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Thnk you, yes, levels of abstraction is precisely what I'm being tedious about. These are very generous explanations and I respect your time on this, please ignore if I veer into a cranky Gish gallop. This distinction between a bidirectional path in a graph vs. a functor between objects is a great clarification, and I can begin to comprehend how I would be confusing the representative directional arrow in a graph with the logical operation of mapping in CT is different - because I'm treating them both as just morphisms between objects. The universality of CT implied a conjecture to me where, either there are theorems of CT that cannot be expressed as graphs, or there are graph theorems that cannot be expressed in CT. "Expressed," is handwavy, but because we're on the edge of notation/encoding and representation vs. whether there a homology between graphs and categories, when I get into the question of a universal modelling languege, it raises the question of what the minimum necessary set of instructions to express theorems in that language (either CT or graphs), and whether the smallest program that can express CT will resolve to a graph. If the Kolmolgorov complexity of the program that produces theorems in CT is greater than the one which produces theorems for graphs, and that program itself reduces to a graph, it implies to me that the less complex program is the more universal modelling language. This was why in terms of modelling I thought that CT seemed more like an ornamented subset of graph theory. However, that's like if someone said math is a just subset of Godel numbering, which well, yes, but that's not very helpful. :) |
Thanks for that; I learned a new phrase ^_^
Yes, admittedly your post throws a bit of a wide net, but I think I see what you're angling for.
> it raises the question of what the minimum necessary set of instructions to express theorems in that language
That's a really good question. Take the NAND gate as an example: we know we can implement any boolean function with only NAND gates. So the circuit language consisting purely of the NAND gate (and wiring rules for connecting gates) is ridiculously compact, even minimal. In some sense, it has a very very low entropy.
However, the circuits we construct are conversely ridiculously complex! Any given circuit is just a wad of NANDs, and trying to extract meaning from that circuit beyond executing it as a black box is going to be very difficult. You could say that these circuits have a very high entropy.
In comparison, suppose we allow ourselves the usual set of {AND, OR, NOT} gates. We know that any logical function implementable by {NAND} is implementable by {AND, OR, NOT}, and vice versa, so there is no loss in functionality. But circuits expressed in this language have far more structure, making them individually much easier to analyze. We can define such notions as "conjunctive normal form", in that any circuit can be transformed into an equivalent three-layer circuit: a layer of NOTs feeding into a layer of ORs feeding into a layer of ANDs. These circuits have far lower entropy in exchange for a modest increase in our circuit language.
So when you say this:
> it implies to me that the less complex program is the more universal modelling language.
I think we have two competing notions of "complexity" at hand, and it's very hard to say which one ought to win out. As a personal preference, I like to imbue my creations with more inherent structure, so I prefer things like {AND, OR, NOT} over things like {NAND}. It makes the constructions easier to understand.
If you're studying the languages and not the things created in those languages, then things like {NAND} may be superior. But you usually need some kind of language in which to describe your languages -- did you notice I was using set-theoretic notation? -- and so you have the same question again at a meta level.
One of the attractions of category theory is that it's also a pretty good language for speaking about category theory!