Logic is reductionist not accidentally but essentially and necessarily: following the route marked out by Frege and Russell, it wants to turn the concept into a function. But this means first of all not only that the function must be defined in a mathematical or scientific proposition but that it characterizes a more general order of the proposition as what is expressed by the sentences of a natural language. Thus a new, specifically logical type of function must be invented. The propositional function “x is human” clearly shows the position of an independent variable that does not belong to the function as such but without which the function is incomplete. The complete function is made up of one or more “ordered pairs.” A relation of dependence or correspondence (necessary reason) defines the function, so that “being human” is not itself the function, but the value of f(a) for a variable x. It hardly matters that most propositions have several independent variables or even that the notion of variable, insofar as it is linked to an indeterminate number, is replaced by that of argument, implying a disjunctive assumption within limits or an interval. The relation of the propositional function to the independent variable or argument defines the proposition’s reference or the function’s truth value (“true” or “false”) for the argument: John is a man, but Bill is a cat. The set of a function’s truth values that determine true affirmative propositions constitutes a concept’s extension: the concept’s objects occupy the place of variables or arguments of the propositional function for which the proposition is true, or its reference satisfied. Thus the concept itself is the function for the set of objects that constitute its extension. In this sense every complete concept is a set and has a determinate number; the concept’s objects are the elements of the set.1
It is still necessary to determine the conditions of reference that provide the limits or intervals into which a variable enters in a true proposition: x is a man, John is a man, because he did this, because he appears in this way. Such conditions of reference constitute not the concept’s comprehension but its intension. They are presentations or logical descriptions, intervals, potentials, or “possible worlds,” as the logicians say, coordinate axes, states of affairs or situations, the concept’s subsets: evening star and morning star. For example, a concept with a single element, the concept of Napoleon I, has for its intension “the victor at Jena,” “the one who was defeated at Waterloo.” There is no qualitative difference between intension and extension here since both concern reference, intension being simply the condition of reference and constituting an endoreference of the proposition, extension constituting the exoreference. Reference is not left behind by ascending to its condition; we remain within extensionality. The question is rather one of knowing how, through these intensional presentations, we arrive at a univocal determination of objects or elements of the concept, of propositional variables, and of arguments of the function from the point of view of exoreference (or of the representation). This is the problem of proper names, and the business of a logical identification or individuation that takes us from states of affairs to the thing or body (object), through operations of quantification that also make possible attribution of the thing’s essential predicates as that which finally constitutes the concept’s comprehension. Venus (the evening star and morning star) is a planet that takes less time than the earth to complete its revolution. “Victor at Jena” is a description or presentation, whereas “general” is a predicate of Bonaparte, “emperor” a predicate of Napoleon, although being named general or holy emperor may be descriptions. The “propositional concept” therefore evolves entirely within the circle of reference insofar as it carries out a logicization of functives that thus become the prospects of a proposition (passage from the scientific to the logical proposition).
Sentences have no self-reference, as the paradox “I lie” shows. Not even performatives are self-referential but rather imply an exoreference of the proposition (the action that is linked to it by convention and accomplished by stating the proposition), and an endoreference (the status or state of affairs that entitles one to formulate the statement: for example, the concept’s intension in the statement “I swear it” may be a witness in court, a child blamed for something, a lover declaring himself, etc.).2 On the other hand, if we ascribe self-consistency to the sentence, this can only reside in the formal noncontradiction of the proposition or between propositions. But this means that propositions do not materially enjoy any endoconsistency or exoconsistency. To the extent that a cardinal number belongs to the propositional concept, the logic of propositions needs a scientific demonstration of the consistency of the arithmetic of whole numbers, on the basis of axioms. Now, according to the two aspects of Gödel’s theorem, proof of the consistency of arithmetic cannot be represented within the system (there is no endoconsistency), and the system necessarily comes up against true statements that are nevertheless not demonstrable, are undecidable (there is no exoconsistency, or the consistent system cannot be complete). In short, in becoming propositional, the concept loses all the characteristics it possessed as philosophical concept: its self-reference, its endoconsistency and its exoconsistency. This is because a regime of independence (of variables, axioms, and undecidable propositions) has replaced that of inseparability. Even possible worlds as conditions of reference are cut off from the concept of the Other person that would give them consistency (so that logic finds itself oddly disarmed before solipsism). The concept in general no longer has a combination but an arithmetical number; the undecidable no longer indicates the inseparability of intensional components (zone of indiscernibility) but, on the contrary, the necessity of distinguishing them according to the requirement of reference, which renders all consistency (self-consistency) “uncertain.” Number itself indicates a general principle of separation: “the concept ‘letter of the word Zahl’ separates Z from a, a from h, etc.” Functions derive all their power from reference, whether this be reference to states of affairs, things, or other propositions: reduction of the concept to the function inevitably deprives it of all its specific characteristics that referred back to another dimension.
Acts of reference are finite movements of thought by which science constitutes or modifies states of affairs and bodies. Historical man may also be said to carry out such modifications, but under conditions of the lived, where functives are replaced by perceptions, affections, and actions. The position is no longer the same with logic: since it considers empty reference in itself as simple truth value, it can only apply it to already constituted states of affairs or bodies, in established scientific propositions or in factual propositions (Napoleon is the one who was defeated at Waterloo) or in simple opinions (“X thinks that …”). All types of propositions are prospects, with an information value. Logic has therefore a paradigm, it is even the third case of paradigm, which is no longer that of religion or science but like the recognition of truth in prospects or informative propositions. The technical expression “metamathematics” clearly shows the passage from scientific statement to logical proposition in a form of recognition. The projection of this paradigm means that logical concepts are in turn only figures and that logic is an ideography. The logic of propositions needs a method of projection, and Gödel’s theorem itself invents a projective model.3 It is like an ordered, oblique deformation of reference in relation to its scientific status. Logic seems to be forever struggling with the complex question of how it differs from psychology. However, we can definitely agree that it sets up as a model an image by right of thought that is in no way psychological (without, for all that, being normative). The question lies rather in the value of this image by right and in what it claims to teach us about the mechanisms of a pure thought.
Of all the finite movements of thought, the form of recognition is certainly the one that goes the least far and is the most impoverished and puerile. From earliest times philosophy has encountered the danger of evaluating thought by reference to such uninteresting cases as saying “hello, Theodore” when Theatetus is passing by. The classical image of thought was not safe from these endeavors that value recognition of truth. It is hard to believe that the problems of thought, in science as well as in philosophy, are troubled by such cases: as the creation of thought, a problem has nothing to do with a question, which is only a suspended proposition, the bloodless double of an affirmative proposition that is supposed to serve as its answer (“Who is the author of Waverley?” “Is Scott the author of Waverley?”). Logic is always defeated by itself, that is to say, by the insignificance of the cases on which it thrives. In its desire to supplant philosophy, logic detaches the proposition from all its psychological dimensions, but clings all the more to the set of postulates that limited and subjected thought to the constraints of a recognition of truth in the proposition.4 And when logic ventures into a calculus of problems, it does so by modeling it, isomorphically, on the calculus of propositions. It is less like a game of chess, or a language game, than a television quiz game. But problems are never propositional.
Instead of a string of linked propositions, it would be better to isolate the flow of interior monologue, or the strange forkings of the most ordinary conversation. By separating them from their psychological, as well as their sociological adhesions, we would be able to show how thought as such produces something interesting when it accedes to the infinite movement that frees it from truth as supposed paradigm and reconquers an immanent power of creation. But to do this it would be necessary to return to the interior of scientific states of affairs or bodies in the process of being constituted, in order to penetrate into consistency, that is to say, into the sphere of the virtual, a sphere that is only actualized in them. It would be necessary to go back up the path that science descends, and at the very end of which logic sets up its camp (the same goes for History, where we would have to arrive at the unhistorical vapor that goes beyond actual factors to the advantage of a creation of something new). But it is this sphere of the virtual, this Thought-nature, that logic can only show, according to a famous phrase, without ever being able to grasp it in propositions or relate it to a reference. Then logic is silent, and it is only interesting when it is silent. Paradigm for paradigm, it is then in agreement with a kind of Zen Buddhism.
By confusing concepts with functions, logic acts as though science were already dealing with concepts or forming concepts of the first zone. But it must itself double scientific with logical functions that are supposed to form a new class of purely logical, or second zone, concepts. A real hatred inspires logic’s rivalry with, or its will to supplant, philosophy. It kills the concept twice over. However, the concept is reborn because it is not a scientific function and because it is not a logical proposition: it does not belong to a discursive system and it does not have a reference. The concept shows itself and does nothing but show itself. Concepts are really monsters that are reborn from their fragments.
Logic itself sometimes allows philosophical concepts to reappear, but in what form and state? As concepts in general have found a pseudorigorous status in scientific and logical functions, philosophy inherits concepts of the third zone that are outside number and no longer constitute clearly demarcated and well-defined sets that can be related to mixtures ascribable as physico-mathematical states of affairs. They are, instead, vague or fuzzy sets, simple aggregates of perceptions and affections, which form within the lived as immanent to a subject, to a consciousness. They are qualitative or intensive multiplicities, like “redness” or “baldness,” where we cannot decide whether certain elements do or do not belong to the set. These lived sets are expressed in a third kind of prospects, which are no longer those of scientific statements or logical propositions but of the subject’s pure and simple opinions, of subjective evaluations or judgments of taste: this is already red, he is nearly bald. However, even for an enemy of philosophy, the refuge of philosophical concepts cannot immediately be found in such empirical judgments. We must isolate the functions, of which these fuzzy sets, these lived contents, are only variables. And at this point we face an alternative: either we will end up reconstituting scientific or logical functions for these variables, which would make the appeal to philosophical concepts definitively useless,5 or we will have to invent a new, specifically philosophical type of function, a third zone in which everything seems to be strangely reversed, since it will be given the task of supporting the other two.
If the world of the lived is like the earth, which must found and support the science and logic of states of affairs, it is clear that apparently philosophical concepts are required to carry out this first foundation. The philosophical concept thus requires a “belonging” to a subject and no longer to a set. Not that the philosophical concept is to be confused with the merely lived, even if it is defined as a multiplicity of fusion or as immanence of a flow to the subject—the lived only furnishes variables, whereas concepts must still define true functions. These functions will have reference only to the lived, as scientific functions have reference to states of affairs. Philosophical concepts will be functions of the lived, as scientific concepts are functions of states of affairs; but the order or the derivation now changes direction since these functions of the lived become primary. A transcendental logic (it can also be called dialectical) embraces the earth and all that it bears, and this serves as the primordial ground for formal logic and the derivative regional sciences. It is necessary therefore to discover at the very heart of the immanence of the lived to a subject, that subject’s acts of transcendence capable of constituting new functions of variables or conceptual references: in this sense the subject is no longer solipsist and empirical but transcendental. We have seen that Kant began to accomplish this task by showing how philosophical concepts are necessarily related to lived experience through a priori propositions or judgment as functions of a whole of possible experience. But it is Husserl who sees it through to the end by discovering, in non-numerical multiplicities or immanent perceptivo-affective fusional sets, the triple root of acts of transcendence (thought) through which the subject constitutes first of all a sensory world filled with objects, then an intersubjective world occupied by the other, and finally a common ideal world that will be occupied by scientific, mathematical, and logical formations. Numerous phenomenological or philosophical concepts (such as “being in the world,” “flesh,” “ideality,” etc.) are the expression of these acts. They are not only liveds that are immanent to the solipsist subject but references of the transcendental subject to the lived; they are not perceptivoaffective variables, but major functions which find in these variables their respective trajectories of truth. They are not vague or fuzzy sets, subsets, but totalizations that exceed all power of sets. They are not merely empirical judgments or opinions but proto-beliefs, Urdoxa, original opinions as propositions.6 They are not successive contents of the flow of immanence but acts of transcendence that traverse it and carry it away by determining the “significations” of the potential totality of the lived. The concept as signification is all of this at once: immanence of the lived to the subject, act of transcendence of the subject in relation to variations of the lived, totalization of the lived or function of these acts. It is as if philosophical concepts get going only by accepting to become special functions and by denaturing the immanence that they still need: as immanence is now only that of the lived it is inevitably immanence to a subject, whose acts (functions) will be concepts relative to this lived—following, as we have seen, the long denaturation of the plane of immanence.
Although it may be dangerous for philosophy to depend on the generosity of logicians, or on their regrets, we might wonder whether a precarious balance cannot be found between scientificological concepts and phenomenological-philosophical concepts. Gilles-Gaston Granger has suggested a distribution in which the concept, being determined first of all as a scientific or logical concept, nonetheless allows a place for philosophical functions in a third but autonomous zone, for functions or significations of the lived as virtual totality (fuzzy sets seem to play the role of a hinge between the two forms of concepts).7 Science has therefore arrogated the concept to itself, but there are nevertheless nonscientific, that is to say, phenomenological concepts that are tolerated in homeopathic doses—hence the strangest hybrids of Frego-Husserlianism, or even Wittgensteino-Heideggerianism, that we see springing up today. This has long been the situation of philosophy in America, with a large department of logic and a very small one of phenomenology, even though the two parts were usually at war. It is like the proverbial lark pie containing one lark and one horse. But the phenomenological lark is not even the most exquisite portion; it is only what the logical horse sometimes leaves for philosophy. The situation is more like the rhinoceros and the bird that lives on its parasites.
There is a long series of misunderstandings about the concept. It is true that the concept is fuzzy or vague not because it lacks an outline but because it is vagabond, nondiscursive, moving about on a plane of immanence. It is intensional or modular not because it has conditions of reference but because it is made up of inseparable variations that pass through zones of indiscernibility and change its outline. It has no reference at all, either to the lived or to states of affairs, but a consistency defined by its internal components. The concept is neither denotation of states of affairs nor signification of the lived; it is the event as pure sense that immediately runs through the components. It has no number, either whole or fractional, for counting things that display its properties, but a combination that condenses and accumulates the components it traverses and surveys. The concept is a form or a force; in no possible sense is it ever a function. In short, there are only philosophical concepts on the plane of immanence, and scientific functions or logical propositions are not concepts.
Prospects designate first of all the elements of the proposition (propositional function, variables, truth value), but also the various types of propositions or modalities of judgment. If the philosophical concept is confused with a function or a proposition, it is not as a scientific or even logical kind but, by analogy, as a function of the lived or a proposition of opinion (third type). Hence a concept must be produced that takes account of this situation: what opinion proposes is a particular relationship between an external perception as state of a subject and an internal affection as passage from one state to another (exo- and endoreference). We pick out a quality supposedly common to several objects that we perceive, and an affection supposedly common to several subjects who experience it and who, along with us, grasp that quality. Opinion is the rule of the correspondence of one to the other; it is a function or a proposition whose arguments are perceptions and affections, and in this sense it is a function of the lived. For example, we grasp a perceptual quality common to cats or dogs and a certain feeling that makes us like or hate one or the other: for a group of objects we can extract many diverse qualities and form many groups of quite different, attractive or repulsive, subjects (the “society” of those who like cats or detest them), so that opinions are essentially the object of a struggle or an exchange. This is the Western democratic, popular conception of philosophy as providing pleasant or aggressive dinner conversations at Mr. Rorty’s. Rival opinions at the dinner table—is this not the eternal Athens, our way of being Greek again? The three characteristics by which philosophy was related to the Greek city were, precisely, the society of friends, the table of immanence, and the confrontation of opinions. One might object that Greek philosophers were always attacking doxa and contrasting it with an episteme as the only knowledge adequate to philosophy. But this is a mixed-up business, and philosophers, being only friends and not wise men, find it very difficult to give up doxa.
Doxa is a type of proposition that arises in the following way: in a given perceptive-affective lived situation (for example, some cheese is brought to the dinner table), someone extracts a pure quality from it (for example, a foul smell); but, at the same time as he abstracts the quality, he identifies himself with a generic subject experiencing a common affection (the society of those who detest cheese—competing as such with those who love it, usually on the basis of another quality). “Discussion,” therefore, bears on the choice of the abstract perceptual quality and on the power of the generic subject affected. For example, is to detest cheese to manage without being a bon vivant? But is being a bon vivant a generically enviable affection? Ought we not say that it is those who love cheese, and all bons vivants, who stink? Unless it is the enemies of cheese who stink. This is like the story, told by Hegel, of the shopkeeper to whom it was said, “Your eggs are rotten old woman,” and who replied, “Rot yourself, and your mother, and your grandmother”: opinion is an abstract thought, and insult plays an effective role in this abstraction because opinion expresses the general functions of particular states.8 It extracts an abstract quality from perception and a general power from affection: in this sense all opinion is already political. That is why so many discussions can be expressed in this way: “as a man, I consider all women to be unfaithful”; “as a woman, I think men are liars.”
Opinion is a thought that is closely molded on the form of recognition—recognition of a quality in perception (contemplation), recognition of a group in affection (reflection), and recognition of a rival in the possibility of other groups and other qualities (communication). It gives to the recognition of truth an extension and criteria that are naturally those of an “orthodoxy”: a true opinion will be the one that coincides with that of the group to which one belongs by expressing it. This is clear to see in certain competitions: you must express your opinion, but you “win” (you have spoken the truth) if you say the same as the majority of those participating in the competition. The essence of opinion is will to majority and already speaks in the name of a majority. Even the man of “paradoxes” only expresses himself with so many winks and such stupid self-assurance because he claims to express everyone’s secret opinion and to be the spokesman of that which others dare not say. This is still only the first step of opinion’s reign: opinion triumphs when the quality chosen ceases to be the condition of a group’s constitution but is now only the image or “badge” of the constituted group that itself determines the perceptive and affective model, the quality and affection, that each must acquire. Then marketing appears as the concept itself: “We, the conceivers. …” Ours is the age of communication, but every noble soul flees and crawls far away whenever a little discussion, a colloquium, or a simple conversation is suggested. In every conversation the fate of philosophy is always at stake, and many philosophical discussions do not as such go beyond discussions of cheese, including the insults and the confrontation of worldviews. The philosophy of communication is exhausted in the search for a universal liberal opinion as consensus, in which we find again the cynical perceptions and affections of the capitalist himself.
EXAMPLE 11
How does this situation concern the Greeks? It is often said that since Plato, the Greeks contrasted philosophy, as a knowledge that also includes the sciences, with opinion-doxa, which they relegate to the sophists and rhetors. But we have learned that the opposition was not so clear-cut. How could philosophers possess knowledge, philosophers who cannot and do not want to restore the knowledge of the sages and who are only friends? And how, since it takes a truth value, could opinion be something entirely for the sophists?9
Furthermore, it seems that the Greeks had a clear-enough idea of science, which was not confused with philosophy: it was a knowledge of the cause, of the definition, a sort of function already. So, the whole problem was, How can one arrive at definitions, at these premises of the scientific or logical syllogism? It was by means of the dialectic: an investigation that aimed, on a given theme, to determine which opinions were the most plausible by reference to the quality they extracted and which opinions were the wisest by reference to the subject who advanced them. Even in Aristotle, the dialectic of opinions was necessary for determining possible scientific propositions, and in Plato “true opinion” was a prerequisite for knowledge and the sciences. Even Parmenides did not pose knowledge and opinion as being two disjunctive pathways.10 Whether or not they were democrats, the Greeks did not so much oppose knowledge and opinion as fight over opinions, as confront and compete against each other in the element of pure opinion. Philosophers blamed the sophists not for confining themselves to doxa but for making a bad choice of the quality to be extracted from perceptions, and of the generic subject to be isolated from affections, so that the sophists could not reach what was “true” in an opinion: they remained prisoners of variations of the lived. Philosophers blamed the sophists for being content with any kind of sensory quality in relation to an individual man, or to mankind, or to the nomos of the city (three interpretations of Man as power or the “measure of everything”). But Platonist philosophers themselves had an extraordinary answer that, they thought, allowed them to select opinions. It was necessary to choose the quality that was like the unfolding of the Beautiful in any lived situation, and to take as generic subject Man inspired by the Good. Things had to unfold in the beautiful, and their users be inspired by the good, for opinion to achieve the Truth. This was not always easy. The beautiful in Nature and the good in minds define philosophy as a function of the variable life. Thus Greek philosophy is the moment of the beautiful; the beautiful and the good are functions whose truth value is opinion. To reach true opinion, perception had to be taken as far as the beauty of the perceived (dokounta) and affection as far as the test of the good (dokimôs): this will no longer be changing and arbitrary opinion but an original opinion, a proto-opinion that restores to us the forgotten homeland of the concept as, in the great Platonic trilogy, the love of the Symposium, the delirium of the Phaedrus, and the death of the Phaedo. Where, on the contrary, the sensory appears without beauty, reduced to illusion, and the mind appears without good, given over to simple pleasure, opinion will remain sophistical itself and false—cheese, perhaps, or mud or hair. However, does not this passionate search for true opinion lead the Platonists to an aporia, the very one expressed in the most astonishing dialogue, the Theatetus? Knowledge must be transcendent; it must be added to and distinguished from opinion in order to make opinion true. But knowledge must be immanent for opinion to be true as opinion. Greek philosophy still remains attached to that old Wisdom ready to unfold its transcendence again, although it now possesses only its friendship, its affection. Immanence is necessary, but it must be immanent to something transcendent, to ideality. The beautiful and the good continue to lead us back to transcendence. It is as if true opinion still demanded a knowledge that it had nevertheless deposed.
Phenomenology can be seen as taking up a similar task. It, too, goes in search of original opinions which bind us to the world as to our homeland (earth). It also needs the beautiful and the good so that the latter are not confused with variable empirical opinion and so that perception and affection attain their truth value. This time it is a question of the beautiful in art and the constitution of humanity in history. Phenomenology needs art as logic needs science; Erwin Straus, Merleau-Ponty, or Maldiney need Cézanne or Chinese painting. The lived turns the concept into nothing more than an empirical opinion as psychosociological type. The immanence of the lived to a transcendental subject, therefore, must turn opinion into a proto-opinion in whose constitution art and culture are involved and that is expressed as an act of transcendence of this subject within the lived (communication), so as to form a community of friends. But does not the Husserlian transcendental subject hide European man whose privilege it is constantly to “Europeanize,” as the Greeks “Greekized,” that is to say, to go beyond the limits of other cultures that are preserved as psychosocial types? Are we not led back in this way to the simple opinion of the average Capitalist, the great Major, the modern Ulysses whose perceptions are clichés and whose affections are labels, in a world of communication that has become marketing and from which not even Cézanne or Van Gogh can escape? The distinction between original and derivative is not by itself enough to get us out of the simple domain of opinion, and the Urdoxa does not raise us to the level of the concept. As in the Platonic aporia, phenomenology is never more in need of a higher wisdom, of a “rigorous science,” than when it invites us to renounce it. Phenomenology wanted to renew our concepts by giving us perceptions and affections that would awaken us to the world, not as babies or hominids but as, by right, beings whose proto-opinions would be the foundations of this world. But we do not fight against perceptual and affective clichés if we do not also fight against the machine that produces them. By invoking the primordial lived, by making immanence an immanence to a subject, phenomenology could not prevent the subject from forming no more than opinions that already extracted clichés from new perceptions and promised affections. We will continue to evolve in the form of recognition; we will invoke art, but without reaching the concepts capable of confronting the artistic affect and percept. The Greeks with their cities, and phenomenology with our Western societies, are certainly right to consider opinion as one of the conditions of philosophy. But, by invoking art as the means of deepening opinion and of discovering original opinions, will philosophy find the path that leads to the concept? Or should we, along with art, overturn opinion, raising it to the infinite movement that replaces it with, precisely, the concept?
Confusing the concept with the function is ruinous for the philosophical concept in several respects. It makes science the concept par excellence, which is expressed in the scientific proposition (first prospect). It replaces the philosophical concept with a logical concept, which is expressed in factual propositions (second prospect). It leaves the philosophical concept with a reduced or defective share that it carves out in the domain of opinion (third prospect) by exploiting its friendship with a higher wisdom or with a rigorous science. But the concept’s place is not in any of these three discursive systems. The concept is no more a function of the lived than it is a scientific or logical function. We discover the irreducibility of concepts to functions only if, instead of setting them against one another in an indeterminate way, we compare what constitutes the reference of one with what produces the other’s consistency. States of affairs, objects or bodies, and lived states form the function’s references, whereas events are the concept’s consistency. These are the terms that have to be considered from the point of view of a possible reduction.
EXAMPLE 12
In contemporary thought such a comparison seems to correspond to Badiou’s particularly interesting undertaking. He proposes to distribute at intervals on an ascending line a series of factors passing from functions to concepts. He takes a base neutralized in relation to both concepts and functions—any multiplicity whatever that is presented as a Set that can be raised to infinity. The first instance is the situation, when the set is related to elements that are doubtless multiplicities but that are subject to a regime of the “counting as one” (bodies or objects, units of the situation). In the second place, situation-states are subsets, always exceeding elements of the set or objects of the situation; but this excess of the state no longer lets itself be hierarchized as in Cantor—it is “inassignable,” following an “errant line,” in conformity with the development of set theory. It must still be re-presented in the situation, this time as “indiscernible” at the same time that the situation becomes almost complete: the errant line here forms four figures, four loops as generic functions—scientific, artistic, political or doxic, and amorous or lived—to which productions of “truths” correspond. But perhaps we then arrive at a conversion of immanence of the situation, a conversion of the excess to the void, which will reintroduce the transcendent: this is the event site that sticks to the edge of the void in the situation and now includes not units but singularities as elements dependent on the preceding functions. Finally, the event itself appears (or disappears), less as a singularity than as a separated aleatory point that is added to or subtracted from the site, within the transcendence of the void or the truth as void, without it being possible to decide on the adherence of the event to the situation in which it finds its site (the undecidable). On the other hand, perhaps there is an operation like a dice throw on the site that qualifies the event and makes it enter into the situation, a power of “making” the event. The fact that the event is the concept, or philosophy as concept, distinguishes it from the four preceding functions, although it takes conditions from them and imposes conditions on them in turn—that art is fundamentally “poem,” that science is set-theoretical [ensembliste], that love is the unconscious of Lacan, and that politics escapes from opinion-doxa.11
By starting from a neutralized base, the set, which indicates any multiplicity whatever, Badiou draws up a line that is single, although it may be very complex, on which functions and concepts will be spaced out, the latter above the former: philosophy thus seems to float in an empty transcendence, as the unconditioned concept that finds the totality of its generic conditions in the functions (science, poetry, politics, and love). Is this not the return, in the guise of the multiple, to an old conception of the higher philosophy? It seems to us that the theory of multiplicities does not support the hypothesis of any multiplicity whatever (even mathematics has had enough of set-theoreticism [ensemblisme]). There must be at least two multiplicities, two types, from the outset. This is not because dualism is better than unity but because the multiplicity is precisely what happens between the two. Hence, the two types will certainly not be one above the other but rather one beside the other, against the other, face to face, or back to back. Functions and concepts, actual states of affairs and virtual events, are two types of multiplicities that are not distributed on an errant line but related to two vectors that intersect, one according to which states of affairs actualize events and the other according to which events absorb (or rather, adsorb) states of affairs.
States of affairs leave the virtual chaos on conditions constituted by the limit (reference): they are actualities, even though they may not yet be bodies or even things, units, or sets. They are masses of independent variables, particles-trajectories or signs-speeds. They are mixtures. These variables determine singularities, insofar as they enter into coordinates, and are held within relations according to which one of them depends upon a large number of others or, conversely, many of them depend upon one. A potential or power is found to be associated with such a state of affairs (the importance of the Leibnizian formula mv2 is due to its introducing a potential into the state of affairs). This is because the state of affairs actualizes a chaotic virtuality by carrying along with it a space that has ceased, no doubt, to be virtual but that still shows its origin and serves as absolutely indispensable correlate to the state of affairs. For example, in the actuality of the atomic nucleus, the nucleon is still close to chaos and finds itself surrounded by a cloud of constantly emitted and reabsorbed particles; but at a further level of actualization, the electron is in relation with a potential photon that interacts with the nucleon to give a new state of the nuclear material. A state of affairs cannot be separated from the potential through which it takes effect and without which it would have no activity or development (for example, catalysis). It is through this potential that it can confront accidents, adjunctions, ablations, or even projections, as we see in geometrical figures: either losing and gaining variables, extending singularities up to the neighborhood of new ones, or following bifurcations that transform it, or passing through a phase space whose number of dimensions increases with supplementary variables, or, above all, individuating bodies in the field that it forms with the potential. None of these operations come about all by themselves; they all constitute “problems.” It is the privilege of the living being to reproduce from within the associated potential in which it actualizes its state and individualizes its body. But an essential moment in every domain is the passage from a state of affairs to the body through the intermediary of a potential or power or, rather, the division of individuated bodies within the subsisting state of affairs. We pass here from mixture to interaction. And finally, the interactions of bodies condition a sensibility, a proto-perceptibility and a proto-affectivity that are already expressed in the partial observers attached to the state of affairs, although they complete their actualization only in the living being. What is called “perception” is no longer a state of affairs but a state of the body as induced by another body, and “affection” is the passage of this state to another state as increase or decrease of potential-power through the action of other bodies. Nothing is passive, but everything is interaction, even gravity. This was the definition Spinoza gave of “affectio” and “affectus” for bodies grasped within a state of affairs, and that Whitehead rediscovered when he made each thing a “prehension” of other things and the passage from one prehension to another a positive or negative “feeling.”* Interaction becomes communication. The (“public”) matter of fact was the mixture of data actualized by the world in its previous state, while bodies are new actualizations whose “private” states restore matters of fact for new bodies.12 Even when they are nonliving, or rather inorganic, things have a lived experience because they are perceptions and affections.
When philosophy compares itself to science, it sometimes puts forward a simplistic image of the latter, which makes scientists laugh. However, even if philosophy has the right to offer an image of science (through concepts) that has no scientific value, it has nothing to gain by attributing limits to science that scientists continually go beyond in their most elementary procedures. Thus, when philosophy relegates science to the “already-made” and reserves for itself the “being-made,” like Bergson or phenomenology, and particularly in Erwin Straus, we not only run the risk of assimilating philosophy to a simple lived but give a bad caricature of science: Paul Klee’s vision was certainly more sound when he said that mathematics and physics, in addressing themselves to the functional, take not the completed form but formation itself as their object.13 Furthermore, when we compare philosophical and scientific multiplicities, conceptual and functional multiplicities, it may be much too simple to define the latter by sets. Sets, as we have seen, are of interest only as actualization of the limit; they depend on functions and not the converse, and the function is the true object of science.
Functions are, first of all, functions of states of affairs and thus constitute scientific propositions as the first type of prospects: their arguments are independent variables on which coordinations and potentializations are carried out that determine their necessary relations. In the second place, functions are functions of things, objects, or individuated bodies that constitute logical propositions: their arguments are singular terms taken as independent logical atoms on which descriptions are brought to bear (logical states of affairs) that determine their predicates. In the third place, the arguments of functions of the lived are perceptions and affections, and they constitute opinions (doxa as third type of prospect): we have opinions on everything that we see or that affects us, to the extent that the human sciences can be seen as a vast doxology—but things themselves are generic opinions insofar as they have molecular perceptions and affections, in the sense that the most elementary organism forms a proto-opinion on water, carbon, and salts on which its conditions and power depend. Such is the path that descends from the virtual to states of affairs and to other actualities: we encounter no concepts on this path, only functions. Science passes from chaotic virtuality to the states of affairs and bodies that actualize it. However, it is inspired less by the concern for unification in an ordered actual system than by a desire not to distance itself too much from chaos, to seek out potentials in order to seize and carry off a part of that which haunts it, the secret of the chaos behind it, the pressure of the virtual.14
Now, if we go back up in the opposite direction, from states of affairs to the virtual, the line is not the same because it is not the same virtual (we can therefore go down it as well without it merging with the previous line). The virtual is no longer the chaotic virtual but rather virtuality that has become consistent, that has become an entity formed on a plane of immanence that sections the chaos. This is what we call the Event, or the part that eludes its own actualization in everything that happens. The event is not the state of affairs. It is actualized in a state of affairs, in a body, in a lived, but it has a shadowy and secret part that is continually subtracted from or added to its actualization: in contrast with the state of affairs, it neither begins nor ends but has gained or kept the infinite movement to which it gives consistency. It is the virtual that is distinct from the actual, but a virtual that is no longer chaotic, that has become consistent or real on the plane of immanence that wrests it from the chaos—it is a virtual that is real without being actual, ideal without being abstract. The event might seem to be transcendent because it surveys the state of affairs, but it is pure immanence that gives it the capacity to survey itself by itself and on the plane. What is transcendent, transdescendent, is the state of affairs in which the event is actualized. But, even in this state of affairs, the event is pure immanence of what is not actualized or of what remains indifferent to actualization, since its reality does not depend upon it. The event is immaterial, incorporeal, unlivable: pure reserve. Two thinkers have gone the farthest into the event—Péguy and Blanchot. Blanchot says that it is necessary to distinguish between, on the one hand, the accomplished or potentially accomplished state of affairs in an at least potential relation with my body, with myself; and, on the other hand, the event, that its own reality cannot bring to completion, the interminable that neither stops nor begins, that remains without relation to myself, and my body without relation to it—infinite movement. Péguy says that it is necessary to distinguish between, on the one hand, the state of affairs through which we, ourselves, and our bodies, pass and, on the other hand, the event into which we plunge or return, that which starts again without ever having begun or ended—the immanent aternal [l’internel].15
Throughout a state of affairs, a cloud or a flow, even, we seek to isolate variables at this or that instant, to see when, on the basis of a potential, new ones arise, into what relations of dependence they can enter, through what singularities they pass, what thresholds they cross, and what bifurcations they take. We mark out the functions of the state of affairs: differences between the local and the global are internal to the domain of functions (for example, depending on whether all independent variables but one can be eliminated). The differences between the physico-mathematical, the logical, and the lived also pertain to functions (depending on whether bodies are grasped in the singularities of states of affairs, or as themselves singular terms, or according to singular thresholds between perception and affection). An actual system, a state of affairs, or a domain of functions are at any rate defined as a time between two instants, or as times between many instants. That is why, when Bergson says that there is always time between two instants, however close to each other they may be, he has still not left the domain of functions and introduces only a little of the lived into it.
But when we ascend toward the virtual, when we turn ourselves toward the virtuality that is actualized in the state of affairs, we discover a completely different reality where we no longer have to search for what takes place from one point to another, from one instant to another, because virtuality goes beyond any possible function. In the conversational words attributable to a scientist, the event “doesn’t care where it is, and moreover it doesn’t care how long it’s been going,” so that art and even philosophy may apprehend it better than science.16 It is no longer time that exists between two instants; it is the event that is a meanwhile [un entre-temps*]: the meanwhile is not part of the eternal, but neither is it part of time—it belongs to becoming. The meanwhile, the event, is always a dead time; it is there where nothing takes place, an infinite awaiting that is already infinitely past, awaiting and reserve. This dead time does not come after what happens; it coexists with the instant or time of the accident, but as the immensity of the empty time in which we see it as still to come and as having already happened, in the strange indifference of an intellectual intuition. All the meanwhiles are superimposed on one another, whereas times succeed each other. In every event there are many heterogeneous, always simultaneous components, since each of them is a meanwhile, all within the meanwhile that makes them communicate through zones of indiscernibility, of undecidability: they are variations, modulations, intermezzi, singularities of a new infinite order. Each component of the event is actualized or effectuated in an instant, and the event in the time that passes between these instants; but nothing happens within the virtuality that has only meanwhiles as components and an event as composite becoming. Nothing happens there, but everything becomes, so that the event has the privilege of beginning again when time is past.17 Nothing happens, and yet everything changes, because becoming continues to pass through its components again and to restore the event that is actualized elsewhere, at a different moment. When time passes and takes the instant away, there is always a meanwhile to restore the event. It is a concept that apprehends the event, its becoming, its inseparable variations; whereas a function grasps a state of affairs, a time and variables, with their relations depending on time. The concept has a power of repetition that is distinct from the discursive power of the function. In its production and reproduction, the concept has the reality of a virtual, of an incorporeal, of an impassible, in contrast with functions of an actual state, body functions, and lived functions. Setting up a concept is not the same thing as marking out a function, although on both sides there is movement, and in each case there are transformations and creations: the two types of multiplicities intersect.
No doubt, the event is not only made up from inseparable variations, it is itself inseparable from the state of affairs, bodies, and lived reality in which it is actualized or brought about. But we can also say the converse: the state of affairs is no more separable from the event that nonetheless goes beyond its actualization in every respect. It is necessary to go back up to the event that gives its virtual consistency to the concept, just as it is necessary to come down to the actual state of affairs that provides the function with its references. From everything that a subject may live, from its own body, from other bodies and objects distinct from it, and from the state of affairs or physico-mathematical field that determines them, the event releases a vapor that does not resemble them and that takes the battlefield, the battle, and the wound as components or variations of a pure event in which there remains only an allusion to what concerns our states. The event is actualized or effectuated whenever it is inserted, willy-nilly, into a state of affairs; but it is counter-effectuated whenever it is abstracted from states of affairs so as to isolate its concept. There is a dignity of the event that has always been inseparable from philosophy as amor fati: being equal to the event, or becoming the offspring of one’s own events—”my wound existed before me; I was born to embody it.”18 I was born to embody it as event because I was able to disembody it as state of affairs or lived situation. There is no other ethic than the amor fati of philosophy. Philosophy is always meanwhile. Mallarmé, who counter-effectuated the event, called it Mime because it side-steps the state of affairs and “confines itself to perpetual allusion without breaking the ice.”19 Such a mime neither reproduces the state of affairs nor imitates the lived; it does not give an image but constructs the concept. It does not look for the function of what happens but extracts the event from it, or that part that does not let itself be actualized, the reality of the concept. Not willing what happens, with that false will that complains, defends itself and loses itself in gesticulations, but taking the complaint and rage to the point that they are turned against what happens so as to set up the event, to isolate it, to extract it in the living concept. Philosophy’s sole aim is to become worthy of the event, and it is precisely the conceptual persona who counter-effectuates the event. Mime is an ambiguous name. It is he or she, the conceptual persona carrying out the infinite movement. Willing war against past and future wars, the pangs of death against all deaths, and the wound against all scars, in the name of becoming and not of the eternal: it is only in this sense that the concept gathers together.
From virtuals we descend to actual states of affairs, and from states of affairs we ascend to virtuals, without being able to isolate one from the other. But we do not ascend and descend in this way on the same line: actualization and counter-effectuation are not two segments of the same line but rather different lines. If we restrict ourselves to the scientific functions of states of affairs, it seems that they cannot be isolated from a virtual that they actualize, but this virtual appears first of all as a cloud or a fog, or even as a chaos—as a chaotic virtuality rather than the reality of an ordered event in the concept. That is why it often appears to science that philosophy covers up a simple chaos, leading science to say, “Your only choice is between chaos and me, science.” The line of actuality lays out a plane of reference that slices the chaos again: it takes from it states of affairs that, of course, also actualize virtual events in their coordinates, but it retains only potentials already in the course of being actualized, forming part of the functions. Conversely, if we consider philosophical concepts of events, their virtuality refers to the chaos, but on a plane of immanence that slices it again in turn, and that extracts from it only the consistency or reality of the virtual. No doubt states of affairs that are too dense are adsorbed, counter-effectuated by the event, but we find only allusions to them on the plane of immanence and in the event. The two lines are therefore inseparable but independent, each complete in itself: it is like the envelopes of the two very different planes. Philosophy can speak of science only by allusion, and science can speak of philosophy only as of a cloud. If the two lines are inseparable it is in their respective sufficiency, and philosophical concepts act no more in the constitution of scientific functions than do functions in the constitution of concepts. It is in their full maturity, and not in the process of their constitution, that concepts and functions necessarily intersect, each being created only by their specific means—a plane, elements, and agents in each case. That is why it is always unfortunate when scientists do philosophy without really philosophical means or when philosophers do science without real scientific means (we do not claim to have been doing this).
The concept does not reflect on the function any more than the function is applied to the concept. Concept and function must intersect, each according to its line. Riemannian functions of space, for example, tell us nothing about a Riemannian concept of space peculiar to philosophy: it is only to the extent that philosophy is able to create it that we have the concept of a function. In the same way, the irrational number is defined by a function as the common limit of two series of rational numbers, one of which has no maximum and the other no minimum. The concept, on the other hand, refers not to series of numbers but to strings of ideas that are reconnected over a lacuna (rather than linked together by continuation). Death may be assimilated to a scientifically determinable state of affairs, as a function of independent variables or even as one of the lived state, but it also appears as a pure event whose variations are coextensive with life: both, very different aspects are found in Bichat. Goethe constructs an imposing concept of color, with inseparable variations of light and shade, zones of indiscernibility, and processes of intensification that show the extent to which there is also experimentation in philosophy; whereas Newton constructed the function of independent variables or frequency. If philosophy has a fundamental need for the science that is contemporary with it, this is because science constantly intersects with the possibility of concepts and because concepts necessarily involve allusions to science that are neither examples nor applications, nor even reflections. Conversely, are there functions—properly scientific functions—of concepts? This amounts to asking whether science is, as we believe, equally and intensely in need of philosophy. But only scientists can answer that question.