Linguistics Using Category Theory (2018)

Linguistics Using Category Theory (2018)
golem.ph.utexas.edu

It has been discussed on Hacker News at least a couple of times previously -- fairly recently, even. You might be interested to look at these discussions:

https://news.ycombinator.com/item?id=20376325

https://news.ycombinator.com/item?id=19701767

Edit to add:

You might also be interested to learn that categorical approaches to linguistics typically take as their starting point monoidal categories, in which there are notions of 'parallel' as well as 'sequential' composition. It turns out that the usual categorical semantics for linguistics shares a lot with the categorical semantics for quantum mechanics: roughly, meanings are vectors, like quantum states. You can read more about doing (finite-dimensional) quantum mechanics entirely using string diagrams (the formal diagrammatic calculus of monoidal categories) in the work of Bob Coecke, who also played a large part in originating these approaches to linguistics.

For example, on the quantum side, an excellent book is 'Picturing Quantum Processes' [0]. And on the linguistics side, the paper linked in the article is a good start: https://arxiv.org/abs/1003.4394

[0] Not freely available, but some slides are at https://www.cs.ox.ac.uk/ss2014/programme/Bob.pdf

Edit, again:

There is also of course Bartosz Milewski's book / blog series 'Category Theory for Programmers', which introduces category theory from the perspective of Haskell and C++ programming: https://bartoszmilewski.com/2014/10/28/category-theory-for-p...

But the best introduction to category theory I have read is Leinster's book, 'Basic Category Theory': https://arxiv.org/abs/1612.09375

And as you might have guessed, I do agree with your statement!

"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 .