Lan Jiang: Infinite Forms and the Spark of Truth—On Materialist Dialectics in Badiou's Immanence of Truths
For the French Marxist philosopher Alain Badiou, the 2018 publication of The Immanence of Truths (L’Immanence des vérités) may well be his final magnum opus. In Being and Event (1988) and Logics of Worlds (2006) (the second volume of the "Being and Event" series), he employed his unique method of mathematical philosophy to attempt to locate the possibilities for moving toward a future communist society. Twelve years after the publication of Logics of Worlds, he has once again released a massive tome of over 700 pages—The Immanence of Truths—noting that this new work constitutes the third volume of the "Being and Event" trilogy. Perhaps Alain Badiou hopes to bring his lifelong philosophical project to a conclusion in this third volume: specifically, how we might change the world, and how such change becomes possible, once we have touched "the infinite"—a true infinite.
I. The Path to the Infinite
"A single spark can start a prairie fire." Badiou is indeed very fond of this famous saying by Mao Zedong [1]. He has referenced it numerous times throughout his oeuvre, from his early work Theory of Contradiction to the 1982 Theory of the Subject, and in the conclusion of Logics of Worlds. We cannot determine the exact extent to which Badiou, when quoting this aphorism, understood the lived reality of the Chinese Workers' and Peasants' Red Army in the Jinggang Mountains [2] in 1930—how they persisted in the ideals of national liberation and communism amidst grueling circumstances, thereby finding hope in a "single spark." Mao Zedong’s writing of A Single Spark Can Start a Prairie Fire was not only a response to the pessimism then pervading the Red Army, but also an expression of an ideal: finding the hope that leads to a future ideal society from within a faint glimmer of light.
Perhaps it is precisely this idealist expression that catalyzed Badiou’s thought. He knows that Mao’s "single spark" was directed toward the arduous revolutionary struggle in China at that time, but that its fundamental core is a philosophical concept: the idea of seeing an infinite future within a finite flame. In fact, Badiou quotes this famous passage again in the introduction to The Immanence of Truths. However, he expresses dissatisfaction with the rendering in the French version of the Selected Works of Mao Zedong—Une seule étincelle peut commencer un feu de prairie—and provides his own French translation: l’étincelle qui jaillit du frottement localisé d’infinités disparates [the spark that gushes forth from the localized friction of disparate infinities].
In the first half of Badiou’s new translation, there is actually little difference, yet the two sentence structures are worlds apart. The former is in the active voice; that is to say, the "single spark" is the cause and the "prairie fire" is the effect. It is precisely because the single spark is lit that the universal spark of the future illuminates the entire world. This is a mode of gradually ascending from the finite to the infinite; in his early thought, Badiou endorsed this approach. In The Immanence of Truths, however, the relationship in this sentence has changed. It can no longer be translated directly as "a single spark can start a prairie fire," but rather as "the single spark is a concrete flash erupted from an omnipresent infinity." In this new structure, the spark is no longer the cause, but the result stimulated by the descent of the infinite into the human realm. This logic is a kind of "inverted romanticism": first there is the form of the prairie fire, and only then is there the concrete single spark.
In Badiou’s view, when we are able to see the spark, it signifies that the world has already changed. For Communists and for revolutionary movements, the task is not to artificially manufacture sparks, but to be faithful to the spark. Faithfulness to the spark means facing a more significant "Event," and an Event is precisely the result stimulated by a new infinity or a new generic procedure. When Badiou quotes Paul Valéry’s poem "The Graveyard by the Sea" (Le Cimetière marin) in Logics of Worlds, what he anticipates is not the quiet and tranquil sea under the sun, but the waves swept up by the storm. This is the famous line from the end of Valéry’s poem:
A fresh breeze, loosed from the sky, gives me back my soul! O salty power! Let’s run into the wave and come out reborn!
Truth and the Absolute are by no means a tranquil sea surface under a gentle breeze. Such tranquility is more like a montage of disguise and concealment; it brings order and peace, allowing people to function year after year, day after day, according to a predetermined apparatus, treating this mode of motion as "natural," "universal," "absolute," and "truth." Badiou wants to tell us: No. Beneath the gentle breeze, the sea also harbors storms, and the poetic imagery of "jagged rocks piercing the sky, terrifying waves crashing on the shore, rolling up a thousand heaps of snow" [3]. If we remain only under the appearance of the tranquil sea, we degenerate into a finite existence. Only through the salty charm of being torn apart by the storm can we see the Real and the infinite.
This is why Badiou calls on us to "run into the wave" in Logics of Worlds. There is another, more important reason: once we touch the Real and the Absolute, once we face those infinities that cannot be faced simply, our lives will be refreshed. The waves bring not just "saltiness," "terrifying billows," or "terror," but a kind of "rebirth." This is why Badiou so admires that "fresh breeze" in "The Graveyard by the Sea." We must not understand this "fresh breeze" simply as romantic sweetness or a triumphalist song of uplifting progressivism; rather, it is a sublimation experienced when facing the true infinite. In this sublimation, "my soul is restored," and when I face the world anew, "to come out will be rebirth."
Whether it is Badiou’s rewritten "single spark" or his final reference to Valéry’s "run into the waves/come out reborn," both are actually expressing a new consciousness to us: if a subject once touches an infinite that is usually untouchable, what will this world—or rather, this world as the appearance of being—become when faced again? When Badiou introduces The Immanence of Truths, he repeatedly emphasizes that it is an "inverted" book. If Being and Event and Logics of Worlds were works that explored the path toward the infinite from finite forms and worlds, then The Immanence of Truths is situated at the other end. That is to say: once we have contacted the Event and encountered Truth, how do we face the world composed of finite beings?
II. Truth in the Cave
Perhaps we can use Plato’s allegory of the cave and the sunlight from The Republic to illustrate the problem Badiou poses in The Immanence of Truths. The sunlight outside the cave represents the existence of the Absolute and Truth, while the vast majority of us can only take the fleeting shadows on the cave wall as reality, becoming subordinated to these shadows and forming "opinion" (doxa). However, once someone leaves the cave and sees the sunlight outside, they immediately feel the difference between truth and opinion. Plato writes:
"Consider, then, what should be the manner of their release and their healing from their bonds and their lack of sense, if such a thing should happen to them. When one was released and suddenly compelled to stand up and turn his neck round and walk and look up toward the light, he’d be pained by doing all these things and be unable on account of the bright ness to see those things whose shadows he’d seen before. What do you think he’d say if someone told him that what he had seen before was an insignificance, but now, being somewhat nearer to what is and having turned toward things that are more, he sees more correctly?" [4]
In this passage from Plato, we perceive two things. First, facing the truth is clearly more painful than facing the shadows on the wall. People do not voluntarily face the truth; only under the urging of a certain procedure or force will they turn around to face the sunlight outside the cave. This is why Badiou has emphasized since Theory of the Subject that the Subject is a "scarcity." Those who can see the shadows of truth are the few, and those willing to let go of the doxa in the cave, voluntarily move toward truth, and merge with the "generic procedure" of that truth are fewer still. Mathematics and philosophy are both such procedures, but those who truly wish to touch the path toward the Absolute and Truth through mathematics are extremely scarce.
In other words, there is no lack of "philosophy" today—that is, the kind of thought that disguises ordinary doxa as philosophy. In Badiou’s view, this "philosophy" is actually a kind of "anti-philosophy." Nietzsche, Wittgenstein, and Lacan are famous masters of anti-philosophy. For example, when Wittgenstein says, "Whereof one cannot speak, thereof one must be silent," Badiou sees this "conclusion" as precisely requiring us to remain limited within the doxa of finitude, suspending all paths leading out of the cave.
In evaluating Wittgenstein’s anti-philosophy, Badiou points out: "The anti-philosophical act consists in letting things show themselves, because it is precisely 'that which is there' that we cannot speak of in true propositions. If Wittgenstein's anti-philosophical act can be appropriately called 'archi-aesthetic,' it is because 'letting be' cannot express its pure existence and clarity propositionally; this clarity of pure existence is unspeakable and can only appear in works of non-thought form (for Wittgenstein, music is of course the most appropriate form)."
Badiou believes that by cancelling the form of philosophical reflection, Wittgenstein argues that philosophy cannot lead us to truth and instead turns that truth into an unspeakable "archi-aesthetic." In fact, these figures are "sophists" under contemporary conditions; they blur the line between opinion and truth, making people maintain a sense of mystery regarding the Absolute and the infinite outside the cave. They preserve this mystery within an original, unspeakable sensibility, thereby cutting off the path to the outside world. Thus, compared to art—a form that people find more intoxicating—people seem to have forgotten Plato’s reminder: the path to the truth outside is painful. In Badiou's view, only mathematics in the strict sense performs this role: "Only mathematics allows us to enter into a unified thought of the visible." This is why Plato famously inscribed "Let no one ignorant of geometry enter" above the entrance to his Academy. Mathematics is painful, but once the path of mathematics is taken and that seemingly untouchable truth is reached...
to obtain greater pleasure.
Second, in The Immanence of Truths, Badiou actually endows Plato's allegory of the cave with a new meaning, which is also the central theme of this work. If Being and Event lay down a formalized path toward the outside of the cave, and Logics of Worlds addressed how finite beings might set foot on this path—realizing the possibility of access to the infinite from a finite perspective—then the final volume, The Immanence of Truths, is about the "return." That is, having attained the infinite, how does one return to the narrative within the cave? Because one has touched the form of truth and come into contact with the infinite, the things within the cave still exist; however, they are no longer "letting themselves be shown" in the Wittgensteinian sense. For Badiou, there is no way for a thing to simply manifest itself. From Being and Event onward, Badiou has maintained that the reason we perceive a thing as a thing—"being as being" (l’être en tant qu’être)—is precisely because it is represented through a structure of the "count-as-one." Real being is "indifferent" [5] (indifférent) being, it is multiplicity and manifold; we can only distinguish things and objects from one another through conceptual tools or functions of the "count-as-one." It is only with these distinguishing functions and concepts that we can understand something as a "thing" (chose) or an "object" (objet), and the one who creates these concepts and functions is the "subject" (sujet).
In this way, Badiou actually reverses the relationship: the emphasis is not on our piecing together bits of experience to obtain a capital-T Totality (Totalité) or the One (UN); rather, the objects in the cave are in fact mediated by concepts, becoming the result of a "count-as-one" operation. Conversely, we can understand why Badiou, in The Immanence of Truths, transforms Mao Zedong's phrase "a single spark can start a prairie fire" [6] into "a single spark is a concrete spark erupted from the ubiquitous infinite." That is to say, as the truth procedure outside the cave changes, the objects within the cave change accordingly, becoming "sparks" oriented toward truth. In other words, once one returns to the cave, the real world hasn't changed; what has changed is the way used to differentiate and architecture the world. That which was called the "state of the situation" in Being and Event, or the "transcendental T" in Logics of Worlds, is transformed by the generic procedure of truth, which alters the entire formal architecture, thereby allowing objects to appear and manifest new states within this new framework.
All objects are immanent to the form, immanent to the newly opened generic procedure of truth. Any existing and appearing object is an object under a generic procedure; in this sense, we can say all objects are "immanent to" (immanent à) truth. This is why Badiou titled his 2018 book The Immanence of Truths and designated it the third volume of the "Being and Event" series. It is not that things and ideas transcend (au-delà) the formal framework; rather, all objects and ideas are architectured by form and are immanent to the framework of truth. Objects are being realized by the formal framework or the generic procedure of truth; they can be represented, counted, and incorporated into existing operational frameworks. Yet there still exists a certain "pure multiple"—a being of the multiple that does not appear through the formal framework. However, this "purity" is not only unsayable (as Wittgenstein suggested) but even imperceptible; even Wittgenstein’s "proto-sensibility" is powerless before such a pure, or indifferent, multiple. Thus, through his own truth procedure—the immanence of truth—Badiou distinguishes objects intervened upon by formal truth procedures from the indistinguishable pure multiple, the pure being of multiplicity. Badiou believes the goal of philosophy cannot be, as Wittgenstein suggested, to remain silent about the unsayable; rather, we must create a form, a generic procedure of truth, so that what was originally unsayable can be presented under a new form. Thus, Badiou points out: "This proves that the immanence of truths, at the most diagrammatic level, in the most formal truth, inevitably provides the possibility for human thought to undergo the ordeal and encounter the infinite." [5] From this perspective, the reason Badiou trusts mathematics so deeply is precisely because it provides us with infinite possibilities, allowing new possibilities to descend into the cave, so that the multiplicities existing there can be liberated from their original "doxa" [7] and attain their own potential.
III. From Language to Mathematics
To achieve this goal, Badiou believes the prestige of philosophy must be re-established. Throughout his life, Badiou has written two "Manifestos for Philosophy." The first Manifesto for Philosophy was completed in 1989, shortly after Being and Event. At that time, postmodernism and post-structuralism were in vogue; thinkers of that era were constantly announcing the "death of philosophy." Lyotard turned philosophy into a "grand narrative" to be denounced; Bourdieu and Baudrillard both claimed to be practicing sociology, avoiding philosophy and metaphysics like the plague. Following the anti-philosopher Nietzsche, philosophy suffered unprecedented skepticism. Badiou went against the current; what he needed to do was rectify the name of philosophy—not only does philosophy exist, but we are already inseparable from it because we exist within different truth procedures.
However, after Logics of Worlds, Badiou once again wrote a Second Manifesto for Philosophy in 2009. By then, the problem of the age had suddenly shifted. People no longer declared the death of philosophy; on the contrary, many believed philosophy was everywhere—there was a "philosophy of stock trading," a "philosophy of cooking," a "philosophy of making money," a "philosophy of managing people," and so on. In Badiou's view, this phenomenon was precisely the "evil fruit" of the previous trend; because postmodernism and post-structuralism had dissolved the legitimacy and justification of philosophy, they allowed "philosophy" to become an empty label that could be pasted anywhere. The distinction between the "philosophical concept" and "doxic opinion" in the Platonic sense no longer existed. The popularity of "anti-philosophy" led people to stop believing in the existence of truth or the fascination with eternity. Philosophical concepts were deconstructed as grand narratives; people valued only what was immediate, fixing their gaze upon the flickering existence of a transient world, valuing only the fleeting glimmer in a "single ladle of water or a bowl of porridge." This way of viewing all things in the world as restricted solely to the "finite," Badiou terms "democratic materialism." In Logics of Worlds, Badiou pointed out quite explicitly that the axiom of democratic materialism is: "There are only bodies and languages." [6] Whether it be the body (as in Merleau-Ponty) or language (as in the later Wittgenstein), both are finite. They cannot directly help us break the bounds of the finite to reach the infinite; instead, they make us linger eternally on "this shore" of fragmented and scattered concrete bodies and linguistic shards. Here we hear the echo of the famous line from Stefan George's poem cited by Heidegger: "Where the word breaks off, no thing may be." [8]
Clearly, Badiou seeks to do the opposite. It is not that he distrusts poetry; in fact, Badiou has a profound literary background and is adept at appreciating and critiquing poetry, possessing deep expertise in the works of Mandelstam, Pessoa, Rimbaud, and Paul Celan. However, Badiou believes that the linguistic fragments of poetry do not lead us toward "unconcealment" (aletheia) or the "infinite absolute" in the Heideggerian sense; they only lead us further astray in the shards of words. Thus, Badiou proposes the path of mathematics to find anew the route to the infinite. In his Second Manifesto for Philosophy, Badiou announced this plan again: "Being is a multiple derived from the empty set, and only mathematics is the thought of being qua being. In short: ontology, and the discourse on being in the etymological sense, has been historically realized in an absolute way only by the mathematics of the multiple." [9] However, the problem for philosophy today is precisely that philosophers are too far removed from mathematicians. While calculus was invented by Newton and Leibniz, and topology appeared as early as the 18th century, modern philosophers are often unfamiliar with set theory following Cantor. This is precisely why Badiou insists on reconstructing philosophy through mathematics, particularly set theory.
In the introduction to The Immanence of Truths, Badiou points out that 20th-century mathematics provided philosophy with three important concepts that have not received sufficient attention from philosophers. The first is the "generic set" (ensemble générique), which lays the foundation for all thought of universality; if one is to reach the infinite, there must be a generic set. However, Badiou believes his discussion of the generic set was completed in Being and Event and is not the task of The Immanence of Truths. The second concept is the "sheaf theory" of the 20th-century German mathematical genius Grothendieck, which Badiou discussed specifically in Logics of Worlds. The final concept—and the one discussed in The Immanence of Truths—is the "large infinite" (grand infini). In Badiou's own words, "this concept of the large infinite, which is an 'immanent approximation' (approximation immanente) of the absolute, will influence us for a long time to come." [5] This "large infinite" involves the absolute and the infinite, which for Badiou are the foundations for reconstructing the entire philosophical ontology.
To reconstruct a philosophical "large infinite" and allow philosophy to once again face "the real" (le réel), a brand-new "dialectical materialism" (dialectique matérialiste) is required. Badiou declares very clearly: "All truths are produced immanently; no one can enclose them within the dominant finite forms. Only within the 'dialectical materialism' of an open philosophy, and only within its 'conditions of truth' (science, politics, love, art)—starting from this concept—can one discover the possibility that does not lead to its opposite, and does not wander in vain in the shadows enclosed by finite scenes." [5] To be precise, this new dialectical materialism fights on two fronts. On one hand, Badiou points the spearhead of his critique toward the "absolute of transcendence"—notions that reduce all finite modes to a transcendence external to the world (in the Middle Ages, this transcendent being was "God"; in modern philosophy, it might be "Substance" or "Nature"). In short, there is an "Absolute One" that serves as the reference frame for philosophy and the various branch sciences of the world. On the other hand, with the rise of postmodernism and the linguistic turn among modern "sophists"...
...the rise of sophistry [10]. With the deconstruction of concepts like the absolute and truth by postmodernism and post-structuralism, philosophy is regarded as a kind of meaningless discourse, thereby confining "meaning" to finite existence ("there are only bodies and languages"). This is what Badiou calls "democratic materialism."
IV. Mathematical Form and the Possibility of Truth
Regarding the first concept, Badiou devotes a large portion of Being and Event to proving that "the One is not" (l’Un n’est pas). Badiou points out that in the original, real world, there are only multiplicities and manifolds—the existence of a non-differential "many." It is only when we invent a structure that "counts-as-one" [11] that we can see the world as "One" (Un). Therefore, Badiou says: "The One is not, there is only the count-as-one. The One, being an operation, is never a presentation. It must be said very seriously: ‘one’ is a number." [19] Since the "one" is a counting structure, this structure eternally exists as an exception relative to the real world. Badiou states: "There will always be an exception relative to the counting structure; it appears in an uncalculable form, a concrete rupture [12], which I call an event." [10]18 This necessarily implies that no count-as-one structure can actually exhaust all of reality; thus, reality possesses an eternal exception relative to the "One."
However, in the history of philosophy and theology, there has been a persistent desire to reduce all things in the world to a single origin—the "Great One" [13]—such as the Christian God or the "Nature" of the natural law school. From the perspective of this transcendental philosophy, the origin of the world is an external "One" outside of which no exception exists. Whether it is Spinoza’s Substance or Hegel’s Absolute Spirit, both are variants of this "Great One"; in reality, neither escapes the relationship Badiou identifies between the real world and the "Great One" as a counting structure. Badiou believes the contribution of the mathematician Cantor lies not only in his proposal of set theory but in his "distinction between the infinite and the One; there can exist an infinity in the form of a 'multiplicity-without-one' (un multiple sans-un)." [10]13 The infinity we pursue is not necessarily the "Great One" or a single source; on the contrary, the infinite or the absolute can also appear in the form of multiplicity and non-differentiation. We cannot view the primordial real world, or the infinite world, as a coherent unity, nor is there a concept of a "constructible universe" [14] (as in Gödel). If we are to reconstruct a materialist dialectic on a new basis, we must abandon this concept of the "Great One." This is because once we assume a primordial "Great One" exists, we have already presupposed an abstract, formalized structure—a structure that allows us to link the real world together into a "Great One" that is understood and grasped within our cognition.
This is a new concept of the infinite, and it is the concept of the "Great Infinite" [15] that Badiou repeatedly emphasizes. This is not the "bad infinity" in the Hegelian sense, but a world full of discontinuities and plural truths (vérités). Relative to our counting structure, this world always possesses an "outside." The purpose of philosophical research is to find the possibility of exceptions outside the structure of our state of the situation—those absolutes that abruptly appear in the architecture of our knowledge and language in a paradoxical form. In this sense, the infinite carries a destructive and devastating significance. The true infinite is a kind of impossibility: "The key point is that it is an impossible paradigm relative to the image of the possible world originally given; it is an unknowable infinite potential, a novelty with devastating repression, and a paradoxical construction." [10]225
The second enemy of the new materialist dialectic is clearly "democratic materialism." Although Badiou critiques "democratic materialism" extensively in Logics of Worlds, rewriting its axiom "there are only bodies and languages" into the axiom of materialist dialectic—"besides bodies and languages, there are also truths" [11]16—the problem remains. Unlike the transcendental idea of the absolute "Great One," democratic materialists do not believe in anything absolute. Consequently, they not only reject concepts like totality, Absolute Spirit, or God as the ultimate "Great One," but they also reject the absolute and the infinite themselves. They believe the only reliable things are our bodies and the language we use to speak. This is why Wittgenstein said "whereof one cannot speak, thereof one must be silent," because language cannot guarantee certainty outside its own limits. For Badiou, "democratic materialism" represents another extreme: locking oneself into a fragmented finitude where one believes in nothing except our finitude, the bodies we can touch, and the languages we can speak.
Badiou believes that in Being and Event, he handled the problem of the "One" and the transcendental well, opening the path for a formal ontology of the event; subsequently, Cantor’s set theory and Cohen’s forcing method resolved the historical problem of the "Great One" found in Gödel’s constructible universe. However, regarding the latter—the critique of "democratic materialism"—this task was not completed in Logics of Worlds. Although Badiou reconstructed a phenomenology of the event from the perspectives of category theory and topology, establishing the possibility of the event within finite things and objects, this phenomenology of the event is not sufficient to thoroughly critique the ideology of "democratic materialism." Here, the problem Badiou needs to solve is: finitude. If one is to address the infinite and the absolute while avoiding falling back into the ruts of transcendentalism and the "Great One," one must conduct a radical ontological critique of the concept of finitude. In this regard, Badiou’s student Quentin Meillassoux provided excellent groundwork in After Finitude, and many of Badiou’s discussions reference Meillassoux's research.
How exactly does Badiou handle the problem of finitude? In the introduction to The Immanence of Truths, Badiou offers a startling claim. Relative to "democratic materialisms" like postmodernism, post-structuralism, and the late Wittgenstein, the most certain things are those closest to us: bodies and languages. We can only touch this certainty; we can neither perceive nor speak of anything beyond it. Even in less radical versions (such as Habermas’s attempt to handle the finitude of bodies and language under communicative rationality to build a public framework through intersubjectivity and discourse ethics), there is a firm belief that what is closest to us is most certain, and we can only obtain greater publicity and universality from these certainties. However, Badiou argues: "Finitude does not exist; it is merely the result of an infinitely multiple operation." [10]15
This is indeed an assertion that diverges vastly from our usual understanding. Furthermore, Badiou even believes that contemporary "democratic materialism," as a trend of anti-philosophy and sophistry, actually establishes an "ideology of finitude." One of the goals of The Immanence of Truths is "a critique of finitude, which has become the most central fetishistic ideology of our era." [10]16
This is a very important judgment. Just as Badiou proposed in Being and Event that the "one" is an operation—a result of the count-as-one—we must understand how finitude is also a result. For example, when we say "there is an apple here," the apple is finite. But we speak of the apple not because it possesses this finite quality inherently, but because within a given classificatory framework (a count-as-one), we view it as an apple separated from its surroundings. In other words, our seeing the apple as an apple is precisely because a framework of representation existed beforehand—a count-as-one framework. Only within this framework can we have knowledge of the apple or perceive its existence. Otherwise, the apple is merged into the given world, unable to be separated and made independent as a finite object. All the bodies we see and the languages we speak rely on an invisible framework, which constitutes the foundation of our understanding and grasp of the world, as well as the basis upon which the world of objects we see and understand operates. In this sense, we must say that the apple as an individual finite object is actually the result of a classification system we have established, which in turn is subject to a higher-order formal framework. Under this framework (the essence of which is the architecture of the count-as-one or the "Great One"), beings are separated from the environment of multiplicity and independent as finite beings.
If finite classification belongs to a given framework (in set-theoretic terms, the elements of the "apple" belong to a part or subset of a situation), then the problem lies not in the elements, but in the "parts" of the classification. How the world is counted and classified becomes our most basic way of grasping and understanding it. Our so-called liberation is not purely a matter of breaking visible boundaries—as Deleuze advocated—nomadically wandering into the "Great Outside" and using lines of flight to break the ordered world. However, Deleuze and Guattari both forgot that the subjects or objects in the contemporary world are actually carved out under the mold of a formal framework. That is to say, their bodies and languages are highly compatible with the schizophrenic apparatus of capitalism. There is no such thing as nomadism outside the territorial jurisdiction of capitalism, because that kind of nomadism would represent breaking individualizing distinctions and the rules of finite counting—it would mean the death of the individual: no uniqueness, only the silence of life.
If the fetishistic ideological form of finitude is to be broken, Badiou believes there is only one path: mathematics—formalized mathematics. [10]24 That is to say, we must first create a form; even if this form has no object, it can become "incarnated" [16] in the existing real world, materializing into objects of a new form. Badiou consistently believes in the revolutionary effect of the event because the event is an absolute exception. Since the original structure of the state of the situation cannot perform a count-as-one calculation on it, it necessarily means a new structure must be created to completely replace the original one. This is revolution in Badiou's sense: a formal revolution. Taking mathematical form as an example again: in ancient Greece, people understood the distinction between rational and irrational numbers and believed they covered the entire range of numbers. However, in mathematics, some problems still could not be solved within the range of all rational and irrational numbers. The most famous was the problem of the solution to the quadratic equation $x^2 + 1 = 0$. The great Indian mathematician Bhāskara believed this equation had no solution because a negative number cannot be an even power of any number. It was not until the 16th century that the Italian mathematician Cardano spoke of a new concept, the "imaginary number" (nombre imaginaire), thereby moving beyond the "real numbers" composed of rational and irrational numbers...
This forms a concept in opposition to real numbers (nombre réel). It is noteworthy that in the natural world, we do not actually discover the concept of "imaginary numbers" because we have found some kind of imaginary numerical object. The emergence of the "imaginary number" concept is, in fact, an infinite formal creation. Before Gerolamo Cardano invented the concept of the imaginary number, we could not imagine the solution to the equation $x^2 + 1 = 0$. Once the form of the "imaginary number" appeared, not only could we find the square roots of negative numbers, but more importantly, a new mathematical form was generated.
In fact, Badiou uses this method to respond to those critics who argue that within the realm of science, he values only mathematics while neglecting physics and chemistry. Indeed, in Badiou's early works, mentions of revolutions in the field of science almost always refer to mathematical revolutions. The key figures of scientific revolution he enumerates are mostly mathematicians such as Descartes, Gauss, D’Alembert, Galois, and Grothendieck; he basically never discusses contributions in physics or chemistry (he mentioned the chemist Lavoisier only once by chance, but he was not discussing Lavoisier's contributions to chemistry, but rather Lavoisier’s fate during the French Revolution). If one starts from Badiou's critique of the "ideology of finitude," it is easy to understand why he dislikes talking about concrete sciences like physics and instead directly discusses mathematics. In The Immanence of Truths, Badiou points out: "We see that for years I have engaged in arduous mathematical work, and I have concluded that we must insist that the root of all problems is the theory of the multiple (set theory); however, we also acknowledge that theories regarding particular worlds, such as physics, achieve formalization through thinking in mathematical categories."[17] To a certain extent, this is both a prejudice toward concrete sciences like physics held by Badiou—who was trained in mathematics and philosophy—and a consequence of the ontology he himself established.
Conclusion
Following the discussions in Being and Event and Logics of Worlds, Badiou reveals his ambition to us step by step. If a future society exists, this society is not an external realm, nor is it a return to Eden like a "religion"—that is, a return to the embrace of the One and the Absolute Spirit—but rather the creation of a new form under the impact of the event as an exception, even if this new form is an empty set. However, because the new form is a generic truth procedure [17] following the event, it constructs a new combination (a generic set), continuously allowing the subject to incorporate into the new set, becoming "the Word made flesh" under the mathematical form of the new set, and concretizing into a new object. We can return once more to Badiou's understanding of Mao Zedong’s "a single spark can start a prairie fire" [18]. The reason a single spark can start a prairie fire is not because it represents a transcendent One in itself, but because it belongs to the generic procedure of a new truth formed by the subject after the event. Because it conforms to the generic procedure, the single spark becomes a concrete representation under this new set and new operation; the expansion and diffusion of this representation can then spread throughout the entire world. This is Badiou's new materialist dialectics, and it is also his way of envisioning a beautiful future society.