For another example take integrals (∫), where the argument for leakiness can be succinctly state. To master the abstraction one needs to understand all of calculus and all that is foundational to calculus (also known as “all of math” until 60 years ago or so).
I’m not sure I follow. I certainly did a lot of integration before I knew how to formalize the concept, and I think the formal details only rarely leak. Certainly, I got through an entire four-year math degree without learning most of the formalisms listed there.
Perhaps this is not “mastering” integrals, but… if integrals are above the bar for leakiness, I’d be surprised if neural nets are below it (though I’m less comfortable with those than I am with integrals).
I mean, the kind of integral one solves in school are rather trivial, essentially edge cases that never come up IRL. (e.g. ∫x^2 or ∫e*x kinda thing)
But even in that case, you still have to correctly define what you want to integrate as a function, you can’t just draw a random geometric shape and integrate it and you have to correctly “use” the integral operator.
Given an arbitrary function it’s not at all obvious what the integral of that function will look like and it sometimes requires a lot of skill to deduce.
Given an arbitrary problem where integrals are needed I find that it’s often non-obvious how to pick the bound, especially when we get into 2d and 3d integrals (or however you call them, I’m referring to ∫∫ and ∫∫∫).
I’m not sure I follow. I certainly did a lot of integration before I knew how to formalize the concept, and I think the formal details only rarely leak. Certainly, I got through an entire four-year math degree without learning most of the formalisms listed there.
Perhaps this is not “mastering” integrals, but… if integrals are above the bar for leakiness, I’d be surprised if neural nets are below it (though I’m less comfortable with those than I am with integrals).
I mean, the kind of integral one solves in school are rather trivial, essentially edge cases that never come up IRL. (e.g. ∫x^2 or ∫e*x kinda thing)
But even in that case, you still have to correctly define what you want to integrate as a function, you can’t just draw a random geometric shape and integrate it and you have to correctly “use” the integral operator.
Given an arbitrary function it’s not at all obvious what the integral of that function will look like and it sometimes requires a lot of skill to deduce.
Given an arbitrary problem where integrals are needed I find that it’s often non-obvious how to pick the bound, especially when we get into 2d and 3d integrals (or however you call them, I’m referring to ∫∫ and ∫∫∫).