The Only Principle, an Analysis of Alan Badiou’s Being and Event II: Logics of Worlds

Sensemaking for the 21st Century

a lee
51 min readDec 29, 2023

Note: this was originally put on Acedemia around 2015–2016. There have been a few edits and clarifications.

Originally, this essay started as a review of Alain Badiou’s beautiful book, Being and Event II: Logics of Worlds. However, his book has plenty of issues with it. There is great beauty in its austere presentation and the structures he outlines deserves study as there is usefulness and merit to many of his ideas. But the issues with this book are many, so a review became a critique and as a critique, this essay suffered from scope creep until it was not only about Badiou’s book but also about the very aesthetics he espouses to produce.

The problem I start with is simple. Let’s being with his introduction: Badiou introduces two terms which could be the same term: “democratic materialism” but does not explain any further. His approach is thusly, if you understand his formalistic procedure then you will understand that he means. In fact this is only so; the formalism isn’t simply how concepts are described, rather, the formalisms are the very foundation that realizes how these concepts are actual. When we understand this basis however, it becomes very clear that there are some particular problems with this logic as determined by the tradition Badiou seeks to encapsulate. Thus, we have the final blow of scope creep. This essay is not only to determine what democratic materialism is, but we are also to understand the limits of transcendental logic which is related to why Badiou felt the need to write a sequel to his Being and Event.

Badiou narrows the differences between the two Being and Events to state that the first book was about onto-logy whereas this book is about onto­-logy. This may seem a trite, clever phrase but it’s actually deeply apt, one that goes back to the split between philosophy and math. Badiou makes it no secret of his love for the Greeks. From his essay “Descartes/Lacan”, Badiou mentions ancient Greek’s logio-mathematical procedure (when melded with Frege and Cantor) “will make it possible to think what this intellectual revolution (a blind return of ontology onto its own essence) conditions in contemporary rationality” (16). While this essay is dated back in 1996 and Being and Event II came out in 2009, you can begin to understand that the revolution around ontology — between ontology and its logic — characterizes Badiou’s methodology.

A great part of Badiou’s genius, especially with Being and Event, relies on his insight that at its root, mathematics is a conceptually based exercise, not a manipulation of pure formalistic symbols. This may seem counter-intuitive since much of our early education approached math as a process based formalism rather than a conceptual exercise. Yet when you look historically at math and philosophy, you’ll gleam that the ancient Greeks were not philosophers or mathematicians. They were neither in the sense that they did not make the distinction between math and science that happened a few decades after the project of modernism started. Thus, we shall soon see, with no great surprise, that the father of modern mathematics is the same figure as the father of modern philosophy: Rene Descartes. And with good reason too, as to why this split happened after Descartes introduced his methodology.

This methodology went a long way to establish the method by which analysis and synthesis proceeded. Irme Lakatos in his essay “The Method of Analysis-Synthesis” points out that Descartes realized the unity of math and philosophy early on and sought to utilize it. Descartes outlines this method in his Rules for Direction. What Lakatos calls the “Cartesian Circuit” was in fact little more than the ancient Greek practice of analysis and synthesis. One starts by decomposing a situation one wishes to explain by analyzing the situation into components. To put the situation back into order as a known situation, one then puts it back together using a synthesis. Lakatos notes that Descartes was so excited by this discovery that he created a “programme [that] was to carry the logic of discovery of Euclidean Mathematics into all domains of human knowledge” (83). This of course, does happen. As we shall see, the result of this method’s proliferation creates what Foucault notes with great confusion in The Order of Things as an epistemological break which happens after the 16th century, in which knowledge starts to become formalized by this approach, and separated into different domains.

The Cartesian Circuit, as illustrated by Irme Lakatos (76)

This break doesn’t happen overnight, nor is this formalized procedure the final form knowledge is to take. In fact, knowledge becomes even more fragmented (formalized into different procedures) towards the end of the 18th century. We can trace this break from the final rejection of the analysis-synthesis method. Before we describe this rejection, let us describe how the analysis-synthesis is to work in general.

Anyone familiar with Euclidean geometry understands that it is built from axioms. From these axioms, we get “proofs” from which our basic shapes are formed, in what is now understood as Euclidean space, space defined by Euclid’s axioms. Lakatos writes that from the fragments of Euclid we have today only demonstration of the synthesis portion of the method. He sums the method as a rule:

Draw conclusions from your conjecture, one after the other, assuming that it is true. If you reach a false conclusion, then your conjecture was false. If you reach an indubitably true conclusion, your conjecture may have been true. In this case reverse the process, work backwards, and try to deduce your original conjecture via the inverse route from the indubitable truth to the dubitable conjecture. If you succeed, you have proved your conjecture. (72–73)

Pretty straight forward. Descartes may call the first leg of the loop induction (in which factual sense-experience is analyzed) and the second leg deduction (where the occult hypothesis as the first principle is the foundation for synthesis to return us to the original context). Consequently Lakatos notes this is why the Greeks were so fond of redutio ad adsurdum. When we produce a contradiction in the breakdown of our conjecture we don’t need to do the second leg of the method: We can throw out the conjecture on analysis alone. Descartes, in his depression and his cosmopolitan travels eventually comes to rediscover this method as he describes in his Rules for Direction. As he is looking for absolute truth, he decides to employ this method here truth is built only from a foundation of truth in all of natural philosophy. For modern philosophy we come to have his a priori cogito ergo sum. For mathematics we gain an introduction to his analytic geometry in which the segmentation of length as quantity is cast into points on a continuum that is analyzable. Descartes also understands that this method isn’t so easy. One must discipline their mind in order to only seek the truths that are self evident. Looking at the figure above, you see the union between induction and deduction as a “first principle”. This becomes a standing problem, one explored later on by Hegel, Kant, Karl Popper and so on.

Going back to Euclid, we find these first principles expressed as axioms. As mentioned before, we have from the ancient Greeks only fragments. They only wrote down the second half of the method, the deductive portion where we follow rules to synthesize truths. The secrets of their philosophies relied on understanding how to get to these axioms. This is the magic for only the initiated, with its reliance on the occult hypothesis, as Lakatos points out. In fact, for Lakatos the failed attempts at refutations of the analysis-synthesis become the starting point for new research programmes for scientific theory.

Today, science and math do not follow the Cartesian Circuit. I don’t want to go on in length about this, but it’s important to note that the interjection of the occult hypothesis was a point many empiricists criticized the method. As one critic said, being able to reason in both directions doesn’t prove truth, only consistency in discourse. And he’s right. This method only produces a greater and greater consistency, with no guarantee of truth in sight. Today however, first principles are still used to describe a given context, accepted axioms in math, or accepted scientific theory by the community of science. But in regards to the Cartesian Circuit, Lakatos is particularly scathing in his explanation as to why analysis synthesis does not work to produce truths.

The truth of conjecture is guaranteed by the full circuit [. . .] As a matter of fact, going several times through the circuit, more and more hidden lemmas emerge, and the conjecture is constantly being improved by critical inspection of the circuit. Proof can be proclaimed only by agreeing by convention, where concept stretching criticism has to stop by giving, for mathematical conjectures, a valid proof in a first- or second- order theory” (102 original italics)

Today, this method has been truncated for modern science, when we examine how modern science and modern math interact at the level of logical relations, we understand that the outline of this method still operates. For this reason, math and science have become increasingly intertwined. Scientific experimentation applied through mathematical reasoning explains how both math and science today and their union as natural philosophy in the past, lie in modern logic’s ability to isolate cause and effect. Causation highlights relations of various quantities by isolating their agency through alternating between experimentation and synthesis. Even still, many phenomena we may want to study continue to elude our ability to bridge a gap between theory and factual proposition simply because, among other things, we cannot find a way to determine those phenomena materially, nor can we, at times, clearly isolate those phenomena functionally to determine how to analyze a situation into its composite units.

It may be no surprise then, that as Badiou wants to bring ontology to math (in this volume, as he brought math to ontology in the first volume) Badiou stumbles upon this very same method. By unifying the synthesis of philosophy with the analysis of mathematics (rather than the synthesis of science with the analysis of mathematics), Badiou has re-forged the analysis synthesis method. There is a difference however. Badiou does not wish to reproduce the Cartesian Circuit — he wishes to highlight the method by which modern logic formulates knowledge as atomizable points that can then be used to synthesize a logical ordination. There does however, remain some parallels with Badiou and other philosophers who have followed the Cartesian Circuit’s aesthetics. Spinoza’s Ethics and Wittgenstein’s Tractatus Logio-Philosophicus have both sought to explicate Truth as a crystalline rigor, without superfluous occult hypothesies. Both these texts follow the form of Euclid’s writings, presenting only the synthesis portion of the text (claiming perhaps, that the analysis portion is already embedded). Badiou does the same, giving us worlds built only from axiomatic first principles. He ends most of his sections on demonstrations of axioms and propositions, preferring to speak only of the application of atomized arrangements in the creation of worlds whose logic is wholly transcendental, showing us through analysis the composite localizations and then later on, their formal interactions[1].

So in passing by first principles, Badiou follows an aesthetic of atomization, much like Descartes’ use of analytic geometry. Badiou’s Book IV on a “Theory of Points” is particular illuminating. Points are how he connects his book not only to his previous Being and Event but also to math. As stated before, Descartes’ analytic geometry was the radical mapping of a point system onto Euclidean space. Descartes reduced what was previously numbers as variable segmented amounts into points that are designated through a logical presentation of ordination. The use of this initial atomization leads us to an analysis of the rate of change of the conceptual curve by Newton and Liebniz, called calculus. This was the start of modern math but that was even exceeded. Badiou’s theory of points goes beyond Cartesian Coordinates into set theory. In the 19th century, math is finally atomized into modern set theory about 200 years after calculus by a mathematician by the name of Weierstrass. He remaps numbers as discrete values: points, further atomizing mathematics but also finally providing a consistency in the conception of math as nothing more than discrete knowable presentation. This brings us back to the foundation of set theory and the last epistemological break before the contemporary era as noted by Foucault in Order of Things. This final break of knowledge into discrete atomization is analogous to the occurrence of what Foucault calls epistemes in today’s fragmented and separate fields of knowledge, each of which has its own different atomized cuts, but more on that in a moment.

To understand the necessity of Badiou’s theory of points it is important to dive a little deeper into math. We may think of math as necessarily being about numbers but conceptual math is not about quantities so much as it is about logical relations that can be grouped, analyzed, and then regrouped. If you know anything about the history of math, math has always been haunted by “monsters,” that is, functions which were unable to provide concordance with existing mathematical conceptions. Zero was one such monster. Some more recent monsters include Hilbert space filling curves, Cantorian dust, fractals, complex numbers and so on. The conquest of these monsters always involved filling in conceptual gaps so that these new functions can be melded with existing “non-monster” functions as a logical consistency. Un-monstering these functions means suturing the domain of math. For Weierstrass, the final steps in transforming continuity into discrete values provided a way for these functions to be conceptualized not only as continuity but also as completely determined values, point for marginal point. Thus, numbers become just another domain of math. Some mathematical domains thus may not even have numbers, wrapping us into topologies and beyond! For example, in Where Mathematics Comes From, George Lakoff and Rafael E Núñez describe Weierstrass’s continuity metaphor with this table (page 311):

Although Weierstrass found his own argument distasteful, he provided it as a way of re-territorializing the entire way mathematics worked. With this, calculus no longer needed space or motion — only points and sets of points. You can see that this allowed an easy transition of mathematics as motion through space, a very inexact conceptualization, to be understood through purely logical relations that did not rely on any vagueness (since we always knew where each motion and each point was). With the atomization of numbers and functions into points, domains can now be understood as mappable from one domain to the next. I won’t go deeply into any examples but as a brief example, Euler’s equation (e^(πi) — 1 = 0) is a masterpiece of mapping several domains onto one another in terms of the points within a singular unit-hood. To understand this equation, one has to be able to freely map between various complex domains such as trigonometry, complex numbers, logarithms, the power series, limits, irrational numbers and calculus[2].

Euler’s Equation mapping across multiple mathematical domains. From here.

At this point, I think you can begin to see how Badiou, by introducing points and atomization, is after remapping philosophical concepts like negation and transcendence onto a discrete mathematical form. The reason for this is actually very simple. A theory of points is also the atomization of difference, recalling Badiou’s concept of pure multiples from his first volume. With a theory of points, Badiou can sidestep the vagueness of his ontological formulations and provide a means to map onto the rest of the world. This addresses a basic criticism of ontology. Ontologies may or may not deal with the rest of the world. Ontologies also do not handle change very well. An ontological substance in the extreme remains separated from the rest of the world, unable to adequately explain its interactions. If you recall, this is one of the Descartes problems with his cogito ergo sum. Individual selves may be, but their interactions remains inexplicable, separated by an insurmountable gulf. With atomized points, however, the key to interaction, like the key to mapping calculus as points, allows different transcendental motions to be mapped via their logical atoms in relation. As Badiou states, change and interaction between objects relies on their parts interacting. How the atoms of objects influence one another is how those objects interact. For example, one army can annihilate another because the individual units relate. These individual units, comprised of individuals, armed with particular tools interact on a certain dimension (perhaps having to do with the hardness and softnesses of material, the interruption of each other’s metabolisms, the disruption of one another’s psychical states) to materially decompose the members of the other’s army. Badiou has some examples, among them are Alexander the Great’s battle with Darius, Quebec’s desire for independence, a protest in Paris, Mao’s writing on politics, Bartók’s opera Ariadne and Bluebeard, and a painting by Hubert Robert, The Bathing Pool. All of these present material relations as logical relations. Using Badiou’s procedural analysis to atomize each of the units we can grasp their logical relations as the primary interface of those atoms with the rest of the world.

Hubert Robert’s The Bathing Pool. From here.

Some critics of Badiou may disagree with his analysis, especially with his political example of Quebec, but Badiou is more concerned with showing how to apply generic relations. As a consistent method, Badiou analyzes each object into various interrelations and then describes their relation using formalizations that localize, dominate and re-ordinate various atoms between each object or within a body as a transcendental. For example, with The Bathing Pool, as seen above, we can understand existence as a matter of the atoms relating to the whole as a transcendental envelope that places each atom and orders them as they interrelate and order each other. In other words, atoms can be functions of each other:

Consider a function of the set which we took as our referent at the outside [the temple] to the transcendental, that is a function which associates a transcendental degree to every element of the set. Such a function may be intuitively interpreted as the degree according to which an element belongs to a component of appearing of the initial set. Let’s suppose, for example, that the function associates to the columns of the temple the maximal value if they are black, and an intermediate value if they are obviously either the one or the other, dissipating their light into the sylvan blue. We can plainly see how this function will separate out, within the apparent-set ‘the columns of the temple,’ the two columns of the foreground as the maximal value of belonging to a component. Conversely the columns on the left will be excluded from this component, their degree being minimal. The columns on the right will be treated as mixed, since they belong to the component ‘to a certain degree,’ just like those in the background, but to a lesser degree (213).

In this manner, Badiou will bring out the relations of atoms as their mode of appearance. Existence is then predicated on a logic of relations between atoms, but only in the dimension upon which they interact. In more formal language, Badiou states:

We will call ‘phenomenal component’ a function of a being-there-in-the-world on the transcendental of this world. If the function has degree p, as its value for an element of being-there under consideration, this means that the elements belong ‘to the p degree’ to the component defined by the function. The elements that ‘absolutely’ compose the phenomenal component are those to which the function assigns the maximal transcendental degree (210, original italics).

For Badiou, this is another way of understanding that discrete knowledge is necessitated by the difference qua atoms which interact and the order upon which they originate themselves. Atoms which do not interact, or have a minimal appearance are “exterior” to a given transcendental, as they are not for that transcendental. In other words, the appearance of an atom accords with the logic of its presentation, the domain of which is a transcendental. Logical relation is existence itself, of one atom to another in their often mutual (but not necessarily equal) appearance. To reverse the statement, “in a given world, a being cannot appear to be more identical to another being than it is to itself. Existence governs difference (210).

In this manner existence is always dependent on a partiality, dependent on interactions that are themselves contingent. If an object has relations in another world, it exists in a different transcendental envelope, and most likely in that order, have a different nature[3]. You can see then how Badiou introduces the transcendental qua domain to determine different envelopes to characterize the completeness of a given world as a synthesis of an atomic function. This fragmentary condition describes how different fields of study today create their immanent logic. Each formulate a transcendental order but only have partial agency (or even no agency) with regards to each other’s domain. Chemistry and biology may overlap to a great degree, but they do not coincide for the most part, that is, biological objects may not exist in chemistry and vis versa. Said in mathematical logic, the two domains are non-injective and non-surjective as organs may be differentiated from another due to their origination and function (in different organisms) although those organs may be chemically identical.

This brings us to the heart of Badiou’s democratic materialism as the formalism of knowledge is the material process by which knowledge is formulated. There is nothing Badiou does not show us if not that the process of atomization and relation is the formulation of knowledge (qua truth values). We can understand that political groups are identified by their different attitudes/identification regarding each other, but these relations may be of completely different appearances, as he exemplifies with his Quebec example. I won’t repeat his example here, as it is quite lengthy, but it’s enough to say that knowledge as a material or political effect relies wholly, for Badiou, on the material order by which their atoms interrelate. What makes a theory of points so effective at designating knowledge is that despite their neutral nomination, a point is no more than an opportunity for a decision to be made.

Although Badiou chooses Kierkegaard as the philosopher who best exemplifies a point, as with Kierkegaard a choice must be made, as a point is an opportunity for the alternative “either/or.” A point is literally a point, as in a game: a mark in the logic of ordering so that we can determine relative positions for contestants. For example, one may win at a preordained point (making it arbitrary) but a point remains the penultimate metric for consideration of ordinance. Points are opportunities for making decisions[4]. So rather than speaking about Kierkegaard, I think it would be more materially expedient to introduce the mathematician Dedekind. His cut of real numbers, as a contemporary of Weierstrauss, also exemplifies the reality of the discrete mathematics program.

Dedekind’s cut is a way for us to define points in relation to each other through the ordination of any arbitrary point. The deployment of a cut provides a real definition of numbers assigning all values, rational to a point, including the point at which a cut was made (which is irrational). This may seem pedantic to non-mathematicians but this cut works materially as an ordination producing knowledge via relations. Lakoff and Núñez provide a metaphor mapping for Dedekind as well (302):

What’s particularly important about Dedekind’s cut is that this conception “guarantees that there will be no ‘gaps’ in the real numbers” (303). This creates completeness so that there are no infinitesimal holes in the number line. By utilizing a geometric metaphor of an Archimedean principle[5], Dedekind is able to eliminate the vagueness of localization (given Descartes’ analytic geometry’s gap in calculus — given the infinitesimal) and present a purely logical relation in its place. Consistent with points as numbers, Dedekind can take any number as the “null” (irrational) point to define this transcendental ordinance so that nothing is left out. Each point becomes an opportunity for any function to orient itself, and thus, a master orientation of the values of all rational numbers can be predicated on any point. Lakoff and Núñez write

It is remarkable that [Dedekind] says, “If space has at all a real existence . . . .” If he can eliminate space from mathematics, maybe it doesn’t exist. He goes on, “If space has at all a real existence, it is not necessary for it to be continuous; many of its properties would remain the same even if it were discontinuous.” Here it is clear that by “continuous” space he means our ordinary concept of naturally continuous space. If he can create a mathematics that does as well as the old mathematics, but is discrete rather than naturally continuous, maybe real space (if it exists!) is also discrete (304, original italics).

The implications for a Dedekind cut are hugely profound. While Badiou isn’t necessarily recasting space per se, this concept is analogous. What Badiou is doing is recasting the interiority of subjectivity, via transcendental logic, into discrete points. The greater the number of points one can have to recompose a world, the greater interiority there is for making decisions, for deciding at each point. And if a point can be apprehended, there is a threshold that is reached and a truth qua difference that can be grasped. While this absolute freedom for Kierkegaard is a Christian venture where one can choose God at each opportunity, Badiou characterizes the interiority of the subject as a topography of points, closing a “circle” (or in his terms, an envelope) and providing a map by which Man can transverse the absolute point by point.

In this sense, Badiou is almost claiming to preform an analogous Dedekind “cut” for subjectivity, or any phenomenon. As we shall see below, this works within capitalist-democracies as well.

This logic of presentation provides for Badiou the absolute territory in which decision making can occur and in which objects can exist for us to be oriented to them. As always, he seems to stay away from creating occult hypothesizes, choosing instead to end each section with a dry exposition of formalism, preferring to end with axioms and tautologies. However, we can follow his logic and recast it in a more explicit material fashion, which is where he gets his term “democratic materialism” from.

The introduction of “democratic materialism” requires an economic foregrounding of marginal values as points.

We understand that behaviors are predicated on animal drives and biological contexts. Normally a biological desire mostly consists of what Badiou may call an “atonic” world, meaning it is devoid of accents or points. A completely atonic world allows no decision to be made since there are no thresholds; nothing can be grasped as there is no difference to hook onto. Following Badiou’s logic, biological desire can only come to be expressed (as a change, or a decision to be made) when a point forms a new relation regarding itself, or with the outside a new function of other atoms. An exterior change may be a temperature drop precipitating a need for warmth, or a metabolic trigger, such as a drop in sugar. Relatively speaking, there are not very many points within an animal’s biology for an animal to make decisions on. Yet when we utilize a Dedekind cut in the economic sense, we can materialize biological desire onto an ordinal plane of utility predicated on material value.

In this sense, money forces a point with each marginal increment by which a subject can make a choice. The larger relation of this choice is easily expressed through the supply and demand curves of neoclassical economics. Given a group valuation in which equilibrium is attained, certain individuals may be barred from buying certain products, or they may be encouraged to buy more of a product depending on where they sit in monetary relation to a situation. As Marx is quick to point out, a change in resource allocation means a change in social relations. When the economy reshuffles, people move. They change jobs. They change spending habits. Society flexes its muscles and yet, in capitalism the total picture doesn’t change because the relations stay the same. This application of points is exactly how money allows other complex machinic assemblages to operate (de/re)composing our social relations. This power of money relies on apprehending minute differences in how each marginal dollar relates to us. Within other domains, the unity type may be a different episteme (not money) but within all domains (of different forms of knowledge), each formalization relies on quantification qua ordinance of points in order to attain its immanence. By overlaying an analytic foundation in order to neutralize any latent content left over from any other domain, a given domain can then compose itself as its own type of expertise.

While this particular approach isn’t taken by Badiou, we can understand his suggestion of a material dialectic is implied by his application of a theory of points onto subjective space as knowledge qua truth. Knowledge is formulated by the formalization of points. Furthermore, as atoms within a logical domain interrelate, knowledge is created and distinctions are formalized. With the fine tuning of material relations, we can understand also, that the material process is in fact discursive in nature. It’s no mistake that Eskimos, with their past uses and reliance on snow have over twenty different terms for snow in order to distinguish the very many and important and different uses of snow types. Each of the terms carries with it a localization of many different points, a host of spots as a unit complexity for decision making so as to better actuate their material environment. Their minute differentiation was essential for their survival before the introduction of modern technology (which now, many rely on). Modern technology however, does also create its own formalizations as various material domains are codified into vast logics, each of which has their own different material domains and their own expertise as a differentiation of different kinds of points, processes and procedures.

Economist John Kenneth Galbraith in his book The New Industrial State, recognizes that classical economics misses accounting for the role technology plays within capitalism. Indeed, technology increases our material agency, and with it, creates a new knowledge predicated on providing more domains of “either/or” thinking. In fact, our entire economy, as Galbraith notes, is a planned economy, one mastered by large corporations run by anonymous committees of administrators, experts whom he calls “technocrats.” Planning for the economy isn’t simply an acknowledgement of the need for experts who can utilize new technology, it’s also an extension of the very logic by which our economy is formulated, modeled and then manipulated. The creation of brands, logos, product lines, distinctions between similar products, distinctions between different populations that use similar products but may need those products to be marketed to, the choices in how to create those products (methods, materials, and each population’s cultural immanence), the decisions on how to manage workers who make products, which workers to hire or fire, and then the management and distribution of those products all rely on the creation of consistencies whose logical relations are measured out in terms of the different worlds populated by various Dedekind type cuts along a chain of capitalization (both material and monetary). Each cut of difference is a threshold for decision making. It’s no mistake cost benefit analysis and various other tools that measure out money, advertisement response and consumer behavior are all predicated on a quantitative mapping so that we can synthesize decisions between them. We measure behavior by each additional margin because the presence of an additional point allows us to weigh alternatives and make decisions. If you wanted to get a house, without money, how would we know which house we could want, given what we do? Without money, how do we realize that this house is beyond our means whereas that house down the street is adequate to our ranking in society? Without money, how do we know which item we value more: a book or a cup of coffee? Without money, how do we decide how many cars we make, how many movie theaters we build, and which method (or person) hinders productivity?

Of course, in terms of Badiou’s terminology, money is but one of the worlds we live in. There are alternate forms of presentation, which may operate through radically different logic. Each of the incompossible academic fields, the cultures (ethnic and corporate), the languages, jargons, genres of music listeners, sports (teams), artists, professions, sexual orientations and so on create their own consistencies populated by various different atomizations alien to one another. A biker gang will have signs immanent to their group (materially and discursively), defining appearance with rank, duty and prestige so that other gangs local to that biker gang and the members within that gang will have foreknowledge of who they are dealing with and where they stand in relation to each other. Likewise, with any military group, religion, or even loose affiliation of students or musicians will also produce new relations and introduce new atomizations. We can thus understand Badiou’s event, in this book, as an agential cut defining a consistency, one that characterizes a subject’s position within an “eternal identity” to become itself (even if it is not). Actively speaking, an event then, is how a logic presents itself to you via the passions you bring (and how you relate to yourself). In a more general sense, knowledge is functionally how you make meaningful the points within a given topography. In fact, topography is nothing if not expressed in minute changes, the vagueness of which is eliminated by a surjective mapping onto points! Knowledge is always supported (actualized) materially, but that material is predicated by a consistency of relation defined by that world’s logic, as relations immanent to material but transcendent as a domain, the navigation of which is political and democratic between agential atoms.

The value Badiou brings to this consideration of democratic materialism is a method by which we can distinguish the ruptures in logic created by our capitalist world, one which produces an excess of material and accordingly, produces an excess of trivial knowledge each of which is its own agential domain[6]. While Badiou only does this through the formalism of his method, in the sense that the world is rich, his book is impoverished because he only explicates the formalism which supports his book.

Now that we have a summation of Badiou’s magnificence, let us evaluate it.

To do so, we need to reflexively echo Badiou against a differing point of reference. Surprisingly Badiou supplies us with this point of reference, as he strives to distinguish himself from another philosopher who is most like him.

Discriminating readers will realize early on that much of what Badiou presents in this book addresses deficiencies of application with his previous work. The patching this book accomplishes brings him closer to the work of Gilles Deleuze and Felix Guattari, mainly through the deployment of the trope: sheaves of transcendentals. This comparison is meaningful as it allows us to detect a weakness in his book as well.

Badiou hides his weakness by delaying his talk on Deleuze until close to 2/3rd of the work, so that by then you may be convinced of what Badiou speaks of, on his own terms. Like Badiou, Deleuze also deals with multiple logics, although the term he favors is “sense”. The title of Deleuze’s second half of his doctoral work Logic of Sense can show us how closely Deleuze and Badiou walk the same territory. I won’t attempt a summation of Deleuze’s work here, but arguably the main difference between Badiou and Deleuze is between their approaches. Their methodology differs vastly as Deleuze forgoes formalization in order to wax with a multi-valence of partial terminologies as machinic assemblages territorialize each other, interacting as a rhizomatic architecture. In a way, Deleuze is closer to poetics and aesthetics than mathematics (as a neutral sounding jargon) as Deleuze’s main impact is through extended metaphors and conceptual imagery (although Deleuze does explore some math, Calculus and Riemann spaces). The impact is different than Badiou, who makes a gesture towards completeness instead of partiality. Deleuze and Guattari seek to create a meta language which is never complete[7]. For them, there is no attempt at being complete and describing all valid moves. Nonetheless, a discriminating reader will notice that despite their different expressions, there is a central spine that would support a very similar core between Deleuze and Badiou. As Badiou acknowledges “As Lyotard might say, this dispute amounts to a differend. For it concerns the fundamental semantic connection of the word ‘event’: on the side of sense of Deleuze, on the side of truth for me” (386). This may be a little confusing, as isn’t Deleuze’s sense much more like Badiou’s logic which designates truth? Recall how careful Badiou is to rely on formalisms to carry relations rather than saying anything conclusive about a particular content. Similarly Deleuze says very explicitly that philosophy is the creation of concepts. Deleuze is less interested in putting forth his own concepts than he is at outlining how concepts can be crafted. So at what point can we use to understand their key difference? The key point relies in their use of negation, as negation is the leap Badiou makes from logic into truth.

While Badiou sums Deleuze surprisingly well (in just a few pages), he introduces four Deleuzian axioms only to claim that his work, Logics of Worlds, reverses these axioms. This is an interesting deployment although at first glance, it doesn’t really appear to be a meaningful distinction as negation only works to focus the object in question, not to create a new logic.

As Badiou specifies throughout Logics of Worlds, negation is a minimal gesture to create an envelope. Unlike Deleuze, whose territories rely on an internal sense regardless of negation, for Badiou, transcendental envelopes have an immanence that is only expressed through negation. Each transcendental has an internal in-existence, a minimal gesture that negates a singular term. Badiou goes on to say that “the existential quantifier, ⱻ, may be interpreted as an envelope in the transcendental ” (179). This means that appearance in the transcendental requires negation to wrestle singularity from indeterminacy. Relating back to political materialism, for Badiou, negation remains the key to expressing any political dimension, for not only are bodies a totality, but this totality “is what gathers together those terms of a site which are maximally engaged in a kind of ontological alliance with the new appearance of an inexistence, which acts as the trace of the event.” This is what Badiou means for the event in logic: it compromises an excess of appearance at a site, predicted on the inexistence of at least one atom. “[N]egation appears because only the nothing grounds the fact that there is appearing” (107). This is further explicated by Badiou’s examination of the political status of Indians in Quebec. For there to be the transcendental domain of politically recognized atomized bodies, there must be at least one atom which is denied existence (negated) in order to posit a positive content of existence.

We can conclude the following: given an object in a world, there exists a single element of this object which in-exists in that world. It is this element that we call the proper inexistent of the object. It testifies, in the sphere of appearance, for the contingency of being-there. In this sense, its (ontological) being has (logical) non-being as its being-there (324).

Recall Dedekind’s cut? What is lost in negation is what was at a point (zero)[8]. Badiou’s relation of points to Kierkegaard is only because of Kierkegaard’s explication of the choice “either/or.” This disjunction focuses as either an establishment of a minimum without maximal limit (qua transcendental) or a maximum without a minimal limit (qua singularity). Negations cut logic by focusing for us this deployment of logical existence. Thus, for Badiou, the event is the establishment of a point, the negation of one side of it, in order to project a transcendental qua world. In other words, what is missing from a “Theory of Points” is the negated point where a point of view is focused from. From this point, a world is formed so that we can then know and experience in its truth the absolute existence of other relations that appear in its logical horizon.

Compared with Deleuze, the event is also the establishment of sense but in a more general way. Readers of Deleuze would be reminded of Deleuze’s distaste for negation as negation adds nothing to a general logic. In other words, Deleuze does not rely on negation for the existence of singularities[9]. This refusal to engage with negation is so that we can understand the ultimate arrangement by which logic territorializes, so that we can recognize a logic even if it is presented as a particular negation or deployment. This also relates to Deleuze’s élan vital, but more on that below as this concept relates to Badiou’s critique of Deleuze, which we will use to critique Badiou.

Badiou plays off the terminology of organs by including Lacan in his Book V “What is a Body?” While bodies need negation structurally, for Badiou bodies are not coherent unless there is a point. Badiou writes: “We then posit that an element x of a body Cε affirms a point φ, if φ (Ex) = 1” (488). In other words, the organ of body can only be understood as a substantial coherency if the synthesis of this body creates points that hold the organ φ in the body Cε but separate from the envelope. All this formalistic apparatii is necessary criteria for Badiou to structure truth. He describes five different arrangements, in concise terms so that a world is neither “atonic, stable, inconsequential, inactive, nor inorganic” (491). Only when these conditions are met, can truth be found, and knowledge can be generated, acted on and of consequence for us to make decisions about, for us to be able to affect change. The inclusion of negation for Badiou leads to a positive content qua truth, but the conceptual apparatus determining truth makes truth a rare condition indeed.

This key distinction of the structural role of negation is why, for Badiou, Deleuze fails. This failure is Badiou’s first critique. While Badiou acknowledges that Deleuze’s rejection of truth is sensible, he ultimately finds this rejection unsustainable as it leads to some “perverse effects” (386). Said in a more concise way, Badiou’s critique of Deleuze is that Deleuze presents worlds which are ultimately atonic. Using again, Dedekind’s cut, this time on the difference between Badiou and Deleuze, Badiou may project his point φ entirely limitless transcendentals for himself, which he does — whole infinite worlds of eternal truth. But for Deleuze, Badiou projects in the opposite direction, limiting Deleuze as a singularity. This is how Badiou recognizes Deleuze as a speaker of the One, as his “latent content” of élan vital, although Deleuze would never state that his philosophies lead to a limited singularity as One[10]. Despite talking of Deleuze as a worshiper of the One, Badiou acknowledges that Deleuze wouldn’t limit his own conceptualizations that way, just as Deleuze would have found his follower’s use of philosophy irrelevant.

I think that Deleuze, often skeptical vis-à-vis his own constructions once they touched on politics, would have laughed up his sleeve about all this pathos. It remains that, having conceptualized before everyone else the place of the event in the multiform procedures of thought, Deleuze was forced to reduce his place to that of what he called ‘the ideal singularities’ that communicate in one and the same Event’ (387).

An effect of this atony is that Deleuze’s lack of focus (something Badiou finds negation and truth would provide) permits people to ramble. For Badiou, this lack of focus forces Deleuze to wax religious about life. Again, applying Dedekind’s cut, we can understand that Badiou puts a cap on how relevant Deleuze can get and thus, no cap prevents us on how irrelevant we would make his philosophy.

If ‘singularity’ is inevitable, the other words are all dubious. [. . .] We have already said why ‘one and the same’ is misleading: it turns the One-effect on bodies of the event’s impact into the absorption of the event by the One of life. [. . .] And that, when [Deleuze] reabsorbs the event into the One of the ‘unlimited’ Aiôn, of the Infinitive in which it subsists and insists’, in the always-there of the Virtual, he has a tendency to dogmatism (387).

Obviously Badiou would not apply the same vector to his own work. Yet even so, how do we know which relation we want? If Badiou is presenting just the method, how do we start with his formalism? Which negation is the right one? This, of course, is a question of value, and one I think Deleuze apprehends but Badiou does not.

Kojin Karatani in Transcritique brings up an interesting point about value. He notes that value is the relation of one domain to a foreign domain. Karatani notes that both Kant and Marx understood this conception of value. Marx provided it through the split nature of commodities and capital: exchange-value is only possible because other commodities have different use-value. Also, merchant capital cannot capture surplus-value without finding new labor and new markets to colonize. As aforementioned, Galbraith’s technostructure economy requires that the comparison between different domains be created and navigated by different experts of those domains. Their interaction as experts assures each domain’s immanence (and their own position as experts) as being valuable within a highly structured economy that presents multiple domains that need navigation[11]. Because these (re/de)territorializations, value is created but only through the interaction with other domains outside the singularity of a corporate culture. Likewise, Deleuze through his partial conceptions, understands the value and creation of concepts as differential territorializations between different plateaus. One machinic assemblage weaves a territory by reterritorializing (or de-territorializing) another’s (such as a new app to present novel information to a client base). Deleuze’s relative lack of focus stems from his desire to outline the creation of concepts “on the fly,” to critique them and avoid the segment regulatory regimes of the State and other oppressive machinic indexes. In the same way, Karatani notes how Kant was able to sidestep the contemporaneous deadlock of metaphysics vs epistemology by speaking between the interstices — between the various faculties. That Kant’s genius was to speak from the interstice. In the exact same way but with more radical inclusion, Deleuze with Guattari also speak from the interstice, where there is no limit on ones’ ability to signify and conceptualize.

To a large extent, Badiou is able to account for connections between envelopes within a transcendental logic, although this really isn’t an interstice between separate worlds. In order to show relations between envelopes, we can turn to his example about Quebec. Here, he presupposes logical completeness by demonstrating that each participant is able to apprehend their relation to the other party. As usual, Badiou introduces some fairly intricate diagrams and with it, some new axioms. He writes that “Broadly speaking, the logical completeness of the world will obtain if it makes sense to say that the Relation between relations is itself in the world” (313). This greater Relation is where Badiou conceptualizes the transcendental as a mathematical logic of existence, connecting him to Kant.

Despite Badiou’s use of Kant’s term “transcendental,” Kant’s use of transcendental is merely a mode of apperception of objects (231) originating from a teleology. Common to both Kant and Badiou, Badiou writes, “the object is what counted as one in appearing” (234) that is “nothing other than the pure capacity for unity” (233). From this point on, Badiou then dismisses Kant for his “inferior” grasp of mathematics with some nice examples of Kant’s “crass” technique (237). Although Badiou takes the word transcendental from Kant, grasping that the transcendental is the field for object’s logical conditioning, he misses Kant’s use of the transcendental as a mode of critique. As noted by Karatani, Kant’s genius was being able to critique both metaphysics and epistemology. He was able to do so from the interstices between faculties in order to reject the immanent solution of a transcendental logic as being founded only on itself, what Kant calls an antinomy. Kant’s Critique of Pure Reason critiques pure reason by showing that reasoning by reason alone will create contradiction.

In some sense, this doesn’t mean that Badiou is incorrect about Kant in some sense, even if Badiou creates an antinomy and then refuses to recognize it as such. Badiou is right in this final analysis that Kant’s separation of form and content is too “academic” as it empties out content to leave the noumenal “ding as such”. By doing this, Kant leaves a space for Hegel and others to come in. In this sense, for Badiou, Kant is an interesting if misguided thinker necessary for the development of philosophy. One who needs to be surpassed in the process to Badiou.

While Badiou prizes Kant’s internal subjective phenomenal space, he finds fault with Kant. Without applying a Leibnizian monad or Descartes analytic geometry, Kant’s subjective space is relatively atonic, one whose points are only noumenal, the “empty forms” of various faculties. For Badiou, this is too academic to be useful! Without points of inflection, Kant remains “pre-Leibnizian, or even pre-Cartesian” (240). While Kant does improve on Hume, Kant’s understanding in his Critique of Pure Reason is that mathematics is synthetic, an understanding Badiou shares. Badiou’s main critique of Kant is that the empty noumenal forms misses the mark, “skirts the truth” (241). Kant’s careful avoidance of the noumenal “intellectual intuition” giving us access to the “in-itself of objects” as being a chimera is not for Badiou. Badiou unifies the phenomenal and the noumenal as “indistinguishable, the point of reciprocity between the logical and the onto-logical” as “[E]very object is the being-there or the being of a being” (241). What Badiou does not recognize what the unity of the synthesized nature of the transcendental means for the eternal truths of math is he after, for it means that math can never be complete because it always leaves out at least one point.

Badiou is quick to point out how closure is achieved. He posits mathematical closure with his Quebec example. He extrapolates from this a closure of Being itself from math, although not from ontology:

The history of the world is nothing but the temporal figure of the universality of its exposition. In the last instance, it is the unfolding of its overabundance of being. The infinite inaccessibility of the ontological support of a world gives rise to the universal exposition of relations and therefore to the logical completeness of that world (321).

This surmise of closure is an interesting point. Early on in Logics of Worlds, Badiou tackles the question of absolute completeness, of everything that can be considered. He is forced by his formalism to reject this possibility as it creates a contradiction (which is the antinomy). Ultimately it’s as simple as being forced to reject negation and non-negation as they are simultaneously incompossible. This is akin to stating that logically, at least half the world is missing from the largest transcendental mathematics can apprehend, solely due to consistency. Badiou doesn’t seem phased by this as he writes that the existence of the whole is too primitive a concept. He differentiates the inaccessible half as being an impossible “ontological” condition not a “mathematical” one. He points out that as with Cantor “the infinite can certainly be local, that it may characterize a singular being, and that it is not only — like Newton’s space — the property of the global place of everything.” This is similar to his first Being and Event where he is forced to reject the set of all sets, the same paradox that Bertrand Russell encountered in his attempt to provide complete totality of logic with mathematics. This is also why Kant’s acknowledgement of mathematics as being a priori synthetic is so stunning. If we treat mathematics as an object, as Badiou does with each relation, we should expect that mathematics on its own should be complete, that the system should have closure the same way that our experience of physical objects have with closure. After all, Badiou claims his Quebec example is a complete world when that world can attain the “Relation between relations is itself in the world.” Thus, if existence of objects is mathematical (and consistent) then we should expect that objects’ consistency in existence should likewise be mirrored in mathematical consistency in existence. Yet this is not so. The expectation of closure reaches an apex with the mathematician Kurt Gödel, in his “incompleteness theorem” which follows that all the properties of natural numbers cannot be summed up axiomatically. This means exactly that math lacks a “Relation” between all its many “relations”. How can logical completeness be attained by logic’s incompleteness? The answer seems to be that Badiou simply defers completeness into the infinite through what cognitive linguist George Lakoff and cognitive scientist Rafael E Núñez call the Basic Metaphor of Infinity. “The reasoning implicit in [the] move from an indefinitely ascending hierarchy to a highest, all-encompassing point in the hierarchy can be seen as an instance of the Basic Metaphor of Infinity, which the result of an indefinitely iterative process of higher categorization results in the highest category.” In other words, these words become infinite (and thus complete, without limit) because we can not only have relations, but also relations of relations, and relations of those and so on. Badiou avoids addressing true closure in the form of a logic that is consistent with itself (forbidden as shown by Gödel) by reliance on this basic metaphor of infinity.

What is the implication of true closure? Closure drives mathematics. We have already looked at various functions as “monsters”, as such functions once caused problems in mathematical mapping. Currently the unproven Riemann hypothesis lacks a formalistic rigor/proof which would allow it to be mapped completely into other domains (as of yet). Likewise many functions (like fractals or Hilbert space filling curves) have no derivative. In mathematical systems like Clifford Algebra, closure is introduced artificially, as a wrap around, so that Clifford Algebra can have closure in various different dimensions. In this sense, much of our increased sophistication in different mathematics (different synthetic forms math can take) results from seeking closure where there is none.

Lakoff and Núñez surmise that closure is impossible because, math is actually a meshwork of various unrelated cognitive metaphors. Because mathematicians can map these domains together we take these metaphors as a single synthesized field. Thus the inability for Badiou to find ontological closure for the appearance of mathematics to itself is wholly problematic as it relates to math’s suppressed inconsistency. This is not an issue that can be sidestepped by recategorizing the problem. The incompleteness and inconsistency of math with itself strongly suggests that mathematics is synthetic in nature, created from various pieces of relations extended and cobbled by human ingenuity.

There is also, the point that Badiou had previously rejected Kant’s noumenal/phenomenal distinction as being academic. If Kant sidesteps ontology by splitting the object, hasn’t Badiou also sidestepped splitting the object? The difference, of course, is that Badiou split ontology to save math from inconsistency by reclassifying the problem as an ontological one. Yet in doing so, Badiou has simply replicated empty form (as negation) vs content and form (as the positive assertion of being-there) through his insertion of negation as logically necessary for being-there to appear. We can also find a similar move with Kant, not only with objects but also with logic. After all, Kantian antinomies are also, as Karatani points out, a kind of bracketing necessary for knowledge to exist, as with these antinomies one cannot decide the either truth because their logic is being extended to areas beyond human experience. This suggests a sophistication with Kant that Badiou does not acknowledge simply because Badiou finds Kant’s treatment of math to be a “childishness in Kant, which is probably the other side of his provincial religiosity.” We can draw from this that Kant’s transcendental is truly an interstice, one that bridges multiple logics, and intuitively conceives of many different worlds as shown by his consideration of antinomies. While we can state that each of these worlds for Badiou can also overlap, highlighting different aspects of the ontological, we can also make this same motion for Kant. The difference though, is that for Badiou the transcendental is not an interstice, but as a world on its own, predicated on an imagined Relation due to Badiou’s use of the Basic Metaphor of Infinity. For Kant, however, the transcendental is the interstice, a non-position that we can then use to critique various logics and uncover other relations. It is not that Kant is stuck in one world/one transcendental, even though he only speaks of a transcendental, it is Badiou who is stuck in the rigid logic of his mathematical method even though he speaks of many transcendentals. This being stuck in a world is problematic since value is only given by application — which means that Badiou needs to connect math to the real world if he wishes people to find it valuable/useful.

So with these two objections to Badiou’s treatment of closure, like mathematicians like Roger Penrose, Badiou still insists on the eternal truth of math. One imagines these truths to be eternal because Badiou assumes are applicable at any time with any position. Here is a Descartes quote Badiou admires:

The mathematical truths which you can call eternal have been laid down by God and depend on him entirely no less than the rest of his creatures. Indeed to say that these truths are independent of God is to talk of him as if he were Jupiter or Saturn and to subject him to the Styx and the Fates (513).

While Badiou does not need God, he agrees with Descartes by “affirm[ing] that all truths without exception are ‘established’ through a subject, the form of a body whose efficacy creates points by point. But, like Descartes, I argue that their creation is but the appearing of their eternity” (513).

Again, this is why closure is so important. As math cannot be consistent with itself, it does not appear to itself. Math cannot talk about itself! Recall from the necessary inexistence of at least one point, the one one speaks from. Thus Badiou is forced to praise math with words instead of with axioms.

Yet even while Badiou praises math as the eternal mechanism to ascertain truth, he scoffs at language, stating that it is only derivative. He writes: “I regard the usual linguistic interpretation of logic as an entirely secondary anthropomorphic subjectivism, which must itself be accounted for” (173). But if math is the more general form that provides a direct link to truth, why is it that only a shadowy derivative is able to express the eternal pragmatic motivation behind why math is great?

The answer is simple. Language is partial, incomplete and the more general way of ascertaining existence (but also vague, by mathematical standards). Going back to our understanding of values, within philosophy Kantian transcendentals are an immanence whose appears is only structural. Since questions of metaphysics and epistemology he addressed are not as pressing, Kant becomes less relevant in those ways. It is in this sense of transcendental as speaking from the interstice that value can be created. Language, being partial to all domains, allows us to float abstractly between them[12]. This floating is also why linguistic arguments are not proofs in mathematics, as mathematicians wish to only address the neutral points of inflection, to dwell within their own domain. Language permits too much ambiguity. Nonetheless, this ambiguity is central to value: Language allows us to create value since language is everywhere and nowhere at once. Language in itself, however, may appear to be meaningless, when we isolate it via its own lingual exception, as Wittgenstein discovered. This is also why Badiou applies this logic of appearance to questions of life and death, the most absolute philosophical quandaries. Badiou recognizes that his own formalism may be off putting. If his subject is left solely as immanent, he may be dismissed as being solely “academic” as how he dismissed Kant. Thus, he makes connections, provides pointed examples, careful to translate the value of mathematics to things of this world because math, left on its own, doesn’t create value. A domain logic is only knowledge (and thus valuable) only when it affords us agency to talk about things outside of its immanence.

So what is our final evaluation? By presenting only logics atomized from logic and worlds synthesized by logic, we can understand that Badiou uses the transcendental to provide for consistency and completeness by eliminating what is indeterminate. The various worlds he mentioned before, as being “atonic, stable, inconsequential, inactive nor organic” are all invalidated, unavailable or in-navigable by Badiou’s standards. What does not fit for math, Badiou continually excludes until he concludes. And what fitting conclusion can there be, but to point out the applicability of these “eternal truths” and relate them to life itself?

The infinite of worlds is what saves us from every finite dis-grace. Finitude, the constant harping on of our mortal being, in brief, the fear of death as the only passion — these are the bitter ingredients of democratic materialism. We overcome all this when we seize hold of the discontinuous variety of worlds and the interlacing of objects under the constant variable regimes of their appearances (514).

In other words, our bewilderment at the excesses of capitalism and the fragmentation of markets, the creation of new markets and new knowledges, the material whirlwind of productive discourses leaves us nothing to hang onto but each immanent “reference-point” one that can make us “disenchanted animals” (514). The value of mathematics then, lies on its ability to preserve our “pure present” to afford us the possibility of a greater truth, which of course, lies in his “point by point” grasp of our ability to make decisions and live a heroic life. Badiou states: “We are open to the infinity of worlds. To live is possible. Therefore, to (re)commence to live is the only thing that matters” (514). Ironically, despite his earlier dismissal of Deleuze’s élan vital, Badiou’s conclusion ends on a plea based on life (qua the big concept of Life). This sentence of Badiou’s, ending his book, is nothing if not the very cry towards and aesthetics of élan vital (although one surjected into points, so that we may not need a contiguous élan vital at all just as Dedekind eagerly dismissed space once he theorized his cut!):

My wish is to make heroism exist through the affirmative joy which is universally generated by following consequences throughout. We could say that the epic heroism of the one who gives his life is supplanted by the mathematical heroism of one who creates life, point by point (514).

This is not the final critique I want to give Badiou however. His method itself is just as suspect. Having now established that his recognition of math’s (or any domain’s) value requires an evaluation that is beyond that domain, to turn its immanence inside out, we should also ask the question, why is math suited for this purpose? While Badiou does not see his own contradiction in his praise of math — that lingual arguments can sometimes (if not often) trump mathematic arguments (and vis versa), we should understand that Badiou’s analysis and synthesis isn’t a neutral supposition with each example.

To this end, we can introduce thinkers like Karen Barad, whose agential realism likewise proposes a materially discursive process by which actionable action and knowledge and both produced. Her theory originates from the understanding that the wave particle duality is possible because both particle and wave are produced through the materially discursive procedures of the scientists performing the experimentation. In turn, the experiment is not just material procedure but discursive procedure. The experience is also an expression of their underlying theories, in material form. Agential realism is not a suggestion of some kind of multiple worlds or that one’s consciousness helps create reality, but a theory that our physical and mental relations are entangled at a productive level that defines time, space while forcing a retroactive immanence through a material-discursive transcendence. I won’t dive at length into this other theory, but it is fruitful to quote her just a little.

The agential realist account does not position human concepts, human knowledge or laboratory contrivances as foundational elements of quantum theory. On the contrary [. . .], agential realism calls on the theory to account for the intra-active emergence of “humans” as a specifically differentiated phenomena, that is, as specific configurations of the differential becoming of the world, among other physical systems. [“H]umans” themselves emerge through specific intra-actions. And measurements are not mere laboratory manipulations but causal intractions of the world in its differential becoming. This means that quantum theory has something to say about ontology of the world, of that world of which we are a part — not as a spectator, not as pure cause, not as mere effect. [T]he human is not as a supplemental system around which the theory revolves but as a natural phenomenon that needs to be accounted for in terms of this relational ontology (352).

In other words, our supposition about being human, or the being-there of anything, signifies the moment we encounter ourselves through our actions (or our actions through ourselves). Despite Žižek’s complaint that Barad does not have a transcendental, here it is, as a material discursive topography that is self actualizing. Recall Badiou’s use of points: each inflective point is a reflection of our discursive-material process. This process maps, point by point, that which then reflects, some aspect of the radical difference we supposed to establish and some aspect of the material we have mapped as that point of difference. Agential realism recognizes that our material processes are discursive in nature and our discursive knowledge is created by material process. It’s no mistake that agential realism was first recognized through the work of Niels Bohr and then refined by Karen Barad, a theoretical particle physicist.

Quantum mechanics is a difficult field because the quantities are so foreign; they are only experiential through very specific equipment, equipment which is designed to pre-suppose the very nature of the objects they are meant to measure. Unlike more traditional fields, we have no independent experience of these objects predicated on our senses. Thus, quantum abstraction is far removed from how we know anything. In this abstraction, fields like mathematics can excel to provide a way to experience these quantities. But in doing so, our experience of these objects becomes specifically narrow, so that our final synthesis comes to reflect the underpinnings of our analysis. Recall Irme Lakato’s work on the Cartesian Circuit? Historically, we rejected the Cartesian Circuit. And yet its underpinnings of induction and deduction come crawling back simply because its truncated form is still in operation within the union of mathematics and science. Both math and science were phylogenetically one with philosophical supposition when all three (math, science, philosophy) were not fragmented into separate fields through their radical immanence, but entangled as natural philosophy.

So in this way, agential realism can be understood as a heuristic to recognize that the creation of specific domains, answering Foucault’s epistemological break. This break expresses as material knowledge, carrying with it the underpinnings of something like first principles. When we apply agential realism as a context to analyze Badiou’s mathematics we can come to the startling conclusion that while Badiou avoided interjecting any occult hypothesizes with each point he made — always careful to illuminate the neutrality of his mathematical supposition, of utilizing negation as the founding of a transcendental of envelopes — the method remains an occult hypothesis as to which point should be the “zero” he ascertains the world from. It is by this structure that he is able to analyze relations, and it is by the projection of a given negation qua event qua neutral that he is also able to ascertain the greater transcendental and come to develop his first principles as axioms. Why negation? Are these the only worlds? Why is truth dependent on this particular structure? What about other decisions that are made, are they not true? What about Badiou’s use of words when math can’t talk about itself? By definition Badiou creates the context so that only this particular localization can be decided as being true because this is what appears according to the logic of appearance, from that localization. Of course, how one projects being within any context is up to you. Deleuze would interject the poet Mallarmé:

[C]onsequences are not subtracted from chance by connecting them with a hypothetical necessity which would tie them to a determinate fragment; on the contrary, they are adequate to the whole of chance, which retains and subdivides all possible consequences. The different throws can then no longer be said to be numerically distinct: each necessarily winning throw entails the reproduction of the act of throwing under another rule which still draws all its consequences from among the consequences of the preceding throw. [. . .] These different throws invent their own rules and compose the unique throw with multiple forms [. . .] which leave them open and never closes them. (283)

True thought is chance, to extract from a moment a new conception, which is not a matter of closing worlds as a matter of category. It is this preservation of the image of the Idea as a set of categories that forces philosophy to follow “rules of sedentary proportionality” (284) so that past and future situations become simulacra of each other, trapped within the image of an Idea as a first principle to which one should be bound in eternity. Again this is the epistemological break reinforcing itself. As each domain of knowledge perpetuates its own agential efficiency, it creates immanent points within itself to enforce its own veracity, so that each field becomes isolated in its own agential realism.

As a human being: you know yourself, what your goals are, what you want to say, how you want the world around you, or how you want you in the world, and how you want to mix with the world, as both you and world have agency. But the larger logics of worlds that Badiou outlines for us, as epic and heroic as it may be, still relies on his own singular “eternal truth,” that he hopes we would recognize as being of value that we would live for an idea, and that this idea would be his (Badiou’s) in form (determining content) alone — what Kant would call a “transcendental chimera” informed by its own teleological necessity. Badiou’s method, while hopefully shown to have merit, is still nothing more than an occult hypothesis, an interjection Badiou would deem to preserve as the eternal truth, blinding him to alternate analysis and synthesis he engages in, just as Barad with Bohr in the interstice of complementarity demonstrates how discursive material processes inform experiment, experimental scientists and theory alike so that one apparatus finds only particles, and another finds only waves.

Whether it is our philosophical/mental abstractions that create the basis for our thought, or the material/discursive assumptions that create the basis for our experiment, such basis will always come back to us in the guise of a ghostly imprint, an excess that colors the answer we produce. We may insist that our basis be neutral, but the baggage of our own enforced neutrality will haunt us as a contextual skew. This skew can both foreclose and create the possibility that we have access to the Truth. After all, even if we, like Badiou, can see truth from a position, in recognition of our antinomy, to see it as the Truth, we must also pretend that the other position(s) don’t really exist, and are somehow illegitimate (or as Badiou said, ontological and not mathematical). In this way, being tied to a foregone conclusion is sometimes the only principle.


[1] As aforementioned, one of Badiou’s greatest strengths relies on his use of mathematical concepts, which are introduced as mere conceptual apparatuses. This appears to differ in method from most thinkers, like Slavoj Žižek, who utilizes more traditional methods. By using familiar language, Žižek quickly becomes entangled with needing to specify exactly what he means different from what others have meant using the same words, which I suppose is part of the fun of reading Žižek. Badiou however, can choose a loftier path, as his concepts are deployed in novel ways, so that their deployment is more clearly how he explains them. Nonetheless, Badiou still reserves the option to interject small chapters on individuals like Hegel, Kant and Leibniz into the fold of his mathematical formalisms, which as we shall see, goes a long way in demonstrating just what kind of philosophy Badiou wants to be, despite (or because of) his radical use of math.

[2] Image from's_formula.svg/2000px-Euler's_formula.svg.png

For a detailed explanation of a mapping this complex, see Where Mathematics Comes From for its excellent case study.

[3] Recall the myth of the blind men groping an elephant. Each man is at a different position of the elephant and thus understands the elephant as having a different nature. “It’s soft here”, “It’s hard here”, “It’s floppy here” as each blind man experiences a different relation than the other. The paradox here, is that each blind man then argues with the other, as they try to come to terms with those differences which emerge as contingencies due to their different localizations of the elephant.

[4] For instance, as the point of time decreases, with each marginal value, or as an opponent’s score increases with each marginal value, we may decide to switch strategies. Additionally, each gaining of a point, in a series of rounds, allows the players to judge the effectiveness of their activities, which is the key to formulating a strategy: “This time we go left, initiating scenario Tau”

[5] Given numbers A and B (where A is less than B) corresponding to the magnitudes of two line segments, there is some natural number n such that A times n is greater than B (298)

[6] As aforementioned, including product lines, technical specifications of products, services, company branding, logos, academic awards and various intra-industry distinctions which unify an industry by making a vector to which it should collectively refine itself. This is the epistemological break Foucault was puzzled over in The Order of Things.

[7] For reasons beyond the scope of this paper, incompleteness in Deleuze and Guattari is a way to resolve problems of domain consistency. Lacan for example, will adopt the moebial strip as a modality to suture completeness and consistency in his theory.

[8] In Žižek’s terms this lost vector would be understood as “less than nothing” or “the universal exception” which is hidden by the negated object by zero.

[9] These singularities could also be called “organs” in which the figure of sense would be a body without organs, a body in which organs slide around.

[10] In fact, in Difference and Repetition, Deleuze criticizes Plato for engaging in mindless discussions about the One vs the many. The main reason for this confusion has to do with Deleuze’s rejection of the image of the Idea. For Deleuze, concepts are constructed from the abstraction of the many, but a given concept is a choice founded on which of the many one differenciates through. Žižek makes this same error with Deleuze when he assumes that all concepts for Deleuze must map to the same Idea, when in fact, for Deleuze the tyranny of the Idea is the thoughtless elimination of thought itself. This is also why Žižek’s analysis often end in a repetition of a few categories of thought (Lacan and Hegel) whereas with Deleuze, there is never that enforcement of a thought.

[11] For instance, different fields of law, different fields of building code, different vehicles of marketing, different markets with different logics, different material sciences, different accounting and banking practices, environmental and social sciences are all examples of various fields of knowledge large corporations commonly navigate. The structure of a large corporation via various departments’ knowledge of given material processes demonstrate the various incompossible agencies a given corporation must cobble together in order to successfully flow merchandise and capital from one domain to another.

[12] This chasing of value and agency is why Deleuze and Guattari provided a partial language instead of a complete philosophical system.


Badiou, Alain. Being and Event II: Logic of Worlds. Bloomsbury Academic, 2013. Alberto Toscano, Trans.

Barad, Karen. Meeting the Universal Halfway: Quantum Physics and the Entanglement of Matter and Meaning. Duke University Press, 2007.

Deleuze, Gilles. Difference and Repetition. Columbia University Press, 1995. Paul Patton, Trans.

Foucault, Michel. The Order of Things: An Archaeology of the Human Sciences. 1st ed. Routeledge, 1994.

Galbraith, John Kenneth. Galbraith: The Affluent Society & Other Writings, 1952–1967: American Capitalism/The Great Crash, 1929/The Affluent Society/The New Industrial State. The Library of America, 1st

Karatani, Kojin. Transcritique: On Kant and Marx. The MIT Press, 2005. Sabu Kohso, Trans.

Lakatos, Irme. “The Method of Analysis-SynthesisMathematics, Science and Epistemology. Cambridge University Press, 1980.

Lakoff, George and Rafael Nuñez. Where Mathematics Come From: How The Embodied Mind Brings Mathematics into Being. Basic Books, 2001.