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.