The apple (or a "mathematical" whole number) in your head is not part of any sort of reality that I meant. The difference between any given apple and, say, the number 3, is that, while both do actually exist in nature, the number 3 exists in a subtly less direct way, it exists as the thing that is common between three apples and the three horses you might want to feed these three apples to; and the fact of the (almost) physical existence of that common (i.e., the number) can be proven by you being bitten by the third horse if you only happen to have two apples instead of three.
(I'll need more time in order to address the rest of your points.)
More time is fine. I am genuinely interested, just not understanding. Because even still what you are describing to me is impossible for me to distinguish from a social construct. Like in the sense the borders are real. But I wouldn't call borders "real" other than "a real agreement" and even then there's lots of arguments. Definitely borders are not inherent to the world/reality other than... our social constructs. So I think where the main point of contention is coming from is me not understanding your definition of "real." It appears like you understand the one I'm using, and is it fair to say that this is the standard one?
I know. What I meant was, these axioms reflect something real, like for example a news report reflects objective events; whereas you allow XYZQJJX to be something that is invented, i.e. to be quite arbitrary, like, to continue with the analogy, a "science fiction" piece (at best).
> you aren't considering language real
What I am saying is that mathematics is more than a language, and using it as if it were just a language, ignoring its positive content, would lead nowhere. I'd say, the content is its most important part, although the language, of course, is nice to have. Moreover, the language (including mathematical notation) is often abused, for various reasons - personal and otherwise. My other point was that even what we consider abstractions does reflect properties of the objective reality - just as do the notions these abstractions generalize. But the more you treat mathematics as merely a language and start playing with words (or symbols, or even concepts), changing their order and "inventing" new combinations of them, the chances that you get anywhere become small (to put it mildly). Yes, you can play with axioms, but notice that even then, as a sane person, you will be testing an idea rather than throwing stuff around or making wild sounds.
The apple (or a "mathematical" whole number) in your head is not part of any sort of reality that I meant. The difference between any given apple and, say, the number 3, is that, while both do actually exist in nature, the number 3 exists in a subtly less direct way, it exists as the thing that is common between three apples and the three horses you might want to feed these three apples to; and the fact of the (almost) physical existence of that common (i.e., the number) can be proven by you being bitten by the third horse if you only happen to have two apples instead of three.
(I'll need more time in order to address the rest of your points.)