i never took the strong position, before, that "math" is all in the brain. it is possibly more useful to imagine that the math already exists and has independent truths and is merely discovered.
this whole thing is recursive. all these are mindset questions. the distinction between "all in the brain" and "exists independently metaphysically" are themselves all in your brain. the question isnt meaningful outside of how you are looking at it.
so i am happy to accept mathematical realism if it helps you do better math, which i would expect it does. what i am really asserting is that "linguistic realism" in the extreme case doesnt help you do better linguistics.
also all human thoughts are contained in the human brain. even if you suppose math exists independently, actual humans doing math are not directly accessing it. suppose you drug a mathematician and he produces some nonsense.
isnt this pretty obvious? you can suppose the real physical world exists, but your thoughts do not contain the physical world. you can "discover" the physical world in the sense that your thoughts correspond to it in some way and you make abstractions. but your thoughts are all in your head unless you want to include sensory input into your thoughts and then declare sensory input a part of the world. which you can do, if you like.
replace the physical world with math and thats a way of looking at discovering math robust to drugged mathematicians.
Math is a language. And your assertion was that all of human consciousness is contained in the brain. Developing tools to access the real, abstract, independent mathematical objects is an act of consciousness. So, your assertions about anything being completely contained in the brain are made invalid.
You admit it is when not just possibly independent of humans, but actually is when you accept the axiom of choice
This claim is tautological. And in any interpretation where it isn't tautological, it's provably not true because the addition of extrinsic axioms literally proves mathematics is real and located in a metaphysical state independent of humans.
No, it is meaningful because it's rigorously defined in formal logic. There is no issue of interpretation of meaning
The former acceptance proves the latter is unfounded. And mathematics is a language.
Wrong. Read the fucking link.
False analogy. The world materially exists. The realism of mathematical objects has no material existence, only a metaphysical one. Yet we can access it from thought processes. This then implies that thought cannot be contained in the human brain, as material methods cannot make any truth-value statements (true, false, or unprovable) about things that are not material. This is limits of empiricism 101 and is part and parcel of the basic idea that makes science consistent
math is a language? then your definition of language is broader than mine. maybe this broader definition helps us do better linguistics. or maybe it doesnt.
the point at which i disagree is "yet we can access it from though processes". i think this is a misunderstanding. you can access it in a dirty, weak way. the material methods in your produces trueness, falseness, and unproveableness as concepts in your head, which are not exactly the mathematical ideals thus postulated.
illustrative example: the drugged mathematician. drugged mathematicians might wrangle lots of concepts and have lots of thoughts that are very convincing to him. but he isn't actually meaningfully accessing "real mathematics". how do you know youre not a drugged mathematician?
i think lots of philosophy tacitly assumes a ideal cognitive unit that is doing thinking. there is no such unit; its just an abstraction. you think you are, but you arent. we can try to look for that unit, but if you hit someone over the head hard enough the unit stops being a good abstraction.
math is done in practice by manipulating symbols in a formal language, because humans are suited to that. i thought we were talking about natural language. i think natural language and formal languages are quite different.
You know this. Mathematics is constructed in a formal language. ZFC defines a formal language with an alphabet and allowable operators. The axioms define allowable predicates. It's not a maybe: it's a fact
a formal language is a mathematical object. its composed of symbols and rules. math is done in part by writing down the symbols and manipulating them because it helps humans think better about the math.
a natural language is an empirical observation. people make noises from their mouths and induce brainstates in each other.
i dont know these things based on formal logic. i only think i do.
for example, marketing studies how to convince people of things. people who get convinced might think they are convinced by formal logic, but the formal logic is not the direct cause, more of a retrospective fitting.
but you can do your best to retroactively analyze your beliefs in the context of formal logic for consistency and such. and if you practice enough you can start believing in things in a way that corresponds better to logic. but thats really hard.