Friday, September 30, 2011

Fernando Zalamea podcast

Fernando Zalamea's seminar for the Versus Laboratory project, "Sheaf Logic & Philosophical Synthesis", is now available as a podcast:

It was a fascinating event, and I'll have more to say about it soon. For now, you can wash your dishes and sheafify your abductions at the same time. What else could you possibly want?

Note: To download the file, go HERE.

Note: The sound quality of the recording has been somewhat improved, thanks to the technical prowess of Julian Rohrhuber. The links have been updated to send you to the improved version of the podcast.

Monday, September 19, 2011


Catarina Dutilh Novaes of M-Phi and NewAPPS fame has just released an early draft of her forthcoming book, Formal Languages in Logic: A Cognitive Perspective, which promises to be extremely interesting. You can find a copy of this draft via the link above.

Reminder for Zalamea Seminar at JVE

TIME: SEPT. 29TH, 14h – 17h

The point of this seminar is not only to acquaint us with the vibrant landscape of contemporary mathematics – and the field of sheaf logic and category theory, in particular – but to show us how this landscape’s powerful new concepts can be deployed in the fields of philosophy and cultural production. Its aim is nothing less than to ignite a new way of thinking about universality and synthesis in the absence of any absolute foundation or stable, pre-given totality – a problem that mathematics has spent the better part of the last fifty years thinking its way through, and which it has traversed by means remarkable series of conceptual inventions – a problem which has also animated philosophical modernity and its contemporary horizon. This marks something of a variation on the theme of antagonism and technique that Versus has taken as its focus for the coming year: rather than seek to fragment philosophical concepts through the prism of non-philosophical disciplines – understood as something like “conditions for philosophy” – we will mobilize mathematical concepts and techniques to synthesize and render continuous what philosophy has fragmented. The crisp dichtomies of realism versus idealism, form versus content, the static versus the dynamic, and so on, are skillfully woven into a complex oscillating fabric that, far from obscuring the polarities in a night in which all cows are black, unleashes a living swarm of powerful conceptual nuances and distinctions from what was, in retrospect, a lazy taxonomy. This labour of synthesis, itself, demonstrates how far real mathematics – the living mathematical practice of the present age – outstrips anything dreamt of in our philosophy.

Our guide in this endeavour will be Fernando Zalamea, a Columbian mathematician, philosopher and novelist whose work seeks to explore the life of contemporary mathematics while redeploying its concepts and forces beyond their native domain. In an incessant, pendular motion, he weaves the warp of post-Grothendieckian mathematics through a heterogeneous weft of materials drawn from architecture and fiction, sculpture and myth, poetry and music.

We see Zalamea’s work as expressing an all-too-rare effort to subject philosophy to the condition of mathematics, and his degree of immersion and care for the latter is perhaps unmatched by any since Albert Lautman. If Lautman was Deleuze’s Virgil through the rings of modern mathematics, we may count on Zalamea’s work to guide us through the contemporary mathematics that we believe any philosophy awake to its own times must traverse. Just as analytic philosophy emerged from the shockwaves of the explosion of classical logic and set theory onto the scene in the early 20th century, the conceptual force of mathematics after Grothendieck holds the potential to spawn a new, ‘synthetic’ vision of mathematically-conditioned philosophy for the present age, one which Zalamea foreshadows under the rubrics of transitory ontology, epistemological sheaves, and universal pragmaticism. Though the seminar will not be fail to be of interest to mathematicians and logicians, who we think will find even their own terrain illuminated by Zalamea’s insights and mediations, we hasten to point out that the seminar will presuppose no prior knowledge of advanced mathematics.

We ask that the seminar participants read the excerpt from Versus Laboratorian Luke Fraser’s translation of Zalamea’s Filosofía Sintética de las Matemáticas Contemporánes (Synthetic Philosophy of Contemporary Mathematics), which is forthcoming from Urbanomic Press and which we have provided for the seminar participants in draft form. Like all Versus Laboratory seminars, this will be a fully participatory event, with plenty of time for a detailed discussion of the concepts and problems at stake.

Readings for the seminar can be found HERE

Saturday, September 17, 2011

Course: Philosophy and Theoretical Computer Science @ MIT

Not technically part of MIT's "OpenCourseWare", but all the readings are available as links & it seems that we may have audio to look forward to. 

Friday, September 2, 2011

Broccoli Logic

This "Broccoli joke" jab at Tarskian semantics turns up at least a dozen times in Girard's writings. Here is a particularly clear form of it, from "Truth, Modality, Intersubjectivity," which should be available in the ARCHIVES.
Truth according to Tarski
Long ago, Tarski gave a definition of truth of the form :
A ∧ B is true iff A is true and B is true. All the other cases being treated in the same way, to sum up :
A is true iff A holds. Such a definition discouraged generations of mathematicians from even thinking at logic ; moreover, logicians developed a sort of esthetics of the meta ensuring that you must be dumb if you don’t understand the depth of such definitions . But the king is naked and one must say it : the arrogant essentialism of the Tarskian approach hides the absence of any interesting idea as to truth. It relies on a fantasy of objectivity reused by logical hustlers to develop systems of their own, typically :
A broccoli B is true iff A is true broccoli B is true.
the logic of broccoli, which is not even edible! This is the triumph of discretionary definitions : the absence of a decent subjective dimension in the logical explanation eventually leads to subjectivism.


On the left-hand side of your screen, you'll see a link (listed under "Pages") to the Form & Formalism ARCHIVE. The idea is the begin something of an AAAAARG-like resource for the sort of material we deal with here. In the future, I'd like to move the archive off to a more secure and anonymous location. For now, it's probably safe and sound in this lonely neck of the woods, but if anyone can offer me any help in setting it up on its own website, that would be lovely.

For now, enjoy the library we've assembled. If you have any suggestions -- or, better, Scribd or links -- for books or articles that you'd like to see in the Archive, please send them to me. You can use the comments section at the end of this post for that purpose.

(Speaking of libraries, I've also started attaching the "Library" label to any posts that contain links to texts, some of which I might not have gotten around to placing in the archive yet.)

Nothingness & Event

Aside from a bit of formatting that remains to be done, I've just finished my article for MonoKL's upcoming special issue on Badiou. It's essentially an extraction (and condensation) from my MA thesis, which I wrote a few years ago on Sartre and Badiou. Though I see it as being, more than anything else, a sort of formal experiment on philosophical materials (I have a hard time drawing anything, I don't know, self-subsistent out of it) it might be of some interest to the readers of this blog.

The gist of the paper is to demonstrate a strict structural similarity between the form of the Badiousian event and that of the Sartrean for-itself, and then use this homology as a means to splice together the two structures, as a way of fleshing out the skeletal theory of the subject in Being and Event with the dynamics of lack that emerge from the immediate structures of the for-itself in Being and Nothingness. Pure scholasticism, really, but I had fun constructing it.

The paper can be found HERE.

Thursday, September 1, 2011

Lawvere on Mathematics and Maoist Dialectic

From The Chevron, Vol. 16, Nº. 31 (Friday, February 6th, 1976) Waterloo, Ontario, Canada:

I haven't yet been able to locate any written trace of the lecture mentioned in this article ("Applying Marxism-Leninism-Mao Tse-Thung Thought to Mathematics & Science"). But, to see how mathematics might appear in the incandescence of the Maoist dialectic, you can read this short but incredibly dense text, "Quantifiers and Sheaves" -- which, it turns out, was first presented in my hometown of Halifax, NS, where Lawvere held a position in the Dalhousie Mathematics Department (before, according to rumour, the university made the reckless decision of firing the militant, effectively flushing Dalhousie -- at the time one of the world centres of category theory -- out of the annals of mathematical history. One bloody-nosed reactionary in an elevator, the rumour goes, and the subject-body is dispersed). Know that everything after page one likely demands some familiarity with category theory in order to follow:

Lawvere - Quantifiers and Sheaves

What interests me, personally, in this line of thought is the particular relation it crystallizes between the dialectic and logico-mathematical thought, something I am just beginning to study for the same of a long-term project (a phd dissertation, as a matter fact, and a research project at the Jan van Eyck Academie). As I try to survey a few of the more interesting -- and explicit -- intersections between mathematical logic and the dialectic over the next several months, I'll try and put together some brief posts on the material for this blog. This will have me looking closely at approaches to the dialectic departing from game semantics, paraconsistent logic, and -- perhaps most interestingly -- Uwe Petersen's patiently articulated substructural sequent calculus with unrestricted abstraction. 

The ultimate objective will be to shed light on the tensions between the formal, the transcendental and the dialectical in Jean-Yves Girard's  research programme, which has for some time been an object of interest for me. 

Some remarks on Lawvere's text, based on quick and superficial first impressions:

PAGE 1 (p. 329):

At the very beginning of the essay, Lawvere situates his problem not only within the theory of the dialectic, but in Mao's theory of contradiction. No sooner have we located a unity of opposites than we have singled out its "leading aspect."

“We first sum up the principal contradictions of the Grothendieck-Giraud-Verdier theory of topos in terms of four or five adjoint functors, significantly generalizing the theory to free it of reliance on an external notion of infinite limit.” (329)

What is at issue here is a sublation, and a passage of the theory from being-in-itself to being-for-itself. Hegel:

“We say that something is for itself inasmuch as it sublates otherness, sublates its connection and community with other, has rejected them by abstracting from them. The other is in it only as something sublated, as its moment; being-for-itself consists in having thus transcended limitation, its otherness; it consists in being, as this negation, the infinite turning back into itself.” (Science of Logic, 126-7) [A pdf of the new Science of Logic translation, by the way, can be found HERE.]

Notice Lawvere's synthesis of the very concepts of ideological combat and scientific rigour:

"When the main contradictions of a thing have been found, the scientific procedure is to summarize them in slogans which one then constantly uses as an ideological weapon for the further development and transformation of the thing.”

A historical-materialist conception of scientific practice as the "development and transformation of the thing" -- a position most clearly articulated by the Althusserian school, in Badiou's Concept of Model, for instance -- is pushed one step further, and, to the extent that it short-circuits the distinction between ideology and science,* in a distinctly non-Althusserian direction: this notion of slogans -- as a means for condensing and steering that process of transformation -- as being at once both scientific instruments and ideological weapons, as the war cries of science!

* Now, I'm not entirely certain that this is a true exception to the Althusserian conception of science, which, after all, explicitly denies that the demarcation between science and ideology (or 'spontaneous philosophy') is visible from the perspective of scientific practice in itself. Drawing the demarcation takes philosophical work, and it is no less an intervention into scientific practice than it is a description... All I want to do here is to underscore the (already obvious?) fact that Lawvere's concept of "slogan" is an intense condensation of a significant epistemological problem -- the question as to whether there is, or can be, an identity between ideological combat and scientific process (an "identity of opposites"?).

Immediately after the previous quotation, Lawvere writes:
“Doing this for ‘set theory’ requires taking account of the experience that the main pairs of opposing tendencies in mathematics take the form of adjoint functors, and frees us of the mathematically irrelevant traces ([EPSILON]) left behind by the process of accumulating ([UNION]) the power set (P) at each stage of a metaphysical ‘construction’. Further, experience with sheaves, permutation representations, algebraic spaces, etc., shows that a ‘set theory’ for geometry should apply not only to abstract sets divorced from time, space, ring of definition, etc., but also to more general sets which do in fact develop along such parameters. For such sets, usually logic is ‘intuitionistic’ (in its formal properties), usually the axiom of choice is false, and usually a set is not determined by its points defined over 1 only.”  (329)
 Here we have one of Lawvere's main philosophical theses: that the dialectical concept of contradiction is either captured by, or at least expresses itself in mathematics as, the concept of pairs of adjoint functors. This is something I still need to explore, but I'll try and prepare a post in the near future which examines this concept and the argument for its association with dialectical contradiction.

This "freeing from irrelevant traces" is, again, an effort of sublation, of passage to the theory-for-itself, by relieving the actual conceptual labour that set theory performs, within the general context of mathematics, from reliance on extrinsic contingencies -- contingencies associated with the (accidental?) placement of this conceptual force by the Zermelo-Fraenkel axiomatic, by the defiles of syntax, etc.

Thus liberated, the concept of set is free for generalization -- a generalization which straightaway marks a departure from what, initially, seemed like the most basic and essential features of what we called 'sets': their discreteness and extensionality.

The internal logic of these 'generalized sets' (toposes or topoi) is, for natural reasons, no longer classical but intuitionistic (another way to put this is that their logic is now organized by a Heyting algebra instead of a Boolean algebra, which, in turn, can be seen as a special case of Heyting algebra). This does not mean that intuitionistic logic -- which can be thought of as classical logic without the law of the excluded middle, or, better, as an asymmetrical version of classical logic in which half of the elegant dualities of the latter (the De Morgan rules, for example) are flattened like a loaf of bread at the bottom of a grocery bag -- is dialectical logic by Lawvere's lights. It is just a local feature of it. When Lawvere makes himself explicit on this point, it is topos theory, or sometimes category theory which gets the title of dialectical logic, or, more precisely, of the "objective dialectic" (which seems to leave open the question of the subjective dialectic...).

I have a fair bit to study before I can comment any further on this piece. In the meantime, I'll be reading Lawvere's kindly pedagogical book, Conceptual Mathematics to catch up on the math.

It would also be a good idea to revisit Mao's On Contradiction, which, it's worth observing, is the first entry in Lawvere's bibliography.

Tuesday, August 30, 2011

"Academic Publishing as Economic Parasitism" post at NEWAPPS

Catarina Dutilh Novaes hits the nail on the head HERE.

I have nothing to add but agreement. Academic presses are, with few exceptions (the few which have made their texts available open access, or which have at least kept their prices low, as Dover has), an obsolete and parasitic farce. 

In a modest effort to exacerbate their obsolescence, or at least remove some of the obstacles to scholarship that they currently present, I would like to use this blog, or perhaps some affiliated website, as something of an AAAAARG.ORG-like resource, gathering together some of the texts that we, the Form & Formalism group, and our readers would like have access to. I'd like to put together something which does for our neck of the woods --- which is what? mostly (but not exclusively) 'continental' philosophy of mathematics and logic, I suppose? -- what AAAAARG is already (and admirably) doing for what seems to be a mostly cultural studies and continental political philosophy community.

Now, I could certainly put together a blog listing links to downloadable pdfs via sites like ifileit, etc., but maybe something more elegant could be built. I don't have much in the way of expertise in these matters, so any ideas or assistance you can offer, dear reader, will be welcome. 

Tuesday, August 23, 2011

Conference Announcement: HISTORY & PHILOSOPHY OF COMPUTING at Ghent

Don't let the 75th anniversary of Church, Post and Turing's 1936 papers pass you by without any philosophical ado:


This essay of Girard's is a rare thing of beauty. It is sheer genius. Bold, sweeping, profound, and yet incredibly accessible and coherent, not just the scattered gnomic gunshots typical of Girard's later writings, which demand the reader to pay for their brilliance with painstaking mathematical and philosophical gap-filling. I plan to write a longer post about this text in the near future, and about the broader programme for a Geometry of Interaction in general, but for now, I ask you to read this beautiful, brilliant essay:

Girard - Towards a Geometry of Interaction

Wednesday, August 17, 2011


So long as academic presses continue to jail our books behind prices like this, I consider anyone who's not making their work available online to be a fool. Paul Livingston, who is not a fool, has just released his excellent book, The Politics of Logic, to you, dear readers. [UPDATE: THE MANUSCRIPT IS NO LONGER AVAILABLE AT THIS URL. PAUL CAN, HOWEVER, BE REACHED HERE.] The book is a fascinating piece of work, which conscripts the conceptual achievements of analytic philosophy -- and, in particular, of that artery of analytic philosophy that has developed a sustained and brilliant reflection on the aporias of structure and language -- to the ends of compiling and illuminating an "orientation of thought" that can compete with Badiou on Badiou's own territory -- what Livingston dubs the "paradoxico-critical orientation". The main gist is something like this: what Gödel's incompleteness theorems throw into dramatic relief is not a simple obligation to accept incompleteness (of any formal system capable of expressing arithmetic, etc.), but the need to make a decision between inconsistency and incompleteness. Badiou's conditioning of his philosophy by mathematics, and principally by the metamathematical and foundational results of Gödel, Skolem, Cohen and others, elides this decision, and so passes over the possibility of the capacity for a rigorous -- and "complete" -- but essentially inconsistent discipline of formal thought to condition philosophy. Against Badiou's vision of the absolutely multiple, Livingston aims to deploy a vision of the paradoxical one, while retaining the ideal of conceiving radical situational change through the lens of formal thought. To this end, the book interweaves a sympathetic and subtle, but at bottom antagonistic reading of Badiou's work with a meditation on the foundations of mathematics and logic, and an invigorating synthesis of Wittgenstein and Agamben, Gödel and Derrida, and others.

Now go and read it [LINK BROKEN] for free.

Tuesday, August 2, 2011

Conference Announcement: Classical Model of Science II

Starting this evening at 20:15 with a lecture by the great philosopher of mathematics Stewart Shapiro [[UPDATE: SHAPIRO'S TALK HAS BEEN MOVED TO TOMORROW (WED.) MORNING AT 9:45. SCHLIESSER WILL BE SPEAKING TONIGHT ON SPINOZA, INSTEAD]], and running through to Friday (when Paolo Mancosu, historian of mathematics and editor of the indispensable anthology, From Brouwer to Hilbert, and Hourya Sinaceur, a brilliant philosopher of mathematics in the tradition of Cavaillès and author of Fields and Models, will both be speaking, among many others), the AXIOM group is hosting its second conference on "the Classical Model of Science", subtitled, "The Axiomatic Method, the Order of Concepts and the Hierarchy of Sciences from Leibniz to Tarski". The conference is taking place at the Vrije Universiteit Amstersdam, in the Philosophy Department. The conference programme, with schedule and abstracts, can be found here in pdf form. 
H/T to NewAPPS blog for the reminder.

Wednesday, July 13, 2011

Fernando Zalamea Seminar in Maastricht, Netherlands

It's official: Fernando Zalamea will be delivering a long-awaited English-language seminar on his recent work at the Jan van Eyck Academie in Maastricht (The Netherlands) this September, through the Versus Laboratory. As some of this blog's readers may know, I'm currently working on a translation of his 2009 book, Filosofía sintética de las matemáticas contemporaneas, for Urbanomic press, a few chapters of which we'll be reading for the seminar. Zalamea is an extraordinarily original philosopher with a vision of mathematical activity as deep and sweeping as any philosopher since Albert Lautman. If the resurgence of the mathematical condition into contemporary 'continental' philosophy is something that interests you, then this seminar is well worth the train to Maastricht. Like all JVE events, admission is free and open to the public. Here is the announcement from the Versus Laboratory page:

Sheaf Logic & Philosophical Synthesis

Thursday 29 September 2011
14h to 17h Auditorium

Fernando Zalamea (Universidad Nacional de Columbia), philosopher and mathematician, will illustrate how today’s sheaf logic carries a potential to revolutionize contemporary philosophical practice in a manner no less radical than the force with which the early 20th Century formalization of classical logic conditioned the practice of Analytic Philosophy. In particular, we will study a series of Grothendieck transformations of mathematical concepts, and come to understand how analogous procedures may be employed to the end of overturning a number of traditional philosophical polarities (realism/idealism, statics/dynamics, form/content, etc.).

Texts: Fernando ZALAMEA, Synthetic Philosophy of Contemporary Mathematics, trans. Z.L. Fraser, London: Urbanomic, 2012. Chapters 4, 8 and 9. Copies of the readings will be made available here at the end of August. 
Information on the book (forthcoming from Urbanomic) can be found here.

* In other Zalamea-related news, Reza Negarestani has recently mentioned plans for a series of introductory posts on Zalamea's work on his own blog, Eliminative Culinarism, which I highly recommend readers of this blog take a look at when they're up. Zalamea also has an article in the most recent volume of COLLAPSE, which I'm still waiting to get my hands on. [[UPDATE: NEGARESTANI'S INTRODUCTORY POST ON ZALAMEA'S PROJECT IS NOW UP, AND CAN BE FOUND HERE.]]

Saturday, June 4, 2011

Georg Cantor's 1883 "Grundlagen" article

Several years ago in Halifax, I found this translation of Cantor's 1883 article "On Infinite Linear Point Manifolds". The text comes from a set of mimeographed typewritten translations of three of Cantor's foundational papers on set theory and transfinite numbers, typewritten and mimeographed, it seems, in Nova Scotia's Annapolis Valley in 1941. The translator we have to thank for this is named George Bingley, though I haven't been able to find out any more information about him.

Of the three texts included in the manuscript, two are already available in English in the decently priced Dover volume. The third, as far as I know, is not [[CORRECTION: as far as I knew it was not available in English. It turns out that it is, in another translationin the 2nd volume of the mammoth and fantastically useful anthology entitled From Kant to Hilbert).]] Milner's French translation of the text appeared in Vol. 10 of the Cahiers pour l'analyse and can be found here.

Cantor - Grundlagen

Saturday, May 21, 2011

Mexico City Seminars on Mathematical Logic, Badiou & Girard

The slides for the seminars (which are predominantly in Spanish, or something resembling Spanish) are available HERE.

The dominant theme in all three seminars was the gap between syntax and semantics, conceived as site for conceptual and mathematical invention rather than as a call to mimicry. This took us from a study of Badiou's Concept of Model (read in the light of Althusser's critique of the "mirror myth of knowledge" in the introduction to Reading Capital, and as a more or less successful attempt to elaborate and refine this critique in order to intervene against the empiricist use of the logico-mathematical theory of models), a case study of the Löwenheim-Skolem theorem (the first really significant theorem dealing with the concept of model, and also the one which opened the abyss between syntax and semantics, and produced the concept of "non-standard" models), in Seminar 1, to a examination, in Seminar 2, of Badiou's category of "forcing", focussing on how it draws on the two mathematical "conditions" of Robinson's method for producing non-standard models for analysis (as analysed in Badiou's early text, "La Subversion infinitesimale"), and, of course, Paul Cohen's "forcing" technique in the controlled production of models for set theory, which was the key condition for the theory of truth and subjectivity in Being and Event. Seminar 3 dealt with Girard's work, and its conceptual and historical context. We looked at his critique of Tarski-style semantics, and his more or less tacit philosophical concepts of blind spot and of logic as essentially 'productive' -- themes which helped to link this material back to Althusser, etc. The shift from a set-theoretic paradigm to a procedural one -- the neglect of which would make Girard's project almost impossible to understand -- was examined in terms of the move from a set-theoretic conception of functions to a conception informed by the lambda calculus. A bit of time, not much, was spent on the parallel difference between Tarskian and denotational semantics (where what is modelled is the dynamics of proofs, not mere 'provability' -- the notion which semantic 'truth' roughly and imperfectly captures, in a garment which leaves nothing to the imagination and yet is far too bulky). We moved from there through the the sequent calculi and the Curry-Howard isomorphism (the isomorphism between proofs (in the sequent calculus, or natural deduction systems) and programmes (in the lambda calculus, the Turing machine formalism, or any actual computer), and so on, to linear logic, looking at the subtle tensions which emerge in our understanding of logic as we direct our attention to its hidden symmetries, its procedural aspects, etc. Another important conceptual distinction that we dealt with was that between "typed" and "untyped" systems -- focussing again on the lambda calculus, but with the intention of asking (in light of the Curry-Howard isomorphism) what an untyped logic might look like.

(Explanation, by way of example: in the untyped lambda calculus, every lambda term -- every programme or function -- can interact with every other lambda term, even if the result is a non-terminating procedure (a 'crash') -- "plus 1", for example, can act not only on the numerals for which it was designed, but even on functions which have nothing to do with numbers. The result's not always pretty, but something always happens. The untyped lambda universe is a wild world, and this leads to some very strange facts -- such as every function possessing a fixed point (for all F there exists an X such that F(X) = X) even if this fixed point is monstrous. In the typed lambda calculus, by contrast, everything is domesticated: the functions are saddled with a "superegoic" apparatus of types (Girard's metaphor, I think, if not Joinet's) which limits interaction, and allows terms to act only upon terms of the appropriate "type". The upshot is that every function eventually "terminates" or reaches "normal form" -- nothing crashes -- in the typed calculus, but the control by which this peace is won seems a bit artificial, or at least superficial, and doesn't really seem to proceed from the deeper structure of the calculus.)

[ADDENDUM: What is a 'type' in logic, you ask? A type is the name of a proposition. "A & B", for example, is a proposition of type A&B. The Curry-Howard isomorphism maps proofs to programmes, and propositions to types of programmes. So the question, "What would an untyped logic look like?" becomes something like "Can we do logic without casting our propositions in types prior to the demonstrative work that explicates them and tests them for consequences?" Can we have a logic where we don't begin with a battery of atomic sentences and pre-fabricated connectives? That's the gist of it.]

Finally, we looked at ludics, which is just such a logic (an untyped logic, that is), and which in Girard's eyes succeeds in sublating the gap between syntax and semantics. This section was pretty much improvised. I'll try and write something more precise about and thorough it soon, and post it here. [ADDENDUM: For now, I'll just say: cut-elimination, the algorithmic procedure by which appeals to lemmas are eliminated from a proof, rendering the proof wholly explicit, without 'subroutines', is the key. Cut-elimination is always possible for classical logic, always yields a unique result for intuitionistic and linear logic, but only in ludics does the dynamic of cut-elimination find its full scope, becoming the real engine of the entire system. In 'pre-ludic' logics, many characteristically 'semantic' properties can be expressed in terms of syntactic properties of cut-free proofs. Ludic 'interaction' -- a generalized form of cut-elimination -- reaches into crannies that ordinary cut-elimination can't.] In the meantime, curious readers can find some of my rough sketches of this subject matter (in English this time) here and here.

I'm happy to say that the seminars went extremely well, better than I could have hoped. I'm incredibly grateful for the boundless hospitality and generosity of Carlos Gomez, the Lacanian psychoanalyst who not only, through some incomprehensible faculty of persuasion, convinced the Department of Mathematics and Physics to invite me to come give the seminars, but ensured that my wife and I received full royal treatment while in the city. (And what a city!)

The participants in the seminars were few, but brilliant, and I left with several loose threads which I hope to follow up soon in my research. Among the most interesting of these concerned the sense that should be read into Girard's project for a "transcendental syntax" -- of which ludics is just one adumbration -- with one participant, named Cristina, pointing out that this sounds like Deleuze's conception of the transcendental more than anything (productive of what it conditions, untyped or 'wild', not already sorted into kinds, not resembling the conditioned -- unlike the Tarskian "meta"). This is something I'll have to look at more closely, so, readers, where should I start for a clear treatment of Deleuze's concept of the transcendental? Deleuze has always been someone I've liked quite a bit, but who I've read more or less casually. I'm thinking that Difference and Repetition would be the key text on this topic, but I welcome other suggestions.

Friday, May 6, 2011

Taller Sobre Badiou y la Lógica Matemática en la Ciudad de México

As I mentioned in an earlier post, I will be holding a three-seminar workshop this Tuesday and Wednesday at La Universidad Iberoamericana, in Mexico City. The seminars will be held in a combination of English and Spanish (I took the trouble to putting together my slides in Spanish, but will probably swing back and forth between the two languages as necessary. It's been a while since I my Spanish was at a fluent conversational level). I'll post my powerpoint slides to the blog after the seminars, if anyone's interested.

1. Sobre El Concepto de Modelo de Alain Badiou, (martes 10 de mayo, de 9am a 10:30 am y de 11 am a 12 pm)

2. El Concepto y La Categoría de Forcing (Forzamiento), de La Subversión Infinitesimal al Ser y el Acontecimiento, (Miércoles 11 mayo, de 9 am a 10:45 am)

3. El Proyecto Lógico de Jean-Yves Girard, como Radicalización Lógico-Matemático de la Critica de el 'Espejo-Mito' de Saber Criticado por Althusser y Badiou, y como una Condición Contemporánea para la Filosofía, (Miércoles 11 mayo, de 11:15am a 1:00pm)


El seminario donde el text se va usar se indica en abrazaderas, en la forma [S#].
Textos citados por negritos son fuertemente recomendados. Los otros son algo opcionales.
Unos de estes textos se pueden encontrar al sitio de Jean-Yves Girard:
o el sitio dedicado a los Cahiers pour l’analyse:

Althusser, Louis. “Prefacio: De El Capital a la filosofía de Marx,” in Para leer el Capital. Buenos Aires: Siglo XXI. [S1]

Althusser, Louis. Curso de filosofía para científicos (introducción: Filosofía y filosofía espontánea de los científicos, 1967). [S1]

Badiou, Alain. 2009 (1969). El Concepto de modelo: Introducción a una epistemología materialista de las matemáticas. Trad. Vera Waksman. Buenos Aires: La Bestia Equilátera. [S1, S2, S3]

———. 1967. La Subversion infinitesimale. En Cahiers pour l’analyse, Vol. 9. [S1]

———. 1968. Marque et manque: à propos de zéro. En Cahiers pour l’analyse, Vol. 10. [S2, S3]

———. 1999 (1988). El Ser y el acontecimiento. Trad. R. Cerdeiras et al. Buenos Aires: Manantial. Meditaciones 31, 33, 34, 35, 36. [S2]

Cohen, Paul. 2008 (1966). Set Theory and the Continuum Hypothesis. Mineola, NY: Dover. [S2]

Miller, Jacques-Alain. 1987 (1967). Acción de la estructura. En Matemas I. Buenos Aires: Manantial. (En francés: Action de la structure. En Cahiers pour l’analyse, Vol. 9.) [S2]

Girard, Jean-Yves. Proofs & Types. Trans. P. Taylor & Y. Lafont. Cambridge: Cambridge University Press, 1989. Vean especialmente Chapters 1-5. [S3]

———. Linear Logic, Theoretical Computer Science, London Mathematical 50:1, pp. 1-102, 1987. Restored by Pierre Boudes. [S3]
———. On the meaning of logical rules I: syntax vs. semantics, Computational Logic, eds Berger and Schwichtenberg, pp. 215-272, SV, Heidelberg, 1999. [S3]
———. Locus Solum, Mathematical Structures in Computer Science 11, pp. 301-506, 2001. (Vean la “Dictionary”, en particular.) [S3]
———. Le fantôme de la transparence, pour les 60 ans de Giuseppe Longo.
Identité, égalité, isomorphie ; ou ego, individu, espèce. D'après une exposé à la réunion LIGC opus 10, Firenze, villa Finaly, 18 Septembre 2009. [S3]
———. La syntaxe transcendantale, manifeste, Février 2011. [S3]
(Todos éstos textos de Girard son para examinar ligeramente. No se preocupen por los detalles muy difícils o técnicos. Están disponible a
Joinet, Jean-Baptiste. 2009. ‘Introduction’ a J-B. Joinet y S. Tronçon (eds.), Ouvrir la logique au monde: Philosophie et mathématique de l’interaction. Paris: Hermann. [S3]

Thursday, April 21, 2011

Everything you needed to know about forcing, but were afraid to ask Alain Badiou

Hat tip to Fabio at Hypertiling and Tzuchien Tho for bringing this to my attention. Wish I could be there. If the reason you can't attend this magnificent, London-based workshop is that you'll be in Mexico City this May, stay tuned...


A Set Theory Postgraduate Workshop for Readers of Being and Event or: Everything You Needed to Know about Forcing but were Afraid to Ask Alain Badiou

24 May, 31 May and 7 June 2011 in Room G03, 28 Russell Square

As part of its ongoing Seminar Series on Mathematics for the Humanities and Cultural Studies, the London Consortium will hold a series of three postgraduate students workshops that aims to discuss and provide a quick overview of some of the mathematics that needs to be known in order to follow the use of set theory
in Alain Badiou’s Being and Event [L'Être et l'Événement]. The main focus of these ‘bootcamps’ will be on introducing and discussing, in a friendly manner, the technicalities concerning the “mathematical bulwarks” mentioned in Badiou’s philosophical masterpiece:
(1) the Zermelo-Fraenkel Axioms of Set Theory
(2) the Theory of Ordinal and Cardinal Numbers
(3) Kurt Gödel and Paul Cohen’s work on Consistency and Independence.

Particular attention will be paid towards providing a sufficiently detailed, rigorous and clear explication of Paul Cohen’s technique of forcing and generic models, a mathematical result that Peter Hallward has called “the single most important postulate” in the whole of Being and Event. If time permits, we might also be touching on some recent mathematical developments in the field as well as trying to understand how Badiou contributes towards the contemporary re-intervention of mathematical thinking into philosophy.

The workshops will be held in room G03 at 28 Russell Square, Bloomsbury, London. The intended audience are postgraduate students or researchers who are interested in understanding Badiou’s philosophy but who lack the mathematical background. There are no assigned compulsory readings for the bootcamps,
and there are no pre-requisites save for some minimal familiarity with mathematics at the pre-university level. The sessions are free and open to the public but please register by sending your name, email and affiliation to so as to give us an idea of the numbers since the classroom size, unfortunately, will be limited.

Workshop Convener: Burhanuddin Baki

Schedule and List of Topics

Session I (Tuesday, 24 May)

2-4pm – Basic Mathematics and Short Introduction to Forcing
Short Introduction to Badiou’s Being and Event; Basic Arithmetic, Abstract Algebra and First-Order Logic;
Idea of Mathematical Proof; Induction; Naïve Set Theory, Cantor’s Theorem and Russell’s Paradox

6-8pm – Zermelo-Fraenkel Set Theory
Zermelo-Fraenkel Axioms plus the Axiom of Choice (ZFC); Peano Axioms of Arithmetic; Gödel’s
Incompleteness Theorems; Being and Event Parts I, II, IV and V; Some Alternative Axiomatizations of
Mathematics and Set Theory

Session II (Tuesday, 31 May)

2-4pm – Cardinal and Ordinal Numbers
Cardinal Numbers; Well-Orderings; Ordinal Numbers; Continuum Hypothesis; Being and Event Part III
and Meditation 26; Introduction to Surreal Numbers

6-8pm – Kurt Gödel and the Constructible Hierarchy
Some Relevant Model Theory; Formal Semantics of Situations; Formal Semantics of Events; Gödel’s
Completeness Theorem; Compactness Theorem; Löwenheim-Skolem Theorem and Paradox; Transfinite
Induction; Cumulative and Constructible Hierarchies; Axiom of Constructibility; Gödel’s Proof of
Consistency; Being and Event Meditation 29; Introduction to Large and Inaccessible Cardinals

Session III (Tuesday, 7 June)

2-4pm – Paul Cohen, Forcing and Generic Models
General Machinery of Forcing; Analogy between Forcing and Field Extensions; Cohen’s Proof of

6-8pm – Beyond Cohen’s Forcing
Cohen’s Proof of Independence (con’t); Being and Event Meditations 33, 34 & 36; Forcing in terms of
Kripkean Semantics; Forcing in terms of Boolean-valued models; Forcing Axioms and Generic
Absoluteness; Introduction to Categorial Logic and Topos Theory; Forcing in terms of Topos Theory;
Short Introduction to Forcing in Badiou’s The Logics of Worlds

Suggested Introductory Reading List

Avigad, J. (2004). “Forcing in Proof Theory”.
Badiou, A. (2007). Being and Event. Translated by Oliver Feltham. London: Continuum.
Badiou, A. (2007). The Concept of Model. Translated by Zachary Fraser & Tzuchien Tho. Victoria: re:press.
Badiou, A. (2008). Number and Numbers. Translated by Robin Mackay: Cambridge: Polity.
Badiou, A. (2009). The Logics of Worlds. Translated by Alberto Toscano. London: Continuum.
Bowden, S. (2005). “Alain Badiou: From Ontology to Politics and Back”.

Chow, T. (2004). “Forcing for Dummies”.
Chow, T. (2008). “A Beginner’s Guide to Forcing”.
Cohen, P. (2002). “The Discovery of Forcing”.

Cohen, P. (2008). Set Theory and the Continuum Hypothesis. New York: Dover.
Crossley, J., et. al. (1991). What is Mathematical Logic?. Mineola: Dover.
Devlin, K. (1991). The Joy of Sets: Fundamentals of Contemporary Set Theory. New York: Springer.
Doxiadis, A., et. al. (2009). Logicomix: An Epic Search for Truth. Bloomsbury Publishing PLC.
Easwaran, K. (2007). ” A Cheerful Introduction to Forcing and the Continuum Hypothesis”.

Fraser, Z. (2006). “The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Kripkean Analyses of
Forcing and the Heyting Calculus”.
Goldblatt, R. (2006). Topoi: The Categorial Analysis of Logic. Mineola: Dover.
Hallward, P. (2003). Badiou: A Subject to Truth. Minneapolis: U. of Minnesota Press.
Halmos, P. (1974). Naive Set Theory. New York: Springer.
Jech, T. (2004). Set Theory: The Third Millennium Edition, Revised and Expanded. New York: Springer-Verlag.
Jech, T. (2008). “What is Forcing?”.
Kanamori, A. “Cohen and Set Theory”. Bull. Symbolic Logic 13(3) (2008), 351-78.
Kunen, K. (1992). Set Theory: An Introduction to Independence Proofs. New York: North Holland.
Norris, C. (2009). Badiou’s Being and Event. London: Continuum.
Pluth, E. (2010). Badiou: A Philosophy of the New. Cambridge: Polity Press.
Potter, M. (2004). Set Theory and its Philosophy. Oxford: Oxford U. Press.
Smullyan, R. & M. Fitting. (2010). Set Theory and the Continuum Problem. Mineola: Dover.
Tiles, M. (1989). The Philosophy of Set Theory. Mineola: Dover.

The London Consortium is a multi-disciplinary graduate programme in Humanities and Cultural Studies. We are a collaboration between five of London’s most dynamic cultural and educational institutions: the Architectural Association, Birkbeck College (University of London), the Institute of Contemporary Arts, the Science Museum, and Tate.


Would you look at that? It looks like I forgot an entry. [[UPDATE: It seems I hadn't forgotten it at all---it was published on this blog back on March 5th. Somehow, due to my clumsy bloghandling, it disappeared, probably while I was trying to do some editing to it. As my wife put it, the suture unravelled somehow...]]

New and original material coming soon, by the way. Probably something like an overview of a series of seminars I'm going to be giving in Mexico City next month (on May 10th and 11th -- details to come!) on Badiou's Concept of Model and use of the forcing concept, before launching into desperate attempt to break out of Badiou's work with some reflection on Jean-Yves Girard's research programme in mathematical logic, the subject of my current research.

For now, though, I bring you SUTURE. Again.

SUTURE entry

The word ‘suture’ takes on three distinct meanings in Badiou’s texts. These do not mark distinct periods in the evolution of a single category so much as three different categories whose association under the same name perhaps signals nothing more interesting than synonymy—though some hesitation in accepting this conclusion is no doubt appropriate. To keep things as clear as possible, we will label these categories ideological suture, ontological suture, and philosophical suture, and we will deal with them in turn.

Ideological Suture

The word ‘suture’, in a sense which Badiou will diagnose as exclusively ideological in scope, first appears in Badiou’s work in late 1960s. It was the subject of an intense debate amongst the members of Le Cercle d’épistémologie (the working group behind Les Cahiers pour l’Analyse), which was polarized by the positions of Jacques-Alain Miller on one side, and Alain Badiou on the other. The first move was Miller’s. His contribution to the first issue Les Cahiers pour l’Analyse, ‘Suture (elements for a logic of the signifier)’, he sought to extract the concept of suture from the implicit state it enjoyed in Lacan’s teachings. By Miller’s reading, Lacan had recourse to the word ‘suture’ on a handful of occasions to name the covering of an essential lack in discourse, by way of an short-circuiting of heterogeneous orders (the imaginary and the symbolic, for instance), an operation that serves to constitute the subject by installing it in a chain of signifiers. Miller’s gambit, above and beyond his effort at exegesis, is to show that the operation of suture is at work even in those discourses where we expect it least, claiming to detect it in Gottlob Frege’s rigorously anti-psychologistic attempt to derive the laws of arithmetic from the foundations of pure logic. The focus of the article is Frege’s definition of zero as ‘the Number which belongs to the concept “not identical with itself”’ which, according to Frege’s earlier definition of a Number as a set of concepts whose extensions are equal, comes down to defining zero as the set of concepts F whose objects can be put in a one-to-one correspondence with the objects describable as ‘not identical to themselves’. On Miller’s reading, it is the exigency to preserve ‘the field of truth’ in which arithmetic must be inscribed that forces Frege to consider the extension of the concept ‘not identical to itself’ to be empty—for this field would suffer ‘absolute subversion’ if a term, being non-self-identical, could not be substituted for itself it the signifying chain (Miller 28-9). (This slippage from an object to the mark that indicates it goes unnoticed by Miller—a thread that Badiou will later seize upon in both MM and NN.) This definition, Miller claims, ‘summons and rejects’ (Miller 32) the non-self-identical subject, whose unconscious effects can be detected in the Fregean operation of succession (the ‘plus one’) that takes us from one number to another. That operation, Miller argues, functions only insofar as it is possible for the non-identical (the subject), lacking from the field of truth, to be ‘noted 0 and counted for 1’ (Miller 31). He grounds this argument on Frege’s definitions of one and the successor: Frege defines one as ‘the number of the extension of the concept: identical with zero’ (Frege §77, p.90), and defines the successor of n as the Number of the concept ‘member of the series of natural numbers ending with n’ (Frege §79, p.92)—a definition which could only yield n itself the zero which belongs to each of these series were not, again and again, counted as one. But this ‘counting of zero for one’, by Miller’s lights, depends entirely on the suturing of the subject that engenders the field of logical truth. It is therefore the subject that makes succession tick—but a subject manifested only in the suturation of its lack and so a condemned to miscognition on logic’s behalf.

Badiou will have none of this. The counterargument he delivers in ‘Mark and Lack: On Zero’ (which appears in the tenth and final volume of the Cahiers) can be condensed as follows:

(1) Scientific discourse in general, and mathematical logic in particular, is not a unitary field of discourse or ‘field of truth’ at all. It must be conceived, instead, in terms of multiple stratified apparatuses of inscription.

(2) At no point does any discursive operation in any of these strata have any occasion or need to invoke a radical, unthinkable ‘outside’. What looks like an invocation of ‘lack’—the statement that the concept ‘not-identical-to-itself’, for example, has an empty extension—is nothing but a referral to an anterior stratum of the discourse. No scientific inscription enjoys the paradoxical status of ‘cancelling itself out’, as Miller supposed to take place in the Fregean invocation of the ‘non-self-identical’. Analysing Frege’s definition of zero, for instance, we should see the inscription, on a particular stratum (which Badiou terms the ‘mechanism of concatenation’, or ‘M2’, and whose task is merely to assemble grammatical expressions), of the predicate ‘x is not identical to x’ as a perfectly stable inscription (which indeed presupposes the self-identity of the mark ‘x’ in a perfectly consistent fashion and without the slightest ambiguity). It is only on another stratum (M3, the ‘mechanism of derivation’, which sorts the output of M2 into theorems and non-theorems) that the system ‘rejects’ the existential quantification of ‘x is not identical to x’ as a non-theorem. In no sense does M3 cancel out the productions of M2, or summon them only to reject them: it receives these productions as its raw material, and operates on them in a fashion altogether different what we find in M2. On a subsequent stratum (M4), the predicate 0 can then be defined in terms of the predicate whose extension was shown to be empty, and so on. What transpires in all of this is not, and cannot be, the ephemeral invocation of the non-self-identical subject, or a wound in discourse obscured by the scar of the letter, but a stable relay between fully positive strata the assemblage of which ‘lacks nothing it does not produce elsewhere’ (MM 151), a rule which, Badiou affirms, holds good for all of science.

(3) Not only does the stratification of the scientific signifier exclude suture from science, it suffices to foreclose the subject from scientific discourse altogether, and this is the secret of science’s universality: Science is a ‘psychosis of no subject, and hence of all: congenitally universal, shared delirium, one has only to maintain oneself within it in order to be no-one, anonymously dispersed in the hierarchy of orders’ (MM 161).

(4) Rigorous stratification and foreclosure of subjective suturation are not just accidental features of science, but what constitute science as science. It is they that give form to the notion of the epistemological break, the continuous struggle by which science separates itself from ideology.

(5) ‘The concept of suture,’ therefore, ‘is not a concept of the signifier in general, but rather the characteristic property of the signifying order wherein the subject comes to be barred – namely, ideology’ (MM 162).

This is not to say that suturation never happens when scientists speak. It occurs repeatedly – but these occurrences are nevertheless extrinsic to science in itself. The suturing of scientific discourse is what occurs in the continual establishment of epistemological obstacles, the destruction of which is the sciences’ incessant task. This dialectic of stratification and suturation, or of science and ideology, is elaborated in the appendix to ‘Mark and Lack,’ in a detailed case study of Gödel’s first incompleteness theorem—a study which implicitly attacks Lacan’s attempt to exploit this theorem in ‘Science and Truth’. (See entry on ideology for more details.)

Ontological Suture

Upon mathematics’ ontological baptism, at the beginning of Being and Event, the word ‘suture’ makes a prominent return. It comes to serve two functions: to name the empty umbilicus that links each situation to being by way of the void that haunts it (‘I term void of a situation this suture to being’ (BE 55)), and to christen being with the ‘proper name’ Ø, the mathematical sign of the empty set. Given the rigour and severity of his attack on Miller’s application of the notion of suture to mathematical discourse, Badiou’s abrupt decision to declare Ø set theory’s ‘suture-to-being’ (BE 66) – in a sense ‘which will always remain enigmatic’ (BE 59) – may strike the reader as surprising. More surprising still is that no link, positive or negative, is drawn between the then-falsified Millerian thesis that the subject’s inconsistency is sutured by the arithmetical 0, and the now-affirmed thesis that being’s inconsistency is sutured by the set-theoretical Ø. Even in Number and Numbers, where Miller’s thesis comes in for a second round of attacks, we find the new metaontological suture-thesis affirmed with innocence throughout the book (see NN Chapter 3).

When pressed on this point, Badiou responds that between these two theses, the word ‘suture’ ‘changes its meaning’: it is no longer a question of invoking the void of the (Lacanian) subject, but the void of being as radical inconsistency. This was not in doubt. But the argument deployed in ‘Mark and Lack’ against applicability of the notion of suture to the Fregean 0 nowhere depends on the identification of lack, or radical inconsistency, with the subject. If the argument is sound then it will remain so under the uniform substitution of ‘being’ for ‘subject’, and one cannot use this substitution to flee the difficulties encountered by Miller: we cannot avoid seeing that ‘the torch which lights the abyss, and seals it up, is itself an abyss.’ If the meaning of ‘suture’ in Being and Event differs from the meaning of ‘suture’ in ‘Mark and Lack’ only with respect to the terms it relates—subject then, being now—then the Badiou of 1988 and after remains hostage to the Badiou of ’69, and the stratified psychosis of mathematics will absolve itself from ontology as relentlessly as it does from ideology, foreclosing being as radically as it does subjectivity.

Philosophical Suture

There is a third sense in which Badiou uses the word ‘suture’, which is not so obscurely entangled nor obviously connected with its older usage, though certain structural similarities can still be observed: here, it names a particular – potentially disastrous – way in which philosophy may relate itself to one of its conditions. The relation of conditioning that the philosopher is charged with maintaining between extra-philosophical disciplines (truth procedures) and her own collapses into a relation of suture when, by way of destratification, the philosopher confuses these two disciplines with one another. It is helpful to make a distinction here, according to which partner in the suture achieves dominance. The dominance of the condition—such as the poetic condition dominates the late Heidegger and his pupil, Gadamer, the political condition dominates certain strains of Marxist thought, the scientific condition dominates Carnap and Hempel, and the amorous condition dominates Levinas and Irigaray—is indicated by its hegemony over the philosophical category of truth and its capture of philosophical rationality. No other modes of truth but those of the condition, sutured in dominance, are recognized, and the philosopher measures her reasoning by strictures proper to the conditioning discipline. This renders philosophy incapable of fulfilling its mandate, which is to construct a systematic compossibilization of heterogeneous truths. The dominant position in a suture may also be occupied by philosophy. When this occurs, philosophy takes itself as producer of truths – the kind of truths, moreover, that ought to be entrusted to an external condition. When this takes place, the threat of disaster looms large, and so I refer the reader to the entry on that concept.

Tuesday, March 8, 2011


This entry completes the series of BADIOU DICTIONARY drafts that Form & Formalism has been bringing you for the last week (unless Tzuchien feels like tossing a few onto the blog), but keep an eye out for the book itself, which should (I think) be coming out at some point within the year. In it, you'll find entries on a slew of Badiousian vocabulary, written by some of the best scholars in the field. If nothing's changed since I last heard, we can expect entries from Nina Power, Tzuchien Tho, Anindya "Bat" Bhattaryya (who I've been waiting to see publish something on Badiou for ages; back in the days of the "Badiou-Dispatch" mailing list, everything he had to say about Badiou's work, and especially its use of mathematics, was profoundly clarifying), Alberto Toscano, both Brunos (Bosteels and Besana), Justin Clemens, Jelica Riha, A.J. Bartlett, Fabien Tarby, Michael Burns, Alenk Zupancic, Dominiek Hoens, Ozren Pupovac, and the editor of the volume, Steve Corcoran, along with a few I've probably forgotten or overlooked. (I welcome corrections on this point.)

That said, here's the entry: last but least, the VOID...


            The word ‘void’ is surprisingly equivocal in Badiou’s writings. Leaving aside the non-ontological, ‘operational’ notion of the void that Badiou discusses in his first Manifesto (where ‘the void’ figures as the empty place in which a philosophy receives the truths of its time), we can discern, in Badiou’s work, at least four distinct senses of ‘void’:
1.     The void as the ultimate ground of ontological identity.
2.     The void as pure, non-self-identical, inconsistent multiplicity.
3.     The void as the emptiness of the count-as-one, itself.
4.     The void as the ‘gap’ between presentation and what-is-presented.
These senses are not always easy to untangle. Sense 1 is the one most readily divined from the empty set (Ø) of ZF. The identity between any two sets is determined by the axiom of extensionality, which simply states that ‘two’ sets are identical if they have all the same elements. Since a set is never anything but a set of sets, this criterion implies a regress: before the identity of a set can be established, we must first establish the identity of its elements by looking at its elements, and so on. The only stopping point to this regress is Ø, whose existence is asserted by the axiom of the empty set, and which, alone, is immediately self-identical. The void, in this sense, figures as the ‘prime matter’ from which presentation is composed; every presentational multiplicity is conceived as ‘a modality-according-to-the-one [selon-de-l’un] of the void itself’ (BE 57).

According to Sense 2, the void is thought as a sheer chaos of self-differentiation – the plethos of being-without-unity that appears in the ‘dream’ evoked at the end of the Parmenides. The void is that aspect of every situation that is still-uncounted, and which inheres in every presentation as the invisible but ineffaceable residuum of inconsistency, a ‘yet-to-be-counted, which causes the structured presentation to waver towards the phantom of inconsistency’ (BE 66). This sense, too, resonates with the set-theoretic Ø – as witnessed by the theorem that Ø is a subset of every set – but at the price of a slight ambiguity: Ø is, mathematically speaking, a perfectly consistent set, neither eluding identification nor threatening the stability of the sets in which it inheres. And so it is that, ‘with the inconsistency (of the void), we are at the point where it is equivocally consistent and inconsistent […] the question of knowing whether it consists or not is split by the pure mark (Ø).’[i]
While these first two senses of ‘void’ seem capable of enjoying a strained harmony as interpretations of the empty set, Sense 3 seems to speak of something else altogether. There is no straightforward set-theoretic formulation of the notion of ‘the emptiness of set-formation’. If there is dissonance here, it is unheard by Badiou, for whom
it comes down to exactly the same thing to say that the nothing [the void] is the operation of the count – which, as source of the one, is not itself counted [Sense 3] – and to say that the nothing is the pure multiple upon which the count operates – which ‘in itself’, as non-counted, is quite distinct from how it turns out according to the count [Sense 2].
            The nothing [the void] names that undecidable of presentation which is its unpresentable, distributed between the pure inertia of the domain of the multiple, and the pure transparency of the operation thanks to which there is oneness [d’où procède qu’il y ait de l’un]. The nothing is as much that of structure, thus of consistency, as that of the pure multiple, thus of inconsistency. (BE 55; emphasis added)

Here, yet another thought of the void emerges, playing on the equivocity of ‘between’ (which can indicate either a distribution or an interval): The void is, now, ‘the imperceptible gap, cancelled then renewed, between presentation as structure and presentation as structured-presentation, between the one as result and the one as operation, between presented consistency and inconsistency as what-will-have-been-presented’ (BE 54; trans. modified): this is Sense 4 in our list. It is not at all clear how this conception of the void is related to the empty set (unless we imagine it to mark a suture in Miller’s sense). Here, Badiou seems to be naming something like the sheer moment of differentiation that must be supposed to hold sway between a set and its elements—a moment which, structurally, recalls nothing so much as the nothingness that, to Sartre’s eyes, interposes itself between impersonal consciousness and its objects, and which arises as an ‘impalpable fissure’ that arises in the heart of consciousness in the event of its reflexive presentation.[ii]
I would argue that it is this notion of the void that allows us to make productive and non-trivial sense of Badiou’s thesis that
for the void to become localisable at the level of presentation, and thus for a certain type of intrasituational assumption of being qua being to occur, there must be a dysfunction of the count, resulting from an excess-of-one. The event will be this ultra-one of chance, on the basis of which the void of a situation may be retroactively discerned (BE 56; trans. modified).

One way of explaining this thesis would be to show how an ‘event’, a moment when presentation folds back on itself, breaking with the axiom of foundation and foiling the extensional regime of ontological identity, formally replicates what Sartre called ‘the immediate structures of the for-itself’.[iii] The void—the ‘nothingness’—which erupts in such circumstances is not described by a set theoretical ontology, but a Sartrean one—one capable of systematically articulating the relation between the ‘void’ exposed by an event and the form of subjectivity to which an event gives rise (including its dimensions of temporality, possibility, normativity, and liberty). What remains lacking is a univocal concept through which these various senses of ‘void’ can be synthesized.

[i] Badiou & Tzuchien Tho, ‘The Concept of Model, Forty Years Later: An Interview with Alain Badiou’, in CM, p. 99.
[ii] Jean-Paul Sartre (1956), Being and Nothingness: An Essay in Phenomenological Ontology, trans. Hazel E. Barnes, New York, Philosophical Library, pp.77-8
[iii] Cf. Sartre, Being and Nothingness, esp. Part II, Chapter I, Section I.