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.

May 14, 2009 at 7:02 am

A book I will be picking up soon that looks very promising (and even has the perfect title given the directions you’ve taken):Tool and Object: A History and Philosophy of Category Theory (Science Networks. Historical Studies)

May 14, 2009 at 8:27 am

i) It’s

bastard hard. Sorry. No royal road, and all that.ii) I’ve heard good things about Lawvere and Schanuel’s

Conceptual Mathematics: A First Introduction to Categories.iii) I have Lawvere and Rosebrugh’s “Sets for Mathematics”, which in spite of the title is in fact substantially about category theory, and is generally slightly easier going than Goldblatt.

iv) There is surprisingly little category theory (in the sense of diagram chasing, or even discussion of categories other than Set), or discussion of the significance of the category-theoretic perspective, in Logics of Worlds.

May 14, 2009 at 2:19 pm

Hey LS,

The Stanford Encyclopedia of Philosophy has an interesting page on Category Theory:

http://plato.stanford.edu/entries/category-theory/#3

As for your post, again very interesting. However, about mid-way through you state:

“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.”

But then you state:

“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.”

So does this mean that an object’s identity can be structured by both its relations or functions with other objects and also its own, internal functions? I wonder to what degree this speaks to a theory of categories and Aristotelian formal characteristics? Can we then say that an object is both Form and Matter, where the functions of the object that point toward another object are functions of that object’s Matter, whereas the self-reflexive functions are a type of Form – Form as self identity?

May 14, 2009 at 2:21 pm

Oops, I see a typo in my last post…

I meant to say in my last paragraph:

“I wonder to what degree this speaks to a theory of KINDS and Aristotelian formal characteristics?”

May 14, 2009 at 2:50 pm

NrG,

I think this is one of the most interesting features of what I’m calling the “self-reflexive identity arrows” which you can see depicted in the category at the beginning of the post. If I have understood some of the concept correctly– and as always I defer to Dominic here –there is no

internalstructure to the object involved. That is, just as the relation between source object A and target object B in the category above has no internal structure, but rather defines the two objects in terms of the morphism or functional relation between the two objects, the identityrelationbetween A and itself depicted by the line that curves outward and that returns to A is itself a purely relational affair withoutinternalrelation. In other words, like an autopoietic system, the object is not something more over and above this self-relating or theactualityof what takes place in this self-relating.May 14, 2009 at 2:55 pm

Dominic,

I noticed that about Logics of Worlds as well. What a shame as the diagrams are really the best part. I find them to be intensely beautiful and fascinating in their own right. Thanks for the recommendations!

May 14, 2009 at 3:44 pm

LS,

Perhaps “internal” was the wrong word, but what I am getting at is that this identity arrow seems to set the boundaries for objects – or its structure or form.

For example, suppose we have object A. Now, it seems to me, since every object must have a source and a target, then when we have an identity arrow or 1A aren’t we simply saying that A –> A = S(A) –> T(A), or the identity arrow from A to A is equal to the function of the Source object A to the Target object A? In this way doesn’t the identity arrow set the boundaries as to what we can say A actually is?

May 14, 2009 at 4:15 pm

NrG,

I honestly don’t have the answer to that question. Arrows or morphisms are more or less functions, so presumably an identity arrow of this sort would be some sort of function whereby outputs of the object also function as inputs for the object. However, I am very likely reading too much into this based on my own ontological commitments.

May 14, 2009 at 4:48 pm

LS,

No problem. I have been trying to work out an understanding of how it is an object becomes a single object, yet at the same time remains a construction – i.e., of wholes and parts – so I was probably reading too much into this, as well.

May 14, 2009 at 5:04 pm

Why was my comment deleted?

May 14, 2009 at 5:13 pm

R.S.,

If it was deleted it was a mistake. There is a comment by you in this thread. Did you post a second comment?

May 15, 2009 at 8:24 pm

[…] post Principles of Onticology (and here, here, here, here, here, here, here, here, here, here, and here). In developing this ontology it should be noted that I proceed experimentally in much the same way […]

May 16, 2009 at 5:19 am

[…] its own differences translating the difference from the source object (cf. my recent post on Category Theory). My skin does not simply transport sunlight as I weed my garden, but rather it translates that […]

July 23, 2010 at 10:27 am

If no one got to you before I did, Lawvere and Scanuel’s ‘Conceptual Mathematics’ is an introduction to category theory that starts very elementary (much more so than Goldblatt) but still gets fairly deep into CT methods. I have a very minimal maths background, and I’m finding it very understandable.

October 20, 2012 at 11:03 am

From a Geometrical Point of View: A Study of the History & Philosophy of Category Theory by Marquisse is a book that raises important questions about certain implications of category theory–eg, the concept of “weak equivalence” along nomological, ontological and epistemological lines. Tool & Object by Krammer, less so, and is more absorbed with technical pragmatics that a PhD type work would raise. Probably, the best way to get to the philosophical morphisms of category theory is to look at Sets for Mathematics by Lawvere and Rosebrugh (2003) and work out for yourself its implications in various modes of thought as Brouwer would have used a philosophy of mind to determine what is mathematically viable or not. BTW although Mac Lane joked that the title “category theory” was “purloined” from Aristotle’s categories, Kant and Carnap, the more serious point is that category theory formalises the substantive-to-teleological intuition of the primary categories of Aristotle by providing a general procedure for obtaining commutative compositions of “said-of” and “being present-in”. Thus, category theory completes the “mapping to” portion of Aristotle’s intended methodology which ushered the first scientific revolution (‘the mapping from included the excluded middle’) and now with category theory, we have a method for unification of sciences (without the need for the principle of excluded middle or even the axiom of choice). Category theory, basically, is a form of Leibniz’ dream of a mathematica universalis. It succeeds where other attempts have not, because it can and does clarify, the morphisms and endomorphisms of element. [Note: a generalised element in the category of abstract sets is the morphism: 1–>A!]