To me it seems a lot like the crypto and web 3.0 promises of "soon there will be all of these amazing applications"... and then there aren't any.
Also, all of the examples of CT I've seen are applicable only to pure functional languages like Haskell, which seems to limit their practical utility.
I usually program in C++ or php, but I do think in monads / functors / applicatives quite often. I then use whatever features of the language I have at my disposal to “encode” my thinking at an appropriate level of abstraction. What level of abstraction that is depends not only on the language, but also how efficient and “communicable” I want my code to be. I don’t mind expanding the `map` of a functor into an explicit for loop, if my junior colleague can more easily deal with it.
In fact, at this point, with functional programming patterns having become so pervasive, most people including beginners have no problem with the concept of functors, and “mapping” functions on things. Monads still have that aura around them, but really, any time you write a line of code that reuses the result of a previous line of code and checks for something, you probably have a monad hiding in there, and “exposing” it might make it easier to factor out your code or find an elegant API.
There was another talk where IIRC Edward Kmett dualizes divide and conquer algorithms to derive a generic bucket sort that works on general algebraic data types[1].
These are generic concepts. Haskell is just convenient to express them in. In some sense, obeying functor or monad laws just means you'll have a "clean", predictable design, for example. The language doesn't need special support; you just need to check whether your code is composable in a sensible way.
- the group of people who don't see the relevance of mathematics "in the real world", and expect any a-ha moment to have come to them by magic already, and;
- the other group.
Differential Equations Statistics / Stochastic Processes Discrete Math / Algorithms / Logic Solver
You do not need CT for all this stuff.
Grothendieck needed CT - ist Algebraic Geometry useful?
I've read the paper. They list many useful applications. CT doesn't seem really necessary for each of them but it looks like CT is (could be) helpful.
It is just very hard to create non-trivial general theories in sich „Applied“ fields, Functional Analysis alone is not very exciting. For an example see for example the work or Hairer (stochastic differential equations) or Costello (QFT).
This is wild stuff:
https://people.math.umass.edu/~mirkovic/0.SEMINARS/1.QFT/C.C...
funny names: a Leonard Cohen writing about an Costello (but Kevin Costello not Elvis Costello)
> Statebox is blending an ACT approach to Petri nets together with blockchain technology to develop a technology stack based on a visual programming language. In addition, they have built a software engine for compositional game-theoretic modeling, a finite state machine oracle.
Hmm.
More seriously:
> Structured cospans of Petri nets were implemented in the software package Julia to develop an SIR model that is compositional in the sense that various cities can each have their own model that can be connected together to form a composite SIR model
SIR is the most common way of modelling infectious diseases. Having said that though - there are plenty of SIR models that don't require category theory.
edit: This question is genuine. There are only abstract concepts included in the video and wiki, as deep as I can see.
My interpretation of "solving a real world problem" means that it can't be infinitely abstract, and must be a relatively good solution, compared to other options. I only see, from my dumb perspective, extreme abstracts.
I'm not being combative. I just don't understand.
In time, that's correct. At the moment, I disagree.