One of my great frustrations is that I lack the mathematical background to understand Category Theory as I think Badiou is really on to something in his most recent work engaging with Category Theory. If someone has a recommendation for a very rudimentary introductory text (and I already have Goldblatt) I’d be eternally grateful. I am something of a peculiarity when it comes to Badiou. When I first began reading him ten years ago I was deeply invigorated by his daring to say “Truth”. Moreover, I was struck by his claim that maths are a form of thought in the context of a philosophical academic space dominated by Heideggerian romanticism and a hostility towards all things mathematical.
Nonetheless, coming from a much more network based and systems theoretical perspective, I’ve never found myself particularly intrigued by his account of the Event or Truth-Procedures, or non-relation and subtraction, being more fascinated by his discussion of situations. I think Badiou just gets it wrong here as to how change takes place. I think Badiou gets things backwards. The question isn’t how we move from non-relation of pure multiplicities qua multiplity without one to relation or being qua appearance, but rather how we move from relation to subtraction. In other words, Badiou places non-relation before relation whereas relation should have ontological primacy. The mystery is not how things come to be related as his most recent work would suggest, but of how something comes to be subtracted from a network of relations. In his most recent work with Logiques des mondes, Badiou seems to move in this direction while still maintaining the ontologically untenable thesis of the primacy of set theory where there are no intrinsic ordering relations among elements of a set and where everything is unrelated to everything else. I can get how this is powerful for thought, but nonetheless find it ontologically untenable (and here I think Badiou’s notion of Truth suffers from an implicit and unstated notion of ontological truth vis a vis the manner in which the elements of an Event reflect the situation of Being in set theory as pure multiplicities uncoded by the encyclopedia. Better had he begun his ontology with the theory of categories and sought non-relation from the relationality there.
If the work on appearance and Category Theory is so exciting, then this is because it conceives objects as pure relations or morphisms. An object’s identity, under this model– and here I’m speaking in a very thumbnail sort of way –is entirely exhausted in its status as a pure source of an action and as an action on another object as its target. In short, the identity of an object is an extended identity, like an underground assemblage spreading out to a variety of other objects, that includes its transformations on other objects in the group. Where a Derridean or Lacanian might argue, for example, that the object is subverted by its “semblable”, mirroring, or doubling in relation to another object that it requires as a prop to “make itself be”, the Category Theorist could argue that this just is (here I’m poking Graham) the identity of the object.
The object’s identity just is these functional morphisms between source and target. Now I know Graham here is having conniptions in this description of objects as it denies objects any sort of internality or withdrawal independent of their relations, but it’s at precisely this point that things become really interesting. For the relational nature of an object in Category Theory, if I’ve understood things properly, can be self-reflexive as well. In other words, we need not have arrows running from a source object to a target object, but can also have arrows or morphisms running from out of an object and back to an object in an identity function. That is, in proper autopoietic fashion, the object can be both its source and target as depicted in the three objects in the category above. In other words, here we would get the sort of “rigid designators” Graham talks about in his vacuum packed withdrawal, while still under a relational account as reflexive self-relation. I’m not sure where I’m going with all this, but it feels important. At the very least, this would be a way of thinking objects as acts rather than substances with predicates.