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 <trucs>. 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 <donnable> 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 <ne fait que calculer>. 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 <joueur> 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.