I don't think that programming and math are the same in a practical sense. I am studying math and CS and they're fairly different. Programs deal with specific things, types and data that you manipulate and see with your eyes. Maths deal with abstract concepts for which finding examples can be pretty difficult.

Also, while programming, you can design your functions and their interfaces before writing them down. In maths, that's impossible: you're constrained by what can/cannot be done.

Scrum or coding practice doesn't apply here. Maths don't have to "keep up", they're already way ahead of their applications. It also doesn't need to be fast or flexible, just rigorous. And while motivation and intuition helps (specially when learning) some concepts cannot be motivated or given an intuition, and even in that case you still have to learn the small details and formalizations.

> Programs deal with specific things, types and data that you manipulate and see with your eyes. Maths deal with abstract concepts for which finding examples can be pretty difficult.

Have you forgotten the pains that you went through trying to understand the difference between values, pointers and references, lexical and dynamic scoping, static and dynamic typing, and the like? Can you see them with your eyes? Have you ever tried fully explaining any of those to a non-CS major in half an hour? :)

Abstractness is pretty subjective. For non-programmers, even the idea of CPU and memory can be abstract. And it doesn't help to open up a computer and point to the hardware; that is like claiming a mathematical paper is not abstract by pointing to the very concrete paper and ink that embodies it.

I'm not saying those concepts are not difficult. But values, pointers and references are something that relate to memory. You can simulate in a paper how is a value or a pointer managed in a program. Same with scopes and static typing. There are specific examples for all of those.

Some concepts in mathematics are way above the abstraction level that you mention. For example, the projective plane, or nowhere-differentiable functions, or geometry in higher dimensions; those are concepts that not only do not have any physical equivalent, but are also very difficult to grasp and imagine in your head.

True. But still, I think the problem lies more in the fact that mathematicians skip too many steps in their prrofs (see another comment of mine) than the inherent abstractness of mathematics.

I've done mathematical research in the past and I actually think a lot of the practical methodology I've learned from software development could be incredibly useful to mathematicians. I'd love to work full-time on a selection of related research problems with a group of coworkers following a sort of "agile" process with quick morning standups, a centralized repository of works and proofs in progress, "proof reviews", Trello boards, and so on.

Would it actually work? I honestly don't know, but I'd seriously love to try it.

Programming is a subset of math, namely, the part of math that deals with algorithms. There are many parts of math that are not contained in programming. For example, floating point numbers are finite representations of real numbers, but since floats are finite, without some math theory beyond just algorithms, there is no way to understand them. There are may similar examples.

The differences you note are what I consider cultural differences.

Math _indeed_ have the idea about interfaces and implementations. They are called something different (exitensisal types from a PL point of view).

I will recommend you to look into type theory, in particular the book "types and programming languages" provides a very good introduction.

In CS we have two areas that make it completely clear that math and CS are coupled, that is complexity theory and programming languages. After having taken a course in each, and pondering a bit, it should indeed be possible to see the that they are the same.