizian. Projective geometry is entirely based on what Poncelet called a completely simple axiom of continuity: if you take an arc of a circle cut at two points by a right angle, if you cause the right angle to recede, there is a moment in which it leaves the circle, no longer touching it at any point. Poncelet's axiom of continuity claims the possibility of treating the case of the tangent as an extreme case, specifically it's not that one of the points has disappeared, both points are still there, but virtual. When they all leave, it's not that the two points have disappeared, they are still there, but both are virtual. This is the axiom of continuity that precisely allows any system of projection, any so-called projective system. Mathematics will keep that integrally, it's a formidable technique.
There is something desperately comical in all that, but that will not bother Leibniz at all. There again, commentators are very odd. From the start, we sink into a domain in which it's a question of showing that the truths of existence are not the same thing as truths of essence or mathematical truths. To show it, either it's with very general propositions full of genius in Leibniz's works, but that leave us like that, God's understanding, infinite analysis, and then what does that amount to? And finally when it's a question of showing in what way truths of existence are reducible to mathematical truths, when it's a question of showing it concretely, all that is convincing in what Leibniz says is mathematical. It's funny, no?
A professional objector would say to Leibniz: you announce to us, you talk to us of the irreducibility of truths of existence, and you can define this irreducibility concretely only by using purely mathematical notions. What would Leibniz answer? In all sorts of texts, people have always had me say that differential calculus designated a reality. I never said that, Leibniz answers, differential calculus is a well-founded convention. Leibniz relies enormously on differential calculus being only a symbolic system, and not sketching out a reality, but designating a way of treating reality. What is this well-founded convention? It's not in relation to reality that it's a convention, but in relation to mathematics. That's the misinterpretation not to make. Differential calculus is symbolism, but in relation to mathematical reality, not at all in relation to real reality. It's in relation to mathematical reality that the system of differential calculus is a fiction. He also used the expression "well founded fiction." It’s a well-founded fiction in relation to the mathematical reality. In other words, differential calculus mobilizes concepts that cannot be justified from the point of view of classical algebra, or from the point of view of arithmetic. It's obvious. Quantities that are not nothing and that equal zero, it's arithmetical nonsense, it has neither arithmetic reality, nor algebraic reality. It's a fiction. So, in my opinion, it does not mean at all that differential calculus does not designate anything real, it means that differential calculus is irreducible to mathematical reality. It's therefore a fiction in this sense, but precisely in so far as it's a fiction, it can cause us to think of existence.
In other words, differential calculus is a kind of union of mathematics and the existent, specifically it's the symbolic of the existent.
It's because it's a well-founded fiction in relation to mathematical truth that it is henceforth a basic and real means of exploration of the reality of existence.
You see therefore what the words "evanescent" and "evanescent difference" mean. It's when the relation continues when the terms of the relation have disappeared. The relation c/e when C and E have disappeared, that is, coincide with A. You have therefore constructed a continuity through differential calculus.
Leibniz becomes much stronger in order to tell us: understand that in God's understanding, between the predicate sinner and the notion of Adam, well, there is continuity. There is a continuity by evanescent difference to the point that when he created the world, God was only doing calculus . And what a calculus! Obviously not an arithmetical calculus.
He will oscillate on this topic between two explanations. Therefore God created the world by calculating . God calculates, the world is created.
The idea of God as player can be found everywhere. We can always say that God created the world by playing, but everyone has said that. It's not very interesting. There is a text by Heraclitus in which it is a question of the player child who really constituted the world. He plays, at what? What do the Greeks and Greek children play? Different translations yield different games. But Leibniz would not say that, when he gives his explanation of games, he has two explanations. In problems of tiling , astride architectural and mathematical problems, a surface being given, with what figure is one to fill it completely? A more complicated problem: if you take a rectangular surface and you want to tile it with circles, you do not fill it completely. With squares, do you fill it completely? That depends on the measurement. With rectangles? Equal or unequal? Then, if you suppose two figures, which of them combine to fill a space completely? If you want to tile with circles, with which other figure will you fill in the empty spaces? Or you agree not to fill everything; you see that it's quite connected to the problem of continuity. If you decide not to fill it all, in what cases and with which figures and which combination of different figures will you succeed in filling the maximum possible? That puts incommensurables into play, and puts incomparables into play. Leibniz has a passion for tiling.
When Leibniz says that God causes to exist and chooses the best of all possible worlds, we have seen, one gets ahead of Leibniz before he has spoken. The best of all possible worlds was the crisis of Leibnizianism, that was the generalized anti-Leibnizianism of the 18th century. They could not stand the story of the best of all possible worlds.
Voltaire was right, these worlds had a philosophical requirement that obviously was not fulfilled by Leibniz, notably from the political point of view. So, he could not forgive Leibniz. But if one casts oneself into a pious approach, what does Leibniz mean by the statement that the world that exists is the best of possible worlds? Something very simple: since there are several worlds possible, only they are not compossible with each other, God chooses the best and the best is not the one in which suffering is the least. Rationalist optimism is at the same time an infinite cruelty, it's not at all a world in which no one suffers, it's the world that realizes the maximum of circles.
If I dare use a non-human metaphor, it's obvious that the circle suffers when it is no more than an affection of the polygon. When rest is no more than an affection of movement, imagine the suffering of rest. Simply it's the best of worlds because it realizes the maximum of continuity. Other worlds were possible, but they would have realized less continuity. This world is the most beautiful, the most harmonious, uniquely under the weight of this pitiless phrase: because it effectuates the most continuity possible. So if that occurs at the price of your flesh and blood, it matters little. As God is not only just, that is, pursuing the maximum of continuity, but as he is at the same time quite stylish, he wants to vary the world. So God hides this continuity. He poses a segment that should be in continuity with that other segment that he places elsewhere to hide his tracks.
We run no risk of making sense of this. This world is created at our expense. So, obviously the 18th century does not receive Leibniz's story very favorably. You see henceforth the problem of tiling: the best of worlds will be the one in which figures and forms will fill the maximum of space time while leaving the least emptiness.
Second explanation by Leibniz, and there he is even stronger: the chess game. Such that between Heraclitus's phrase that alludes to a Greek game and Leibniz's allusion to chess, there is all the difference that there is between the two games at the same moment in which the common formula "God plays" could make us believe that it's a kind of beatitude. How does Leibniz conceive of chess: the chess board is a space, the pieces are notions. What is the best move in chess, or the best combination of moves? The best move or combination of moves is the one that results in a determinate number of pieces with determinate values holding or occupying the maximum space. The total space being contained on the chess board. One must place ones pawns in such a way that they command the maximum space.
Why are these only metaphors? Here as well there is a kind of principle of continuity: the maximum of continuity. What does not work just as well in the metaphor of chess as in the metaphor of tiling? In both cases, you have reference to a receptacle. The two things are presented as if the possible worlds were competing to be embodied in a determinate receptacle. In the case of tiling, it's the surface to be tiled; in the case of chess, it's the chess board. But in the conditions of the creation of the world, there is no a priori receptacle.
We have to say, therefore, that the world that passes into existence is the one that realizes in itself the maximum of continuity, that is, which contains the greatest quantity of reality or of essence. I cannot speak of existence since there will come into existence the world that contains not the greatest quantity of existence, but the greatest quantity of essence from the point of view of continuity. Continuity is, in fact, precisely the means of containing the maximum quantity of reality.
Now that's a very beautiful vision, as philosophy.
In this paragraph, I have answered the question: what is infinite analysis. I have not yet answered the question: what is compossibility. That's it.






Today we must look at some amusing and recreational, but also quite delicate, things.
Answer to a question on differential calculus: It seems to me that one cannot say that at the end of the seventeenth century and in the eighteenth century, there were people for whom differential calculus is something artificial and others for whom it represents something real. We cannot say that because the division is not there. Leibniz never stopped saying that differential calculus is pure artifice, that it’s a symbolic system. So, on this point, everyone is in strict agreement. Where the disagreement begins is in understanding what a symbolic system is, but as for the irreducibility of differential signs to any mathematical reality, that is to say to geometrical, arithmetical and algebraic reality, everyone agrees. A difference arises when some people think that, henceforth, differential calculus is only a convention, a rather suspect one, and others think that its artificial character in relation to mathematical reality, on the contrary, allows it to be adequate to certain aspects of physical reality. Leibniz never thought that his infinitesimal analysis, his differential calculus, as he conceived them sufficed to exhaust the domain of the infinite such as he, Leibniz, conceived it. For example, calculus. There is what Leibniz calls calculus of the minimum and of the maximum which does not at all depend on differential calculus. So differential calculus corresponds to a certain order of infinity. If it is true that a qualitative infinity cannot be grasped by differential calculus, Leibniz is, on the other hand, so conscious of it that he initiates other modes of calculus relative to other orders of infinity. What eliminated this direction of the qualitative infinity, or even simply of actual infinity tout court, Leibniz wasn’t the one who blocked it off. What blocked this direction was the Kantian revolution. This was what imposed a certain conception of the indefinite and directed the most absolute critique of actual infinity. We owe that to Kant and not to Leibniz.
In geometry, from the Greeks to the seventeenth century, you have two kinds of problems: those in which it’s a question of finding so-called straight lines and so-called rectilinear surfaces. Classical geometry and algebra were sufficient. You have problems and you get the necessary equations; it’s Euclidean geometry. Already with the Greeks, then in the Middle Ages of course, geometry will not cease to confront a type of problem of another sort: it’s when one must find and determine curves and curvilinear surfaces. Where all geometries are in agreement is in the fact that classical methods of geometry and algebra no longer sufficed. The Greeks already had to invent a special method called the method by exhaustion. It allowed them to determine curves and curvilinear surfaces in so far as it gave equations of variable degrees, to the infinite limit, an infinity of various degrees in the equation. These are the problems that are going to make necessary and inspire the discovery of differential calculus and the way in which differential calculus takes up where the old method by exhaustion left off. If you already connect a mathematical symbolism to a theory, if you don’t connect it to the problem for which it is created, then you can no longer understand anything. Differential calculus has sense only if you place yourself before an equation in which the terms are raised to different powers. If you don’t have that, then it’s non-sensical to speak of differential calculus. It’s very much about considering the theory that corresponds to a symbolism, but you must also completely consider the practice. In my opinion, as well, one can’t understand anything about infinitesimal analysis if one does not see that all physical equations are by nature differential equations. A physical phenomenon can only be studied ? and Leibniz will be very firm: Descartes only had geometry and algebra, and what Descartes himself had invented under the name of analytical geometry, but however far he went in that invention, it gave him at most the means to grasp figures and movement of a rectilinear kind; but with the aggregate of natural phenomena being after all phenomena of the curvilinear type, that doesn’t work at all. Descartes remained stuck on figures and movement. Leibniz will translate: it’s the same thing to say that nature proceeds in a curvilinear manner, or to say that beyond figures and movement, there is something that is the domain of forces. And on the very level of the laws of movement, Leibniz is going to change everything, thanks precisely to differential calculus. He will say that what is conserved is not MV, not mass and velocity, but MV2. The only difference in the formula is the extension of V to the second power. This is made possible by differential calculus because differential calculus allows the comparison of powers and of rejects . Descartes did not have the technical means to say MV2. >From the point of view of the language of geometry and of arithmetic and algebra, MV2 is pure and simple non-sense.
With what we know in science today, we can always explain that what is conserved is MV2 without appealing to any infinitesimal analysis. That happens in high school texts, but to prove it, and for the formula to make any sense, an entire apparatus of differential calculus is required.

< Intervention by Comptesse.>

Gilles: Differential calculus and the axiomatic certainly have a point of encounter, but this is one of perfect exclusion. Historically, the rigorous status of differential calculus arises quite belatedly. What does that mean? It means that everything that is convention is expelled from differential calculus. And, even for Leibniz, what is artifice? It’s an entire set of things: the idea of a becoming, the idea of a limit of becoming, the idea of a tendency to approach the limit, all these are considered by mathematicians to be absolutely metaphysical notions. The idea that there is a quantitative becoming, the idea of the limit of this becoming, the idea that an infinity of small quantities tends toward the limit, all these are considered as absolutely impure notions, thus as really non-axiomatic or non-axiomitizable. Thus, from the start, whether in Leibniz’s work or in Newton’s and the work of his successors, the idea of differential calculus is inseparable and not separated from a set of notions judged not to be rigorous or scientific. They themselves are quite prepared to recognize it. It happens that at the end of the nineteenth and the start of the twentieth century, differential calculus or infinitesimal analysis would receive a rigorously scientific status, but at what price? We hunt for any reference to the idea of infinity; we hunt for any reference to the idea of limit, we hunt for any reference to the idea of tendency toward the limit. Who does that? An interpretation and a rather strange status of calculus will be given because it stops operating with ordinary quantities, and its interpretation will be purely ordinal. Henceforth, that becomes a mode of exploring the finite, the finite as such. It’s a great mathematician, Weierstrass, who did that, but it came rather late. So, he creates an axiomatic of calculus, but at what price? He transformed it completely. Today, when we do differential calculus, there is no reference to the notions of infinity, of limit and of tendency toward the limit. There is a static interpretation. There is no longer any dynamism in differential calculus, but a static and ordinal interpretation of calculus. One must read Vuillemin’s book, La philosophie de l’algèbre [Paris: PUF, 1960, 1962].
This fact is very important for us because it must certainly show us that the differential relations ? Yes, but even before the axiomatization, all mathematicians agreed in saying that differential calculus interpreted as a method for exploring the infinite was an impure convention. Leibniz was the first to say that, but still in that case, one would have to know what the symbolic value was then. Axiomatic relations and differential relations, well no. They were in opposition.
Infinity has completely changed meaning, nature, and, finally, is completely expelled. A differential relation of the type dy/dx is such that one extracts it from x and y.
At the same time, dy is nothing in relation to y, it’s an infinitely small quantity; dx is nothing in relation to x, it’s an infinitely small quantity in relation to x.
On the other hand, dy/dx is something else.
But it’s something completely different from y/x.
For example, if y/x designates a curve, dy/dx designates a tangent.
And what’s more, it’s not just any tangent.
I would say therefore that the differential relation is such that it signifies nothing concrete in relation to what it’s derived from, that is, in relation to x and to y, but it signifies something else concrete [autre chose de concret], and that is how it assures [the] passage to limits. It assures something else concrete, namely a z.
It’s exactly as if I said that differential calculus is completely abstract in relation to a determination of the type a/b. But on the other hand, it determines a C. Whereas the axiomatic relation is completely formal from all points of view, if it is formal in relation to a and b, it does not determine a c that would be concrete for it. So it doesn’t assure a passage at all. This would be the whole classical opposition between genesis and structure. The axiomatic is really the structure common to a plurality of domains.
Last time, we were considering my second topic heading, which dealt with Substance, World, and Compossibility.
In the first past, I tried to state what Leibniz called infinite analysis. The answer was this: infinite analysis fulfills the following condition: it appears to the extent that continuity and tiny differences or vanishing differences are substituted for identity.
It’s when we proceed by continuity and vanishing differences that analysis becomes properly infinite analysis. Then I arrive at the second aspect of the question. There would be infinite analysis and there would be material for infinite analysis when I find myself faced with a domain that is no longer directly governed by the identical, by identity, but a domain that is governed by continuity and vanishing differences. We reach a relatively clear answer. Hence the second aspect of the problem: what is compossibility? What does it mean for two things to be compossible or non compossible? Yet again, Leibniz tells us that Adam non-sinner is possible in itself, but not compossible with the existing world. So he maintains a relation of compossibility that he invents, and you sense that it’s entirely linked to the idea of infinite analysis.
The problem is that the incompossible is not the same thing as the contradictory. It’s complicated. Adam non-sinner is incompossible with the existing world, another world would have been necessary. If we say that, I only see three possible solutions for trying to characterize the notion of incompossibility.
First solution: we’ll say that one way or another, incompossibility has to imply a kind of logical contradiction. A contradiction would have to exist between Adam non-sinner and the existing world. Yet we could only bring out this contradiction at infinity; it would be an infinite contradiction. Whereas there is a finite contradiction between circle and square, there is only an infinite contradiction between Adam non-sinner and the world. Certain texts by Leibniz move in this direction. But yet again, we know that we have to be careful about the levels of Leibniz’s texts. In fact, everything we said previously implied that compossibility and incompossibility are truly an original relation, irreducible to identity and contradiction. Contradictory identity.
Furthermore, we saw that infinite analysis, in accordance with our first part, was not an analysis that discovered the identical as a result of an infinite series of steps. The whole of our results the last time was that, far from discovering the identical at the end of a series, at the limit of an infinite series of steps, far from proceeding in this way, infinite analysis substituted the point of view of continuity for that of identity. Thus, it’s another domain than the identity/contradiction domain.
Another solution that I will state rapidly because certain of Leibniz’s texts suggest it as well: it’s that the matter is beyond our understanding because our understanding is finite, and hence, compossibility would be an original relation, but we could not know what its roots are. Leibniz brings a new domain to us. There is not only the possible, the necessary and the real. There is the compossible and the incompossible. He was attempting to cover an entire region of being.
Here is the hypothesis that I’d like to suggest: Leibniz is a busy man, he writes in all directions, all over the place, he does not publish at all or very little during his life. Leibniz has all the material, all the details to give a relatively precise answer to this problem. Necessarily so since he’s the one who invented it, so it’s him who has the solution. So what happened for him not to have put all of it together? I think that what will provide an answer to this problem, at once about infinite analysis and about compossibility, is a very curious theory that Leibniz was no doubt the first to introduce into philosophy, that we could call the theory of singularities.
In Leibniz’s work, the theory of singularities is scattered, it’s everywhere. One even risks reading pages by Leibniz without seeing that one is fully in the midst of it, that’s how discreet he is.
The theory of singularities appears to me to have two poles for Leibniz, and one would have to say that it’s a mathematical-psychological theory. And our work today is: what is a singularity on the mathematical level, and what does Leibniz create through that? Is it true that he creates the first great theory of singularities in mathematics? Second question: what is the Leibnizian theory of psychological singularities?
And the last question: to what extent does the mathematical-psychological theory of singularities, as sketched out by Leibniz, help us answer the question: what is the incompossible, and thus the question what is infinite analysis?
What is this mathematical notion of singularity? Why did it arrive [tomb?]? It’s often like that in philosophy: there is something that emerges at one moment and will be abandoned. That’s the case of a theory that was more than outlined by Leibniz, and then nothing came afterwards, the theory was unlucky, without follow-up. Wouldn’t it be interesting if we were to return to it?
I am still divided about two things in philosophy: the idea that it does not require a special kind of knowledge, that really, in this sense, anyone is open to philosophy, and at the same time, that one can do philosophy only if one is sensitive to a certain terminology of philosophy, and that you can always create the terminology, but you cannot create it by doing just anything. You must know what terms like these are: categories, concept, idea, a priori, a posteriori, exactly like one cannot do mathematics if one does not know what a, b, xy, variables, constants, equation are. There is a minimum. So you have to attach some importance to those points.
The singular has always existed in a certain logical vocabulary. "Singular" designates what is not difference, and at the same time, in relation to "universal." There is another pair of notions, it’s "particular" that is said with reference to "general." So the singular and the universal are in relation with each other; the particular and the general are in relation. What is a judgment of singularity? It’s not the same thing as a judgment called particular, nor the same thing as a judgment called general. I am only saying, formally, "singular" was thought, in classical logic, with reference to "universal." And that does not necessarily exhaust a notion: when mathematicians use the expression "singularity," with what do they place it into relation? One must be guided by words. There is a philosophical etymology, or even a philosophical philology. "Singular" in mathematics is distinct from or opposed to "regular." The singular is what is outside the rule.
There is another pair of notions used by mathematicians, "remarkable" and "ordinary." Mathematicians tell us that there are remarkable singularities and singularities that aren’t remarkable. But for us, out of convenience, Leibniz does not yet make this distinction between the non-remarkable singular and the remarkable singular. Leibniz uses "singular," "remarkable," and "notable" as equivalents, such that when you find the word "notable" in Leibniz, tell yourself that necessarily there’s a wink, that it does not at all mean "well-known"; he enlarges the word with an unusual meaning. When he talks about a notable perception, tell yourself that he is in the process of saying something. What interest does this have for us? It’s that mathematics already represents a turning point in relation to logic. The mathematical use of the concept "singularity" orients singularity in relation to the ordinary or the regular, and no longer in relation to the universal. We are invited to distinguish what is singular and what is ordinary or regular. What interest does this have for us? Suppose someone says: philosophy isn’t doing too well because the theory of truth in thought has always been wrong. Above all, we’ve always asked what in thought was true, what was false. But you know, in thought, it’s not the true and the false that count, it’s the singular and the ordinary. What is singular, what is remarkable, what is ordinary in a thought? Or what is ordinary? I think of Kierkegaard, much later, who would say that philosophy has always ignored the importance of a category, that of the interesting! While it is perhaps not true that philosophy ignored it, there is at least a philosophical-mathematical concept of singularity that perhaps has something interesting to tell us about the concept "interesting."
This great mathematical discovery is that singularity is no longer thought in relation to the universal, but is thought rather in relation to the ordinary or to the regular. The singular is what exceeds the ordinary and the regular. And saying that already takes us a great distance since saying it indicates that, henceforth, we wish to make singularity into a philosophical concept, even if it means finding reasons to do so in a favorable domain, namely mathematics. And in which case does mathematics speak to us of the singular and the ordinary? The answer is simple: concerning certain points plotted on a curve. Not necessarily on a curve, but occasionally, or more generally concerning a figure. A figure can be said quite naturally to include singular points and others that are regular or ordinary. Why a figure? Because a figure is something determined! So the singular and the ordinary would belong to the determination, and indeed, that would be interesting! You see that by dint of saying nothing and marking time, we make a lot of progress. Why not define determination in general, by saying that it’s a combination of singular and ordinary, and all determination would be like that? Perhaps? I take a very simple figure: a square. Your very legitimate requirement would be to ask me: what are the singular points of a square? There are four singular points in a square, the four vertices a, b, c, d. We are going to define singularity, but we remain with examples, and we are making a childish inquiry, we are talking mathematics, but we don’t know a word of it. We only know that a square has four sides, so there are four singular points that are the extremes. The points are markers, precisely that a straight line is finished/finite [finie], and that another begins, with a different orientation, at 90 degrees. What will the ordinary points be? This will be the infinity of points that compose each side of the square; but the four extremities will be called singular points.
Question: how many singular points do you give to a cube? I see your vexed amazement! There are eight singular points in a cube. That is what we call singular points in the most elementary geometry: points that mark the extremity of a straight line. You sense that this is only a start. I would therefore oppose singular points and ordinary points. A curve, a rectilinear figure perhaps, can I say of them that singular points are necessarily the extremes ? Maybe not, but let us assume that at first sight, I can say something like that. For a curve, it’s ruined. Let’s take the simplest example: an arc of a circle, concave or convex, as you wish. Underneath, I make a second arc, convex if the other is concave, concave if the other is convex. The two meet one another at a point. Underneath I trace a straight line that, in accordance with the order of things, I call the ordinate. I trace the ordinate. I draw a line perpendicular to the ordinate. It’s Leibniz’s example, in a text with the exquisite title, "Tantanem analogicum", a tiny little work seven pages long written in Latin, which means "analogical essays." Segment ab thus has two characteristics: it’s the only segment raised from the ordinate to be unique. Each of the others has, as Leibniz says, a double, its little twin. In fact, xy has its mirror, its image in x’y’, and you can get closer through vanishing differences of ab, there is only ab that remains unique, without twin. Second point: ab can also be considered a maximum or a minimum, maximum in relation to one of the arcs of the circle, minimum in relation to the other. Ouf, you’ve understood it all. I’d say that AB is a singularity.
I have introduced the example of the simplest curve: an arc of a circle. It’s a bit more complicated: what I showed was that a singular point is not necessarily connected, is not limited to the *extremum*. It can very well be in the middle, and in that case, it is in the middle. And it’s either a minimum or a maximum, or both at once. Hence the importance of a calculus that Leibniz will contribute to extending quite far, that he will call calculus of maxima and of minima. And still today, this calculus has an immense importance, for example, in phenomena of symmetry, in physical and optical phenomena. I would say therefore that my point a is a singular point; all the others are ordinary or regular. They are ordinary and regular in two ways: first, they are below the maximum and above the minimum, and second, they exist doubly. Thus, we can clarify somewhat this notion of ordinary. It’s another case; it’s a singularity of another case.
Another attempt: take a complex curve. What will we call its singularities? The singularities of a complex curve, in simplest terms, are neighboring points of which ? and you know that the notion of neighborhood, in mathematics, which is very different from contiguity, is a key notion in the whole domain of topology, and it’s the notion of singularity that is able to help us understand what neighborhood is. Thus, in the neighborhood of a singularity, something changes: the curve grows, or it decreases. These points of growth or decrease, I will call them singularities. The ordinary one is the series, that which is between two singularities, going from the neighborhood of one singularity to another’s neighborhood, of ordinary or regular character.
We grasp some of these relations, some very strange nuptials: isn’t "classical" philosophy’s fate relatively linked, and inversely, to geometry, arithmetic, and classical algebra, that is, to rectilinear figures? You will tell me that rectilinear figures already include singular points, OK, but once I discovered and constructed the mathematical notion of singularity, I can say that it was already there in the simplest rectilinear figures. Never would the simplest rectilinear figures have given me a consistent occasion, a real necessity to construct the notion of singularity. It’s simply on the level of complex curves that this becomes necessary. Once I found it on the level of complex curves, now there, yes, I back up and can say: ah, it was already an arc of a circle, it was already in a simple figure like the rectilinear square, but before you couldn’t.

Intervention: xxx [missing from transcript].

Gilles groans: … Too bad [Piti?]… My God… He caught me. You know, speaking is a fragile thing. Too bad… ah, too bad … I’ll let you talk for an hour when you want, but not now … Too bad, oh l? l? … It’s the blank in memory [trou].
I will read to you a small, late text by Poincar? that deals extensively with the theory of singularities that will be developed during the entire eighteenth and nineteenth centuries. There are two kinds of undertaking by Poincar?, logical and philosophical projects, and mathematical ones. He is above all a mathematician. There is an essay by Poincar? on differential equations. I am reading a part of it on kinds of singular points in a curve referring to a function or to a differential equation. He tells us that there are four kinds of singular points: first, crests , which are points through which two curves defined by the equation pass, and only two. Here, the differential equation is such that, in the neighborhood of this point, the equation is going to define and going to cause two curves and only two to pass. The second type of singularity: knots, in which an infinity of curves defined by the equation come to intersect. The third type of singularity: foci , around which these curves turn while drawing closer to them in the form of a spiral. Finally, the fourth type of singularity: centers, around which curves appear in the form of a closed circle. And Poincar? explains in the sequel to the essay that, according to him, one great merit of mathematics is to have pushed the theory of singularities into relationship with the theory of functions or of differential equations.
Why do I quote this example from Poincar?? You could find equivalent notions in Leibniz’s works. Here a very curious terrain appears, with crests, foci, centers, truly like a kind of astrology of mathematical geography. You see that we went from the simplest to the most complex: on the level of a simple square, of a rectilinear figure, singularities were extremum; on the level of a simple curve, you have singularities that are even easier to determine, for which the principle of determination was easy. The singularity was the unique case that had no twin, or else was the case in which the maximum and minimum were identified. There you have more complex singularities when you move into more complex curves. Therefore it’s as if the domain of singularities is infinite, strictly speaking. What is the formula going to be? As long as you are dealing with problems considered as rectilinear, that is, in which it’s a question of determining right angles or rectilinear surfaces, you don’t need differential calculus. You need differential calculus when you find yourself faced with the task of determining curves and curvilinear surfaces. What does that mean? In what way is the singularity linked to differential calculus? It’s that the singular point is the point in the neighborhood of which the differential relation dy/dx changes its sign . For example: vertex, relative vertex of a curve before it descends, so you will say that the differential relation changes its sign. It changes its sign at this spot, but to what extent? To the extent that it becomes equal; in the neighborhood of this point, it becomes equal to zero or to infinity. It’s the theme of the minimum and of the maximum that you again find there. All this together consists in saying: look at the kind of relationship between singular and ordinary, such that you are going to define the singular as a function of curvilinear problems in relation to differential calculus, and in this tension or opposition between singular point and ordinary point, or singular point and regular point. This is what mathematicians provide us with as basic material, and yet again if it is true that in the simplest cases, the singular is the extremity, in other simple cases, it’s the maximum or the minimum or even both at once. Singularities there develop more and more complex relations on the level of more and more complex curves.
I hold onto the following formula: a singularity is a distinct or determined point on a curve, it’s a point in the neighborhood of which the differential relation changes its sign, and the singular point’s characteristic is to extend [prolonger] itself into the whole series of ordinary points that depend on it all the way to the neighborhood of subsequent singularities. So I maintain that the theory of singularities is inseparable from a theory or an activity of extension .
Wouldn’t these be elements for a possible definition of continuity? I’d say that continuity or the continuous is the extension of a remarkable point onto an ordinary series all the way into the neighborhood of the subsequent singularity. With this, I’m very pleased because at last, I have an initial hypothetical definition of what the continuous is. It’s all the more bizarre since, in order to reach this definition of the continuous, I used what apparently introduces a discontinuity, notably a singularity in which something changes. And rather than being the opposite, it’s the discontinuity that provides me with this approximate definition. Leibniz tells us that we all know that we have perceptions, that for example, I see red, I hear the sea. These are perceptions; moreover, we should reserve a special word for them because they are conscious. It’s perception endowed with consciousness, that is, perception perceived as such by an "I" , we call it apperception, as a-perceiving. For, indeed, it’s perception that I perceive. Leibniz tells us that consequently there really have to be unconscious perceptions that we don’t perceive. These are called minute perceptions, that is, unconscious perception. Why is this necessary? Why necessary? Leibniz gives us two reasons: it’s that our a-perceptions, our conscious perceptions are always global. What we perceive is always a whole. What we grasp through conscious perception is relative totalities. And it is really necessary that parts exist since there is a whole. That’s a line of reasoning that Leibniz constantly follows: there has to be something simple if there is something composite , he builds this into a grand principle; and it doesn’t go without saying, do you understand what he means? He means that there is no indefinite, and that goes so little without saying that it implies the actual infinite. There has to be something simple since there is something composite. There are people who will think that everything is composite to infinity, and they will be partisans of the indefinite, but for other reasons, Leibniz thinks that the infinite is actual. Thus, there has to be something . Henceforth, since we perceive the global noise of the sea when we are seated on the beach, we have to have minute perceptions of each wave, as he says in summary form, and moreover, of each drop of water. Why? It’s a kind of logical requirement, and we shall see what he means.
He pursues the same reasoning on the level of the whole and the parts yet again as well, not by invoking a principle of totality, but a principle of causality: what we perceive is always an effect, so there have to be causes. These causes themselves have to be perceived, otherwise the effect would not be perceived. In this case, the tiny drops are no longer the parts that make up the wave, nor the waves the parts that make up the sea, but they intervene as causes that produce an effect. You will tell me that there is no great difference here, but let me point out simply that in all of Leibniz’s texts, there are always two distinct arguments that he is perpetually trying to make coexist: an argument based on causality and an argument based on parts. Cause-effect relationship and part-whole relationship. So this is how our conscious perceptions bathe in a flow of unconscious minute perceptions.
On the one hand, this has to be so logically, in accordance with the principles and their requirement, but the great moments occur when experience comes to confirm the requirement of great principles. When the very beautiful coincidence of principles and experience occurs, philosophy knows its moment of happiness, even if it’s personally the misfortune of the philosopher. And at that moment, the philosopher says: everything is fine, as it should be. So it is necessary for experience to show me that under certain conditions of disorganization in my consciousness, minute perceptions force open the door of my consciousness and invade me. When my consciousness relaxes, I am thus invaded by minute perceptions that do not become for all that conscious perceptions. They do not become apperceptions since I am invaded in my consciousness when my consciousness is disorganized. At that moment, a flow of minute unconscious perceptions invades me. It’s not that these minute perceptions stop being unconscious, but it’s me who ceases being conscious. But I live them, there is an unconscious lived experience . I do not represent them, I do not perceive them, but they are there, they swarm in these cases. I receive a huge blow on the head: dizziness is an example that recurs constantly in Leibniz’s work. I get dizzy, I faint, and a flow of minute unconscious perceptions arrives: a buzz in my head. Rousseau knew Leibniz, he will undergo the cruel experience of fainting after having received a huge blow, and he relates his recovery and the swarming of minute perceptions. It’s a very famous text by Rousseau in the Reveries of a Solitary Stroller , which is the return to consciousness.
Let’s look for thought experiences: we don’t even need to pursue this thought experience, we know it’s like that, so through thought, we look for the kind of experience that corresponds to the principle: fainting. Leibniz goes much further and says: wouldn’t that be death? This will pose problems for theology. Death would be the state of a living person who would not cease living. Death would be catalepsy, straight out of Edgar Poe, one is simply reduced to minute perceptions.
And yet again, it’s not that they invade my consciousness, but it’s my consciousness that is extended, that loses all of its own power, that becomes diluted because it loses self-consciousness, but very strangely it becomes an infinitely minute consciousness of minute unconscious perceptions. This would be death. In other words, death is nothing other than an envelopment, perceptions cease being developed into conscious perceptions, they are enveloped in an infinity of minute perceptions. Or yet again, he says, sleep without dreaming in which there are lots of minute perceptions.
Do we have to say that only about perception? No. And there, once again, appears Leibniz’s genius. There is a psychology with Leibniz’s name on it, which was one of the first theories of the unconscious. I have already said almost enough about it for you to understand the extent to which it’s a conception of the unconscious that has absolutely nothing to do with Freud’s which is to say how much innovation one finds in Freud: it’s obviously not the hypothesis of an unconscious that has been proposed by numerous authors, but it’s the way in which Freud conceived the unconscious. And, in the lineage from Freud some very strange phenomena will be found, returning to a Leibnizian conception, but I will talk about that later.
But understand that he simply cannot say that about perception since, according to Leibniz, the soul has two fundamental faculties: conscious apperception which is therefore composed of minute unconscious perceptions, and what he calls "appetition", appetite, desire. And we are composed of desires and perceptions. Moreover, appetition is conscious appetite. If global perceptions are made up of an infinity of minute perceptions, appetitions or gross appetites are made up of an infinity of minute appetitions. You see that appetitions are vectors corresponding to minute perceptions, and that becomes a very strange unconscious. The drop of sea water to which the droplet corresponds, to which a minute appetition corresponds for someone who is thirsty. And when I say, "my God, I’m thirsty, I’m thirsty," what do I do? I grossly express a global outcome of thousands of minute perceptions working within me, and thousands of minute appetitions that crisscross me. What does that mean?
In the beginning of the twentieth century, a great Spanish biologist fell into oblivion; his name was Turro. He wrote a book entitled in French: The Origins of Knowledge (1914), and this book is extraordinary. Turro said that when we say "I am hungry" ? his background was entirely in biology -- and we might say that it’s Leibniz who has awakened-- and Turro said that when one says, "I am hungry," it’s really a global outcome, what he called a global sensation. He uses his concepts: global hunger and minute specific hungers. He said that hunger as a global phenomenon is a statistical effect. Of what is hunger composed as a global substance? Of thousands of minute hungers: salt hunger, protein substance hunger, grease hunger, mineral salts hunger, etc. . . . When I say, "I’m hungry," I am literally undertaking, says Turro, the integral or the integration of these thousands of minute specific hungers. The minute differentials are differentials of conscious perception; conscious perception is the integration of minute perceptions. Fine. You see that the thousand minute appetitions are the thousand specific hungers. And Turro continues since there is still something strange on the animal level: how does an animal know what it has to have? The animal sees sensible qualities , it leaps forward and eats it, they all eat minute qualities. The cow eats green, not grass, although it does not eat just any green since it recognizes the grass green and only eats grass green. The carnivore does not eat proteins, it eats something it saw, without seeing the proteins. The problem of instinct on the simplest level is: how does one explain that animals eat more or less anything that suits them? In fact, animals eat during a meal the quantity of fat, of salt, of proteins necessary for the balance of their internal milieu. And their internal milieu is what? It’s the milieu of all the minute perceptions and minute appetitions.
What a strange communication between consciousness and the unconscious. Each species eats more or less what it needs, except for tragic or comic errors that enemies of instinct always invoke: cats, for example, who go eat precisely what will poison them, but quite rarely. That’s what the problem of instinct is.
This Leibnizian psychology invokes minute appetitions that invest minute perceptions; the minute appetition makes the psychic investment of the minute perception, and what world does that create? We never cease passing from one minute perception to another, even without knowing it. Our consciousness remains there at global perceptions and gross appetites, "I am hungry," but when I say "I am hungry," there are all sorts of passages, metamorphoses. My minute salt hunger that passes into another hunger, a minute protein hunger; a minute protein hunger that passes into a minute fat hunger, or everything mixed up, quite heterogeneously. What causes children to be dirt eaters? By what miracle do they eat dirt when they need the vitamin that the earth contains? It has to be instinct! These are monsters! But God even made monsters in harmony.
So what is the status of psychic unconscious life? It happened that Leibniz encountered Locke’s thought, and Locke had written a book called An Essay Concerning Human Understanding. Leibniz had been very interested in Locke, especially when he discovered that Locke was wrong in everything. Leibniz had fun preparing a huge book that he called New Essays on Human Understanding in which, chapter by chapter, he showed that Locke was an idiot . He was wrong, but it was a great critique. And then he didn’t publish it. He had a very honest moral reaction, because Locke had died in the meantime. His huge book was completely finished, and he put it aside, he sent it to some friends. I mention all this because Locke, in his best pages, constructs a concept for which I will use the English word, "uneasiness." To summarize, it’s unease , a state of unease. And Locke tries to explain that it’s the great principle of psychic life. You see that it’s very interesting because this removes us from the banalities about the search for pleasure or for happiness. Overall, Locke says that it’s quite possible to seek one’s pleasure, one’s happiness, perhaps it’s possible, but that’s not all; there is a kind of anxiety for a living person. This anxiety is not distress . He proposes the psychological concept of anxiety. One is neither thirsting for pleasure, nor for happiness, nor distressed; he seems to feel that we are, above all, anxious. We can’t sit still. And Leibniz, in a wonderful text, says that we can always try to translate this concept, but that finally, it’s very difficult to translate. This word works well in English, and an Englishman immediately sees what it is. For us, we’d say that someone is nervous. You see how he borrows it from Locke and how he is going to transform it: this unease of the living, what is it? It’s not at all the unhappiness of the living. Rather, it’s when he is immobile, when he has his conscious perception well framed, it all swarms: minute perceptions and minute appetitions invest the fluid minute perceptions, fluid perceptions and fluid appetites ceaselessly move, and that’s it. So, if there is a God, and Leibniz is persuaded that God exists, this ‘uneasiness’ is so little a kind of unhappiness that it is just the same as the tendency to develop the maximum perception. And the development of the maximum perception will define a kind of psychic continuity. We again find the theme of continuity, that is, an indefinite progress of consciousness.
How is unhappiness possible? There can always be unfortunate encounters. It’s like when a stone is likely to fall: it is likely to fall along a path that is the right path , for example, and then it can meet a rock that crumbles it or splits it apart. It’s really an accident connected to the law of the greatest slope. That doesn’t prevent the law of the greatest slope from being the best. We can see what he means.
So there is an unconscious defined by minute perceptions, and minute perceptions are at once infinitely small perceptions and the differentials of conscious perception. And minute appetites are at once unconscious appetites and differentials of conscious appetition. There is a genesis of psychic life starting from differentials of consciousness.
Following from this, the Leibnizian unconscious is the set of differentials of consciousness. It’s the infinite totality of differentials of consciousness. There is a genesis of consciousness. The idea of differentials of consciousness is fundamental. The drop of water and the appetite for the drop of water, specific minute hungers, the world of fainting. All of that makes for a very funny world.
I am going to open a very quick parenthesis. That unconscious has a very long history in philosophy. Overall, we can say that in fact, it’s the discovery and the theorizing of a properly differential unconscious. You see that this unconscious has many links to infinitesimal analysis, and that’s why I said a psycho-mathematical domain. Just as there are differentials for a curve, there are differentials for consciousness. The two domains, the psychic domain and the mathematical domain, project symbols . If I look for the lineage, it’s Leibniz who proposed this great idea, the first great theory of this differential unconscious, and from there it never stopped. There is a very long tradition of this differential conception of the unconscious based on minute perceptions and minute appetitions. It culminates in a very great author who, strangely, has always been poorly understood in France, a German post-Romantic named Fechner. He’s a disciple of Leibniz who developed the conception of differential unconscious.
What was Freud’s contribution? Certainly not the unconscious, which already had a strong theoretical tradition. It’s not that, for Freud, there were no unconscious perceptions, [but] there were also unconscious desires. You recall that for Freud, there is the idea that representation can be unconscious, and in another sense, affect also can be unconscious. That corresponds to perception and appetition. But Freud’s innovation is that he conceived the unconscious ? and here, I am saying something very elementary to underscore a huge difference -- he conceived the unconscious in a conflictual or oppositional relationship with consciousness, and not in a differential relationship. This is completely different from conceiving an unconscious that expresses differentials of consciousness or conceiving an unconscious that expresses a force that is opposed to consciousness and that enters into conflict with it. In other words, for Leibniz, there is a relationship between consciousness and the unconscious, a relation of difference to vanishing differences, whereas for Freud, there is a relation of opposition of forces. I could say that the unconscious attracts representations, it tears them from consciousness, and it’s really two antagonistic forces. I could say that, philosophically, Freud depends on Kant and Hegel, that’s obvious. The ones who explicitly oriented the unconscious in the direction of a conflict of will, and no longer of differential of perception, were from the school of Schopenhauer that Freud knew very well and that descended from Kant. So we must safeguard Freud’s originality, except that in fact, he received his preparation in certain philosophies of the unconscious, but certainly not in the Leibnizian strain.
Thus our conscious perception is composed of an infinity of minute perceptions. Our conscious appetite is composed of an infinity of minute appetites. Leibniz is in the process of preparing a strange operation, and were we not to restrain ourselves, we might want to protest immediately. We could say to him, fine, perception has causes, for example, my perception of green, or of any color, that implies all sorts of physical vibrations. And these physical vibrations are not themselves perceived. Even though there might be an infinity of elementary causes in a conscious perception, by what right does Leibniz conclude from this that these elementary causes are themselves objects of infinitely minute perceptions? Why? And what does he mean when he says that our conscious perception is composed of an infinity of minute perceptions, exactly like perception of the sound of the sea is composed of the perception of every drop of water?
If you look at his texts closely, it’s very odd because these texts say two different things, one of which is manifestly expressed by simplification and the other expresses Leibniz’s true thought. There are two headings: some are under the Part-Whole heading, and in that case, it means that conscious perception is always one of a whole, this perception of a whole assuming not only infinitely minute parts, but assuming that these infinitely small parts are perceived. Hence the formula: conscious perception is made of minute perceptions, and I say that, in this case, "is made of" is the same as "to be composed of." Leibniz expresses himself in this way quite often. I select a text: "Otherwise we would not sense the whole at all". . . if there were none of these minute perceptions, we would have no consciousness at all. The organs of sense operate a totalization of minute perceptions. The eye is what totalizes an infinity of minute vibrations, and henceforth composes with these minute vibrations a global quality that I call green, or that I call red, etc. . . . The text is clear, it’s a question of the Whole-Parts relationship. When Leibniz wants to move rapidly, he has every interest in speaking like that, but when he really wants to explain things, he says something else, he says that conscious perception is derived from minute perceptions. It’s not the same thing, "is composed of" and "is derived from". In one case, you have the Whole-Parts relationship, in the other, you have a relationship of a completely different nature. What different nature? The relation of derivation, what we call a derivative. That also brings us back to infinitesimal calculus: conscious perception derives from the infinity of minute perceptions. At that point, I would no longer say that the organs of sense totalize. Notice that the mathematical notion of integral links the two: the integral is what derives from and is also what operates an integration, a kind of totalization, but it’s a very special totalization, not a totalization through additions. We can say without risk of error that although Leibniz doesn’t indicate it, it’s even the second texts that have the final word. When Leibniz tells us that conscious perception is composed of minute perceptions, this is not his true thinking. On the contrary, his true thinking is that conscious perception derives from minute perceptions. What does "derive from" mean?
Here is another of Leibniz’s texts: "Perception of light or of color that we perceive, that is, conscious perception ? is composed of a quantity of minute perceptions that we do not perceive, and a noise that we do not perceive, and a noise that we do perceive but to which we give no attention becomes a-perceptible, i.e. passes into the state of conscious perception, through a minute addition or augmentation."
We no longer pass minute perceptions into conscious perception via totalization as the first version of the text suggested; we pass minute perceptions into global conscious perception via a minute addition. We thought we understood, and suddenly, we no longer understand a thing. A minute addition is the addition of a minute perception; so we pass minute perceptions into global conscious perception via a minute perception? We tell ourselves that this isn’t right. Suddenly, we tend to fall back on the other version of the text, at least that was more clear. More clear, but insufficient. Sufficient texts are sufficient, but we no longer understand anything in them. A wonderful situation, except if we chance to encounter an adjoining text in which Leibniz tells us: "We must consider that we think a quantity of things all at once. But we pay attention only to thoughts that are the most distinct . . ."
For what is "remarkable" must be composed of parts that are not remarkable ? there, Leibniz is in the process of mixing up everything, but on purpose. We who are no longer innocent can situate the word "remarkable," and we know that each time that he uses "notable", "remarkable", "distinct", it’s in a very technical sense, and at the same time, he creates a muddle everywhere. For the idea that there is something clear and distinct, since Descartes, was an idea that circulated all over. Leibniz slides in his little "distinct" , the most distinct thoughts. Understand "the distinct," "the remarkable," "the singular." So what does that mean? We pass from minute unconscious perception to global conscious perception through a minute addition. So obviously, this is not just any minute addition. This is neither another conscious perception, nor one more minute unconscious perception. So what does it mean? It means that your minute perceptions form a series of ordinaries, a series called regular: all the minute drops of water, elementary perceptions, infinitesimal perceptions. How do you pass into the global perception of the sound of the sea?
First answer: via globalization-totalization. Commentators answer: Fine, it’s easy to say. One would never thinking of raising an objection. You have to like an author just enough to know that he’s not mistaken, that he speaks this way in order to proceed quickly.
Second answer: I pass via a minute addition. This cannot be the addition of a minute ordinary or regular perception, nor can it be the addition of a conscious perception since at that point, consciousness would be presupposed. The answer is that I reach a neighborhood of a remarkable point, so I do not operate a totalization, but rather a singularization. It’s when the series of minute perceived drops of water approaches or enters into the neighborhood of a singular point, a remarkable point, that perception becomes conscious.
It’s a completely different vision because at that moment, a great part of the objections made to the idea of a differential unconscious falls away. What does that mean? Here appear the texts by Leibniz that seem the most complete. From the start, we have dragged along the idea that with minute elements, it’s a manner of speaking because what is differential are not elements, not dx in relation to an x, because dx in relation to an x is nothing. What is differential is not a dy in relation to a y because dy in relation to a y is nothing.
What is differential is dy/dx, this is the relation.
That’s what is at work in the infinitely minute.
You recall that on the level of singular points, the differential relation changes its sign. You recall that on the level of singular points, the differential relation changes its sign. Leibniz is in the process of impregnating Freud without knowing it. On the level of the singularity of increases or decreases, the differential relation changes it sign, that is, the sign is inverted. In this case of perception, which is the differential relation? Why is it that these are not elements, but indeed relations? What determines a relation is precisely a relationship between physical elements and my body. So you have dy and dx. It’s the relation of physical excitation to my biological body. You understand that on this level, we can no longer speak exactly of minute perceptions. We will speak of the differential relation between physical excitation and the physical state by assimilating it frankly to dy/dx, it matters little. And perception becomes conscious when the differential relation corresponds to a singularity, that is, changes its sign.
For example, when excitation gets sufficiently closer.
It’s the molecule of water closest to my body that is going to define the minute increase through which the infinity of minute perceptions becomes conscious perception. It’s no longer a relation of parts at all, it’s a relation of derivation. It’s the differential relation between that which excites and my biological body that is going to permit the definition of the singularity’s neighborhood. Notice in which sense Leibniz could say that inversions of signs, that is, passages from consciousness to the unconscious and from the unconscious to consciousness, the inversions of signs refer to a differential unconscious and not to an unconscious of opposition.
When I alluded to Freud’s posterity, in Jung, for example, there is an entire Leibnizian side, and what he reintroduces, to Freud’s greatest anger, and it’s in this that Freud judges that Jung absolutely betrayed psychoanalysis, is an unconscious of the differential type. And he owes that to the tradition of German Romanticism which is closely linked also to the unconscious of Leibniz.
So we pass from minute perceptions to unconscious perception via addition of something notable, that is, when the series of ordinaries reaches the neighborhood of the following singularity, such that psychic life, just like the mathematical curve, will be subject to a law which is that of the composition of the continuous.
There is composition of the continuous since the continuous is a product: the product of the act by which a singularity is extended into the neighborhood of another singularity. And that this works not only upon the universe of the mathematical symbol, but also upon the universe of perception, of consciousness, and of the unconscious.
From this point onward, we have but one question: what are the compossible and incompossible? These derive directly from the former. We possess the formula for compossibility. I return to my example of the square with its four singularities. You take a singularity, it’s a point; you take it as the center of a circle. Which circle? All the way into the neighborhood of the other singularity. In other words, in the square abcd, you take *a* as center of a circle that stops or whose periphery is in the neighborhood of singularity *b*. You do the same thing with *b*: you trace a circle that stops in the neighborhood of the singularity *a* and you trace another circle that stops in the neighborhood of singularity *c*. These circles intersect. You go on like that constructing, from one singularity to the next, what you will be able to call a continuity. The simplest case of a continuity is a straight line, but there is also precisely a continuity of non-straight lines. With your system of circles that intersect, you will say that there is continuity when the values of two ordinary series, those of *a* to *b*, those of *b* to *a*, coincide. When there is a coincidence of values of two ordinary series encompassed in the two circles, you have a continuity. Thus you can construct a continuity made from continuity. You can construct a continuity of continuity, for example, the square. If the series of ordinaries that derive from singularities diverge, then you have a discontinuity. You will say that a world is constituted by a continuity of continuity. It’s the composition of the continuous. A discontinuity is defined when the series of ordinaries or regulars that derive from two points diverge. Third definition: the existing world is the best? Why? Because it’s the world that assures the maximum of continuity. Fourth definition: what is the compossible? An set of composed continuities. Final definition: what is the incompossible? When the series diverge, when you can no longer compose the continuity of this world with the continuity of this other world. Divergence in the series of ordinaries that depend on singularities: at that moment, it can no longer belong to the same world.
You have a law of composition of the continuous that is psycho-mathematical. Why isn’t that evident? Why is all this exploration of the unconscious necessary? Because, yet again, God is perverse. God’s perversity lies in having chosen the world that implicates the maximum of continuity, in composing the chosen world in this form, only by dispersing the continuities since these are continuities of continuities. God dispersed them. What does that mean? It seems, says Leibniz, that there are discontinuities in our world, leaps, ruptures. Using an admirable term, he says that it seems that there are musical descents . But in fact, there are none. To some among us, it seems that there is a gap between man and animal, a rupture. This is necessary because God, with extreme malice, conceived the world to be chosen in the form of the maximum of continuity, so there are all sorts of intermediary degrees between animal and man, but God held back from making these visible to us. If the need arose, God placed them on other planets of our world. Why? Because finally, it was good, it was good for us to be able to believe in the excellence of our domination of nature. If we had seen all the transitions between the worst animal and us, we would have been less vain, so this vanity is still quite good because it allows man to establish his power over nature. Finally it’s not a perversity of God, but that God did not stop breaking continuities that God had constructed in order to introduce variety in the chosen world, in order to hide the whole system of minute differences, of vanishing differences. So God proposed to our organs of sense and to our stupid thinking, presented on the contrary a very divided world . We spend our time saying that animals have no soul (Descartes), or else that they do not speak. But not at all: there are all sorts of transitions, all sorts of minute definitions. In this, we grasp a specific relation that is compossibility or incompossibility. I would say yet again that compossibility is when series of ordinaries converge, series of regular points that derive from two singularities and when their values coincide, otherwise there is discontinuity. In one case, you have the definition of compossibility, in the other case, the definition of incompossibility.
Why did God choose this world rather than another, when another was possible? Leibniz’s answer becomes splendid: it’s because it is the world that mathematically implicates the maximum of continuity, and it’s uniquely in this sense that it is the best of all possible worlds. A concept is always something very complex. We can situate today’s meeting under the sign of the concept of singularity. And the concept of singularity has all sorts of languages that intersect within it. A concept is always necessarily polyvocal. You can grasp the concept of singularity only through a minimum of mathematical apparatus: singular points in opposition to ordinary or regular points, on the level of thought experiences of a psychological type: what is dizziness, what is a murmur, what is a hum , etc. And on the level of philosophy, in Leibniz’s case, the construction of this relation of compossibility. It’s not a mathematical philosophy, no more than mathematics becomes philosophy, but in a philosophical concept, there are all sorts of different orders that necessarily symbolize. It has a philosophical heading, it has a mathematical heading, and it has a heading for thought experience. And it’s true of all concepts. It was a great day for philosophy when someone brought this odd couple to general attention, and that’s what I call a creation in philosophy. When Leibniz proposed this topic, the singular, there precisely is the act of creation; when Leibniz tells us that there is no reason for you simply to oppose the singular to the universal. It’s much more interesting if you listen to what mathematicians say, who for their own reasons think of "singular" not in relation to "universal," but in relation to "ordinary" or "regular." Leibniz isn’t doing mathematics at that point. I would say that his inspiration is mathematical, and he goes on to create a philosophical theory, notably a whole conception of truth that is radically new since it’s going to consist in saying: don’t pay too much attention to the matter of true and false, don’t ask in your thinking what is true and what is false, because what is true and what is false in your thinking always results from something that is much deeper.
What counts in thinking are the remarkable points and the ordinary points. Both are necessary: if you only have singular points in thinking, you have no method of extension , it’s worthless; if you have only ordinary points, it’s in your interest to think something else. And the more you believe yourself [to be] remarkable (special), the less you think of remarkable points. In other words, the thought of the singular is the most modest thought in the world. It’s there that the thinker necessarily becomes modest, because the thinker is the extension onto the series of ordinaries, and thought itself explodes in the element of singularity, and the element of singularity is the concept.






The last time, we ended with the question: what is compossibility and what is incompossibility? What are these two relationships, the relationship of compossibility and incompossibility? How do we define them?

We saw that these questions created all kinds of problems and led us necessarily to the exercise, however cursory, of infinitesimal analysis. Today, I would like to create a third major rubric that would consist in showing the extent to which Leibniz organizes in a new way and even creates some genuine principles. Creating principles is not a fashionable task of late. This third major introductory chapter for a possible reading of Leibniz is one I will call: Deduction of principles, precisely because principles are objects of a special kind of deduction, a philosophical deduction, which does not go without saying.
There is such a rich abundance of principles in Leibniz's work. He constantly invokes principles while giving them, when necessary, names that did not previously exist. In order to situate oneself within his principles, one has to discover the progression [cheminement] of Leibnizian deduction.

The first principle that Leibniz creates with a rapid justification is the principle of identity. It is the minimum, the very least that he provides. What is the principle of identity? Every principle is a reason. A is A. A thing, it's a thing, that is what a thing is. I have already made some progress. A thing is what it is is better than A is A. Why? Because it shows what is the region governed by the principle of identity. If the principle of identity can be expressed in the form: a thing is what it is, this is because identity consists in manifesting the proper identity between the thing and what the thing is.
If identity governs the relationship between the thing and what the thing is, namely what thing is identical to the thing, and the thing is identical to what it is, I can say: what is the thing? What the thing is, everyone has called it the essence of the thing. I would say that the principle of identity is the rule of essences or, what comes down to the same thing, the rule of the possible. In fact, the impossible is contradictory. The possible is the identical so that, to the extent that the principle of identity is a reason, a ratio, then which ratio? It is the ratio of essence or, as the Latins used to say, or the Middle Age terminology long before: ratio essendi. I choose that as a typical example because I think that it is very difficult to do philosophy if you do not have a kind of terminological certainty. Never tell yourself that you can do without it, but also never tell yourself that it is difficult to acquire. It is exactly the same as scales on the piano. If you do not know rather precisely the rigour of concepts, that is, the sense of major notions, then it is very difficult. One has to approach that like an exercise. It is normal for philosophers to have their own scales, it is their mental piano. One must change the tune of the categories. The history of philosophy can only be created by philosophers, yet alas, it has fallen into the hands of philosophy professors, and that's not good because they have turned philosophy into examination material and not material for study, or for scales.

Each time that I speak of a principle according to Lebiniz, I am going to give it two formulations: a vulgar formulation and a scholarly one. This is a beautiful procedure on the level of principles, the necessary relation between pre-philosophy and philosophy, this relationship of exteriority in which philosophy needs a pre-philosophy. The vulgar formulation of the principle of identity: the thing is what the thing is, the identity of the thing and of its essence. You see that, in the vulgar formulation, there are lots of things already implied.

The scholarly formulation of the principle of identity: every analytical proposition is true. What is an analytical proposition? It is a proposition in which the predicate and the subject are identical. An analytical proposition is true, A is A, is true. By going into the detail of Leibniz's formulae, one can even complete the scholarly formulation: every analytical proposition is true in two cases: either by reciprocity or by inclusion.

An example of a proposition of reciprocity: the triangle has three angles. Having three angles is what the triangle is. Second case: inclusion: the triangle has three sides. In fact, a closed figure having three angles envelopes, includes, implies having three sides.
We will say that analytical propositions of reciprocity are objects of intuition, and we will say that analytical propositions of inclusion are objects of demonstration.
Therefore, the principle of identity, the rule of essences, or of the possible, ratio essendi: what question does it answer? What cry does the principle of identity answer? The pathetic cry that constantly appears in Leibniz's works, corresponding to the principle of identity, why is there something rather than nothing? It is the cry of the ratio essendi, of the reason for being [raison d'être]. If there were no identity, no identity conceived as identity of the thing and what the thing is, then there would be nothing.

Second principle: principle of sufficient reason.
This refers us back to the whole domain that we located as being the domain of existences. The ratio corresponding to the principle of sufficient reason is no longer the ratio essendi, the reason of essences or the reason for being, it is now the ratio existendi, the reason for existing. It is no longer the question: why something rather than nothing, since the principle of identity assured us that there was something, namely the identical. It is no longer: why something rather than nothing, but rather it is why this rather than that? What would its vulgar formulation be? We saw that every thing has a reason. Indeed, every thing must have a reason. What would the scholarly formulation be? You see that we apparently are completely outside the principle of identity. Why? Because the principle of identity concerns the identity of the thing and what it is, but it does not state whether the thing exists. The fact that the thing exists or does not exist is completely different from what it is. I can always define what a thing is independently of the question of knowing if it exists or not. For example, I know that the unicorn does not exist, but I can state what a unicorn is. Thus, a principle is indeed necessary that makes us think of the existent [lâexistant]. So just how does a principle, that appears to us as vague as "everything has a reason," make us think of the existent? It is precisely the scholarly formulation that will explain it to us. We find this scholarly formulation in Leibniz's works in the following statement: every predication (predication means the activity of judgment that attributes something to a subject; when I say "the sky is blue," I attribute blue to sky, and I operate a predication), every predication has a basis [fondement] in the nature of things. It is the ratio existendi.

Let us try to understand better how every predication has a basis in the nature of things. This means: everything said about a thing, the entirety of what is said about a thing, is the predication concerning this thing. Everything said about a thing is encompassed, contained, included in the notion of the thing. This is the principle of sufficient reason. You see that the formula which appeared innocent a short while ago - every predication has a basis in the nature of things, taking it literally - becomes much stranger: everything said about a thing must be encompassed, contained, included in the notion of the thing. So, what is everything said about a thing? First, it is the essence. In fact, the essence is said about the thing. Only one finds at that level that there would be no difference between sufficient reason and identity. And this is normal since sufficient reason includes all the properties [tout l'acquis] of the principle of identity, but is going to add something to it: what is said about a thing is not only the essence of the thing, it is the entirety of the affections, of the events that refer or belong to the thing.

Thus, not only will the essence be contained in the notion of the thing, but the slightest of events, of affections concerning the thing as well, that is, what is attributed truthfully to the thing is going to be contained in the notion of the thing.
We have seen this: crossing the Rubicon, whether one likes it or not, must be contained in the notion of Caesar. Events, affections of the type "loving" and "hating" must be contained in the notion of that subject feeling these affections.
In other words, each individual notion -- and the existent is precisely the object, the correlate of an individual notion -- each individual notion expresses the world. That is what the principle of sufficient reason is.
"Everything has a reason" means that everything that happens to something must be contained forever in the individual notion of the thing.
The definitive formulation of the principle of sufficient reason is quite simple: every true proposition is analytical, every true proposition, for example, every proposition that consists in attributing to something an event that really occurred and that concerns the something. So if it is indeed true, the event must be encompassed in the notion of the thing.

What is this domain? It is the domain of infinite analysis whereas, on the contrary, at the level of the principle of identity, we were only dealing with finite analyses. There will be an infinite analytical relationship between the event and the individual notion that encompasses the event. In short, the principle of sufficient reason is the reciprocal of the principle of identity. Only, what has occurred in the reciprocal? The reciprocal has taken over a radically new domain, the domain of existences. It was sufficient merely to reciprocate, to reverse the formula of identity in order to obtain the formula of sufficient reason; it was enough to reciprocate the formula of identity that concerns essences in order to obtain a new principle, the principle of sufficient reason concerning existences. You will tell me that this was not complicated. Yet it was enormously complicated, so why? The reciprocal, this reciprocation was only possible if one were able to extend the analysis to infinity. So the notion, the concept of infinite analysis is an absolutely original notion. Does that consist in saying that this takes place uniquely in the understanding [l'entendement] of God, which is infinite? Certainly not. This implies an entire technique, the technique of differential analysis or infinitesimal calculus.

Third principle: is it true that the reciprocal of the reciprocal would yield the first? It is not certain. Everything depends, there are so many viewpoints. Let us try to vary the formulation of the principle of sufficient reason. For sufficient reason, I left things off by saying that everything that happens to a thing must be encompassed, included in the notion of the thing, which implies infinite analysis. In other words, for everything that happens or for every thing there is a concept. I insisted earlier that what matters is not at all a way for Leibniz to recover a famous principle. On the contrary, he does not want that at all; this would be the principle of causality. When Leibniz says that everything has a reason, that does not at all mean that everything has a cause. Saying everything has a cause signifies A refers to B, B refers to C, etc. ... Everything has a reason means that one must account for reason in causality itself, namely that everything has a reason means that the relationship that A maintains with B must be encompassed in one way or another in the notion of A. Just like the relationship that B maintains with C must be encompassed one way or another in the notion of B. Thus, the principle of sufficient reason goes beyond the principle of causality. It is in this sense that the principle of causality states only the necessary cause, but not the sufficient reason. Causes are only necessities that themselves refer to and presuppose sufficient reasons.

Thus, I can state the principle of sufficient reason in the following way: for every thing there is a concept that takes account both of the thing and of its relations with other things, including its causes and its effects.
For every thing, there is a concept, and that does not go without saying. Lots of people will think that not having a concept is the peculiarity [le propre] of existence. For every thing there is a concept, so what would the reciprocal be? Understand that the reciprocal does not at all have the same meaning. In Aristotle's work, there is a treatise of ancient logic that deals solely with the table of opposites. What is the contradictory, the contrary, the subaltern, etc. ... You cannot say the contradictory when it is the contrary, you cannot just say anything. Here I use the word reciprocal without specifying. When I say for every thing there is a concept (yet again, this is not at all certain), assume that you grant me that. There I cannot escape the reciprocal. What is the reciprocal ?

For a theory of the concept, we would have to start again from the bird song [chant dâoiseaux]. The great difference between cries and songs. The cries of alarm, of hunger, and then the bird songs. And we can explain acoustically what the difference is between cries and songs. In the same way, on the level of thought, there are cries of thought and songs of thought [chants de pensée]. How to distinguish these cries and these songs? One cannot understand how a philosophy as song or a philosophical song develops if one does not refer it to coordinates that are kinds of cries, continuous cries. These cries and songs are complex. If I return to music, the example that I recall again and again is the two great operas of [Alban] Berg; there are two great death cries, the cry of Marie and the cry of Lulu. [TN: cf. "N as in Neurology" in LâAbécédaire de Gilles Deleuze] When one dies, one does not sing, and yet there is someone who sings around death, the mourner. The one who loses the loved one sings. Or cries, I do not know. In Wozzeck, it is a si-, it is a siren. When you put sirens into music, you are placing a cry there. It is strange. And the two cries are not the same type, even acoustically: there is a cry that flits upward and there is a cry that skims along the earth. And then there is the song [chant]. Luluâs great woman friend sings death. It is fantastic. It is signed Berg. I would say that the signature of a great philosopher is the same. When a philosopher is great, although he writes very abstract pages, these are abstract only because you did not know how to locate the moment in which he cries. There is a cry underneath, a cry that is horrible.

Let us return to the song of sufficient reason. Everything has a reason is a song. It is a melody, we could harmonize. A harmony of concepts. But underneath there would be rhythmic cries: no, no, no. I return to my sung formulation of the principle of sufficient reason. One can sing off key in philosophy. People who sing off key in philosophy know it very well, but it [philosophy] is completely dead. They can talk interminably. The song of sufficient reason: for every thing there is a concept. What is the reciprocal ? In music, one would speak of retrograde series. Let us look for the reciprocal of "every thing has a concept." The reciprocal is: for every concept there is one thing alone.

Why is this the reciprocal of "for every thing a concept"? Suppose that a concept had two things that corresponded to it. There is a thing that has no concept and, in that case, sufficient reason is screwed [foutue]. I cannot say "for every thing a concept". As soon as I have said "for every thing a concept," I have necessarily said that a concept had necessarily one thing alone, since if a concept has two things, there is something that has no concept, and therefore already I could no longer say "for every thing a concept." Thus, the true reciprocal of the principle of sufficient reason in Leibniz will be stated like this: for every concept, one thing alone. It is a reciprocal in a very funny sense. But in this case of reciprocation, sufficient reason and the other principle, notably "for every thing, a concept" and "for every concept, one thing alone," I cannot say one without saying the other. Reciprocation is absolutely necessary. If I do not recognize the second, I destroy the first.

When I said that sufficient reason was the reciprocal of the principle of identity, it was not in the same sense since, if you recall the statement [énoncé] of the principle of identity ö namely, every true proposition is analytical, there is in this no necessity. I can say that every analytical proposition is true without necessarily preventing any true proposition from being analytical. I could very well say that there are true propositions that are something other than analytical. Thus, when Leibniz created his reciprocation of identity, he made a master stroke because he had the means to make this master stroke, that is, he let out a cry. He had himself created an entire method of infinite analysis. Otherwise, he could not have done so.

Whereas in the case of the passage from sufficient reason to the third principle that I have not yet baptized, there reciprocation is absolutely necessary. It had to be discovered. What does it mean that for every concept there is a thing and only one thing? Here it gets strange, you have to understand. It means that there are no two absolutely identical things, or every difference is conceptual in the last instance. If you have two things, there must be two concepts, otherwise there would not be two things. Does that mean that there are no two absolutely identical things as far as the concept goes? It means that there are no two identical drops of water, no two identical leaves. In this, Leibniz is perfect, he gets delirious with this principle. He says that obviously you, you believe that two drops of water are identical, but this is because you do not go far enough in your analysis. They cannot have the same concept. Here this is very odd because all of classical logic tends rather to tell us that the concept, by its very nature, encompasses an infinite plurality of things.

The concept of drops of water is applicable to all drops of water. Leibniz says, of course, if you have blocked off analysis of the concept at a certain point, at a finite moment. But if you push the analysis forward, there will be a moment in which the concepts are no longer the same. This is why the ewe recognizes its lamb, one of Leibniz's examples: how does the ewe recognize its little lamb? They [Eux] think it is via the concept. A little lamb does not have the same concept as the same individual concept, it is in this manner that the concept extends to the individual, another little lamb. What is this principle? There is but a single thing; there is necessarily one thing per concept and only one. Leibniz names it the principle of indiscernibles. We can state it this way: there is one thing and only one thing per concept, or every difference is conceptual in the final instance.

There is only conceptual difference. In other words, if you assign a difference between two things, there is necessarily a difference in the concept. Leibniz names this the principle of indiscernibles. And if I make it correspond to a ratio, what is this? You sense correctly that it consists in saying that we only gain knowledge through the concept. In other words, the principle of indiscernibles seems to me to correspond to the third ratio, the ratio as ratio cognoscendi, the reason as reason for knowing [raison de connaitre].

Let us look at the consequences of such a principle. If this principle of indiscernibles were true, namely that every difference is conceptual, there would be no difference except the conceptual. Here Leibniz asks us to accept something that is quite huge. Let us proceed in order: what other kind of difference is there other than conceptual? We see it immediately: there are numerical differences. For example, I say a drop of water, two drops, three drops. I distinguish the drops by the number alone [solo numero, that Deleuze translates as par le nombre seulement]. I count the elements of a set [ensemble], one two three four, I neglect their individuality, I distinguish them by the number. This constitutes a first type of very classic distinction, the numerical distinction. Second type of distinction: I say "take this chair"; some obliging person takes a chair, and I say, "not that one, but this one." This time, it is a spatio-temporal distinction of the here-now type. The thing that is here at a particular moment, and this other thing that is there at a particular moment. Finally, there are distinctions of figure and of movement: roof that has three angles, or something else. I would say that these are distinctions by extension and movement. Extension and movement.

Understand that this commits Leibniz to a strange undertaking, merely with his principle of indiscernibles. He has to show that all these types of non-conceptual distinctions - and in fact, all of these distinctions are non-conceptual since two things can be distinguished by the number even though they have the same concept. You focus on the concept of a drop of water, and you say: first drop, second drop. It is the same concept. There is the first and there is the second. There is one that is here, and another that is there. There is one that goes fast, and another that goes slowly. We have now nearly completed the set of non-conceptual distinctions.

Leibniz arrives and calmly tells us, no no. These are pure appearances, that is, these are only provisional ways of expressing a difference of another nature, and this difference is always conceptual. If there are two drops of water, they do not have the same concept. What of any great import does this mean? It is very important in problems of individuation. It is very well known, for example, that Descartes tells us that bodies are distinguished from one another by figure and by movement. Lots of thinkers have appreciated that. Notice that in the Cartesian formula, what is conserved in movement (mv) (the product of mass times movement) depends strictly on a vision of the world in which bodies are distinguished by the figure and movement. What does Leibniz commit himself to when he tells us no? It is absolutely necessary that to all these non-conceptual differences there correspond conceptual differences; they only cause it to be imperfectly translated. All non-conceptual differences only cause a basic conceptual difference to be imperfectly translated. Leibniz commits himself to a task of physics. He has to find a reason for which a body is either in a particular number, or in a particular here and now, or has a particular figure and a particular velocity. He will translate that quite well in his critique of Descartes when he says that velocity is a pure relative. Descartes was wrong, he took something that was purely relative for a principle. It is therefore necessary that figure and movement be surpassed [se dépassent] toward something deeper. This means something quite enormous for philosophy in the seventeenth century.

Specifically, that there is no extended substance or that extension [l'étendue] cannot be a substance. That extension is a pure phenomenon. That it refers to something deeper. That there is no concept of extension, that the concept is of another nature. It is therefore necessary that figure and movement find their reason in something deeper. Henceforth, extension has no sufficiency. It is not by chance that this is precisely what makes a new physics, he completely recreates the physics of forces. He opposes force, on one hand, to figure and extension, on the other, figure and extension being only manifestations of force. It is force that is the true concept. There is no concept of extension because the true concept is force.
Force is the reason of figure and movement in extension.
Hence the importance of this operation that appeared purely technical when he said that what is conserved in movement is not mv, but mv2. Squaring velocity is the translation of the concept of force, which is to say that everything changes. It is physics that corresponds to the principle of indiscernibles. There are no two similar or identical forces, and forces are the true concepts that must take account of or justify everything that is figure or movement in extension.

Force is not a movement, it is the reason for movement. Hence the complete renewal of the physics of forces, and also of geometry, of kinematics [la cinématique]. Everything passes through this, merely by the squaring of velocity. MV2 is a formula of forces, not a formula of movement. You see that this is essential.

To sum up generally, I can also say that figure and movement must move forward toward force. Number must move forward toward the concept. Space and time must also move forward toward the concept.

But this is how a fourth principle develops quite slowly, one that Leibniz names the law of continuity. Why did he say law? That is a problem. When Leibniz speaks of continuity that he considers to be a fundamental principle and one of his very own great discoveries, he no longer uses the term "principle," but uses the term "law." We have to explain that. If I look for a vulgar formulation of the law of continuity, it is quite simple: nature does not make a jump [la nature ne fait pas de saut]. There is no discontinuity. But there are two scholarly formulations. If two causes get as close as one would like, to the point of only differing by a difference decreasing to infinity, the effects must differ in like manner. I immediately say what Leibniz is thinking about because he has it in for Descartes so much. What are we told in the laws of the communication of movement? Here are two cases: two bodies of the same mass and velocity meet each other; one of the two bodies has a greater mass or a greater velocity, so it carries off the other. Leibniz says that this cannot be. Why? You have two states of the cause. First state of the cause: two bodies of the same mass and velocity. Second state of the cause: two bodies of different masses. Leibniz says that you can cause difference to decrease to infinity, you can act so these two states approach one another in the causes. And we are told that the two effects are completely different: in one case, there is a repulsion [rebondissement] of the two bodies, in the other case, the second body is dragged off by the first, in the direction of the first. There is a discontinuity in the effect whereas one can conceive of a continuity in the causes. It is in a continuous manner that we can pass from different masses to equal masses. Thus, it is not possible for there to be discontinuity in the facts/acts [faits] if there is possible continuity in the cause. That leads him again into a whole, very important physical study of movement that will be centered on the substitution of a physics of forces for a physics of movement. I was citing this to refresh our memory. But the other scholarly formulation of the same principle, and you will understand that it is the same thing as the preceding one: in a given case, the concept of the case ends in the opposite case.

This is the pure statement of continuity. Example: a given case is movement, the concept of movement ends in the opposite case, that is in rest. Rest is infinitely small movement. This is what we saw from the infinitesimal principle of continuity. Or I might say that the last possible scholarly formulation of continuity is: a given singularity extends itself [se prolonge] into a whole series of ordinaries all the way to the neighborhood of the following singularity. This time it is the law of the composition of the continuous. We worked on that the last time. But right when we thought we had finished there arises a very important problem. Something impels me to say that, between principle three and principle four, there is a contradiction, that is between the principle of indiscernibles and the principle of continuity, there is a contradiction. First question: in what way is there a contradiction? Second question: the fact is that Leibniz never considered there to be the slightest contradiction. Here we are in that situation of liking and profoundly admiring a philosopher, yet of being disturbed because some texts seem contradictory to us, and he did not even see what we might tell him. Where would the contradiction be if there was one? I return to the principle of indiscernibles, every difference is conceptual, there are no two things having the same concept. At the limit I might say that to every thing corresponds a determined difference, not only determined but assignable in the concept. The difference is not only determined or determinable, it is assignable in the very concept. There are no two drops of water having the same concept, that is the difference one, two must be encompassed in the concept. It must be assigned in the concept. Thus every difference is an assignable difference in the concept. What does the principle of continuity tell us? It tells us that things proceed by vanishing differences [différences évanouissantes], infinitely small differences, that is unassignable differences. That gets really awful. Can one say that every thing proceeds by unassignable difference and say at the same time that every difference is assigned and must be assigned in the concept? Ah! Doesn't Leibniz contradict himself? We can move forward a small bit by looking at the ratio of the principle of continuity since I found a ratio for each of the first three principles. Identity is the reason of essence or ratio essendi, sufficient reason is the reason of existence or the ratio existendi, the indiscernibles are the reason for knowing or the ratio cognoscendi, and the principle of continuity is the ratio fiendi, that is, the reason for becoming. Things become through continuity. Movement becomes rest, rest becomes movement, etc. The polygon becomes a circle by muliplying its sides, etc·. This is a very different reason for becoming from the reasons of being or of existing. The ratio fiendi needed a principle, and it is the principle of continuity.

How do we reconcile continuity and indiscernibles? Moreover, we have to show that the way in which we will reconcile them must take account of this at the same time: that Leibniz was right to see no contradiction at all between them. In this we have the experience of thought. I return to the proposition: each individual notion expresses the whole world. Adam expresses the world, Caesar expresses the world, each of you expresses the world. This formula is very strange. Concepts in philosophy are not a single word. A great philosophical concept is a complex, a proposition, or a prepositional function. One would have to do exercises in philosophical grammar. Philosophical grammar would consist of this: with a given concept, find the verb. If you have not found the verb, you have not rendered the verb dynamic, you cannot live it. The concept is always subject to a movement, a movement of thought. A single thing counts: movement. When you do philosophy, you are looking only at movement, only it is a particular kind of movement, the movement of thought. What is the verb? Sometimes the philosopher states it explicitly, sometimes he does not state it. Is Leibniz going to state it? In each individual notion that expresses the world, there is a verb, this is expressing. But what does that mean? It means two things at once, as if two movements coexisted.

Leibniz tells us at the same time: God does not create Adam the sinner, but creates the world in which Adam sinned. God does not create Caesar crossing the Rubicon, but creates the world in which Caesar crosses the Rubicon. Thus, what God creates is the world and not the individual notions that express the world. Second proposition by Leibniz: the world exists only in the individual notions that express it. If you privilege one individual notion over the other . . . If you accept that, what results is like two readings or two complementary and simultaneous ways of understanding, but two understandings of what? You can consider the world, but yet again the world does not exist in itself, it exists only in the notions that express it. But you can make this abstraction, you consider the world. How do you consider it? You consider it as a complex curve. A complex curve has singular points and ordinary points. A singular point extends itself into the ordinary points that depend on it all the way to the neighborhood of another singularity, etc. etc. . . . and you compose the curve in a continuous manner like that, by extending singularities into series of ordinaries. For Leibniz, that is what the world is. The continuous world is the distribution of singularities and regularities, or singularities and ordinaries that constitute precisely the set chosen by God, that is the set that unites the maximum of continuity. If you remain in this vision, the world is governed by the law of continuity since continuity is precisely this composition of singulars insofar as they extend into the series of ordinaries that depend on them. You have your world that is literally laid out [déployé] in the form of a curve in which singularities and regularities are distributed. This is the first point of view that is completely subject to the law of continuity.

Only here we are, this world does not exist in itself, it exists only in the individual notions that express this world. That means that an individual notion, a monad, that each one encompasses a small determined number of singularities. It encloses a small number of singularities. It is the small number of singularities·. You recall that individual notions or monads are points of view on the world. It is not the subject that explains [explique] the point of view, it is the point of view that explains the subject. Hence the need to ask oneself, what is this point of view? A point of view is defined by this: a small number of singularities drawn from the curve of the world. This is what is at the basis of an individual notion. What makes the difference between you and me is that you are, on this kind of fictional curve, you are constructed around such and such singularities, and me around such and such singularities. And what you call individuality is a complex of singularities insofar as they form a point of view. There are two states of the world. It has a developed state . . .






I would like to finish these meetings on Leibniz by presenting the problem that I wanted to consider. I return to this question that I asked from the start, specifically: what does this image mean that good sense often creates about philosophy, what does this image mean that good sense sometimes produces about philosophy, like a kind of locus of discussion in which philosophers are fundamentally not in agreement? A kind of philosophical atmosphere in which people dispute, fight among themselves, whereas at least in science, they know what they are talking about. We are told as well that all philosophers say the same thing, they all agree or all hold opposite views. It’s in relation to Leibniz that I would like to select some very precise examples. What does it mean that two philosophies do not agree? Polemics, like a certain state of things that traverses certain disciplines, I do not find that there are more polemics in philosophy than there are in science or in art. What is a philosopher who critiques another philosopher? What is the function of critique? Leibniz offers us this example: what does the opposition between Kant and Leibniz mean, once we have said that it was a fundamental opposition in the history of philosophy? What does it mean for Kant to undertake a critique of Leibniz? I would like to number what I want to tell you. An initial task: to localize the oppositions. There are two fundamental oppositions from the point of view of knowledge. They function like thesis and antithesis. When we manage to trace the great philosophical oppositions, on the level of the concepts used by one philosopher or another, we also have to evaluate their relations to these oppositions. They [the oppositions] are not of equal value. Perhaps one does happen to have greater weight than another, to be more decisive. If you fail to organize the oppositions, I think that you are no longer able to understand what the subject is in a polemic.

First opposition between Leibniz and Kant, from the point of view of knowledge. I will let Leibniz speak. A Leibnizian proposition: all propositions are analytical, and knowledge can proceed only by analytical propositions. You recall that we call “analytical proposition” a proposition in which one of the two terms of the proposition is contained in the concept of the other. It’s a philosophical formula. We can already sense that there is no point in arguing at this level. Why? Because there is already something implied, specifically that there is a certain model of knowledge. What is presupposed, but in science as well, there are also presuppositions; what is presupposed is a certain ideal of knowledge, specifically knowing is discovering what is included in the concept. It’s a definition of knowledge. We are pleased to have a definition of knowledge, but why this one rather that something else?
From the other side, Kant arises and says: there are synthetic propositions. You see what a synthetic proposition is: it’s a proposition in which one of the terms is not contained in the concept of the other. Is this a cry? Is this a proposition? Against Leibniz, he says, “no”; he says that there are synthetic propositions and that knowledge exists only through synthetic propositions. The opposition seems perfect. At this point, a thousand questions assail me: What would that mean to argue, to argue about who is right, who is right about what? Is this provable, are we in the domain of decidable propositions? I say simply that the Kantian definition must interest you because, if you consider it closely, it also implies a certain conception of knowledge, and it happens that this conception of knowledge is very different from Leibniz’s. When one says that knowledge proceeds only through synthetic propositions, that is, a proposition such that one of its terms is not contained in the concept of the other, there is therefore a synthesis between the two terms. Someone who says this can no longer base knowledge on the Leibnizian conception.
He will tell us, on the contrary, that to know is not at all to discover what is included in a concept, that knowledge necessarily means leaving behind one concept in order to affirm something else. We call “synthesis” the act through which one leaves a concept behind in order to attribute to it or to affirm something else. In other words, to know is always to go beyond the concept. Knowing is to go beyond [connaître c’est dépasser]. Understand all that is in play here. In the first conception, to know is to have a concept and discover what is contained in the concept. I would say about that knowledge that it is based on a particular model which is one of passion or of perception. To know is finally to perceive something, to know is to apprehend, a passive model of knowledge, even if many activities depend on it. In the other case, to the contrary, it means leaving the concept behind in order to affirm something, and is a model of the knowledge-act [un modèle de la connaissance acte].

I return to my two propositions. Let us suppose that we are referees. We find ourselves faced with these two propositions, and we say: what do I choose? First when I say: is it decidable? What would that mean? It could mean that it’s a question of fact. One has to find the facts that allow one to say that one or the other is right. Obviously, it’s not that. Philosophical propositions, to some extent, aren’t justifiable on the basis of a verification of facts. That is why philosophy has always distinguished two questions, and Kant especially will take this distinction up again. This distinction was formulated in Latin: quid facti, what is derived from fact [qu'en est-il du fait], and quid juris, what is derived from principle [qu'en est-il du droit]. And if philosophy is concerned with principle, it is precisely because it poses questions that are called de jure questions [questions de droit]. What does it mean that my two paradoxical propositions, Leibniz’s and Kant’s, are not justifiable on the basis of a factual response? It means that in fact, there is no problem because all the time we encounter phenomena that are synthetic phenomena. Indeed, in my simplest judgments, I pass my time operating syntheses. I say, for example, that this straight line is white.

It is quite obvious that with this, I am affirming about a straight line something that is not contained in the concept of straight line. Why? Every straight line is not white. That this straight line is white is obviously an encounter in experience; I could not have made such a statement beforehand. I therefore encounter in experience straight lines that are white. It’s a synthesis, and we call this kind of synthesis *a posteriori*, that is, given in experience. Thus, there are syntheses of fact, but that does not resolve the problem. Why? For a very simple reason: this straight line [that] is white does not constitute knowledge. It’s a protocol of experience. Knowledge is something other than tracing protocols of experience.
When does one know? One knows when a proposition appeals to a principle [se réclame d’un droit]. What defines a proposition’s principle is the universal and the necessary. When I say that a straight line is the shortest path from one point to another, I maintain a proposition in principle (une proposition de droit). Why? Because I don’t need to measure each straight line to know that, if it’s straight, it’s the shortest path. Every straight line, beforehand, a priori, that is, independently of experience, is the shortest path from one point to another, otherwise it would not be a straight line. Thus, I would say that the proposition “a straight line is the shortest path” constitutes indeed a proposition of knowledge. I do not await experience to discover that a straight line is the shortest path; to the contrary, I determine the experience since the shortest path from one point to another is my way of tracing a straight line experientially. Any straight line is necessarily the shortest path from one point to another. This is a proposition of knowledge and not a protocol proposition. Let us take this proposition, it’s an a priori proposition. In this, are we going to be able finally to pose the question of separation between Leibniz and Kant, specifically is it an analytical proposition or is it a synthetic proposition?

Kant says something very simple: it’s necessarily, a priori, a synthetic proposition. Why? Because when you say that the straight line is the shortest path from one point to another, you are leaving behind the concept “straight line.” Isn’t it the content in a straight line to be the shortest path from one point to another? It goes without saying that Leibniz would say that it is the content in “straight line.” Kant says no, the concept “straight line”, according to the Euclidian definition is: line ex aequo in all of its points. You won’t draw from this the shortest path between one point and another. You have to leave the concept behind to affirm something else about it. We’re not convinced. Why does Kant say that? Kant would answer, I suppose, that the shortest path to another is a concept that implies a comparison, the comparison of the shortest line with other lines that are either broken lines or curvilinear lines, that is, curves. I cannot say that the straight line is the shortest path from one point to another without implying a comparison, the relation of the straight line to curved lines. For Kant this suffices to say that a synthesis lies therein; you are forced to leave the “straight line” concept in order to reach the “curved line” concept, and it’s in the relation of straight lines to curved lines that you say the straight line is the shortest path from one point to another. . . It’s a synthesis, thus knowledge is a synthetic operation. Would Leibniz be disturbed by that? No, he would say that obviously you have to keep in mind the “curved line” concept when you say that the straight line is the shortest path from one point to another, but Leibniz is the creator of a differential calculus through which the straight line is going to be considered as the limit of curves. There is a process to the limit. Hence Leibniz’s theme: it’s an analytical relation, only it’s an infinite analysis. The straight line is the limit of the curve, just as rest is the limit of movement. Does this advance us? So either one can no longer resolve this, or they mean the same thing. [If] they say the same thing, what would this be? It would mean that what Leibniz calls infinite analysis is the same thing as what Kant calls finite synthesis. Henceforth, it’s only a question of words. In this perspective, at that point, we would say that they agree in order to establish a difference in nature, one of them between finite analysis and infinite analysis, the other between analysis and synthesis. It comes down to the same thing: what Leibniz calls infinite analysis, Kant will call finite synthesis.

You see the good sense idea that, simultaneously, a philosophical dispute is inextricable since we cannot decide who is right, and at the same time, knowing who is right is without any importance since they both say the same thing. Good sense can conclude just as well: the only good philosophy is me. Tragic situation. Because if good sense achieves the goals of philosophy, better than philosophy itself does it, then there is no reason to wear yourself out doing philosophy. So?
Let’s look for a kind of bifurcation since this first great opposition between Leibniz and Kant, even if it now seems obvious too us, isn’t this because, in fact, this opposition moves well beyond itself toward a deeper opposition, and if we don’t see the deeper opposition, we can understand nothing. What would this second, deeper opposition be?
We saw that there was a great Leibnizian proposition, called the principle of indiscernibles, notably that any difference, in the final instance, is conceptual. Any difference is in the concept. If two things differ, they cannot simply differ by number, by figure, by movement, but rather their concept must not be the same. Every difference is conceptual. See how this proposition is truly the presupposition of Leibniz’s preceding proposition. If he is right on this point, if every difference is conceptual, it is quite obvious that it’s by analyzing concepts that we know, since knowing is knowing through differences. Thus, if every difference, in the final instance, is conceptual, the analysis of the concept will make us know the difference, and will therefore cause us quite simply to know. We see into which quite advanced mathematical task this drew Leibniz, [a task] which consisted in showing the differences between figures, the differences between numbers, referring to differences in the concepts. Ok, what is Kant’s proposition in opposition to the second Leibnizian proposition? Here again, this is going to be pretty odd [un drôle de truc]. Kant maintains a very strange proposition: if you look closely at the world presented to you, you will see that it is composed of two sorts of irreducible determinations: you have conceptual determinations that always correspond to what a thing is, I can even say that a concept is the representation of what the thing is. You have determinations of this sort, for example, the lion is an animal that roars; that’s a conceptual determination. And then you have another kind of determination altogether. Kant proposes his great thing [son grand truc]: he says that it’s no longer conceptual determinations, but spatio-temporal determinations. What are these spatio-temporal determinations? It’s the fact that the thing is here and now, that it is to the right or to the left, that it occupies one kind of space or another, that it describes a space, that it lasts a certain time. And so, however far you push the analysis of concepts, you will never arrive at this domain of spatio-temporal determinations by analyzing concepts. Although you might push your analysis of the concept to infinity, you will never find a determination in the concept that takes this into account for you: that this thing is on the right or on the left.

What does he mean? He selects examples for himself that initially seem very convincing. Consider two hands. Everyone knows that two hands don’t have exactly the same traits, nor the same distribution of pores. In fact, there are no two hands that are identical. And this is a point for Leibniz: if there are two things, they must differ through the concept, it’s his principle of indiscernibles.
Kant says that, in fact, it is indeed possible, but that’s not important. He says that it’s without interest. Discussions never pass through the true and the false, they pass through: does it have any interest whatsoever, or is it a platitude? A madman is not a question of fact, he’s also a question quid juris. It’s not someone who says things that are false. There are loads of mathematicians who completely invent absolutely crazy theories. Why are they crazy? Because they are false or contradictory? No, they are determined by the fact that they manipulate an enormous conceptual and mathematical apparatus [appareillage], for example, for propositions stripped of all interest.
Kant would dare to tell Leibniz that what you are saying about the two hands with their different skin features [différences de pores] has no interest since you can conceive quid juris, in principle but not in fact, you can conceive of two hands belonging to the same person, having exactly the same distribution of pores, the same outline of traits. This is not logically contradictory, even if it does not exist in fact. But, says Kant, there is something nonetheless that is very odd: however far you push your analysis, these two hands are identical, but admire the fact that they cannot be superposed. You have your two absolutely identical hands, you cut them in order to have a radical degree of mobility. You cannot cause them to coincide; you cannot superpose them. Why? You cannot superpose them, says Kant, because there is a right and a left. They can be absolutely identical in everything else, there is still one that is the right hand and the other the left hand. That means that there is a spatial determination irreducible to the order of the concept. The concept of your two hands can be strictly identical, however far you push the analysis, there will still be one of them that is my right hand and one that is my left hand. You cannot cause them to be superposed. Under what condition can you cause two figures to be superposed? On the condition of having access to a dimension supplementary to that of the figures . . . It’s because there is a third dimension of space that you can cause two flat figures to be superposed. You could cause two volumes to be superposed if you have access to a fourth dimension. There is an irreducibility in the order of space. The same thing holds for time: there is an irreducibility in the order of time. Thus, however far you push the analysis of conceptual differences, an order of difference will always remain outside of the concepts and the conceptual differences. This will be spatio-temporal differences.

Does Kant again gain the stronger position? Let’s go back to the straight line. [Regarding] the idea of synthesis, we are going to recognize that it was not a matter of mere words for Leibniz. If we stopped at the analysis-synthesis difference, we didn’t have the means of finding [more]. We are in the process of discovering the extent to which this is something more than a matter of words. Kant is saying: as far as you go in your analysis, you will have an irreducible order of time and space, irreducible to the order of the concept. In other words, space and time are not concepts. There are two sorts of determinations: determinations of concepts and spatio-temporal determinations. What does Kant mean when he says that the straight line is the shortest path from one point to another, that it’s a synthetic proposition? What he means is this: [the] straight line is indeed a conceptual determination, but the shortest path from one point to another is not a conceptual determination, but a spatio-temporal determination. The two are irreducible. You will never be able to deduce one from the other. There is a synthesis between them.
And what is knowing? Knowing is creating the synthesis of conceptual determinations and spatio-temporal determinations. And so he is in the process of tearing space and time from the concept, from the logical concept. Is it by chance that he himself will name this operation Aesthetics? Even on the most vulgar level of aesthetics, the best known word – the theory of art --, won’t this liberation of space and time in relation to logical concepts be the basis of any discipline called aesthetics?
You see now how it is that, at this second level, Kant would define synthesis. He would say that synthesis is the act through which I leave behind all concepts in order to affirm something irreducible to concepts. Knowing is creating a synthesis because it necessarily means leaving behind all concepts in order to affirm something extra-conceptual in it. The straight line, concept, I leave it behind, it’s the shortest path from one point to another, a spatio-temporal, extra-conceptual determination. What is the difference between this second Kantian proposition and the first? Just admire the progress Kant made. Kant’s first definition – when he was saying that knowing means operating through synthesis – this is issuing synthetic propositions, Kant’s first proposition amounted to this: knowing means leaving behind a concept in order to affirm about it something that was not contained in it. But at this level, I could not know if he was right. Leibniz arrived and said that, in the name of an infinite analysis, what I affirm about a concept will always be contained in the concept. A second, deeper level: Kant no longer tells us that knowing means leaving a concept behind in order to affirm something that would be like another concept. Rather [he says that] knowing means leaving one concept in order to leave behind all concepts, and to affirm something about it that is irreducible to the order of the concept in general. This is a much more interesting proposition.

Yet again, they react [on rebondit]. Is this decidable? One of them tells us that every difference is conceptual in the last instance, and therefore you can affirm nothing about a concept that might go outside the order of the concept in general; the other one tells us that there are two kinds of differences, conceptual differences and spatio-temporal differences such that knowing necessarily means leaving behind the concept in order to affirm something about it that is irreducible to all concepts in general, specifically something that concerns space and time.
At this point, we realize that we haven’t left all that behind because we realize that Kant, quietly – and he wasn’t obligated to say it, even since he could say it a hundred pages later – Kant can only maintain the proposition he just suggested about the irreducibility of spatio-temporal determinations in relation to conceptual determination, he can only affirm this irreducibility because he dealt a master stroke [coup de force]. For his proposition to make sense, he had to change radically the traditional definition of space and time. I hope that you are becoming more sensitive [to this]. He gives a completely innovative determination of space and time. What does that mean?
We arrive at a third level of the Kant-Leibniz opposition. This opposition is stripped of any interest if we do not see that the Leibnizian propositions and the Kantian propositions are distributed in two completely different space-times. In other words, it’s not even the same space-time about which Leibniz said: all of these determinations of space and time are reducible to conceptual determinations; and this other one about which Kant told us that the determinations [of space-time] are absolutely irreducible to the order of the concept. This is what we have to show in a simple way; take note that this is a moment in which thought reels. For a very, very long time, space was defined as, to some extent, the order of coexistences, or the order of simultaneities. And time was defined as the order of successions. So, is it by chance that Leibniz is the one who pushes this very ancient conception to its limit, all the way to a kind of absolute formulation? Leibniz adds and states it formally: space is the order of possible coexistences and time is the order of possible successions. By adding “possible,” why does he push this to the absolute? Because it refers to his theory of compossibility and of the world. Thus, he captures in this way the old conception of space and time, and he uses it for his own system. At first glance, that seems rather good. In fact, it’s always delicate when someone tells me: define space, define time; if I don’t say by reflex that space is the order of successions and space is the order of coexistences, at least that’s something [c’est quand même un petit quelque chose]. What bothers Kant can be found in his most beautiful pages. He says: but not at all. Kant says that this just won’t do, he says that, on the one hand, I cannot define space as the order of coexistences, on the other hand, I cannot define time as the order of successions. Why? Because “coexistences,” after all, belong to time. Coexistence means, literally, at the same time. In other words, it’s a modality of time. Time is a form in which occur not only that which succeeds something, but also that which is at the same time. In other words, coexistence or simultaneity is a modality of time. At some far distant date when there will be a famous theory called the theory of simultaneity, of which one of the fundamental aspects will be to think simultaneity in terms of time, I don’t at all say that Kant invented relativity, but that such a formula, particularly what we already found comprehensible in it, would not have had this comprehensible element if Kant hadn’t been there centuries before. Kant is the first one to tell us that simultaneity does not belong to space, but belongs to time.

This is already a revolution in the order of concepts. In other words, Kant will say that time has three modalities: what lasts through it is called permanence; what follows after something else within it is called succession; and what coexists within it is called simultaneity of coexistence. I cannot define time through the order of successions since succession is only a modality of time, and I have no reason to privilege this modality over the others. And another conclusion at the same time: I cannot define space through the order of coexistences since coexistence does not belong to space. If Kant had maintained the classical definition of time and space, order of coexistences and of successions, he couldn’t have, or at least there wouldn’t have been any interest in doing so, he couldn’t have criticized Leibniz since if I define space through the order of coexistences and time through the order of successions, it goes without saying, whereas space and time refer in the last instance to that which follows something else and to that which coexists, that is, to something that one can enunciate within the order of the concept. There is no longer any difference between spatio-temporal differences and conceptual differences. In fact, the order of successions receives its raison d’être from that which follows, the order of coexistences receives it raison d’être from that which coexists. At that point, it’s conceptual difference that is the last word, on all differences. Kant couldn’t break with classical concepts, pushed to the absolute by Leibniz, if he didn’t propose to us another conception of space and time. This conception is the most unusual and the most familiar. What is space? Space is a form. That means that it’s not a substance and that it does not refer to substances. When I say that space is the order of possible coexistences, the order of possible coexistences is clarified in the last instance by things that coexist. In other words, the spatial order must find its reason in the order of things that fill space. When Kant says that space is a form, that is, is not a substance, that means that it does not refer to things that fill it. It’s a form, and how must we define it? He tells us that it’s the form of exteriority. It’s the form through which everything that is exterior to us reaches us, OK, but that’s not all it is; it’s also the form through which everything that is exterior to itself occurs. In this, he can again jump back into tradition. Tradition had always defined space as partes extra partes, one part of space is exterior to another part. But here we find that Kant takes what was only a characteristic of space in order to make it the essence of space. Space is the form of exteriority, that is the form through which what is exterior to us reaches us, and through which what remains exterior to itself occurs. If there were no space, there would be no exteriority.

Let’s jump to time. Kant is going to provide the symmetrical definition, he hits us with time as form of interiority. What does that mean? First, that time is the form of that which happens to us as interior, interior to ourselves. But it does not mean only that. Things are in time, which implies that they have an interiority. Time is the way in which the thing is interior to itself.
If we jump and if we make some connections [rapprochements], much later there will be philosophies of time, and much later time will become the principal problem of philosophy. For a long time, things were not like that. If you take classical philosophy, certainly there are philosophies greatly interested in the problem of time, and they appeared unusual. Why are the so-called “unforgettable” pages on time by Saint Augustine always shown to us? The principal problem of classical philosophy is the problem of extension [étendue], and notably what the relation is between thought and extension, once it is said that thought is not part of extension.
And it is well known that so-called classical philosophy attaches a great importance to the corresponding problem, the union of thought and extension, in the particular relation of the union of soul and body. It is therefore the relation of thought to that which appears most opaque to thought, specifically extension [l’étendue]. In some ways, some people find the source of modern philosophy in a kind of change of problematic, in which thought commences to confront time and no longer extension. The problem of the relationship between thought and time has never ceased to cause difficulties for philosophy, as if the real thing that philosophy confronted was the form of time and not the form of space. Kant created this kind of revolution: he ripped space and time from the order of the concept because he gave two absolutely new determinations of space and time: the form of exteriority and the form of interiority. Leibniz is the end of the seventeenth century, start of the eighteenth, while Kant is the eighteenth century. There is not much time between them. So what happened? We must see how everything intervenes: scientific mutations, so-called Newtonian science, political events. We cannot accept that when there was such a change in the order of concepts that nothing happened in the social order. Among other things, the French revolution occurred. Whether it implied another space-time, we don’t know. Mutations occurred in daily life. Let us say that the order of philosophical concepts expressed it [the revolution] in its own way, even if [this order] comes beforehand.

Yet again, we have started from an initial Leibniz-Kant opposition, and we have said that it is undecidable. I cannot decide between the proposition “every proposition is analytical,” and the other proposition in which knowledge proceeds by synthetic propositions. We had to step back. First step back, I have again two antithetical propositions: every determination is conceptual in the last instance, and the Kantian proposition: there are spatio-temporal determinations that are irreducible to the order of the concept. We had to step back again in order to discover a kind of presupposition, notably [that] the Leibniz-Kant opposition is valid only to the extent that we consider that space and time are not at all defined in the same way. It’s odd, this idea that space is that which opens us to an outside; never would someone from the Classical period have said that. It is already an existential relationship with space. Space is the form of what comes to us from outside.

If, for example, I look for the relationship between poetry and philosophy, what does that imply? It implies an open space. If you define space as a milieu of exteriority, it is an open space, not an enclosed space [espace bouclé]. Leibnizian space is an enclosed space, the order of coexistences. Kant’s form is a form that open us up, opens us to an x, it is the form of eruptions. It is already a Romantic space. It is an aesthetic space since it is emancipated from the logical order of the concept. It is a Romantic space because it is the space of overflows. It is the space of the open [l’ouvert]. And when you see in works of certain philosophers who came much later, like Heidegger, a kind of grand song on the theme of the open, you will see that Heidegger calls on Rilke who himself owes this notion of the Open to German Romanticism. You will better understand why Heidegger feels the need to write a book about Kant. He will deeply valorize the theme of the Open. At the same time, poets are inventing it as a rhythmic value or aesthetic value. At the same time, researchers are inventing it as a scientific species.

At this point of my thinking about it, it is very difficult to say who is right and who is wrong. One might like to say that Kant corresponds better to us, goes better with our way of being in space, space as my form of opening. Can we say that Leibniz has been left behind? It is not that simple.
A fourth point. It is perhaps at the farthest extreme of what is new that, in philosophy, occurs what we call the return to [le retour à]. After all, it is never up to an author to push himself as far as he can. It is not Kant who is going as far as is possible for Kant; there will always be post-Kantians who are the great philosophers of German Romanticism. They are the ones who, having pushed Kant as far as possible, experience this strange thing: making a return to Leibniz. [end of the tape] ... I am looking for the deep changes that Kantian philosophy was to bring about both in relation to so-called Classical philosophy and in relation to the philosophy of Leibniz. We have seen a first change concerning space-time. There is a second change, this time concerning the concept of the phenomenon. You are going to see why one results from the other. For quite a long time, the phenomenon was opposed to what, and what did it mean? Very often phenomenon is translated as appearance, appearances. And appearances, let’s say that it is the sensible [le sensible]. The sensible appearance. And appearance is distinguished from what? It forms a doublet, a couple with the correlative of essence. Appearance is opposed to essence. And Platonism will develop a duality of appearance and essence, sensible appearances and intelligible essences. A famous conception results from this: the conception of two worlds. Are there two worlds, the sensible world and the intelligible world? Are we prisoners, through our senses and through our bodies, of a world of appearances? Kant uses the word “phenomenon,” and the reader gets the impression that when he [the reader] tries to situate the old notion of appearances under the Kantian word, it doesn’t work. Isn’t there going to be as important a revolution as for time and space, on the level of the phenomenon? When Kant uses the word “phenomenon,” he loads it with a much more violent meaning: it is not appearance that separates us from essence, it is apparition, that which appears insofar as it appears. The phenomenon in Kant’s work is not appearance, but apparition. Apparition is the manifestation of that which appears insofar as it appears. Why is it immediately linked to the preceding revolution? Because when I say that what appears insofar as it appears, what does the “insofar” [en tant que] mean? It means that that which appears does so necessarily in space and time. This is immediately united to the preceding theses. Phenomenon means: that which appears in space and in time. It no longer means sensible appearance, it means spatio-temporal apparition. What reveals the extent to which this is not the same thing? If I look for the doublet with which apparition is in relation. We have seen that appearance is related to essence, to the point that there are perhaps two worlds, the world of appearances and the world of essences. But apparition is related to what? Apparition is in relation to condition. Something that appears, appears under conditions that are the conditions of its apparition. Conditions are the making-appear of apparition. These are the conditions according to which what appears, appears. Apparition refers to the conditions of the apparition, just as appearance refers to essence. Others will say that apparition refers to meaning [sens]. The doublet is: apparition and meaning of the apparition. Phenomenon is no longer thought as an appearance in relation to its essence, but as an apparition in relation to its condition or its meaning. Yet another thunderclap: there is no longer only one world constituted by that which appears and the meaning of that which appears. What appears no longer refers to essences that would be behind the appearance; that which appears refers to conditions that condition the apparition of what appears. Essence yields to meaning. The concept is no longer the essence of the thing, it is the meaning of the apparition. Understand that this is an entirely new concept in philosophy from which will unfold philosophy’s determination under the name of a new discipline, that of phenomenology. Phenomenology will be the discipline that considers phenomena as apparitions, referring to conditions or to a meaning, instead of considering them as appearances referring to essences. Phenomenology will take as much meaning as you want, but it will at least have this unity, specifically its first great moment will be with Kant who pretends to undertake a phenomenology, precisely because he changes the concept of the phenomenon, making it the object of a phenomenology instead of the object of a discipline of appearances. The first great moment in which phenomenology will be developed as an autonomous discipline will be in Hegel’s famous text, Phenomenology of Spirit. And the word is very peculiar. The Phenomenology of Spirit being precisely the great book that announces the disappearance of the two worlds, there is no more than a single world. Hegel’s formula is: behind the curtain, there is nothing to see. Philosophically that means that the phenomenon is not a mere appearance behind which an essence is located; the phenomenon is an apparition that refers to the conditions of its apparition. There is but one single world. That is the moment when philosophy breaks its final links to theology. Phenomenology’s second moment will be the one in which Husserl renews phenomenology through a theory of apparition and meaning. He will invent a form of logic proper to phenomenology. Things are obviously more complex than that.

I will offer you an extremely simple schema. Kant is the one who broke with the simple opposition between appearance and essence in order to establish a correlation [between] the apparition and conditions of apparition, or apparition-meaning [apparition-sens]. But separating oneself from something is very difficult. Kant preserves something from the former opposition. In Kant, there is a strange thing, the distinction between the phenomenon and the thing in itself. Phenomenon-thing in itself, for Kant, preserves something from the former apparition. But the really innovative aspect of Kant is the conversion of another set of notions, apparition-conditions of the apparition. And the thing in itself is not at all a condition of apparition, but something completely different. And a second correction is this: from Plato to Leibniz, we were not simply told that there are appearances and essences. Moreover, already with Plato there appears a very curious notion that he calls well-founded appearance, that is, essence is hidden from us, but in some ways, appearance expresses it as well. The relation between appearance and essence is a very complex one that Leibniz will try to push in a very strange direction, specifically: he will create from it a theory of symbolization. The Leibnizian theory of symbolization quite singularly prepares the Kantian revolution. The phenomenon symbolizes with essence. This relation of symbolization is no longer that of appearance with essence.

I am trying to continue: there occurs a new disturbance at the level of the conception of the phenomenon. You will see just how it immediately links up with the disturbance of space-time. Finally there is a fundamental disturbance at the level of subjectivity.
There again it’s a strange story. When does this notion of subjectivity take off? Leibniz pushes the presuppositions of classical philosophy as far as he can, down the paths of genius and delirium. From a perspective like that of Leibniz, one really has very little choice. These are philosophies of creation. What does a philosophy of creation mean? These are philosophies that have maintained a certain alliance with theology, to the point that even atheists, if indeed they are that, will pass by way of God. Obviously, that does not take place on the level of the word. As a result of this alliance that they have with theology, they pass by way of God to some extent. That is, their point of view is fundamentally creationist. And even philosophers who do something other than creationism, that is, who are not interested or who replace the concept of creation with something else, they fight against creation according to the concept of creation. In all cases, the point that they start from is infinity. Philosophers have an innocent way of thinking starting from infinity, and they give themselves to infinity. There was infinity everywhere, in God and in the world. That let them undertake things like infinitesimal analysis. An innocent way of thinking starting from infinity means a world of creation. They could go quite far, but not all the way. Subjectivity. To move in this direction, a completely different aggregate was necessary. Why couldn’t they go all the way toward a discovery of subjectivity? Still they went very far.

Descartes invents his own concept, the famous “I think, therefore I am,” notably the discovery of subjectivity or the thinking subject. The discovery that thought refers to a subject. A Greek would not even have understood what was being said with the idea of a thinking subject. Leibniz will not forget it, for there is a Leibnizian subjectivity. And generally we define modern philosophy with the discovery of subjectivity. They could not go all the way through this discovery of subjectivity for a very simple reason: however far they might go in their explorations, this subjectivity can only be posited as created, precisely because they have an innocent way of thinking starting from infinity.
The thinking subject, insofar as the finite subject can be thought of as created, created by God. Thought referring to the subject can only be thought as created: what does that mean? It means that the thinking subject is substance, is a thing. Res. It is not an extended thing, as Descartes says it’s a thinking thing. It is an unextended thing, but it is a thing, a substance, and it has the status of created things, it is a created thing, a created substance. Does that block them? You will tell me that it’s not difficult, they had only to put the thinking subject in the place of God, no interest in exchanging places. In that event, one has to speak of an infinite thinking subject in relation to which finite thinking subjects would themselves be created substances. Nothing would be gained. Thus, their strength, specifically this innocent way of thinking according to infinity, leads them to the threshold of subjectivity and prevents them from crossing through.

What does Kant’s rupture with Descartes consist in? What is the difference between the Kantian cogito and the Cartesian cogito? For Kant, the thinking subject is not a substance, not determined as a thinking thing. It is going to be pure form, form of the apparition of everything that appears. In other words, it is the condition of apparition of all that appears in space and in time. Yet another thunderclap. Kant undertakes to find a new relation of thought with space and time.
Pure form, empty form, there Kant becomes splendid. He goes so far as to say of the “I think” that it is the poorest thought. Only, it is the condition of any thought about any one thing. “I think” is the condition of all thought about any one thing that appears in space and in time, but itself is an empty form that conditions every apparition. That becomes a severe world, a desert world. The desert grows. What has disappeared is the world inhabited by the divine, the infinite, and it became the world of men. What disappeared is the problem of creation, replaced by an completely different problem that will be the problem of Romanticism, specifically the problem of founding [fondement]. The problem of founding or of foundation [fondation]. Now there arises a clever thought, puritanical, desert-like, that wonders, once it’s admitted that the world exists and that it appears, how to found it?
The question of creation has been rejected, but now the problem of founding arrives. If there is really a philosopher who spoke the discourse of God, it was Leibniz. Now the model philosopher has become the hero, the founding hero. He is the one who founds within an existing world, not the one who creates the world. What is foundational [fondateur] is that which conditions the condition of what appears in space and in time. Everything is linked there. A change in the notion of space-time, a change in the notion of the subject. The thinking subject as pure form will only be the act of founding the world such as it appears and knowledge of the world such as it appears. This is an entirely new undertaking.

A year ago, I tried to distinguish the Classical artist from the Romantic artist. The Classical and the Baroque are two poles of the same enterprise. I was saying that the Classical artist is one who organized milieus and who, to some extent, is in the situation of God, this is creation. The Classical artist never stops undertaking creation anew, by organizing milieus, and never ceases to pass from one milieu to another. He passes from water to earth, he separates the earth and the waters, exactly God’s task in creation. He poses a kind of challenge to God: they are going to do just as much, and that is what the Classical artist is. The Romantic at first glance would be less crazy; his problem is that of founding. It is no longer the problem of the world, but one of the earth. It is no longer the problem of milieu, but one of territory. To leave one’s territory in order to find the center of the earth, that’s what founding is. The Romantic artist renounced creating because there is a much more heroic task, and this heroic task is foundation. It is no longer creation and milieu; it’s: I am leaving my territory. Empedocles. The founding is in the bottomless [Le fondement est dans le sans fond]. All post-Kantian philosophy from Schelling on will arise around this kind of abundant concept or the bottom, the fundament founding, the bottomless. That is always what the lied is, the tracing of a territory haunted by the hero, and the hero leaves, departs for the center of the earth, he deserts. The song of the earth. Mahler. The opposition maintained between the tune about the territory and the song of the earth.

The musical doublet territory-earth corresponds exactly to what in philosophy is the phenomenon apparition and the condition of apparition. Why do they abandon the point of view of creation?
Why is the hero not someone who creates, but someone who founds, and why isn’t it the final word? If there were a moment in which Western thought was a bit tired of taking itself for God and of thinking in terms of creation, the seed must be here. Does the image of heroic thought suit us still? All that is finished. Understand the enormous importance of this substitution of the form of the ego [forme du moi] by the thinking subject. The thinking substance was still the point of view of God, it’s a finite substance, but created according to the infinite, created by God.
Whereas when Kant tells us that the thinking subject is not a thing, he well understands a created thing, a form that conditions the apparition of all that appears in space and in time, that is, it is the form of founding. What is he in the process of doing? He institutes the finite ego [le moi fini] as first principle.
Doing that is frightening. Kant’s history depends greatly on the reform. The finite ego is the true founding. Thus the first principle becomes finitude. For the Classics, finitude is a consequence, the limitation of something infinite. The created world is finite, the Classics tell us, because it is limited. The finite ego founds the world and knowledge of the world because the finite ego is itself the constitutive founding of what appears. In other words, it is finitude that is the founding of the world. The relations of the infinite to the finite shift completely. The finite will no longer be a limitation of the infinite; rather, the infinite will be an overcoming [dépassement] of the finite. Moreover, it is a property of the finite to surpass and go beyond itself. The notion of self-overcoming [auto-dépassement] begins to be developed in philosophy. It will traverse all of Hegel and will reach into Nietzsche. The infinite is no longer separable from an act of overcoming finitude because only finitude can overcome itself. Everything called dialectic and the operation of the infinite to be transformed therein, the infinite becoming and become the act through which finitude overcomes itself by constituting or by founding the world. In that way, the infinite is subordinated to the act of the finite. What results from this? Fichte has an exemplary page for the Kantian polemic with Leibniz. Here is what Fichte tells us: I can say A is A, but this is only a hypothetical proposition. Why? Because it presupposes “if there is A.” If A is, A is A, but if there is nothing, A is not A. This is very interesting because he is in the act of overthrowing the principle of identity. He says that the principle of identity is a hypothetical rule. Hence he launches his great theme: to overcome hypothetical judgment to go toward what he calls “thetic” judgment (le jugement thétique]. To go beyond hypothesis toward thesis. Why is it that A is A, if A does exist because finally the proposition A is A is not at all a final principle or a first principle? It refers to something deeper, specifically that one must say that A is A because it is thought. Specifically, what founds the identity of things that are thought is the identity of the thinking subject. Moreover, the identity of the thinking subject is the identity of the finite ego. Thus the first principle is not that A is A, but that ego equals ego. German philosophy will encumber its books with the magic formula: ego equals ego. Why is this formula so bizarre? It is a synthetic identity because ego equals ego marks the identity of the ego that thinks itself as the condition of all that appears in space and in time, and [illegible] that appears in space and in time itself. In this there is a synthesis that is the synthesis of finitude, notably the thinking subject, primary ego, form of all that appears in space and time, must also appear in space and in time, that is ego equals ego. Hence the synthetic identity of the finite ego replaces the infinite analytic identity of God.

I will finish with two things: what could it mean to be Leibnizian today? It’s that Kant absolutely created a kind of radically new conceptual aggregate. These are completely new philosophical conceptual coordinates. But in the case of these new coordinates, Kant in one sense renews everything, but there are all sorts of things that are not elucidated in what he proposes. An example: what exact relation is there between the condition of the phenomenon itself insofar as it appears?
I will review: The thinking ego, the finite ego, conditions, founds the phenomenal apparition. The phenomenon appears in space and in time. How is this possible? What does this relation of conditioning mean? In other words, the “I think” is a form of knowledge that conditions the apparition of all that appears.
How is this possible, what is the relation between the conditioned and the condition? The condition is the form of “I think.” Kant is quite annoyed. He says that this is a fact of reason, he who had so demanded that the question be elevated to the state quid juris, now he invokes what he himself call a factum: the finite ego is so constituted that what appears for it, what appears to it, conforms to the conditions of the apparition such that its very own thought determines it. Kant will say that this agreement of the conditioned and the condition can only be explained by a harmony of our faculties, specifically our passive sensibility and our active thought. What Kant does is pathetic; he is in the process of sneaking God in behind our backs. What guarantees this harmony? He will say it himself: the idea of God.
What will the post-Kantians do? They are philosophers who say above all that Kant is inspired [genial], but still, we cannot remain in an exterior relation of the condition and conditioned because if we remain in this relation of fact, specifically that there is a harmony between the conditioned and the condition and that’s that, then we are obliged to resuscitate God as a guarantee of harmony.
Kant still remains in a viewpoint which is that of exterior conditioning, yet he does not reach a true viewpoint of genesis. It would require showing how conditions of apparition are at the same time genetic elements of what appears. What is necessary to show that? One has to take seriously one of the Kantian revolutions that Kant left aside, notably that the infinite is truly the act of finitude insofar as it overcomes itself. Kant had left that aside because he was content with a reduction of the infinite to the indefinite.
To return to a strong conception of the infinite, but in the manner of the Classics, one has to show that the infinite is an infinite in the strong sense, but as such, it is the act of finitude insofar as it overcomes itself, and in so doing, it constitutes the world of apparitions. This is to substitute the viewpoint of genesis for the viewpoint of the condition. Moreover, doing that means returning to Leibniz, but on bases other than Leibniz’s. All the elements to create a genesis such as the post-Kantians demand it, all the elements are virtually in Leibniz. The idea of a differential of consciousness, at that point the “I think” of consciousness must bathe in an unconscious, and there must be an unconscious of thought as such. The Classics would have said that there is only God who goes beyond thought. Kant would say that there is thought as a form of the finite ego. In this, one must almost summon an unconscious to thought that would contain the differentials of what appears in thought. In other words, which performs the genesis of the conditioned as a function of the condition. That will be Fichte’s great task, taken up again by Hegel on other bases.

You see henceforth that at the limit, they can rediscover all of Leibniz. And us? A lot has taken place. So I define philosophy as an activity that consists in creating concepts. To create concepts is as creative as art. But like all things, the creation of concepts occurs in correspondence with other modes of creation. In which sense [do] we need concepts? It’s a material existence, and concepts are spiritual animals [bêtes spirituelles). How do these kinds of appeals to concepts occur? The old concepts will serve, provided that they are taken up within new conceptual coordinates. There is a philosophical sensibility which is the art of evaluating the consistency of an aggregate of concepts. Does it work? How does it function? Philosophy does not have a history separate from the rest. Nothing, never is anyone overcome [dépassé]. We are never left behind in what we create. We are always left behind in what we do not create, by definition. What happened in our contemporary philosophy? I believe that the philosopher ceased taking himself for a founding hero, in the Romantic manner. What was fundamental in what we can call, generally, our modernity, was this kind of bankruptcy of Romanticism in our estimation. Hölderlin and Novalis no longer work for us and only work for us within the framework of new coordinates. We are finished taking ourselves for heroes. The model of the philosopher and artist is no longer God at all insofar as he [or she] proposes to create the equivalent of a world. This is no longer the hero insofar as he [or she] proposes to found a world, for it has become something else. There is a small text by Paul Klee in which he tries to say how he sees his own difference even from earlier painters. One can no longer go towards the motif. There is a kind of continuous flow, and this flow has twists and turns. Then the flow no longer passes in that direction. The coordinates of painting have changed.

Leibniz is infinite analysis, Kant is the grand synthesis of finitude. Assuming that today we are in the age of the synthesizer, that is something else entirely.