I'm... not sure I'm following most of what you're saying (in this and your other comment).
> Why is 1 + 1 = 2? This isn't really a mathematics problem
Umm, so first of all, depending on what you mean, "why is 1+1=2" is definitely a maths problem. It's exactly in the foundations of math, as you said. Or rather, it makes more sense to think of it as "what is our system for deriving true statements, that lets us do things like 1+1=2?".
> (I know about axiomatic systems, they were invented relatively late in the timeline of mathematical history, invented as a fiction to avoid getting to the bottom of the messier things, i.e. the social context of how mathematics was actually invented and is used)
That's not really why axiomatic systems were developed. It was part of trying to give better foundations to math, because mathematicians ran into a lot of actual problems, mostly beginning with calculus. They managed to reach paradoxes, weird conclusions, etc, and wanted to start being more rigorous in how they treated everything in math, which has led to lots of flourishing in maths, so it was probably a good idea.
> [...] 1+1=2 presumes a reductionist and somewhat capitalist view of the world...
This I don't understand at all. What does this have to do with capitalism? The rest of your comment is equally weird to me. Mathematicians like to invent concepts and play with them, sure, I'm fine with that description, but how does this have anything to do with the capitalist system?
> [From your other comment] I mean, to cite a specific example, while the exercise is probably entertaining to some, anyone seriously believing 2+2=4 requires thousands of lines of "proof" is probably not very good at getting to the bottom of things in a general sense. (ref: http://us.metamath.org/mpegif/mmset.html#trivia )
I think you're confusing two things here. For the most part, most mathematicians don't need to bother with "foundations" questions, or reach a point where they're proving 1+1=2. It can be helpful to "peak under the hood" of various definitions to understand how they are defined rigorously, and more importantly, it can be fun. But most mathematicians use numbers as a black box, never needing the detailed definition of how numbers work.
As for taking 2000 lines to prove this, that's probably only when you want to rigorously use a computer proof or something from first first principles. Given a set of axioms (e.g. Peano), I'm pretty sure it's easy to prove 1+1=2 in only a few steps. And again, this is mostly of interest to people who actually want to work on this kind of foundational thing, not the average mathematician I wouldn't think.
> Why is 1 + 1 = 2? This isn't really a mathematics problem
Umm, so first of all, depending on what you mean, "why is 1+1=2" is definitely a maths problem. It's exactly in the foundations of math, as you said. Or rather, it makes more sense to think of it as "what is our system for deriving true statements, that lets us do things like 1+1=2?".
> (I know about axiomatic systems, they were invented relatively late in the timeline of mathematical history, invented as a fiction to avoid getting to the bottom of the messier things, i.e. the social context of how mathematics was actually invented and is used)
That's not really why axiomatic systems were developed. It was part of trying to give better foundations to math, because mathematicians ran into a lot of actual problems, mostly beginning with calculus. They managed to reach paradoxes, weird conclusions, etc, and wanted to start being more rigorous in how they treated everything in math, which has led to lots of flourishing in maths, so it was probably a good idea.
> [...] 1+1=2 presumes a reductionist and somewhat capitalist view of the world...
This I don't understand at all. What does this have to do with capitalism? The rest of your comment is equally weird to me. Mathematicians like to invent concepts and play with them, sure, I'm fine with that description, but how does this have anything to do with the capitalist system?
> [From your other comment] I mean, to cite a specific example, while the exercise is probably entertaining to some, anyone seriously believing 2+2=4 requires thousands of lines of "proof" is probably not very good at getting to the bottom of things in a general sense. (ref: http://us.metamath.org/mpegif/mmset.html#trivia )
I think you're confusing two things here. For the most part, most mathematicians don't need to bother with "foundations" questions, or reach a point where they're proving 1+1=2. It can be helpful to "peak under the hood" of various definitions to understand how they are defined rigorously, and more importantly, it can be fun. But most mathematicians use numbers as a black box, never needing the detailed definition of how numbers work.
As for taking 2000 lines to prove this, that's probably only when you want to rigorously use a computer proof or something from first first principles. Given a set of axioms (e.g. Peano), I'm pretty sure it's easy to prove 1+1=2 in only a few steps. And again, this is mostly of interest to people who actually want to work on this kind of foundational thing, not the average mathematician I wouldn't think.