Linguistics Using Category Theory (2018)

The definition I like for sandwich is “A set of materials in the configuration ABA, mated along their longest side”. It holds up in contexts outside of food, that the word “sandwich” is sometimes used in, and is valid in dimensionalities other than 3.

The sub issue remains though. Semantically, it could be argued that subs are sandwiches that have not had their end fully cut.

so?

"Once category theory was developed and used, in particular when the central theoretical role played by adjoint functors was understood, a fascinating process of reversal of perspective, a gestalt switch, took place: what was seem as a useful tool in organizing and guiding mathematical thought became a theoretical framework that revealed the basic or fundamental principles underlying mathematical concepts, theories and theorems. Thus, the Stone duality theorem is indeed more perspicuously presented in the context of categories and functors—it is organized nearly and the basic consequences of the result are transparent—but once it is seen as a special case of a very general adjoint situation, a theoretical understanding of the phenomenon becomes available. Category theory is not applied to Stone's theorem, it is the latter that becomes a specific instance of a general, universal conceptual situation.

Although it might in the end be more obscure than what I have said so far, I dare at this stage put forward a slogan that, I believe, sums up the core of what I have been presenting: category theory is the architectonic of mathematics. Category theory is, indeed, as in the philosophical sense of the expression "architectonic", the systematization of mathematical knowledge. Mathematical knowledge is systematic. Mathematics is a conceptual system. That much is indubitable."

Note that it's -tectonic (from the Greek for carpenter) not -ectronic (from electron). And it's just "the architectonic of mathematics", not concepts in general. There are obviously architectonics of other things as well, like (spatial) architecture itself, or music, or cooking. So while perhaps CT is the ultimate theory of algebraic abstractions, but it's not the only kind of design system that exists. It represents the mathematical aspect of design.

As a mathematician, I feel that I have a reasonable understanding of category theory, but have never heard the word 'architectronic' (and Google, or at least the first page of Google results, knows it only as the name of a Red Snapper song). What does it mean?

EDIT: Ah, notfashion (https://news.ycombinator.com/item?id=20772313) clarifies that it is architectonic (just one 'r'): https://en.wiktionary.org/wiki/architectonics#English .

So usually we "design" one possible system out of many to solve a problem in a solution space we do not completely understand.

Once a formal axiomatic theory is in place it could very well become "deriving" the best system to solve a problem in a solution space we completely understand.

Of course it's likely not that simple and likely there is no singular "best" solution but a theory, in my opinion, would quantify all possible tradeoffs between a subset of "best" solutions out of all possible solutions.

Category theory seems to be the closest thing to such a theory I have found. I'm a beginner in learning this stuff though. So far I'm guessing that it's more than likely that there are no algorithms within category theory that can be used to optimize systems or even derive one... but it is the closest thing to such a theory that I have uncovered? My question in the initial post was asking whether or not anyone has found anything better....

Some posters have remarked that the movement of category theory into engineering (a field which is largely involved with designing solutions more-so than deriving solutions) is starting to happen.