It is an implementation detail when you go to the extreme. Most people are not segmented this rigidly though so of course they worry about execution as well as the high level mathematics. But most of the high level math and designs give a high level view of everything.
Category theory and type theory don't fit cleanly with assembly language (the implementation detail).
Additionally at the very fundamentals of mathematical CS: the Church–Turing thesis doesn't describe anything about the real world. It says that the implementation of a true lambda machine vs. a turing machine are essentially the same thing, the thesis does not describe how it's basically impossible or extremely hard to build a true lambda machine.
To be a mathematician computer programmer, you don’t have to use the Church-Turing mental framework all the time. Sometimes it’s useful and sometimes not.
Of course you don't. I never said you did. In fact I would argue you never think about this isomorphism 99% of the time.
I'm just saying that a big portion of the math doesn't describe the real world implementation details.
As I said the Church side of the theory doesn't have a real world machine equivalent. We have machines that move things and save things but we don't have a physical machine that represents the concept of a function call. What we instead have is a Turing machine that emulates a function call, not a true simulation of it.
Category theory and type theory don't fit cleanly with assembly language (the implementation detail).
Additionally at the very fundamentals of mathematical CS: the Church–Turing thesis doesn't describe anything about the real world. It says that the implementation of a true lambda machine vs. a turing machine are essentially the same thing, the thesis does not describe how it's basically impossible or extremely hard to build a true lambda machine.