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Since the mid twentieth century, digital computers have increasingly defined everyday life. Digital devices operate using discrete elements, combining and changing the elements using logic and mathematics. Often small, sometimes imperceptible, these discrete elements may be found in the innumerable pixels illuminating digital images. They may be found in the genetic codes of all biological creatures. And, if one is to believe the so-called Digital Philosophers, discretization may also be found in the most fundamental elements of nature. This essay is an investigation into the mathematical economies of existence, particularly through the digital and the analog. My focus is on the work of Alain Badiou, a philosopher who has closely intertwined mathematics with questions of existence. But what is the relationship between digitality and philosophy? And is Badiou ultimately a philosopher of the digital?

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"Since its very origins," Alain Badiou wrote in Being and Event, "philosophy has interrogated the abyss that separates numerical discretization from the geometrical continuum. . . . From Plato to Husserl, passing by the magnificent developments of Hegel's Logic, the strictly inexhaustible theme of the dialectic of the discontinuous and the continuous occurs time and time again."1

Badiou does not deploy vocabulary like "digital" and "analog" but we may furnish them ourselves: the strictly inexhaustible theme of the digital and the analog occurs time and time again.2 Indeed such themes pervade the history of thought and culture, no matter which terms are used—discontinuous and continuous, number and geometry, quantity and quality, formal abstraction and real materiality, digital and analog, or some other terms entirely—so much so that it would not be an anachronism to label figures like Democritus or G. W. Leibniz digital thinkers, the former with his discrete atoms, the latter with his discrete monads, or someone like Gilles Deleuze an analog thinker, what with his penchant for topology, calculus, qualitative intensity, and real continuity.3

In his masterwork from 1971, System and Structure, Anthony Wilden describes the digital and the analog not just as two modes of mediation but as two varieties of difference:

Analog differences are differences of magnitude, frequency, distribution, pattern, organization, and the like. Digital differences are those such as can be coded into DISTINCTIONS and OPPOSITIONS, and for this, there must be discrete elements with well-defined boundaries.4

These discrete elements with their well-defined boundaries provide some insight into the economies of existence that define daily life in the new millennium. Such economies are ancient, to be sure, as old as exchange, as old as language, as old as the symbolic order itself. These economies structure the relationship between society and the human, between the physical networks of quantitative exchange that scaffold human life and the very experience of that life, from the smallest psychic impulses to the largest cultural phenomena, in short, between the base and the superstructure.5 Yet since the mid-twentieth century, and increasing up to the present day, such economies of existence have been redefined explicitly through computation and digitality. The digital age is thus an age of "discrete elements with well-defined boundaries." Often small, sometimes imperceptible, these discrete elements may be found in the alphanumeric symbols that undergird all digital computation. They may be found in the innumerable pixels illuminating digital images. They may be found in the genetic codes of all biological creatures. And, if one is to believe the so-called digital philosophers, discretization may also be found in the most fundamental elements of nature.6

The question today is thus not so much when and how mathematics has seeped into everyday life, whether in actuarial tables, clustering algorithms, or artificial intelligence. Rather, the question concerns a migration from one kind of technology to another, from analog technology to digital technology, from the mathematics of the continuous real to the mathematics of the discrete symbol. Thus a new scrutiny of math and tech is necessary today, but not just any math and tech: not geometry, calculus, or topology, those [End Page 97] mathematical fields bent on capturing purely analogical phenomena like curves and continuous rates of change; but rather that most elementary branch of mathematics, arithmetic, and indeed that most basic miracle of the human logos, counting 1 . . . 2 . . . 3 . . ..

The shift from analog reality to digital rationality has its disadvantages, to be sure, and many commentators have pitted themselves against the overbearing dominance of rationality in all its avatars. Recall for instance the way in which quantification guided Marx's analysis of the value form, so much so that his famous description of the circulation of capital as Money-Commodity-Money (abbreviated in the formula M-C-M) could easily have been written Quantity-Quality-Quantity or even Digital-Analog-Digital, anachronism notwithstanding.7 And, more generally, recall the way in which the strict devotion to logical operations and mechanistic rationality tends to produce undesirable outcomes, as when, in his Doctor Faustus, Thomas Mann admonishes the folly of "acting foolishly with reason."8 Other commentators strive to undo the binarism altogether, showing that the real is always bound up within the rational and the rational always within the real.

As regards the political potency of digital rationality, the computer age is built around a central contradiction. Following Jean-François Lyotard, postmodernism signaled the general collapse of grands récits. But, following Alan Turing, it was also marked by the rise of "universal machines," bound together through global protocols.9 What results is not so much the collapse of meta-narratives as the creation of a new one, a new infrastructure with a new culture, a new International Style to replace the old. Even Deleuze, that great champion of schizo culture, felt the need to caution future generations against the "control societies" that had emerged by the end of the twentieth century, warning as he did against a kind of generalized political paralysis concurrent with the age of rational machines.10 Is it any surprise, then, that Badiou, the greatest thinker of militancy in our times, emerged in the anglophone world just after the turn of the new millennium precisely at the point when postmodernity outgrew its utility, Badiou's project formulated on the basis of a reinvigorated modernism in which all subjects are militants of some form or another?11 The break is everything in Badiou; and everything, to the extent that it deviates from the state of the situation, is a break.

Hence this is a story about breaks, a story about breaking, a description of a generalized infrastructure of discrete breaks. Sometimes, as in the case of Badiou, it is even necessary to break with the generalized infrastructure of breaks itself—an infrastructure of discretization—in order to reinvigorate the staid, logical world with newfound truth. The prize, then, is discretization as such. The prize is arithmetic digitality as such. What is it? How does it work? What are the key technologies of discretization? Is there a philosophy of the discrete? If so, is there also a philosophy of continuity? These questions and others will [End Page 98] help reveal the nature of contemporary rationality and rationalization, the nature of measurement, what counts as countable, and how to account for what is accountable. Further, the questions help reveal all that gets lost in the counting, all that remains unassimilable to logic, all that is excluded from the symbolic order, whether by accident or by design.

Indeed the generalized infrastructure of discretization may be characterized as much through the negative as through the positive, as much through the many attempts to deviate from it or otherwise invalidate it. Thus alongside Jean-Joseph Goux and his examination of the "general equivalent," one might also consult the various writings on the potlatch or the gift as forms of economic circulation that do not simply reduce to quantification or accumulation, even as they produce and sustain their own symbolic regimes.12 Likewise one might turn toward more appealing sorts of economies, as David Graeber suggests in his book on the mutual exchange of debt.13

On the other hand, contemporary discourse is defined by a series of attempts to complicate or eliminate symbolic exchange altogether, in effect to discard the digital in favor of the analog. Deleuze is the key thinker of pure analogicity, as he attempted to eliminate discrete symbolization in favor of a thoroughly analog landscape of assemblages, affects, and desiring machines.14 But a number of other interesting options also present themselves, from queer and feminist digitality (Homay King), to the "measures" of blackness (Katherine McKittrick), alternative forms of feminist abstraction (Laboria Cuboniks), speculative algorithms (Ezekiel Dixon-Román), or experiments in counter-hegemonic rationality (Eve Tuck and K. Wayne Yang).15 Or if not to problematize the digital in favor of an alternative logic that flows through it, others have proposed a radical suspension of exchange altogether, whether it be Frank Wilderson's particular version of Afro-pessimism in which blackness and whiteness are never "analogs," or François Laruelle's non-philosophy built around irreversibility and what he calls the "unidirectional" axiom of the one.16

In other writings I have attempted to suspend or otherwise problematize the digital. Several years ago I coauthored a "Liberated Computer Language" consonant with some of the aforementioned explorations of counter-hegemonic rationality. In a recent study of Laruelle, I tried to describe a theoretical mode in which digitality does not exist.17 But today's task is a different one. The goal here is not to problematize or suspend digital rationality, but rather to confront it directly. The goal is not to discard the digital in favor of the analog, at least not entirely. I thus refrain from focusing on real continuity and immeasurable reality, or on affect, immanence, experience, aesthetics, empiricism, and ethics. These are the paradigmatic domains of the analog. Instead the focus is on the digital itself and its attendant technologies, from discretization, decision, and distinction, to an identity "with common measure" (symmetry), to rationality and ratio (logos), and ultimately to the transcendental itself. These are the paradigmatic domains of the digital. Thus, in this examination of the economies of existence, I seek no calculus of real continuity, but rather a simple mathematics of discrete number. No geometry, only arithmetic. No curves, only counting.

For that, we return to where we began, to Alain Badiou, whose work issues from an elementary fidelity to arithmetic and digitality. A perfect inversion of Deleuze, Badiou [End Page 99] begins with discrete number not real continuity. He is a rationalist not an empiricist. His prize is the transcendental not immanence. Badiou is an ideal chaperone for any examination of the economies of calculation and digital rationality.

Yet, before continuing to scrutinize Badiou in detail, and with an eye on the larger context, it will be important to address two fundamental truths about him. First, Badiou is undoubtedly one of the most significant thinkers of our times. Consider his numerous books on topics ranging from poetry to politics to ontology; his deep knowledge of Western philosophy, complemented by a thorough understanding of mathematics; his bold interpretations of Plato, Heidegger, or Lacan; his frequent interventions into contemporary politics and social justice; not to mention his own literary and dramatic works. Over a career of several decades, Badiou has displayed an extraordinary ability to act as both public intellectual and fastidious theorist of philosophical esoterica, never sacrificing one for the other.

At the same time Badiou is not universally loved, far from it. "I'd say he's the worst of them all. What does he want us to believe, that 'we' made the Long March together?" Such was Guy Debord's sarcastic rejection of what he called the "latest effronteries of Badiou the Maoist" in a 1982 letter. "Too bad he hasn't been picked up with the rest of the trash, all those critics I intend to crush!"18 "Trash" is a harsh term, and Debord was notoriously stingy in his compliments, yet the larger problem rings true: Badiou has not been received with the same excitement and interest as others in contemporary philosophy and theory. The American academy, for instance, has lived through the Age of Derrida and the Age of Deleuze, each sustained by legions of acolytes. But beyond a loyal inner circle, Badiou does not incite the same level of fascination. Despite Badiou's being widely translated and cited, academic departments in the United States are not overhauling their curricula to make room for Badiou studies, as they once did for deconstruction or affect theory. Today there is no School of Badiou.19

As to why, a few answers come to mind, some merely circumstantial, others more foundational. While many thinkers on this side of the Atlantic focus on cultural difference and nonhuman multiplicity, Badiou is a white European philosopher advocating universal truth, complete with detailed mathematical proofs. In an intellectual climate dominated by pragmatism and empiricism, Badiou is an inveterate rationalist. And in a time when even the most progressive college campuses are rife with anti-Marxism, bolstered by a general skepticism from the media and culture at large, Badiou is an unreconstructed Marxist who stubbornly remains faithful even to his Maoist past, as Debord and others have been keen to point out.20 Any one of these traits might taint Badiou as undesirable. Taken together they present significant challenges.

Thus Badiou is unappealing to many, without a doubt. But for others Badiou is one of the key philosophers of our age. Perhaps the very elements that make Badiou so proud—his Platonism, his idealism, his penchant for mathematics—also expose a series of flaws, both intellectual and political. And perhaps such flaws are themselves, mutatis mutandis, the raw materials for a new configuration. If so, can we not claim, as Marx said of Hegel, that Badiou is standing on his head, our job being merely to set him right again? [End Page 100]

Discrete Points

Is Badiou a digital thinker, and, if so, how exactly? A particularly vivid example of Badiou's yen for discretization may be found in his "theory of points" from Logics of Worlds.21 At the heart of this analysis is the concept of a point. But what is a point? As Peter Hallward described it, "a point is a place in which participation in a world may polarize into a simple yes or no, for or against, backwards or forwards and so on."22 Or, as Badiou himself put it, "a point is a transcendental testing-ground for the appearing of a truth . . . a point is the crystallization of the infinite in a figure—which Kierkegaard called 'the Alternative'—of the 'either/or,' what can also be called a choice or a decision."23 In essence, a point is a place where choice or decision becomes possible.

Consider some of the physical qualities of a point. Less than a plane and less than a line, the point has no extension. A point may have a location in space, but it has no dimension. Such qualities are "decisive" for Badiou; they embody decision as such. Pointing is a choice. Through pointing, one identifies a single point at the exclusion of all else. And thus, writ large, any world in which point locations may be identified is a world in which decision pertains (in which choice pertains). In this way, points provide a geography. Points inflect a landscape with a spatial or geometrical truth structure. For Badiou, pointing allows truths to be localized in a space.

At the same time, a point is never just a point. Decisive and exclusionary, a single point will always produce another point, whatever has been excluded or decided against. Hence the theory of points entails not one point but two: one point or the other, yes or no, for or against, backwards or forwards. In this way, points transform a complex, heterogenous universe by galvanizing and constricting it around two points. Badiou calls these two points the anchor points, because they impose an ordering on the world and thus anchor it. Of course such point-making is a necessarily reductive if not also violent act, since the two points will distill all of the many divergent possibilities of the world down to just two possibilities. But such is their appeal for Badiou. (His is a political theory, after all, not an ethical one.) Confronted with two points—and only two points—a person must choose. Either/or, with nothing in between.

Consider a scenario in which there are only two points on a field. Consider a strike by organized labor, or a demonstration where protestors confront police. Respecting a picket line means not crossing it. Standing with demonstrators on one side of the barricade means not standing on the other side (with the police). These are "pointy" landscapes for Badiou. Physicality is a key part of the pointy landscape, because if subjects occupy one point, they cannot simultaneously occupy another point. The picket line or barricade thus anchors the world by deliberately reducing it, sacrificing complexity but gaining decision. [End Page 101]

Badiou's theory of points makes sense intuitively and would sound familiar to anyone interested in political encounters. Yet, while not unfamiliar, it is worth noting how unfashionable this way of thinking remains today. For instance, a common criticism of Badiou's theory of points is that such deliberate reduction, such deliberate dumbing down, betrays what some see as the very essence of politics, that is, a full recognition of the complex, messy, and overlapping relations between people, as captured perhaps most vividly by theories of subjectivity and identity, be they intersectional or otherwise. Aren't points false reductions, caricatures even, of a more full political reality? Wouldn't the goal of politics be to rebel against points rather than construct them? Shouldn't the goal be to respond to such digitality via recourse to a rich and full analogicity? Or, perhaps most profoundly, what about all those who cannot or do not have the ability to decide? Is this just liberalism all over again? Indeed many figures in subaltern studies, feminism, critical race theory and other fields have focused on this very question, whether confronting Badiou or not. Still, Badiou approaches the situation in reverse: for him the only genuinely political position is the position of the point; such choice-bound pointing provides the conditions for truth; and thus the true political subject is the subject who sits on a point.

To render Badiou's digital profile in greater detail, consider a second source, that section from Being and Event where Badiou marshals all his poetic and persuasive powers. Consider the "impasse of ontology" described in meditations 26 and 27, the crux of the book, if not the crux of Badiou's project overall.

By what path did Badiou arrive at the impasse? It all began with a query. "Is being intrinsically quantifiable? . . . Is there thus an essential numerosity of being?"24 The query is innocent enough. Is it always possible to compare two things quantitatively? Is it always possible to assert that something is larger than something else? Is there a concept of "larger than" from which to construct quantity or numerosity, and, if so, is there a concept of "larger than" in thought overall? The path to the impasse begins just like that, because the simple numerosity of being, the simple notion that everything is intrinsically quantifiable and therefore relatable via the operation of "larger than"—this simple reality collapsed in the late nineteenth century, at least in mathematical circles.

The technical details, often quite onerous in Badiou, will be glossed over for the time being. They involve the mathematician Georg Cantor and his discoveries concerning the nature of infinity, specifically that there are two sizes to infinity (and perhaps even more than two).25 More important than these particulars, however, is the impasse itself, an impasse that so captured Badiou's imagination he structured Being and Event almost entirely around it, around what he called the errancy or the unmeasure of ontology. In Badiou's view, Cantor [End Page 102] unearthed "two regimes," mandating an "arbitrary decision" between them. (Per the previous discussion, these two regimes might be understood as two points requiring an either/or decision.) In meditation 26, Badiou labels this arbitrary decision a "wager" (pari) beyond the effectivity of known calculation. By "wager," Badiou meant a kind of gamble precipitated by the failure of numerical logic. After such a failure, one is obligated to make a choice, if not a leap of faith then a leap of faithfulness (fidélité). "A chasm opens" in the wake of Cantor, a chasm that requires "a conceptless choice."26

If this sounds like existentialism, it should; Badiou is, in a sense, rewriting existentialism for a new age. His is not simply a theory of points or decisions, but a much larger theory about subjects and how they are formed. "Being . . . is unfaithful to itself," Badiou observes. And, as a result, "quantity . . . lead[s] to pure subjectivity."27 It is an astounding if not radical claim. Begin with quantity, with mathematical concepts; pursue their consequences far enough by following all the innovations of modern mathematics; and the result will be subjectivity. In short, quantity leads to subjectivity.

Following Badiou's version of events, any investigation into pure quantity will ultimately derail at Cantor's impasse. A yawning void will eventually open at the heart of mathematics, a void within mathematics, to be sure, but the consequence of mathematics nonetheless. And at that point, the person investigating will encounter a decision, a point that is not quantifiable, a point that does not follow the succession of numbers. Such is subjectivity for Badiou; the subject appears from out of the abyss of the conceptless choice. The pursuit of quantity is, in essence, subject formation. Or, for short, math makes subjects.

Does discretization fail at the impasse? In one sense, Badiou appears to abandon his commitment to discretization, given the failure of simple arithmetic. But in another sense, Cantor's impasse constitutes a form of "bigger" quantification. If not a continual sequence of small steps, the encounter with the void introduces a break, a break that precipitates a quantum leap. Ultimately the failure of number produces a larger arithmetic.

Badiou's Principle

To be sure, some readers of Badiou object to this sort of "mathification." The more romantic among them object to mathification on the grounds that it dehumanizes the subject, that it turns subjects into objects, the objects of a logical if not mechanical process. For these skeptics, "math makes subjects" is simply shorthand for a kind of soulless, life-killing structure of reification in which the magical spark of human life is replaced by a clean, formal logic. Yet from a certain perspective Badiou is even more romantic than his most romantic critics. Digital thinking, for Badiou, is not the sterile enemy of quality, sensation, or poetic experience—all those attractive virtues of an analog subjectivity beyond the confines of the rational logos—but, in fact, digital thinking is the very condition of such "conceptless" existence.

Such critics also tend to excoriate Badiou for arguing that "ontology is spoken in the language of mathematics," as he has argued consistently for many years.28 But what does [End Page 103] Badiou mean by this notorious claim? Does he mean that the world is, at root, mathematical? Does he mean that the universe is a gigantic computer? On the contrary, Badiou simply means that if one were to pursue ontology, if a thinker were to explore the branch of philosophy that deals with being and existence (ontology), such a person would necessarily encounter numerical concepts such as "the one" or the distinction between the one and the multiple. Indeed, these are some of the oldest questions in philosophy: What is the one? What is not one? What is the difference between the one and the multiple? To answer such questions requires an elemental concept of number, argues Badiou, and thus to speak of "one" or "multiple" means to speak in the language of mathematics.

Such is the strong interpretation of Badiou's claim that mathematics is ontology: to examine ontology requires an elemental concept of number. But it goes further too, because Badiou has constructed a dense thicket of interconnection between ontological claims and mathematical claims, particularly those furnished by set theory29 Still, one need not broach the question of set theory at all in order to derive the strong interpretation. The concepts of the one, the two, and the multiple are mathematical enough.

A weak interpretation exists as well. Less commonly discussed, the weak interpretation is perhaps ultimately more persuasive. This interpretation attempts to understand math through its Ancient Greek etymology in terms like mathēsis (μάθησις) and mathēmata (μαθήματα). Such Greek terms do not necessarily mean the same as "math" in today's parlance. For the Greeks mathēsis simply meant education or learning; mathēmata referred to a school lesson, that is, something learned. Mathēsis did not mean numerical learning exclusively—arithmetic and geometry for the ancients, or calculus and topology for the moderns—but rather the act of learning as a whole, the act of gaining knowledge in general. The English term "polymath" captures the original meaning nicely; a polymath may boast of knowledge across many subjects, not simply mathematics.

Badiou allows himself to be misunderstood on this point, I suspect, because to say "ontology is spoken in the language of mathematics" is to say, in essence, that philosophy takes place through the education of the mind. Badiou's notorious claim loses its controversial air when rendered such. If mathēsis is simply education or learning—simply the cultivation of abstraction, abstraction of whatever kind—then to say that philosophy is spoken in the language of mathematics is to say that philosophy is spoken in the language of the cultivate of abstraction. Who would contest such a claim?

Here Badiou sounds a bit like Heidegger, particularly Heidegger's notion that "being furnishes itself to thought."30 Under the weak interpretation, mathification means nothing more than this: being furnishes itself to abstraction, to the cultivation of abstraction. Or indeed if not Heidegger, then Badiou sounds like his great hero Socrates, particularly the Socratic notion that philosophy is the sincere pursuit of truth. Thus, via the weak interpretation of mathification, it is not necessary to assent to set theory or anything so technical. One need only acknowledge learning, and mathification follows.

So when Badiou uses the term mathēsis he benefits from an ambiguity in meaning: number, but also training. One may interpret it as the cultivation of the capacity for abstraction, or one may interpret it in a political context as commitment or militancy. [End Page 104] And when Badiou opposes mathematical ontologies to poetic ontologies—the matheme to the poem—the true distinction he seeks is that between militancy and romanticism.31

But to utter the phrase "cultivation of abstraction" is sure to invite controversy. Of the many challenging terms in thought and culture, "abstraction" is certainly one of the most difficult to define. Furthermore it is difficult to know where the digital and the analog come down on the question of abstraction. Speaking very generally, there are two schools of thought concerning how, or if, the digital and the analog are modes of abstraction.

One school—sometimes called "metaphysical"—claims that the analog means the real, while the digital means the symbolic. The analog is thus "outside" or "prior to" abstraction, while the digital is synonymous with abstraction (in the form of language, code, or symbol). In other words, for the metaphysical school, analogicity is not representation or abstraction at all. And when analog media are duplicated—in sculptural molding, for instance, or during wave transmission—they do so in a way that is qualitatively and substantively different from digital media.

A second school of thought—most simply labeled "physical"—claims that analogicity and digitality are both equally abstract, even if their modes of abstraction are different. Hence for the physical school, the analogical mimesis of, say, wave duplication is just as abstract as the kind of digital mimesis that takes place during discrete sampling and symbolic encoding. Thus, for the second school, analogicity and digitality may be differentiated aesthetically or formally, but not ontologically.

Deleuze is a good ambassador for the second school, the physical school. And, by contrast, Badiou is a fitting representative of the first. For Badiou, the digital is abstract in a way that the analog is not. For him, the analogical mode of abstraction is dumb, direct, and rote; "monotonous" was Badiou's word for Deleuze.32 But the digital mode of abstraction is superlative, complex, and generative. For Badiou, digital abstraction quite literally creates subjects.

Recall the citation provided at the outset, where Badiou described "numerical discretization," the "geometrical continuum," and the "abyss" that separates the two. If Badiou is correct, then we have arrived at the true heart of the matter, a major discovery, or shall we say rediscovery, of a distinction that has existed since the ancients, rediscovered by Cantor, rediscovered again by Badiou, and now reiterated here in slightly different language: digitality and analogicity are elemental modes of philosophy (even if Badiou favors the former). Digitality and analogicity are two elemental modes, yes—but the claim is even stronger than that, for digitality and analogicity are the only two modes, and there is nothing in between them. Their twoness is crucial; equally important, however, is the "only-ness" of the two, only two and nothing between. In other words, what Cantor discovered (and asserted in his famous continuum hypothesis) is not simply that there are two sizes to infinity, but that there are two entirely distinct modes of abstraction, the mode of abstraction evident in the natural numbers and the mode evident in the real numbers. "Natural abstraction" is the abstraction of discretization; "real abstraction" is the abstraction within continuity.33 The former transpires under purview of a "natural" philosophy, while the latter a "real" philosophy. [End Page 105]

On this score, Badiou is more natural than real. And while he is greatly influenced by Cantor, Badiou does not give real abstraction (the analog type) the same value as natural abstraction (the digital type). Given more time it will be necessary to have a longer discussion of the relative merits of digital and analog abstraction. But for now one might simply pause to admire what Badiou gains from his unbridled enthusiasm for the digital. Abstraction (as mathēsis) becomes the passage through the impasse, the method by which one traverses the gap between analogicity and digitality To move from analogicity to digitality is to enter abstraction, to enter mathēsis, to enter mathematics as such. This is why Badiou remains so steadfastly aligned with mathematics, and why, for him, math is synonymous with subjectivity, but also synonymous with digitality.

But the final consequence of Badiou's system is even more dramatic and requires an even greater leap. Badiou's most provocative intervention is not simply to claim that mathematics is ontology—an ambitious and perhaps even radical claim—but, beyond that, Badiou's intervention is to render a definition of mathematics itself. Badiou provides not just a way to think about ontology (as "the thing spoken in the language of mathematics"). Badiou also provides a meta-mathematical claim, in which mathification is defined as the difference between the real numbers and the natural numbers. In other words, if a mathētēs (a student, disciple, or learner) comprehends the difference between the real and the natural, such a student necessarily understands mathematics as such.

Corollary: given how the analog maps onto the real numbers and the digital maps onto the natural numbers, mathification may also be defined as comprehension of the difference between the analog and the digital. Likewise, to enter mathematics means to move from the analog to the digital.

Taken together these theses constitute what might be called Badiou's Principle:

Mathification is defined as the difference between the real and the natural, that is, between the analog and the digital.

Conclusion: Badiou, Or Idealism

Badiou's economies of existence are located not so much within the real—however debased and profaned it may be—as they are standing apart from it, breaking with the real (breaking in the direction of the natural). For Badiou, he that great modernist, the subject is constituted through a break, through an impasse, through what was discussed previously as a conceptless choice. And Badiou's break always follows the same line of advancement: away from "real" continuity and toward "natural" discretization (again borrowing the precise definition of these two descriptors from the real numbers and natural numbers).

This produces a delightfully counter-intuitive conclusion. Mathification must be understood in precisely the opposite way from how it was characterized by, say, Theodor Adorno or Max Weber. In Badiou, mathification does not signal the triumph of industrial capitalism or rationalized modernity, but rather, taken in reverse, it signals the [End Page 106] capacity to break with the staid routine of this mundane purgatory in favor of a newly enlivened subjective experience. Mathematical rationality poses no threat to the subject. Indeed the opposite is the case. Such mathematical rationality provides the very conditions for the subject to emerge, and to emerge as true. So while Badiou is most certainly a leftist, his political theory does not derive from anti-rationalism, that frequent ally of progressive theory. Indeed Badiou's version of Marxism is an unusual one. His is not an historical materialism so much as a rational idealism, proud and unapologetic. As ironic as it may seem, Badiou's Marxism is an idealism.

In other words, Badiou's complaint is not with abstraction, form, or idea, as it was for Marx and many Marxists, all those who have decried abstraction as a form of false mystification of real conditions. For Badiou, form and abstraction are absolutely crucial. After all, things like abstraction, idea, form, and essence are the very bedrock of pure mathematics. So while Marxists are often criticized for being overly reductive, and such reductivism is likewise often construed as a kind of false or even romantic idealism, Badiou brandishes his idealism with gusto.34 What else should one expect from a Platonist?

How unsatisfying, then, for today's conversation. For the old debates spring forth anew, and those same tired criticisms of Marxism reemerge from the sidelines, trotted out once again like some ragged old warhorse: Marxism is just thinly veiled idealism; Marxism is just thinly veiled essentialism; or, worst of all, Marxism is just thinly veiled romanticism. But isn't that the problem? To borrow a classic phrase from Gayatri Chakravorty Spivak, perhaps a bit of "strategic essentialism" is precisely what is needed at this particular point in time. Criticality is so thoroughly disempowered today, the velocity of co-optation so rapid, the inversions of political desire so complete, perhaps the only truly revolutionary act available today is to promulgate a kind of strategic essentialism—as if to say that no sort of direct, rational process will ever yield results, but that they will arise instead from those irrational processes, those untranscendable horizons, that abyss and that impasse that fix the very coordinates of nature itself.

Is this not what Hegel meant by "objective thoughts," by concepts becoming objects? Hegel was fond of citing an old maxim from Anaxagoras that "nous governs the world" (pantōn nous kratei), arguing that nature is "a system of unconscious thought . . . a petrified intelligence."35 When the stress falls on nous, this is undoubtedly a form of idealism. But when the stress falls on kratei (to rule over, to govern), a more kinetic logic takes over. Mind governs the world; mind incites the world; mind is insurrectional. Badiou's theory of the subject is more or less identical.36 [End Page 107]

"Yes or no, does human life make sense, and does man have a destiny?" Such was the decisive question for Maurice Blondel in 1893, and it is not difficult to discern a thread running through the French Hegelians, from Blondel to Michel Henry to Badiou himself, in which human experience, even when conceived in the most rigorously idealist fashion, hinges on the subject's ability to realize an insurrectional mandate, religious or otherwise. (This is not the occasion to pursue the subsequent question of whether or not all Hegelianism forces a return to theology in some form.) I suspect that the apogee of idealism, or at least the point where it ineluctably transgresses its own logic, remains the condition of subjective transformation if not self-annihilation, wherein names become so suspended they appear nameless, faces so defaced they become blank, the world so remote it withdraws, like Dante's first glimpse of the great abyss, "dark and deep and filled with mist . . . [and] though I gazed into its pit, I was unable to discern a thing."37 Was it not Husserl, that master of suspension, who summoned phenomenology to transcend the seemingly "anonymous" nature of the life-world, or Henry who plumbed the deeps of the ego only to find the "facelessness" of essence, to say nothing of Descartes and his hyperbolic doubt?38 Idealism, one will recall, is not so much the science of forms or abstractions, much less concepts or notions, but the science of subjectivity. The subject is always both the stake and the site of any kind of idealism. And idealism's own special ironic condition is one in which the invigorated potency of a pure subjective stance is, as it were, so potent that the subject is transformed in the wake of its own stubborn fidelity. This is the lesson of Badiou, a strange dialectical creature comprising equal parts radical idealist and radical materialist—shall we not simply agree to call him a radicalist—and it is the most convincing rationale he provides for Plato's ongoing relevance today, in short, that Socrates and Rimbaud speak in one voice: il faut changer la vie! [End Page 108]

Alexander R. Galloway

Alexander R. Galloway teaches media studies at New York University and is author of several books on digital media and critical theory, including The Interface Effect (Polity, 2012). A 2019 Guggenheim Fellow, Galloway is currently finishing a new manuscript on the deep history of cybernetics.


1. Badiou, Being and Event, 281; emphasis added. Comprising dozens of books, Badiou's bibliography is too large to enumerate here much less summarize. During a period of peak productivity in the late 1980s, Badiou produced the large treatise L'être et l'événement (Being and Event) followed by a series of aftershocks: the pendant Manifeste pour la philosophie (Manifesto for Philosophy), which aimed to popularize the claims made in the larger treatise, aping the Prolegomenon that followed Kant's first Critique; a primer on mathematics called Le nombre et les nombres (Number and Numbers); and Conditions, which explore the four truth procedures described in the 1988 treatise, although under slightly different headings. In fact L'être et l'événement was simply the first volume of a trilogy, including Logiques des mondes (Logics of Worlds) and L'immanence des vérités (The Immanence of Truths). Groundwork for the 1988 treatise was in some ways laid by Badiou's Théorie du sujet (Theory of the Subject). Readers of Badiou will also wish to consult two invaluable secondary texts, Peter Hallward, Badiou: A Subject to Truth, and Bruno Bosteels, Badiou and Politics.

2. Badiou's archive of writings on mathematics is extensive and varied. The archive would include the previously cited works on core ontology, riddled as they are with mathematical endeavors of various kinds. There are also volumes that examine a particular aspect of mathematics such as the early text Le concept de modèle (The Concept of Model) on mathematical formalization, or his foray into category theory, Mathematics of the Transcendental. Badiou has also published casual texts on mathematics such as his dialogue with Gilles Haéri titled Éloge des mathématiques (In Praise of Mathematics).

3. An exploration of the discrete (digital) and continuous (analog) in Aristotle and others is provided in Michael Eldred, The Digital Cast of Being. I thank Erick Felinto for bringing this unique book to my attention.

4. Wilden, System and Structure, 169; capitalization in original.

5. To be sure, many are dubious of the base/superstructure distinction, which in its most vulgar form assigns determination to the base. For instance in his preface to Fredric Jameson's The Geopolitical Aesthetic, Colin MacCabe identifies three basic problems in "granting a primacy to the forms of economic activity in an understanding of cultural forms." The first is causality, or how precisely the base and superstructure touch and effect each other. Second is the dubious autonomy of the base, or the fact that the base is ultimately defined using concepts borrowed from the superstructure. Finally, MacCabe identifies what might be called the monotony of determinism, the dreary fact that for a vulgar Marxist "all cultural forms end up with the same content" (x).

6. For basic admittance into this school of thought the uninitiated may Fredkin's "An Introduction to Digital Philosophy," the various shortcomings of which are discussed in Luciano Floridi's "Against Digital Ontology."

7. Regarding quantity and quality, Marx says as much himself in his writings on the "General Formula for Capital." See Marx, Capital, 248–57. Although the formula deduced from this, Q-Q-Q, leaves something to be desired.

8. Mann, Doctor Faustus, 75. The reference is to Terence's The Eunuch: "Incerta haec si tu postules / Ratione certa facere, nihilo plus agas, / Quam si des operam, ut cum ratione insanias" ("You can do no better to act rationally with such uncertain things than to act foolishly with reason," 1.61–63).

9. See Lyotard, The Postmodern Condition, and Turing, "On Computable Numbers."

10. Deleuze, "Postscript on Control Societies." The concept of control is central to cybernetics, computer science, and several other adjacent disciplines.

11. Badiou's L'être et l'événement was published at Éditions du Seuil in 1988 at the height of anglophone interest in French theory. The English edition appeared seventeen years later in 2005.

12. See Goux, Symbolic Economies and, among others, Marcel Mauss, The Gift, and Jacques Derrida, Given Time 1.

13. Graeber, Debt: The First 5,000 Years.

14. Not entirely, as Deleuze peppered his writings with references to the discrete and the symbolic even as he aimed to rewrite such concepts using a strictly analogical vocabulary of affect, intensity, desire, experience, and expression. The main volume explicitly devoted to language and rationality is Deleuze's The Logic of Sense.

15. See King, Virtual Memory; McKittrick, "Diachronic Loops/Deadweight Tonnage/Bad Made Measure"; Laboria Cuboniks, "Xenofeminism"; Dixon-Roman, "Algo-Ritmo"; and Tuck and Yang, "Decolonization Is Not a Metaphor."

16. See in particular Wilderson, "The Ruse of Analogy" in Red, White, and Black, 35–53, and Laruelle, Principles of Non-philosophy.

17. See Galloway, Laruelle: Against the Digital. The Liberated Computer Language was published as an appendix in Galloway and Thacker, The Exploit: A Theory of Networks, 159–66, and is available online at http://R-S-G.ORG/LCL.

18. Debord, Correspondance, vol. 6, 212–13. Debord was responding to Badiou's article "Guy Debord—« In girum imus nocte et consumimur igni »: Un homme qui ne cède pas." Badiou's short piece would later be reprinted in Debord's Ordures et décombres, 137–41, a compendium of press clippings following the release of Debord's film In girum imus nocte et consumimur igni (1978). An English translation appeared in chapter 6 of Badiou, Cinema, 50–53.

19. An engaging analysis of Badiou's potential shortcomings is found in Denise Ferreira da Silva, "Fractal Thinking." Fredric Jameson also cast a skeptical eye in his essay "Badiou and the French Tradition."

20. Laruelle's unambiguously titled Anti-Badiou opens with an attack on Badiou's Maoism. Alluding to the Chinese Cultural Revolution, Laruelle accuses Badiou of trying to "reeducate" philosophy by forcing it to submit to the yoke of mathematics.

21. Badiou, Logics of Worlds, 399–447. While the "either/or" logic of the theory of points is binary and disjunctive, Logics of Worlds overall is, among other things, an attempt to address the shortcomings of Badiou's previous volume Being and Event, where various phenomena (situations, conditions, events, etc.) are presented as clean and separate. In Logics of Worlds, Badiou introduces more gradual and complex processes, for instance events that transpire across several temporal arrangements, or more nuanced forms of subjectivity that do not simply exist in one of two binary states (pre-evental and post-evental). In this sense, Logics of Worlds mollifies Badiou's more strictly arithmetical tendencies, revealing a profile that is less digital and more dialectical. See in particular Logics of Worlds, 1–40.

22. Hallward, "Order and Event," 106–7.

23. Badiou, Logics of Worlds, 399, 400.

24. Badiou, Being and Event, 265.

25. For more on Cantor, his continuum hypothesis, as well as Richard Dedekind and others from the mathematics of real continuity, see Benjamin Lee Buckley, The Continuity Debate, and Mary Tiles, The Philosophy of Set Theory. Also recommended is David Foster Wallace's Everything and More, an engaging romp through the mathematics of infinity (and, with it, continuity). For assistance in understanding Badiou's particular use of mathematics, see Paul Livingston, The Politics of Logic.

26. Badiou, Being and Event, 280.

27. Badiou, 280.

28. "Mathematics accomplishes ontology," wrote Badiou in Being and Event, 10. "Mathematics is the science . . . of everything that is, insofar as it is" (Badiou, 7).

29. This prompts a whole series of secondary concerns, in particular the question of completism: do all claims in mathematics correlate to something in ontology? Do the two discourses overlap with complete conterminousness, or only partial? And if only partial then by what criteria shall the ontological math be differentiated from the non-ontological math?

30. For one instance among others, see Heidegger's meditation on unconcealedness in Poetry, Language, Thought, particularly pp. 49–54.

31. In Being and Event, Badiou contrasts the poem of Parmenides to the mathematical formulations (mathemes) of Plato: "The Platonic matheme must be thought here precisely as a disposition which is separated from and forgetful of the preplatonic poem, of Parmenides' poem" (125).

32. Badiou, Deleuze, 14.

33. This synchronizes with Floridi's argument as well, where he suggests that digitality and analogicity are both modes of representation, their qualities becoming most legible when viewed at different levels of abstraction. "Reality is experienced, conceptualised and known as digital or analogue depending on the level of abstraction" (Floridi, "Against Digital Ontology," 160).

34. A version of this criticism is found in Bruno Latour, Reassembling the Social, 83–86. See also note 5 above.

35. Hegel, The Science of Logic, 75.

36. See Véronique Bergen's Résistances philosophiques for an illuminating tripartite typology in which Badiou's "axiomatic" form of resistance is contrasted with the vitalism of Deleuze and the dialectical position of Marx or Sartre.

37. Dante, The Divine Comedy: Inferno, 72 (canto 4, lines 10–12).

38. "The life-world can be disclosed as a realm of subjective phenomena which have remained 'anonymous,'" wrote Husserl. "Does philosophy fulfill the sense of its primal establishment as the universal and ultimately grounded science if it leaves this realm to its 'anonymity'?" See Husserl, The Crisis of European Sciences and Transcendental Phenomenology, 111, 112. "The essence remains hidden in its very revelation," Henry observed in a cryptic section of his Essence of Manifestation full of references to facelessness, obscurity, and the night (Michel Henry, The Essence of Manifestation, 440; emphasis removed).

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IMAGE: Mike Nelson


Installation view, Tate Britain, 2019

Courtesy the artist and 303 Gallery, New York; Galleria Franco Noero, Turin; Matt's Gallery, London; and neugerriemschneider, Berlin.

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