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and then all these books are just listing out what that looks like, or is there really a separate body of ideas about something that isn't already covered everywhere else?
(Not to discount the potential strength of the idea of doing equations in 2D, it's just, everyone seems to be claiming it's more than that, but I don't see it, and I am trying).
This is really not my field, but I feel that the power of category theory comes from the unification of seemingly different ideas in math. Whether that counts as useful or not is subjective.
It counts as useful. This process lets mathematicians use techniques from previously unrelated areas to solve previously intractable problems. From https://plato.stanford.edu/entries/category-theory/ …pic.twitter.com/euMU4NH459
I am currently writing a chapter about category theory and PL design. Basically the strength of it comes to using it as a "universal language" to tie different areas of programming together. Here is an example of it https://parlab.eecs.berkeley.edu/sites/all/parlab/files/The%20Compiler%20Forest.pdf …
As you can see in this paper, in section 6 they re-frame their discovery in category theory. This allows you to draw connections between other things that have interpretation in category theory, like effect systems, parallel programming, state updates, programing languages, etc…
I know what you mean. I keep trying to find in set theory anything more than just an alternative notation for predicate logic.
In modern type systems you can define a type like this (example is in Haskell): data X = X (X -> Bool) X is the type of functions that take an X and return a Boolean.
The diagrams aren't the powerful idea, they're just a convenient reasoning technique. The powerful idea is that it's another level "up" in abstraction. (1/n)
In group theory we talk about SO(3) as an abstract idea independent of how it is represented (e.g. matrices vs quaternions). The same group shows up with different implementations. (2/n)
When first introduced in 1948, physicists didn't see how Feynman diagrams were more than just a way to represent the interactions of elementary particles in 2D. In fact, they are only approximations of field equations, and yet, physicists can now see:https://www.quantamagazine.org/why-feynman-diagrams-are-so-important-20160705 …
i think this is the paper you are looking for: https://arxiv.org/abs/1803.05316
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