But since mine is easier, I'll just lay some stuff out. Which is real? Finite ZF? ZF? ZFC? Are you calling one of these math and the others not? What distinguishes them? Are infinities real? Are they not? No matter your choice we're going to lead to a decision where math isn't simply due to our ability to have mathematical systems in either form. Similarly I can mathematically describe geometries that cannot exist in our universe or create algebras which are not in fact useful. I lay out a lot more in the other comments. But I really don't appreciate the "trust me bro" responses. They are intellectually lazy and I have no reason to actually trust you.
Trust has little if anything to do with knowledge. I know calculus not because I trust my professor. Trust no one, only yourself - after you have read the book, of course.
What we are talking about here is a rather deep (and obviously confusing) philosophical topic, and HN is not the place to be discussing it in full, but the gist of it is this. Mathematical reality is somewhat different from the physical reality ("our universe"). The focus of mathematics is on abstractions and generalization. But abstractions are direct reflections of the objective (i.e. what is outside our mind).
You are asking about ZF, ZFC, etc. Try asking yourself about XYZQJJX (it's language, after all, which you are free to invent, right?) and you won't be getting anywhere. You better trust me on this one.
> Mathematical reality is somewhat different from the physical reality ("our universe").
I think in general when we're referring to "reality" we're being synonymous with physical reality. Include the reality beyond observable (like String Theory predicts) if you want. With your usage I honestly have no concept of what you mean by "reality" that I can differentiate from any other abstract concept. If the apple in my head is part of reality (excluding the literal physical state of my brain analogous to bit state of computer memory) then I feel this discussion is rather moot and we can go ahead and conclude that nothing is invented and everything is discovered. But then why not discuss with the definition of reality I'm using (and what I'm assuming is fairly status quo)?
And at what point is math "real?" Is any axiomatic system real? If so we can create arbitrary reality and go back to the above point. If not, then not all math is real. I don't see how we can have something that is self consistent here. Unless you're saying that "math" is whatever rules the universe operates on and not these other inaccurate set theories/systems that we use to approximate this thing." But then again the latter is what we call math and I guess we need to come up with a new definition for many things that exist within recreational mathematics. Sorry happy numbers, idk what you are but I guess not math?
> You are asking about ZF, ZFC, etc. Try asking yourself about XYZQJJX (it's language, after all, which you are free to invent, right?) and you won't be getting anywhere.
Sorry, you're going to have to define it if you want me to determine if it goes anywhere or not.
I'm referring to Zermelo–Fraenkel set theory and Zermelo–Fraenkel Choice set theory respectively[0]. (I figured it wasn't hard to Google, but you're right, I should clarify) These are for all intents and purposes different "maths." They have different axioms. Different rules. Different conclusions.
I'm unconvinced that utility has anything to do with something being real or not. Your usage suggests you aren't considering language real but it's one of the most useful things animals have invented (especially spoken and written language). Nor is the an arbitrary computer image "real" in the same sense. There's a lot of things that belong to this same categorical set.
And while we're at it, just give me anything, 1 or more, that wouldn't be considered real. I'll represent them with a set and I can do math with them... After all, math is the study of relationships between arbitrary objects. See Poincare or ... Category Theory
To be clear, we can create arbitrary set theories that will go "nowhere" and not have utility in physics or other common things. So if you don't want to call this math (I'm not sure who would say that. It looks like math, acts like math, and even mathematicians call it math) what would you call it?
Here, I'll provide an example:
∀ u(u ⊂ X≡u ⊂ Y)⇒X≠Y. : (Axiom of non-Extensionality) When x and y have the same members they are not the same set
Idk we can have the rest of ZF if we want here but it isn't really going to make sense because we destroyed uniqueness. But we can still do math from here. This is a more extreme version of what I was suggesting with Finite ZF vs ZF vs others. They have different axioms and they lead to different things. ZFC is in contention specifically because the axiom of choice leads to nonsensical things (again, see Banach-Taraki paradox). Are these both math? Back to paragraph 1.
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.
But since mine is easier, I'll just lay some stuff out. Which is real? Finite ZF? ZF? ZFC? Are you calling one of these math and the others not? What distinguishes them? Are infinities real? Are they not? No matter your choice we're going to lead to a decision where math isn't simply due to our ability to have mathematical systems in either form. Similarly I can mathematically describe geometries that cannot exist in our universe or create algebras which are not in fact useful. I lay out a lot more in the other comments. But I really don't appreciate the "trust me bro" responses. They are intellectually lazy and I have no reason to actually trust you.