By the way: logicians sometimes pedantically draw a distinction between ‘recursion’ and ‘induction’: induction is for proving things; recursion is for defining things.
And it’s not just terminological: the difference becomes very important when you realise one needs to prove that recursion is possible (that is, function definitions by recursion on a well-ordered set are well-defined) by induction. So one actually comes logically before the other.
I guess to truly determine whether they’re distinct concepts you’d have to exhibit some model of set theory that supports recursion but not induction, or something. But that’s beyond me.
That's kind of what annoys me though; it's almost like the word "math" is toxic to people. When I try explaining a correspondence between something that they're doing via code, and how maybe learning a bit of the mathematics behind it might be useful, people kind of just roll their eyes, acting like I asked them to go get three PhDs, and acting like it's just one of those weird "Tombert things".
I don't know anything about neural networks yet (though I really need to get on that), but I have noticed the other two examples you mentioned as well; recursion is more or less applied inductive proofs, a lot of crypto boils to number theory.
My dream is to some day convince a manager to give me budget to spend a few weeks designing a new system and proving correctness with TLA+. I'm not saying it's terribly likely, but a man can dream.
An unwelcome comparison: For certain people you can’t just teach Buddhist meditation using Buddhist terminology and context because that’s “woo”. However if you dress it up in vaguely “neuroscience” terms then it’s fine. And preferably you have some scientific studies at hand to prove that meditation has an effect. (Although no one demands biomechanic studies from their intro tennis teacher.)
Compare with all the moaning and complaining that comes from some people when you talk about functional programming and how it might relate to computer science.[1] Then you might have a better chance inventing ad hoc words for basics like “map” (preferably very prosey) and claiming that Martin Fowler invented it.
[1] This goes triply for anything having to do with proofs, at least proof software associated with FP.
It is easy to tell if a tennis instructor is giving “good” instruction, where good means instructions that help you be a better tennis player. You just look at their students. If you want to know if they give good fundamentals for long term growth, you look at their much older students.
How do you know if a meditation teacher is giving good instructions? Seems much harder to me.
It is interesting that many of the traditional religions have hard to quantify benefits. Buddhism offers a personal inward help, and the abrahamic religions often benefit communities.
But schools typically teach these things as a "theory part" and as "practice/application part" (for example by "syncing/sequencing" courses if possible) which helps with both relevance and "getting it".
Recursion? Oh you mean proof by induction?
Cryptography? Oh you mean number theory?
Neural networks? Oh you mean calculus?
Almost everything I learn while programming can be associated with some theory I learnt before for math.