Two dozen arrows:
See also “Pierce’s Playground: Zalamea on Sheaf Theory on Dialogue” here.
Following the call for a universalist mathematics, theory of sheaves (a term from Middle English meaning “two dozen arrows”) may be useful.
The following is an overview of the various areas this blog will be heading, beginning with sheaves, moving on through towards topos theory, to Grothendieck topology, category theory, to “quantum set theory”, and so forth.
This sketch is obviously absolutely incomplete, obviously subject to much error, and obviously highly Fragmented in kind. Yet, as we have revealed, even Fragments do have a positive efficacy of their own. My aim here is to creatively synthesize a lot of threads in complex mathematics so as to provide a narrow directory for others who are interested in this general project of universalist mathematics. These threads should not be taken for more than what they are: Fragments.
What is a Sheaf?
According to Wikipedia, “In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.”
J. Benavides, in Sheaf Logic, Quantum Set Theory and the Interpretation of Quantum Mechanics, writes the following (see here):
The notion of a Sheaf of structures has its origins on the study of the continuation of analytic functions in the XIX century, but the modern definition was introduced just few years after the end of the second world war, by Leray, Cartan and Lazard, in the context of Algebraic topology and Algebraic Geometry. However, the idea of a sheaf of structures is naturally contained in the conception of spacetime which derives from the Galilean relativity principle. [...] This is the case of the sheaves of structures, which have a logic based on the next contextual-truth paradigm: If a property for an extended object holds in some point of its domain then it has to hold in a neighbourhood of that point. [...] This proves that the laws of the intuitionistic logic, which are those that assume the value 1 in a Heyting algebra, are forced in each node of a Sheaf of Structures.
In the Logic of Worlds, Alain Badiou is operating on the need to translate what a complete Heyting algebra looks like in terms of dialectical materialism.
Peter Hallward has a telling footnote in Badiou: A Subject to Truth, page 418:
To clarify further, putting Badiou’s project into its larger scope, Paul M. Livingston writes in his review of B&EII (see here) that:
Whereas Being and Event theorized the overarching structure of Being (at least insofar as it is speakable) as modeled by the axioms of standard set theory, Logics of Worlds turns instead to category theory to model the domain of appearing. In general, a category can be understood as a structure of relations; the identity of the objects thus structured is irrelevant, as long as this structure of relations is preserved. Most significantly for Badiou’s project, however, it is also possible to use a special kind of categorical structures, known as topoi, to model logical ones; for instance, we can use topos theory to model algebraically all of the axioms and relations of standard, classical propositional logic. In fact, it is a consequence of this categorical method that the logics modeled need not be classical ones; indeed, we can use topoi to model any number of non-classical logics, including intuitionist and many-valued ones. These non-classical logics can uniformly be understood as determined by total algebraic structures called Heyting algebras (pp. 173-190).
Now, Antti Veilahti rather carefully draws out the implications of Badiou’s aim, in his article Alain Badiou’s Mistake — Two Postulates of Dialectic Materialism (see here), using further insights from topos theory:
Topos theory — generally a theory of the categories of the so called sheaves — intersects the surface amidst set-theoretic problematics and categorical designation. In effect, one needs the theory of sheaves in order to effectuate the notion of ’space’ in categorical language. A sheaf of functions (or arrows) is an object not directly related to a particular function but a class of functions with varying domains. It is a functorial composition of different sections in order to give rise to effects similar to those pertinent to functions defined in set-theoretic sense. [...]
It would be interesting to contrast topos theory — as a real mathematical change — to his own event philosophy. Badiou’s fidelity in set theory makes him blind to precisely this new, topos-theoretic frontier that withholds Cohen’s procedure by other, ’diagrammatic’ means. It is another mathematical grammar that doesn’t designate the analytic plane of causation but internalises the ’dialectic impasses’ pertinent to such structural ’torsion’ that obstructs morphisms and ultimately the mathematical truth itself from being split. It is an alternative, inscriptive approach to mathematics that seems to falsify at least one of Badiou’s two arguments: (1) that the Event cannot be approached mathematically except by regulating its consequences; and (2) that formal logic is the only meta-structural foundation of such regulatory intervention of truth. Any one of these two propositions can survive only on the detriment of the other.”
According to Veilahti, Badiou has mistakenly neglected the weak postulate of materialism in favor of the strong one in his project of “democratic materialism”. This second postulate of materialism is much weaker, and it is the pre-condition for the first, stronger postulate of this atomistic materialism. The weak postulate reads as follows:
8.1. Postulate (Weak Postulate of Materialism). Categorically the weaker version of the postulate of materialism signifies precisely condition which makes an elementary topos a so called Grothendieck-topos.
In other words, Badiou achieves only the appearance of materialism when viewed from the outside, but he does not have an internal materialism to which his Logic of Worlds is committed because of its (lack of) use of a Grothendieck-topos.
What is Grothendieck topology?
In category theory, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. (Wikipedia)
Of note, Grothendieck’s political views were radical and pacifist, and he was the son of two anarchist parents. While I should not like to be the one to make the joke of Leftist infighting — e.g. Badiou the pseudo-Maoist (categorically, this time) pestering the anarchists once more — it does appear as if there is a certain “weak” sensitivity to the state of affairs that those in the anarchist tradition have historically carried, one which is often neglected by “strong” Marxist praxis.
This “weaker” anarchist ethos which pertains, in my view, to the notion of identity and identification when it comes to structural organization that is found wanting in communist-structured organizations. I think that Levi Bryant, on this subject in particular, is more or less correct in repeatedly touching upon this problem (see here).
It is interesting to note that the prevailing attitude of diagrams, mapping, cartographies, and so forth in the SR/OOO crowd seems also to be at odds with what Badiou’s Logic of Worlds requires. The move to diagrams seems definitely a positive move, and a more or less necessary one coming out of the “linguistic turn”, following Lacan’s mathemes. Now that we are oriented better with respect to the impetus given to us by Badiou, we must turn to consider his encounter with Deleuze in the spirit of sheaves.
What is an Envelope?
Henri Bergson wrote, in Matter and Memory, p.99, that
Life is tendency, and the essence of a tendency is to develop in the form of a sheaf, creating, by its very growth, divergent directions along which its impetus is divided.
Deleuze likely had this idea of a sheaf in mind, at least following his reading in Bergsonism. He is without a doubt striving for an ethos which remedies these problems in that it becomes increasingly more inclusive to multiplicity. In works such as The Fold, we find that “there is no one substance, only an always-differentiating process, an origami cosmos, always folding, unfolding, refolding.”
Thus, it is not surprising that Deleuze writes, on page 23-24, the following:
Now, recall the brief note of Hallward, wherein he translates Badiou in terms of the ‘envelope’.
Furthermore, in Logic of Worlds, Book II, Section 3, Subsection 6, Badiou requires that the envelope of any given subset of T (defined by a certain property P) must obey the following two properties:
- It is greater than or equal than all the elements q of T that have property P.
- It is lesser than or equal to every element t which, like itself, is greater than or equal to all the elements q with property P (least upper bound)
Furthermore, in his remarks upon Marcel DuChamp (see here), Badiou notes: “Of course, there is a superior irony with respect to the envelope. The object that envelops the point is particularly without particularity.”
The differences are now made clear through these many Fragments, insofar as Deleuze seems to be mistaken here in his positing of “inclusion” as the final cause of a fold. Is it possible to answer Deleuze’s question otherwise? Certainly, this is true, as we have for our example Badiou’s formal act of folding which seems to be performing an inclusion and an exclusion simultaneously.
Enter Francois Laruelle, who notes in I, the Philosopher, Am Lying: A Response to Deleuze (see here):
Deleuze has discovered a secret — the secret or the property of philosophy, a secret which gives us the impression that it is very old and that it has been lost. He discovers the philosophical idiom, which now becomes alien to itself, but which remains an idiom precisely because it has become the language of the infinite. The language of the good news is absolutely private and absolutely universal. Their coincidence is the peak of the self-contemplation of the philosophical community. Hence the horror displayed towards transcendent artifacts like consensus and communication.
The authors at Fractal Ontology continue by noting that in the work of Deleuze & Guattari “…the ashes of a critique of communication end up communicating only the reasons for abandoning communication.”
This insight appears to hold true, and so we find ourselves with a choice: quantum mechanics, or set theory? How do we possibly decide upon the undecidable?
Daniel Tutt, at Spirit is a Bone, notes appropriately (see here) that:
It is Badiou’s contention that the end point of the multiple is the empty set. For Deleuze, to realize the virtual, you must construct a plane of immanence, which is pure chaos where simulacra and the virtual precede all thought. Because science does not attain to the ground of its own truth and it passes through the plane of immanence, science does not realize the virtual. For this reason, Deleuze can’t understand Badiou’s reason for choosing set theory, because the pure multiple can never be a set (CB, 48).
Now, for all the clamor that has occurred since the publishing of The Clamor of Being, with all kinds of positive reviews (see Roffe’s review here) and reviews of reviews (see Williams’ review of Roffe .pdf here), we find ample evidence of Badiou’s reason for choosing ZFC set theory. In particular it is, among many other things, related to the need to cut back on drastically on the distracting noise of the post-structuralist “linguistic turn”.
He accomplishes this through a new emphasis on formalization and a re-birth of Cartesian rationalism. However, we have seen that provided the way in which Badiou carelessly throws out this “weak” postulate of materialism, the underlying mentality of Deleuze must be kept even if Deleuze clearly had his share of mistakes, too. In this way, is there ground for a bringing-together of both quantum mechanics and set theory? Yes, we need not decide between the two names of Being — we may take both.
Insofar as Badiou is committed to sheaves, and Deleuze is likewise committed to sheaves, we may likely find the beginnings of a pluralistic sheaf theory in the non-standard work of Francois Laruelle. It is Laruelle, in the end, who must articulate something like a “quantum set theory” in order to unite these two loose ends of relationality. According to recent lectures by Anthony Paul Smith, it sounds as if Laruelle is dabbling in quantum physics. Perhaps he is simultaneously dabbling in alternate forms of category theory, too.
In the mean time, let us attempt a creative synthesis…
Quantum set theory:
“Since quantum theory is inherently blind to the existence of such state-space geometries, the analysis here suggests that attempts to formulate unified theories of physics within a conventional quantum-theoretic framework are misguided, and that a successful quantum theory of gravity should unify the causal non-Euclidean geometry of space time with the atemporal fractal geometry of state space” (see here)
We may resort to a post- or otherwise non-Cantorian notion of set theory.
To develop a post- or non-Cantorian set theory, we can assume the “negation” so to speak of the Continuum Hypothesis. As we have seen in the previous post “Continuous integration” (see here), we may posit the actual infinite as woven into the fabric of reality itself. The route for a quantum set theory remains open. This route seems to be unconventional, if not entirely non-standard.
Along the way, let us not forget to remember the radicality of Cantor’s proposal at the time, as recalled in Saburo Matsumoto’s Call for a Non-Euclidean, Post-Cantorian Theology (see .pdf here):
Often this type of mathematical unorthodoxy is compared to theological heresy. Kronecker accused Cantor of being a “corrupter of youth” for teaching transfinite numbers (Aczel, 2000, p. 132). Again, Burton describes the attitude of Cantor’s contemporaries vividly (2003, p. 629):
Cantor became a heretic. The outcry was immediate, furious, and extended. Cantor was accused of encroaching on the domain of philosophers and of violating the principles of religion. Yet, in this bitter controversy, he had the support of certain colleagues, most notably Dedekind, Weierstrass, and Hilbert. Hilbert was later to refer to Cantor’s work as “the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.”
On the whole, in its theological content, it is striking to find that this paper contains many elements articulated in Laruelle’s Future Christ, albeit with a far less heretical feeling. It is also interesting to note that it is by way of theology that these calls for non-Euclidean and post-Cantorian models are being called forward. Perhaps there is, after all, something to the experience of the Wilderness which imbues the Wanderer with an abundance of Life.
What does a quantum set theory look like?
My proposal is that a quantum set theory has everything to do with relationality. In other words, it is a matter of using our sheaves (arrows) “properly”, as though to have the Vision of a skilled archer.
Despite all of their differences, Badiou and Deleuze seem to come to similar conclusions on the aspect of relationality, even prompting the term “brothers in relational arms” (see here):
In summary then, Badiou employs two different regimes: being/the ontological/set theory and appearing/logic/category theory. Within the latter domain, whilst at first it might appear as if relationality is key, in fact this is constrained to self-relationality, with relations between elements being strictly a secondary derived property of this. Perhaps I am confusing things here, but it certainly seems that in this regard Badiou comes very close to Deleuze, in that relations between elements are always secondary to the relations of those elements to themselves (cf: difference-in-itself). This comes close to Deleuze’s position:
“… the differential relation between bodies/forces/etc, in which the two terms in the relation are brought together, their degree of power measured against one another, and given specific qualities on the basis of this. (like, in the determination of forces two forces come into contact and the one that is of a greater quantity of force is designated active, the other reactive…).”
How convenient that they would be close to each other, in a striking “family resemblance”, on the question of relationality.
Moreover, it seems much could be explained about the wild and violent encounter between them by how close they really are. One may jokingly view their feud as an instance of an otherwise loving, sibling rivalry. To put it in a far less tongue-and-cheek way, John Mullarkey very carefully notes the following in his Badiou and Deleuze (see here):
Any possible ad hominem dimension to this observation should be set aside, however, for the more interesting implication it contains: that Badiou needed to demonstrably create some distance between his own thought and Deleuze’s, not because of their opposition, but because of what they hold in common. Again, this is not an ad hominem argument I’m making (that might involve some kind of ‘anxiety of influence’), but simply to note the shared objectives that animate Badiou’s and Deleuze’s philosophies. It is their methodologies that fundamentally differ rather than their core concepts and attitudes… [...]
To think about relationality in this way is to bring us back immediately to our previous search for a transfer principle as full as possible.
Badiou, Alain. Deleuze: The Clamor of Being. Minneapolis: University of Minnesota Press, 2000. Print.
Badiou, Alain. Being and Event. London: Continuum, 2005. Print.
Badiou, Alain. Logics of Worlds: Being and Event, 2. London: Continuum, 2009. Print.
Deleuze, Gilles. The Fold: Leibniz and the Baroque. Minneapolis: University of Minnesota Press, 1993. Print.
Hallward, Peter. Badiou: A Subject to Truth. Minneapolis, MN: University of Minnesota Press, 2003. Print.