In category theory as practiced, there is this phrase "generalized abstract nonsense" (GAN), which is often used to cover sections of a derivation/judgment(?) that the audience is meant to either understand or accept on the "authority" (not quite) of the lecturer. I haven't read enough actual category-theory essays to say definitively, but there is another phrase "too simple to be simple" that seems as if it might pertain to the kind of cases in which it is permissible to reason from/through GAN:
Quite often, classical references will define ‘simple’ (or an analogous term) in naïve way, so that a ‘trivial’ object is simple, but later it will become clear that more sophisticated theorems (especially classification theorems) work better if the definition is changed so that the trivial object is not simple. Usually this can be done by changing ‘if’ to ‘iff’ (or sometimes changing ‘or’ to ‘xor’) in the classical definition.
Now, I was trying to see if I could say anything worthwhile by attempting to put my deontic questions/theories into a category-theoretic, as opposed to a set/type-theoretic, format, but so I was trying to come up with my own category theory as a whole, then. My point of departure was the general-particular distinction and the encoding-exemplifying distinction (from Edward Zalta): e.g., a particular term whose predications were true through the encoding relation was a trope, whereas encodable generalities were attributes of substances, say, or exemplifiable generalities were concrete universals (a phrase I've seen here and there, most perniciously in the work of Cornelius van Til, a Reformed epistemologist (not well-known in the way that Plantinga is, say)). Eventually, the scheme spiraled out of control and so also for other reasons did I more or less give up on trying to neatly package my categories in some simplified range.
Now, I have kept the following from the precursor to the above scheme, though. Let F be generality, f particularity. Then:
- F > f: if a term is more general than particular, it refers to an abstract object.
- F < f: the term covers a concrete object.
- F = f: "equimorphic" objects, e.g. divine simplicity, maybe the Tao (as symbolized by ☯ instead of 道).
- F ≹ f: the object's degree of generality is incommensurate with its degree of particularity: it is not more or less particular, per its particularity, than it is general over its own generalities. This kind of object, even if logically possible, is absurd.
My gloss, so far, of the "too simple to be simple" phenomenon is that a more complex scheme S can have a simpler base instance i, while some simpler scheme T will have a more complex base case j. But this makes it look like i or j have "absurd" degrees of simplicity/complexity. In fact, I did wonder if the mereological relation would be equimorphic, i.e. parts and wholes are equally particular and general, with a whole being a part considered on the generalized end of the local terminology, a part being a whole too, though, considered on the particular end of things, here. And simplicity/complexity pertain to mereology, often enough, even on an abstract level.
One might think that GAN-talk could not be put as "generalized abstract absurdity," in that one might think e.g. that the "N" in "GAN" has a Wittgensteinian meaning. So though Wittgenstein distinguished nonsense from senselessness in some sense(!), he did not mean to say that nonsensical (or senseless) things are "absurd." (Or did he? I actually don't know if Wittgenstein ever offered substantive commentary on the word "absurdity.") Yet, taking stock of all the whimsical terms that appear in higher mathematics, one might get the impression that category theorists, with GAN-talk, are indicating a kind of surreal pattern of reasoning. Is the phrase "too simple to be simple" an example of GAN-talk under the absurditarian(!) interpretation?
Partial option for sufficient answers: reference-requests for Wittgenstein or others on the concept of absurdity, from an analytic-philosophy or analytic-adjacent POV perhaps (though I think the issue appears in continental philosophy at greater length, maybe).
EDIT: some contention has arisen over my apparent conflation of Aristotle/Kant-themed category-talk, and the use of the word "category" in the name of mathematical category theory. This PhilosophySE post addresses that specific issue, and shows the continuity (historical, but also to some extent conceptual) between old-school category talk and the newer-wave use of the word.
And given Kant's deduction of his own categories, it seems like it would be easy (and I imagine, has already been done!) to translate his deduction into modern talk of morphisms, objects, etc. The 12 first-order categories fall under four next-order headings, and Kant says something that could easily be phrased as "functions g and f, as categories under a heading, are composed to yield the third category under each heading." So the four headings are like metacategories, and these then get compressed into Kant's distinction between mathematical and dynamical categories, which is a third-order category. And so the similarities between second-order categories, as well as their differentiae, presumably pertain to concepts like isomorphisms (or the lack thereof, with some weaker, if similar, relation the only one to be found). Furthermore, if Paul Corazza can talk about a "category theory of large cardinals", or if this peer-reviewed, published essay on deontic logic involves applying category theory to the categorical imperative (no less!), I really don't see why it would be impossible to compare the impossibility of finitely listing proper ontological categories, to the open-endedness of mathematical category theory (which was the interval of my apparent conflation of the two domains of discourse). One of the major themes of category theory is finding connections between zones of concepts that might seem relatively unrelated and so finding a connection, which in the historical, academic origination of category theory was referred to anyway, between the ontological use of the word "category" and the use of the word in this modern mathematical discipline/framework, seems to be what we should expect (if we take the mathematical framework itself seriously, here).