Lan Jiang: Infinite Forms and the Spark of Truth — On Materialist Dialectics in Badiou's *The 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 (1989) and Logics of Worlds (2006)—the second volume of the Being and Event trilogy—he employed his unique mode of mathematical philosophy in an attempt to locate the possibilities for moving toward a future communist society. Twelve years after Logics of Worlds, he released this 700-plus-page volume, identifying it as the third installment of the series. Perhaps Badiou hopes that in The Immanence of Truths, he can bring his lifelong philosophical project to a conclusion: namely, having touched the "infinite"—a true infinite—how can we effect change in the world, and how is such change possible?
I. The Path to the Infinite
"A single spark can start a prairie fire." Badiou is indeed very fond of this famous quote from Mao Zedong. He has referenced it numerous times throughout his oeuvre, from his early work Theory of Contradiction to Theory of the Subject (1982), and into the conclusion of Logics of Worlds. It is difficult to determine to what extent Badiou, when citing this aphorism, understands the historical reality of the Chinese Workers’ and Peasants’ Red Army in 1930—how they persisted in their ideals of national liberation and communism amidst the grueling environment of the Jinggang Mountains [1], thereby finding hope within a "single spark." Mao’s creation 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 idealistic expression: finding hope for a future ideal society within a faint glimmer of light.
It was perhaps this idealistic expression that catalyzed Badiou’s philosophical sensibilities. He recognizes that while Mao’s "single spark" was directed at a specific environment of arduous revolutionary struggle, it essentially constitutes a philosophical concept—the idea of perceiving an infinite future from within a finite spark. In the introduction to The Immanence of Truths, Badiou cites this famous passage once more. However, expressing dissatisfaction with the translation in the French edition of the Selected Works of Mao Zedong—Une seule étincelle peut commencer un feu de prairie—he offers his own translation: l’étincelle qui jaillit du frottement localisé d’infinités disparates [2].
Badiou’s new translation does not differ much in its first half, yet the two sentence structures are worlds apart. The former uses the active voice, where the single spark is the cause and the prairie fire is the effect; it is the ignition of the spark that illuminates the entire world with a future universal flame. This is a mode of ascending from the finite to the infinite—a mode Badiou endorsed in his earlier thought. In The Immanence of Truths, however, the relationship in this sentence has shifted. It can no longer be translated directly as "a single spark can start a prairie fire," but rather as "the spark is a concrete flare erupting from an omnipresent infinity." In this new formulation, the spark is no longer the cause but the result triggered when the infinite descends into the human realm. This logic is a kind of "inverted romanticism": the form of the prairie fire exists first, and only then does the concrete spark appear.
In Badiou’s view, when we are able to see the spark, it signifies that the world has already changed. For Communists and revolutionary movements, the task is not to artificially manufacture sparks, but to be faithful to the spark. To be faithful to the spark is to confront a more significant event, and the event is precisely the result triggered 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, tranquil sea under the sun, but the waves swept up by a violent storm. This is the famous line from the end of Valéry’s poem:
Fresh air! A power from the sea’s deep springs Restores my soul... O salt-fragranced might! Let’s run to the wave and leap back reborn! [3]
Truth and the Absolute are never the tranquil surface of a sea under a gentle breeze. Such tranquility is more like a montage-like mask or concealment; it brings order and peace, allowing people to function year after year, day after day, according to established apparatuses, treating this mode of movement as natural, universal, absolute, and true. Badiou wants to tell us: no, beneath the sun and breeze, the sea also holds storms and the poetic power of "jagged rocks piercing the sky, frightening waves crashing on the shore, churning up a thousand heaps of snow" [4]. If we remain only beneath the surface of the tranquil sea, we degenerate into a finite existence. Only within the "salt-fragranced might" torn asunder by the storm can we see the Real and the infinite. This is why Badiou calls us to "run to the wave" in Logics of Worlds.
There is a more vital reason: once we touch the Real and the Absolute, once we face those infinities that cannot be easily confronted, our lives are made entirely new. The waves bring not only "saltiness," "tempestuous surges," or "terror," but a "rebirth." This is why Badiou appreciates the "fresh air of the sea" in The Graveyard by the Sea. We must not understand this "fresh air" simply as romantic sweetness or a triumphalist song of aspirational progressivism, but as a sublimation when facing the true infinite. In this sublimation, "my soul is restored," and when I face the world again, "leaping back will be a rebirth."
Whether it is Badiou’s rewritten "single spark" or the closing of Valéry’s "The Graveyard by the Sea," both express a new consciousness: if the subject once touches the infinity that is normally inaccessible, how does the world—or rather, the world as the representation of being—appear when facing it again? Introducing The Immanence of Truths, Badiou repeatedly emphasized that it is an "inverted" book. If Being and Event and Logics of Worlds were works exploring the path toward the infinite from the perspective of finite forms and worlds, The Immanence of Truths stands 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 sunlight from The Republic to illustrate the questions Badiou raises 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 treat the flickering shadows on the cave walls as reality, becoming subordinate to these shadows and forming "doxa" (opinion). However, once someone leaves the cave and sees the sunlight outside, they immediately sense the distinction 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... When one was freed and suddenly compelled to stand up and turn his neck around and walk and lift up his eyes toward the light, and in doing all this felt pain, and because of the dazzle was unable to see the things whose shadows he had seen before—what do you think he would say if someone told him that what he had seen before was an illusion, but that now, being somewhat nearer to reality and turned toward more real beings, he saw more correctly?" [5]
In this passage from Plato, we perceive two things.
First, facing Truth is clearly more painful than facing the shadows on the wall. People do not voluntarily face Truth; they can only turn around and face the sunlight outside the cave under the pressure of certain procedures and forces. This is why Badiou, since Theory of the Subject, has consistently emphasized that the subject is "scarce." Those who can see the shadows of truth are a small minority, and those willing to let go of the cave's doxa, to voluntarily march toward Truth, and to unite with that generic procedure of truth, are even rarer. Mathematics and philosophy are both such procedures, but those willing to use mathematics to touch the path leading to the Absolute and Truth are truly scarce.
In other words, there is no shortage of "philosophy" today—the kind of thought that disguises ordinary doxa as philosophy. In Badiou’s view, this "philosophy" is actually a form of anti-philosophy; Nietzsche, Wittgenstein, and Lacan are famous masters of anti-philosophy. For instance, when Wittgenstein says, "Whereof one cannot speak, thereof one must be silent," Badiou sees this "conclusion" as precisely requiring us to remain confined within finite doxa, suspending all paths leading out of the cave. In evaluating Wittgenstein’s anti-philosophy, Badiou notes: "The anti-philosophical act consists in letting things show themselves, because 'that there is' [the 'there is'] is precisely what prevents us from speaking with true propositions. If Wittgenstein’s anti-philosophical act can be appropriately called 'archi-aesthetic,' it is because 'letting be' cannot express its pure being and clarity through propositions; this clarity of pure being is something ineffable, appearing only in works of non-thought forms (for Wittgenstein, music is certainly the most appropriate form)." [6]
Badiou believes Wittgenstein abolishes the form of philosophical reflection, arguing that philosophy cannot lead us to Truth and instead turns that Truth into an ineffable "archi-aesthetic." This effectively makes him a contemporary "sophist" who confuses opinion with truth and makes people maintain a sense of mystery regarding the Absolute and the infinite outside the cave. By preserving this mystery within a primordial, ineffable sensibility, the path to the outside world is severed. Thus, compared to the more intoxicating form of art, people seem to forget Plato’s warning: the path toward the external Truth is painful. In Badiou's view, only mathematics in the strict sense serves this function: "Only mathematics allows us to enter into a unified thought of the visible." This is why Plato supposedly inscribed "Let no one ignorant of geometry enter here" above his Academy's entrance. Mathematics is painful, but once one touches that seemingly inaccessible truth through the mathematical path, a greater pleasure is obtained.
Second, in The Immanence of Truths, Badiou actually endows Plato’s cave allegory with a new meaning, which is the primary theme of the book. If Being and Event lay down a formal path to the outside of the cave, and Logics of Worlds dealt with how to make finite beings walk this path—realizing the possibility of moving toward the infinite from a finite perspective—then finally The Immanence of Truths is the story of the return: how to return to the cave after having attained the infinite. Because the forms of truth have been touched and the infinite has been contacted, the things in the cave still exist, but they are no longer "letting themselves show themselves" in the Wittgensteinian sense. For Badiou, there is no way for things to "show themselves." Since Being and Event, Badiou has maintained that the reason we see a thing as a thing—"being as being" (l’être en tant qu’être)—is precisely because it is represented by a structure of "count-as-one." Real being is "indifferent" being; it is multiplicity and manifold. We can only distinguish things and objects from one another through the conceptual tools or functions of the "count-as-one." It is only with these functions and concepts of distinction that we can understand certain things as "objects."
) understood as objects (objet), and the human who creates these concepts and functions is the subject (sujet). In this way, Badiou actually reverses this 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, all mediated by concepts, becoming the result of a certain "count-as-one" operation. Conversely, we can understand why Badiou, in The Immanence of Truths, transforms Mao Zedong’s "a single spark can start a prairie fire" [7] into "the single spark is a concrete flare erupting from an omnipresent infinity." That is to say, as the truth procedure outside the cave changes, the objects within the cave change accordingly, becoming "single sparks" facing the truth. In other words, once one returns to the cave, the real world has not changed; what has changed is the way used to differentiate and structure the world—that which was called the "state of the situation" in Being and Event and the "transcendental T" in Logics of Worlds. The generic procedure of truth alters the entire formal framework, thereby allowing objects to appear within a new architecture and manifest in a new state. All objects are immanent to form, immanent to the newly opened generic procedure of truth. Any object that exists and appears is an object under a generic procedure; in this sense, we can say all objects are immanent to (immanent à) truth. This is precisely why Badiou titled his 2018 book The Immanence of Truths and defined it as 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 structured by form, immanent to the framework of truth. The object is being realized by the formal framework or the generic procedure of truth; it can be represented, it can be counted, and it can be incorporated into existing operational frameworks. However, there remains a certain pure multiplicity—that is, the being of multiplicity that does not appear through the formal framework. This purity is not only unspeakable (as Wittgenstein suggested) but even imperceptible; even Wittgenstein’s "proto-sensibility" is powerless before such pure multiplicity or undifferentiated multiplicity. Thus, through his own truth procedure—the immanence of truth—Badiou distinguishes objects intervened upon by the formal truth procedure from the indiscernible pure multiplicity, the manifestation of pure multiplicity. Badiou believes the goal of philosophy certainly cannot be, as Wittgenstein suggested, to remain silent about the unspeakable. Instead, we must create a form, create a generic procedure of truth, so that what was originally unspeakable can appear under a new form. Thus, Badiou notes: "This proves that the immanence of truths, at the most schematic level and within the most formal truth, inevitably offers the possibility for human thought, through ordeal and suffering, to encounter the infinite." From this perspective, the reason Badiou trusts mathematics so deeply is precisely that mathematics can provide us with infinite possibilities, allowing new possibilities to descend into the cave, so that the manifolds existing there can be liberated from original doxa [8] and attain their own potential.
III. From Language to Mathematics
To achieve this goal, Badiou believes it is necessary to re-establish the prestige of philosophy. In 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 "grand narratives" to be attacked, while Bourdieu and Baudrillard both claimed to be conducting sociological research, avoiding philosophy and metaphysics like the plague. Since the anti-philosopher Nietzsche, philosophy has suffered unprecedented skepticism. Badiou went against the current; what he needed to do was to vindicate 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 wrote another book, Second Manifesto for Philosophy, in 2009. By then, the problems of the age had suddenly changed. People no longer proclaimed 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 governing people, and so on. In Badiou's view, this phenomenon is precisely the "evil fruit" of the former: because the legitimacy and justification of philosophy were dissolved by postmodernism and post-structuralism, philosophy became an empty label that could be pasted anywhere. The Platonic distinction between philosophical ideas and common opinion (doxa) no longer exists. The prevalence of anti-philosophical trends has led people to stop believing in the existence of truth or the obsession with eternity. The concept of philosophy is deconstructed as a grand narrative, and people place more value on the immediate, fixing their gaze on the fleeting world of existence, valuing only the ephemeral glimmer in "a ladle of water or a bowl of porridge." [9] This way of viewing all things in the world as strictly limited to the finite is what Badiou calls "democratic materialism." In Logics of Worlds, Badiou explicitly pointed out that the axiom of "democratic materialism" is that "there are only bodies and languages." Whether it be bodies (as in Merleau-Ponty) or language (as in the late Wittgenstein), both are finite. They cannot directly help us break through the boundaries of the finite toward the infinite; instead, they leave us eternally lingering on "this shore" of fragmented and scattered concrete bodies and linguistic shards. Here once again echoes the famous line from Stefan George's poem cited by Heidegger: "Where the word breaks off, no thing may be."
Clearly, Badiou attempts to do the opposite. It is not that he distrusts poetry; in fact, Badiou has profound literary roots and is highly skilled at appreciating and critiquing poetry. He possesses deep expertise in the poetry of Mandelstam, Pessoa, Rimbaud, and Paul Celan. However, Badiou believes that the linguistic fragments of poetry do not lead us toward aletheia (unconcealment) or the absolute of the infinite in the Heideggerian sense; they only lead us further astray among the fragments of words. Thus, Badiou proposes a mathematical path to rediscover the route to the infinite. In the 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 historically been fully realized only through the mathematics of the multiple." However, the problem for philosophy today is precisely that philosophers are too far removed from mathematicians. Yet, while calculus was invented by Newton and Leibniz and topology appeared in the 18th century, contemporary philosophers are often unfamiliar with set theory post-Cantor. This is why Badiou insists on reconstructing philosophy through mathematics, especially set theory.
In the introduction to The Immanence of Truths, Badiou points out that 20th-century mathematics provided three important concepts for philosophy that have not received sufficient attention from philosophers. The first concept is the "generic set" (ensemble générique), which establishes the foundation for all universalist thought; to access the infinite, one must have the generic set. However, Badiou believes he already completed the discussion of the generic set in Being and Event; this 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, an immanent approximation (approximation immanente) of the absolute, will influence us for a long time." For Badiou, this large infinite involves the absolute and the infinite, serving as the foundation for reconstructing the entire philosophical ontology.
To reconstruct the philosophical large infinite and make philosophy face "the real" (le réel) once again, a brand-new dialectical materialism is required. Badiou explicitly declares: "All truths are produced immanently; no one can enclose them within dominant finite forms. Only within the dialectical materialism of an open philosophy, and only within its conditions of truth (science, politics, love, art), can one discover, starting from this concept, the possibility of not turning into its opposite, of not wandering in vain in the shadows enclosed by finite scenes." More precisely, this new dialectical materialism fights on two fronts simultaneously. On one hand, Badiou directs his critique at the "absolute of transcendence." This perspective reduces all finite modes to a transcendence external to the world [in the Middle Ages, this transcendent being was God (Dieu); in modern philosophy, this transcendent being might be Substance (Substance) or Nature (Nature)]. In short, there is an "absolute One" acting as the frame of reference for philosophy and the various specialized sciences of the world. On the other hand, with the rise of modern "sophistry" (sophistique) such as the postmodern and linguistic turns, and the deconstruction of concepts like the absolute and truth by postmodernism and post-structuralism, philosophy is viewed as a 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 perspective, Badiou spent a great deal of space in Being and Event 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, only the being of undifferentiated multiplicity. Only when we invent the structure of the "count-as-one" can we perceive the world as "One" (Un). Therefore, Badiou says: "There is no One, only the count-as-one. The One, as an operation, is never a presentation. It must be said very seriously: 'One' is a number." Since the One is a counting structure, this structure eternally possesses an "exception" relative to the real world. Badiou says: "There will always be an exception to the counting structure; it appears in an uncomputable form, a concrete rupture that I call an 'event'." This necessarily means that any count-as-one structure cannot, in fact, exhaust all of the real; thus, the real stands as an eternal exception to the "One." However, in the history of philosophy and theology, there has always been a desire to reduce all things in the world to a single origin—the One—such as the Christian God or the "Nature" of the Natural Law school. In this transcendental philosophy, the origin of the world is an external One, and outside this One, no exception exists. Whether it be Spinoza’s "Substance" or Hegel’s "Absolute Spirit," both are variants of this One, but in reality, they have not escaped the relationship Badiou identifies between the real world and the One as a counting structure. Badiou believes that the contribution of the mathematician Cantor lies not only in his proposal of set theory but in his "distinguishing the infinite from the One, allowing for the existence of a 'multiplicity-without-one' (un multiple sans-un)."
) "infinite form." The infinity we pursue is not necessarily the One, nor a singular source; on the contrary, the infinite or the absolute can also appear in the form of the multiple and the undifferentiated. We cannot view the original real world, or the infinite world, as a coherent unity, nor is there a concept of a constructible universal set like that of Gödel. If we are to construct a dialectical materialism on a new foundation, we must discard this concept of the One. This is because, in assuming the existence of an original One, we have already presupposed an abstract formalized structure—a structure that allows us to link the real world together into a One that is understood and grasped in our cognition. This is a new concept of infinity, also the concept of the great infinite 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, there is always an exterior to this world. The purpose of philosophical research is to find the possibility of the exception outside the structure of our state of the situation—those absolutes that appear abruptly in paradoxical forms within our frameworks of knowledge and language. Thus, infinity carries a sense of subversion and destruction; true infinity is an impossibility, i.e., "the point is that it is an impossible paradigm relative to the initially given image of the possible world; it is an unknowable infinite potential, a novelty with devastating pressure, and a paradoxical construction."
The second enemy of the new dialectical materialism 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 dialectical materialism—"except that there are also truths"—the problem with democratic materialism persists. Unlike the transcendental idea of the absolute One, democratic materialists do not believe in anything absolute. Consequently, they not only reject the concept of the One as the ultimate absolute (such as totality, Absolute Spirit, or God), but they also reject the absolute and the infinite itself. They believe the only reliable things are our bodies and the languages we use to speak. This is why Wittgenstein spoke of "keeping silent," because language cannot guarantee certainty beyond its own limits. For Badiou, democratic materialism represents the other extreme: locking oneself into a fragmented finitude. Beyond our finitude, beyond the bodies we can touch and the languages we can speak, they believe in nothing else.
Badiou believes he handled the problem of the One and the transcendental well in Being and Event, opening the path for a formal ontology of the event, where Cantor’s set theory and Cohen’s forcing method resolved the problem of the One in the history of philosophy (represented by Gödel’s constructible universe). However, the critique of the latter—democratic materialism—was not completed in Logics of Worlds. Although Badiou reconstructed a phenomenology of the event from the perspective of category theory and topology, constructing the possibility of the event within finite things and objects, this phenomenology 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 a relapse into the ruts of transcendentality and the One, one must conduct a radical ontological critique of the concept of finitude. In this regard, Badiou’s student Quentin Meillassoux performed excellent groundwork in After Finitude, and many of Badiou’s arguments refer to Meillassoux's research. How, then, does Badiou handle the problem of finitude?
In the introduction to The Immanence of Truths, Badiou offers a startling claim. Relative to postmodernism, post-structuralism, and the "democratic materialism" of the late Wittgenstein, these schools believe that the most certain things are those finite objects closest to us: bodies and languages. We can only touch this certainty; regarding anything outside of it, we can neither perceive nor speak. Even in less radical versions (such as Habermas attempting to handle the finitude of body and language through communicative rationality to construct a framework of publicity that transcends the finitude of individual bodies and languages), there is a firm belief that what is closest to us is most certain, and we can only obtain greater publicity and universality from the most certain bodies and languages. However, Badiou argues: "Finitude does not exist; it is merely the result of an operation on the infinite multiple." This is indeed a claim that differs vastly from our usual understanding. Furthermore, Badiou even believes that contemporary democratic materialism, as an anti-philosophical trend and sophistry, actually establishes a "finitism ideology." One of the goals of The Immanence of Truths is "the critique of finitude, which has become the most central fetishistic ideology of our era."
This is a vital judgment. When Badiou proposed in Being and Event that the "One" is an operation—a result of the count-as-one—how are we to understand finitude as also being a result? For example, when we say there is an apple here, the apple is finite. However, we speak of the apple not because the apple itself possesses this finite quality, but because we regard it as an apple separated from its surroundings within a given classificatory framework (a count-as-one). In other words, the reason we see the apple as an apple is precisely because there exists a prior representational framework—a count-as-one framework. Only within this framework can we have any knowledge of this apple, and only then can we see or perceive its existence; otherwise, the apple is fused into the given world, unable to be separated from that world and stand independently as a finite object. All the bodies we see and the languages we speak depend on an invisible framework, which constitutes the basis for our understanding and grasping of the world, and the foundation upon which the world of objects we see and understand functions. In this sense, we must say that the apple as an individual finite object is actually the result of the 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 count-as-one, or the architecture of the One), beings are separated from the environment of the multiple and stand independently as finite beings.
If finite classification belongs to a given framework (in set-theory terms, the elements of the apple belong to a part or subset of the situation), then the problem lies not with the elements, but with the part that classifies. How the world is counted and classified becomes the most fundamental way we grasp and understand it. Our so-called liberation is not purely a matter of breaking visible boundaries—as Deleuze advocated—to go "nomadizing" in the Great Outdoors or using lines of flight to break the ordered world. However, both Deleuze and Guattari forgot that 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 nomadism outside the territoriality of capitalism, because such nomadism would represent the breaking of individualized 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. That is, we must first create a form; even if this form has no object, it can be "incarnated" in the actual, existing world, becoming concrete as an object of a new form. Badiou consistently believes in the revolutionary effect of the event because the event is an absolute exception. The structure of the original state of the situation cannot perform a count-as-one calculation upon it, which necessarily means a new structure must be created to thoroughly replace the old one. This is revolution in Badiou's sense: a formal revolution. Taking the form of mathematics 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. Yet, in mathematics, there were still difficulties that could not be solved within the range of rational and irrational numbers, the most famous being 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 the 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), forming a concept in opposition to the "real numbers" (nombre réel) composed of rational and irrational numbers. Notably, in the natural world, we did not discover the concept of "imaginary numbers" because we found some kind of imaginary object. The appearance of the concept was, in fact, an infinite formal creation. Before Cardano invented the concept, we could not imagine a solution to $x^2 + 1 = 0$. Once the form of the "imaginary number" appeared, we could not only find the root of a negative number, but more importantly, a new mathematical form was produced.
In fact, Badiou uses this method to respond to critics who argue that within the realm of science, he values only mathematics while ignoring physics and chemistry. Indeed, in Badiou’s early works, when he spoke of revolutions in the field of science, he almost always referred to mathematical revolutions. The important figures of scientific revolution he listed were mostly mathematicians like Descartes, Gauss, D'Alembert, Galois, and Grothendieck. He basically never discussed contributions in physics or chemistry (he mentioned the chemist Lavoisier only once by chance, and even then, he discussed Lavoisier’s fate during the French Revolution rather than his chemical contributions). From Badiou’s critique of the "ideology of finitude," it is easy to understand why he dislikes discussing concrete sciences like physics and prefers to discuss mathematics directly. In The Immanence of Truths, Badiou points out: "We see that for years I have been 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), although we also acknowledge that theories regarding particular worlds, such as physics, are formalized through the thinking of mathematical categories." To some extent, this is both a prejudice against concrete sciences held by Badiou (who was trained in mathematics and philosophy) and a result of the ontology he has established.
Conclusion
Following the discussions in Being and Event and Logics of Worlds, Badiou reveals his ambition step by step. If a future society exists, it is not an "exterior," nor is it a return to Eden like "religion"—that is, a return to the embrace of the One and the Absolute Spirit. Rather, it is the creation of a new form under the impact of the event-as-exception, even if this new form is an empty set. Because the new form is a generic truth procedure following the event, it constructs a new combination (a generic set), constantly allowing the subject to incorporate into the new set, becoming "incarnated" under the mathematical form of the new set and becoming concrete as a new object. We can return again to Badiou’s understanding of Mao Zedong’s phrase, "A single spark can start a prairie fire." [10] 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 an event. Because it conforms to the generic procedure, the single spark becomes a concrete appearance under this new set and new operation; the expansion and diffusion of this appearance can then pervade the entire world. This is Badiou’s new dialectical materialism and his way of envisioning a beautiful future society.