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.
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.