29/04/1980
Charles J. Stivale, C_Stivale@wayne.edu

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.

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