I’m a generalist that likes dipping down to where my brain hurts and then coming back up again. Category theory really hits all the right notes with me.
While a lot of it is over my head, it _does_ do the magic thing of loosely tying lots of diverse things together using a consistent language.
So even if I don’t understand their peculiar way of writing, simply seeing two different examples of a category together in the same place tells me that if I know one, I know the other.
This is what happened with mathematical morphology. I saw these features in this graphics program that’s used for medical imaging. [I’m not a doctor but I love playing with images and trying new things]. – ImageJ.
I wondered, “What is “erode”? What is “dialate?” What is “watershed function?” “what is Laplacian of Gaussian?” and discovered mathematical morphology.
Things like Infimum and supremum fascinated me. It actually made logical sense and I could experiment visually.
I found it it was a complete lattice. Not knowing what that was, I looked into that. Then I saw other examples of complete lattices on nLab (the Category theory site). — and it’s as if a whole bunch of math I didn’t know before, I could now at least partially wrap my brain around because it should work similarly to what I’m learning – and it does.
I’m lucky in a way: I don’t worry about Gödel’s incompleteness and never did. I started with computing, which “sidestepped” Gödel by introducing steps and time and a different way to think of functions.
Church, Turing, Von Neumann, with Lambda Calculus in the 1930s and 1940s, then Claude Shannon in 1948 with Information Theory. — this is the kind of family I was raised with. I didn’t know about set theory or Gödel’s incompleteness until my 20s, long after I’d already worked with computing-type things.
My own mathematics abilities stopped at Trig. I got a D in Calculus because I didn’t care about slicing circles and could not understand why I couldn’t simply use rough estimates under the curves as I was used to pixels and grids not smooth things.
So, it never became a dilemma for me. But the discontinuity between ways-to-think-of-math always hummed in the background of my mind and it was only a few years ago that I really paid attention to what was bothering me.