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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:

  1. F > f: if a term is more general than particular, it refers to an abstract object.
  2. F < f: the term covers a concrete object.
  3. F = f: "equimorphic" objects, e.g. divine simplicity, maybe the Tao (as symbolized by ☯ instead of 道).
  4. Ff: 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).

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    I think you have a misunderstanding about what category theory is. Your second paragraph links to the SEP article on ontological categories. Category theory in mathematics has nothing to do with ontological categories. Putting your ideas in categories is not "category-theoretic." Category theory is the study of mathematical structures consisting of "objects" and "arrows" that satisfy certain axioms.
    – causative
    Mar 9, 2023 at 10:03
  • @causative the SEP article on CT says the name comes from the tradition incl. Aristotle and Kant. Category theorists were not concerned to introduce some zone of concepts entirely de novo seeing as mathematicians as a community are less prone to the illusion of disagreement. So now CTists don't have to suppose that there are a finite range of categories but can suppose their common form can be copied without end. Mar 9, 2023 at 12:45
  • 1. Where the name for CT comes from doesn't matter - it's an arbitrary name. They might as well have called it "Thingamabob Theory." Categories in mathematics are structures with very specific properties, that shouldn't be confused with categories in natural language or ontological categories. 2. Yes, you can talk about your own categories. But they aren't the categories of category theory unless you have objects, arrows, and a composition of arrows operation, that obey the category theory axioms. Just organizing ideas in ontological categories does not achieve this.
    – causative
    Mar 9, 2023 at 15:58
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    @jd Not so far as I can discern.
    – user4894
    Mar 9, 2023 at 22:32
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    "Abstract nonsense" does not indicate acceptance on authority or "generality in general" or "surreality", it is more specific. It refers to arrow chasing in diagrams, as it makes explicit relying on "pure form/structure" devoid of "content". Wittgenstein's "nonsense" does mean senselessness, it is something devoid of both form and content that only has impact (perlocutionary force, to use Austin's term). He did not distinguish it from absurdity, but see Kind, p.39 on his "patent" vs "disguised" nonsense.
    – Conifold
    Mar 9, 2023 at 23:07

1 Answer 1

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Is the phrase "too simple to be simple" an example of GAN-talk under the absurditarian(!) interpretation?

No. First, a few clarificatory remarks.

  • Later LW in PI used the term "Unsinn" (nonsense, absurdity) in contradiction to an alternate German word "sinnloss" (not possessing sense). He clearly talked about the conceptual difference. See my inventory in PI of these terms here.
  • Generalized abstracted nonsense is a tounge-in-cheek neologism for a very real psycholinguistic experience. It's possible to look at a mathematical proof, computer code, or formal logic, recognize every symbol (the letters, the relations, grouping) and have absolutely NO idea what it means. I always treat formalisms as cryptograms personally. Math papers are rigorous, but they are often not very good examples of clear, explanatory exposition. So, strictly speaking, the passages are meaningful, however not clear since the resulting syntax that results from abstractions upon abstractions (topoi derived from categories abstracted from set-theory from the logic ZFC itself rooted in natural language). It's a useful term. Lots of academics make their bread and butter reading, writing, and publishing GAN. GAN is an experience or property in regards to very difficult, complicated abstraction.
  • Now, "too simple to be simple" is also meaningful and is also useful. It might be helpful to think of it applying to first instances from which a sequence derives, but the rest of the sequence obeys the rules, where the very first item, the small item, otherwise creates problems and is complicated. Think in abstract algebra. A group has three requirements, associativity, an identity, and an inverse. Many things which a group can represent often have rules that apply WITH THE EXCEPTION of the identity itself. For instance, making one prime or composite should be simple, since 1 is, among naturals literally the simplest; except its not because it's so simple, it creates problem. When one says "number times 1", and number refers to 1, we're good, but now we have a prime with only one factor and all of our theorems might apply to primes of 2 factors. Easiest just to throw the case out than to deal with ad hoc statements in the axiomatic method.

Now, what is the relationship between GAN and TSTBS? GAN and TSTBS are both dialetheia. It may feel like TSTBS should be recognized as GAN because of the paradox inherent in the language makes it difficult to apprehend at first glance. After all, isn't it absurd to be so simple implies its complicated? It tickles our contradiction catcher.

But I think you're being too literal by interpreting the absurdity (nonsense) in GAN literally. What is meant by GAN is Unsinn, not sinloss which are easy to confuse. GAN is something that appears to be nonsensical because its so semantically dense, it overwhelms the viewer at first, but math proofs are VERY meaningful. Too complicated to be complicated. (In fact, one can indeed parse GAN in small pieces simply.) And TSTBS is something that seems so semantically simple, but then turns out to entail so much work in semantic grounding, that underwhelms the viewer at first. But both are dialetheia in their formulation and set off our contradiction "spider senses", both are pragmatic uses of language to describe a psycholinguistically very real experience when dealing semantic intuitions derived from complexity and simplicity of syntax.

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  • "Many things which a group can represent often have rules that apply WITH THE EXCEPTION of the identity itself. For instance, making one prime or composite should be simple, since 1 is, among naturals literally the simplest; except its not because it's so simple, it creates problem. When one says "number times 1", and number refers to 1, we're good, but now we have a prime with only one factor and all of our theorems might apply to primes of 2 factors. Easiest just to throw the case out than to deal with ad hoc statements in the axiomatic method." is a bit obscure
    – Frank
    Mar 10, 2023 at 1:27
  • What's the relationship between group, almost groups with no identity (?) and 1 being a prime or not??
    – Frank
    Mar 10, 2023 at 1:28
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    @Frank 1 not prime illustrates TSTBS as per ncatlab.org/nlab/show/too+simple+to+be+simple, and I was trying to suggest that there are perhaps certain types of primitives of formal systems that systematically appear to qualify for TSTBS. The choice to exclude one as prime is a concrete example, and the choice to consider the identity element of a group is an abstract example since the same reference includes the trivial group regarding the cardinality of the set with the identity as the sole element. I guess my wording doesn't communicate that explicitly.
    – J D
    Mar 10, 2023 at 4:15
  • The reference to dialetheias makes this answer. I do want to add that I am implicitly differentiating absurdity from nonsense/senselessness, I want to find a way for "surd beings" to be logically possible and substantive, albeit highly unusual from our normal POV (an object whose general features are not more or less general than its particular features, but not equally general or particular either, might be inconsistent, but I'm not sure). Mar 10, 2023 at 8:11
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    @Frank Horse's mouth has spoken. You're right of course about incommensurability. I'm just shaking the tree and seeing what comes out. :D
    – J D
    Mar 10, 2023 at 16:49

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