Mathematics: For a modern, the transcendental is confused with ontology itself, i.e. as Kant would translate from its Latin definition, the “science of the most universal properties of all things.” It is with what François Regnault called his “stroke of genius”, i.e. the equation mathematics=ontology, that Badiou (and a few others, constituting what is typically called a “philosophical generation”) hit me for such a long time on the head. We had to admit in fact that this simple but devastating equation constituted the mightiest answer of the history of metaphysics to the ontological question. In fact, since science is the art of appropriation of the “eternal” Laws of Nature, i.e. of the being-necessary of everything, what discipline of thought could lay claim to the rank of ontology more than mathematics? All in all, Badiou’s “discovery” is so powerful that it appears, after a (long, very long…) digesting period, somewhat trivial. We ask ourselves why metaphysics did not think of it before, and for such a long time sought “ontology” in other places than where it was obviously to be found. Mathematics is the science of “the most universal properties of all things” in such an obvious manner that everything should have been said on that point. And yet, it would seem that Badiou alone discovered it.
However, even this point is not so self-assured, if we decide to take a more serious look at it. I will give the irrefutable proofs in one of my two as yet unpublished, “posthumous” books, but even the equation mathematics=ontology eventually seemed to me not only suspect, but finally null and void—which is by no means a simple “anti-Badiou” polemic, but a real conclusion which, on the pretext of the latter, constitutes a perspective turning point embracing the entire history of metaphysics and its false paths. I explained above why, at the end of the day, and in agreement with the rest of the tradition on this point alone, I confirmed the superiority of the logical paradigm over the mathematical (see Logic). Here I will argue, with more specific examples, why I think the equation of mathematics with ontology is nothing more than deceptive: like the previous chapter’s conclusion, this too will show how the heavily interested position of “anti-humanism” openly falls into its own trap. The demonstration will illustrate, en abîme, the concept of irony (see above), by laying bare the semblance of semblance of the naked king, which is the match without remainder (see Ontological Differend) between being and mathematical thought, just as the police prefect had to wait for Dupin to steal the stolen letter of the Minister. Even more en abîme, it is pure analytical logic, again in the sense theorized by Dupin, that dismantles the metaphysical postulate of an “inhumanist” equivalence of ontology and mathematics.
And if Logic alone is a legitimate transcendental, i.e. the empty ontology of all possible link between the beings [étants], without taking account of any singular properties, but mathematics is delegitimized with respect to the claim to say its contents, Ontological differend (see above) oblige, then the necessary outcome sanctions, collaterally, what the philosophical twentieth century sought for the most part, that is to say, the institution of a thought of being which no longer lays claim to any “ontology.” It is therefore rather in the original fruit of metaphysics that the deadly worm must be sought.
The sciences—every science—divide things up into their components that are inaccessible to animal perception. Etymologically, “to divide” means: to introduce the void. From the moment we remain on the level of the pure practice of pure sciences; it cannot do any harm to say, with dear Badiou, that these sciences are “innocent.” But as Hegel said with his solid Lutheran sense, only the stone is innocent. For the void is literalized. It is all the forms of Evil. It is the perversion of predatory cruelty into tormenting jouissance. It is the most literal decomposition of the very matter of things by mathematized physics. The latter in turn literalizes the fact that “ontology,” by claiming to apply to “the in-itself” of things, the in-itself itself created as it were from scratch, had always lied. In short: the equivalence mathematics=ontology “triumphantly” completes metaphysics. It says the very last word, as a truism. Yes, mathematics is ontology, and that is the whole problem.
I think mathematics refines the metaphysical lie par excellence: the equality formula, x=x. This should be enough to call into question the whole of what I call “Platonic fiduciarism,” that is, the blind confidence in the epistemological, i.e. absolutory-ontological status of mathematics. Only in the light of this formula so fundamental to itself does mathematics as a whole appear as what it is at its origin: the ultimate sublimation of the mimetic impulse through which the human animal (self-)appropriated things, in instrumentalizing them. The formula of equality is in a way “the absolute weapon” which made man “Master and possessor of Nature,” as Descartes, Platonic fiduciarism’s second great historial kick-off, thought. And nothing is less innocent than a sublimation that lays claim, precisely, to the most immaculate innocence. We know well that in love, the domain reserved for the inspection of the operation of sublimation, there is just one step between the fact of saying that Helen is divine and the fact of realizing that she is nothing but a bag of guts and excrement. The amorous countersublimation should enlighten us about all the other types of sublimation, which are always pleonastically metaphysical, and therefore never more innocent than the lover interested in his sublimation who, when disappointed, becomes the worst slanderer of what he had only just deified.
Yet this is not the only argument against the equation mathematics=ontology. Once again, it is the unpublished book on Meillassoux that will get to the bottom of things. But I will make do with just one striking example here. It is about the famous “point of excess” discovered by the greatest paranoid of the twentieth century, Kurt Gödel (whose rivals were not few in that), and which Badiou, as you might expect, makes into a primordial “ontological” category, since all that is mathematical is ipso facto ontological.
All that is material is structured by what mathematics calls “membership” or “belonging,” the relation denoted by ∈. (Incidentally, perhaps this choice of the word “belonging” alone, to designate the essential structure on which modern mathematics is based, already betrays the pleonectic prejudice the human mind can never help applying to things.) Crudely put, my body “belongs to”/“is a member of” this room, which “belongs” to this house, which “belongs” to this village, which “belongs” to this country, which “belongs” to this planet, which “belongs” to this solar system … and so on to infinity, without any end point. No end point “bottom-up”; but no end point “top-down” either: everything is indefinitely subdividable, decomposable into always smaller atoms. Even with quarks, physics will someday discover smaller subdivisions, and this is why mathematics transcends the other sciences, literally and in every sense.1 The question is, as usual, what transcendentalization will cost sooner or later, in the face of the compulsory euphoria of professional metaphysicians.
On top of this comes the relation of inclusion, denoted by ⊂. It indicates the doubling of the relation of belonging into representation of a “belonging of belonging.” It is materially that my arm belongs to my body, but it is immediately a matter of representation (what twentieth-century philosophy, for a long time, determined transcendentally as “language”) when we say: “my arm.” Thus I separate (I divide …) artificially the universal intrication of material “belongings” into parts represented as “separate.” By what do I do this? By the pure void of this representation itself: my arm is only ideally [idéallement] separate from my body. Naturally, a virtuoso of the chainsaw, an instrument we owe to the appropriation of the void called science, can come to literalize this separation and cut my arm off, or other less avowable parts of me. Yet it changes nothing in the structure of things, which is universally governed by “belonging.” I put the word in quotation marks in order to refer to the parenthesis that intervenes in the preceding paragraph. It is very likely that the sole fact of calling this structure “belonging” is already a more than interested representation, betraying the inveterate pleonectic “second nature” of the technomimetic animal. Then it is perhaps a poor choice of words, but we must obviously agree that “belonging,” in fact, coincides universally with the structure of things—as logic spells out the universal structure of their relations.
Is this the case with inclusion? What Gödel’s theorem—certainly one of the most admirable logico-mathematical discoveries of the twentieth century—tells us, is that the set of the parts of a given being always exceeds the structural set of its members. The representation of a being, which divides the being into infinite—infinite because arbitrary—parts (I can, in an intelligible manner, cut my body up into as many odd parts as I will, for instance), is always superior to the simple structural presentation of the being in question. Very well.
Because of Badiolism, that is, accomplished metaphysics, mathematics is ontological, then inclusive, and the ensuing representational excess, is “ontological.”
Must it be conceded? Obviously not. No being [étant], in its compact materiality, overflows its own representation. The parts of a planet do not spend their time exceeding the elements that compose them, the inclusions that are supposed to ontologically haunt my coffee maker do not seem to be in a hurry to overflow in every direction. The representation of a tree does not pounce on my face with all the might of its excess, and even a rabid dog, if he jumps at my throat, does not do so because of a point of excess, but rather because of its singular material structure, which is after all still too nicely pleonectic compared to any human being, even communist. Conclusion: it really seems that the “ontological” scope of the famous point of excess, which Badiou applies to the political situation for instance, is caught in the anthropologizing “correlationism,” where human thinking plates [plaque], on the very inside of things, a process that shows merely his own power to (self-)appropriate them. It is us who divide the being into “excessive” parts (it is even a particularly eloquent example of the techno-appropriative operation), and not things that subdivide themselves into partitive representation that exceeds their universally elementary structure.
Therefore the equation mathematics=ontology is worthless, but not because ontology would have to be sought in a safer place. It is because ontological propensity itself, the scientific universalism that sublimates the ferocity of technomimetically appropriative animal, is the oldest of metaphysical illusions—practically the definition of the latter all by itself.
Thus we see by this single example that mathematics is not ontology. It is rather “the ontology”… of the noetic-noematic link that unites the technomimetic animal with the being [l’être] it appropriates. The keyword here is the one that is indiscernible from every “ontological problem” since Kant: transcendental. And with Logics of Worlds, it must be admitted that Badiou got quite closer to what he wanted to reach with the first version of Being and Event; but as if despite himself, and as if on the flip side of what he says. If logic is the transcendental of appearing, then logic alone is properly ontological.
But it is so, precisely, transcendentally: it says nothing of the singulars (see below) it subsumes, in reducing them to their “most general properties.”2 Whereas mathematics claims to apply to those singularities themselves, and this is the disastrous error where Plato, Descartes, and Badiou join each other. Such is the price to pay for it to be what it is: the “purest,” i.e. the most purifying of sciences. We know to what sweet engagements some were led by the “passion of purification” … But such is especially the—mostly disastrous—price that will have to be paid by the beings to which these purifications will be applied, by mistaking representational excess, principle of identity, indiscernibles, or relation of biunivocity for material structure, pleonectic-evental exception, or alimentary or reproductive instincts.
We can henceforth formulate this disastrous error in several oxymoronic ways:
•Mathematics “is” ontology … but there is no ontology. The critique of metaphysics launched by Kant and Heidegger must be dramatized “after Badiou,” and after the many effects of neo-dogmatism it gives rise to at the contemporary philosophical university (what we call “Speculative Realism”).
•Logic is ontology, but provided that it never says anything on the singular contents, of which it merely enacts the transcendental laws of relation.
•Logic prevails over mathematics in conceptual dignity, precisely because, according to Badiou, it describes “appearing” [l’apparaître] and not “being” [l’être]. But there is quite simply no being outside appearing: this naive Platonism, or positivized Kantianism (which holds that the in-itself of things can be appropriated, in Kantian terms, being God3), does not take account of the exacerbation of the contemporary critique of metaphysics by Nietzsche: “with the true world, we also abolished the world of appearances.” For transcendentalizing the appearing into empty eternal and universal rules, Logic is in fact “ontology,” because in (self-)appropriating these empty rules, it does not pretend to create an inaccessible world “in-itself”: these are strictly correlational rules.
We saw earlier how mathematics, in its turn, perpetrated the metaphysical overturning par excellence: by applying a typically anthropological operation to the being [l’étant], whether it be the principle of equality, identity, or excess of parts over members, it takes this instrumental operation for the in-itself and claims to have discovered the “ground” [fonds] of things. From then on the terrain is leveled for all forms of religion, which a Badiou will replace “advantageously” with all forms of “laicisized” egalitarian fanaticism. In appearing, which is being [l’être] itself, there is nothing but the elementary: neither parts in excess—except in the anthropological closure that creates this excess (and this is … “ontological” capitalism)—not representation (see below).
From this point of view, yes, mathematics is much more “evental” than Logic, for the fact that it adds to being [l’être] things that are not there. Only Logic is really faithful to the things outside humanity; whereas mathematics contaminates these things with the anthropological ability to represent, that is, technologically (self-)appropriate them. The excess of representation lies nowhere else than in ourselves. This contamination, it should go without saying at the point we are now, becomes, in the middle of the twentieth century, more literal than it has ever been, with the nuclear fission we claim to see “in” things. The problem is that things dreadfully show that they themselves do not recognize themselves therein.
Then the history of mathematics is by no means “the history of eternity” as Badiou sees it.4 It is a “bank” of phenomenal appropriations, and the greatest scientific “conquests” of the human animal are based, in fact, on the mathematized sciences. Whereas Logic is much less generous in terms of appropriative “gifts,” but this means that it is much more “respectful” of the being outside-humanity [l’être hors-humanité] than is mathematics. Which comes down to rethinking the one and the other on completely different bases than those laid out a first time by Plato, a second time by Descartes, and a third time (this poisonous credit should be granted to him) by Badiou.
So how do “we pass” from the infinite approximations of logic (see above), appearing, and the “identical in the nth degree,” to absolute (=God) identity5 that founds mathematics? Because it is this principle that is essentially false, and it is from this falsity that the set of “eternal ontological truths” of mathematics are deduced, which lead to nuclear fission for instance. The mimetic decomposition of physis through the most archaic technai, sublimated by the “pure sciences” of logic and mathematics, is accomplished, after the second Galileo-Cartesian break with nature, in a literal decomposition of the latter. The Neolithic and Paleolithic technique, despite its violence, merely “diverted” nature into new outlets, like efflorescence into agriculture; it is by way of this first violence which is then sublimated by pure sciences that we entered history itself, and it is this beginning that would then be recapped by Platonism, i.e. second degree sublimation of appropriative violence. In the second, Galileo-Cartesian beginning, sublimation, having reached an apocalyptic stage that could not have occurred in any way even to Galileo and Descartes, does not take more than two or three centuries to display the new mode of its destructive violence. In any event, what is confirmed is the anthropological illusion par excellence, through which we will have sublimated and made “disinterested” our selfish interest and our literally unlimited pleonectic voracity: the identitarian illusion, i.e. the mimetic astuteness which made us, for a couple of thousands of years, “stronger” than our planetary rivals. On this point, Hölderlin is very much ahead not only of Fichte and his colleagues from the Stift, but still today of Badiou: the question of being must be radically distinguished from the principle of identity. Heidegger or Derrida will understand something of this. Others, still today, turn a deaf ear to it.
If being thinks itself [se pense] effectively as event, then all “ontology” is done for. Forever. Schürmann is the guiding thinker of the coming century. Badiou is not only useless and uncertain, but potentially disastrous, if we take him at his hyper-thetical word. It is moreover because of this that Badiou’s theory of event, particularly concerning arts and politics, but also at bottom the mathematical formula he gives of the event, as well as his falsely bland theory of love, is, for those who can read, perfectly lame and false. What is false is what I call the “masculine” side (see Appendix) of the philosophical process: the tightness of the “truth procedures” between themselves. It is basically absurd to think that the truths of science, politics, love, and art follow a course independent from the other procedures: as such economically giving themselves the means to maximize the segregation, by putting the “Good” itself in a bureaucratic isolation room.
The minutest detail of the SoN shows that it is strictly impossible to think such tightness, and that, on the contrary, the procedures do not stop interacting with each other. A name for this perpetual interaction, “perpetual movement” as they say, when it is foreclosed, is the one Deleuze and Guattari turn into a philosophical category: schizophrenia. It is the assumption of this objective schizophrenia of the technomimetic animal that opens the path for a properly modern Wisdom. Conversely, the rococo philosophical promotion of all the rational nec plus ultra of all domains, from Cantor to Pol Pot, is the safest path of thetic raving with a funnel for a headpiece. The more metaphysics hardens its pretension to founding rationality on supposedly irrefutable bases, the more it predisposes humanity to all forms of irrationalism and superstition; prompted by always singular (therefore pleonectic) motives, the more it claims to sincerely believe in the existence of principles, archetypes, eidetic and subsuming conceptions of the being [l’étant] modeled on scientific and ultimately mathematical purification, the more it encourages “the entire humanity” to chase after phantoms.
Then let us be clear on this: the polemic against mathematics is essentially aimed less at mathematics, and its admirable discoveries, than at the paradigmatic appropriation philosophy makes of it right in its Platonic envoi, commiting us to two and a half thousand years of horrors. I share with the majority of philosophical judgement (Kant, for instance) the idea that mathematics probably consitutes the nec plus ultra of what a human mind is capable of thinking technically. The great mathematical inventions are to the intellect what athletic or acrobatic exploits are to the body. In fact, many mathematical propositions well and truly have an ontological scope (such as the fundamental question of the zero). Yet, in this respect, there are obviously things to take or leave—especially to leave when you take a closer look, and especially when you look at the effects. The task is henceforth that of an infinite weight of being which will use mathematics, among others and always critically (the actual ontological scope, for instance, of real numbers). As to the history of ontologies, it is behind us for good: we should give our thanks to Badiou for unintentionally having us get rid of it.
In terms of scholastic protocol, this translates as follows: the thesis of “the univocity of being,” which is merely the narcissistic projection of the philosopher-king as old as Plato himself, must be abandoned. Being is impure, muddy, plurivocal: even the Deleuzian virtual reabsorbs somewhat too much the differential influx of the being [l’étant] in the univocity of Bergsonian duration. Heidegger, Schürmann, a part of Derrida are safer invocations: no point of Sirius to the treatment of the question of being, which is said in more than one way, and is distributed in regions as impure as they are non-communicating, dissymmetrical. I am simply urging to think differently the situation of mathematics, and perhaps in the respect I just mentioned, in the heart of human thought, and thus to place it in an entirely different manner within philosophical strategies.
As long as this change of paradigm does not gain the upper hand, we will have to go on launching attacks against what I have also called “Platonic fiduciarism,” of which Badiou will have given the ultimate historical expression with his equation ontology=mathematics. Because ontology will not recover from it. And this is not as bad of news as it sounds.