Super interesting, thank you. I forsee these topics distracting me well into old age.
This idea of adding the structure of AND, OR, NOT over a field of NAND gates, without losing information sounds like the the beginnings of where Categories become more meaningful than graphs. The analogy would be that AND would be like a functor category, where again, it is represented as an 'arrow' but it's really it's more like a lossless abstraction, in a strict information or logical sense.
This circuit that is a just wad of NANDs is not unlike a generalized universal intermediate representation of a program, or even an esolang program like BF, because while they aren't efficient for human reasoning or computation at all, they are consistent universal forms for symbolic representation.
The problem in code is always, "this would just be compute/work problem if only this thing were made of <consistent-data>," and what attracted me to the idea of CT was that it seemed to be a tool for reasoning 'different' things into compositions of their consistent abstractions.
I still think that's true, but now have a better sense of why.
This idea of adding the structure of AND, OR, NOT over a field of NAND gates, without losing information sounds like the the beginnings of where Categories become more meaningful than graphs. The analogy would be that AND would be like a functor category, where again, it is represented as an 'arrow' but it's really it's more like a lossless abstraction, in a strict information or logical sense.
This circuit that is a just wad of NANDs is not unlike a generalized universal intermediate representation of a program, or even an esolang program like BF, because while they aren't efficient for human reasoning or computation at all, they are consistent universal forms for symbolic representation.
The problem in code is always, "this would just be compute/work problem if only this thing were made of <consistent-data>," and what attracted me to the idea of CT was that it seemed to be a tool for reasoning 'different' things into compositions of their consistent abstractions. I still think that's true, but now have a better sense of why.
I will revisit the short course that led me here as well (https://applied-compositional-thinking.engineering/lectures/)