> Turing Machine is real, and it either halts, or it doesn't.

*for computable functions. There's plenty of non-computable mathematics. Ideally everything would be computable but that's just not the case (pun specifically intended).

> but that it is a mathematical construct I know the exact definition of, and the mathematical object behind that definition wouldn't change even if you would not exist.

I don't think this is true. Wouldn't the definition change were we to consider finite ZF set theory? Since you cannot have an infinitely long tape?

There are algorithms where it is not possible to determine if they halt or not (Godel says "True v False is not accurate, it's True v False v Indeterminate"). There are also algorithms that halt in exactly infinite time. We can make that countable steps on uncountable. (Finite ZF is going to definitely change things here) But then there's hypercomputation and we run into the same problem.

But maybe we're discussing this the wrong way. Let's do this differently. Certainly we can operate a "Turing Machine" purely through spoken language, right? No mathematical symbols or equations required. Now I understand you might just say "yes, but that's still math." Which is a valid point. But here we have the conundrum of distinguishing any arbitrary communication from math.

And while we're at it, let's figure out this problem. What is the relationship between {math} and {language}. Is math a superset of language? Subset? Are they completely disjoint? If they intersect then what is in the set of math that is not in the set of language? And what is in the intersection? Fuck, we're doing math right now!

> These "pretty useless algebras" exist nevertheless as themselves, you don't need to find some other, physical objects in the world that correspond to them.

Sure, but now we're at circular logic. Go back to how you started. You said math is real (claim, no proof). Then you follow by giving an example of Turing Machines stating (incorrectly) that they either halt or don't (damn tricky indeterminate option!) and then state that you know the precise definition. Your evidence here is one of utility. I'm also not entirely convinced you know the __exact__ (overly pedantic) definition as we'd need to describe that starting with our choice of preferred axioms/set theory. Yes, you can wave your hand and say that there is an exact definition in ZF, ZFC, FZF, NBG, MK, KFC, or Godelsk's-Fucked-up-ZF Set Theory, or whatever. But this arbitrariness just makes a more compelling case for math being something humans made up. We invented math to be very precise and highly consistent description of relationships between things but I'm not even sure what you're saying math "is". I don't even know what you mean by "just". As if there's something disgraceful about language that tarnishes the purity of math. Which really just means you haven't looked under the covers because it's pretty fucked up under there.

Why would math be any lesser were it a made up thing? I want to ask, what would be __wrong__ with it? If it is made up what would it change? Would it really be anything other than our perspective of it? Honestly, I think there is something wrong if the quality of being "invented" tarnishes something, with that specific quality in isolation (being precise here). So if we can't address the above can we at least address this? What would be wrong with math being invented? Honestly I think that just would speak more to the ingenuity of humans.

Side note: I'm also highly convinced that were we to meet an alien species we'd find a lot of similarities between our maths. But that would not convince me that this is real, but rather that there's just a convergent solution space. The whole process, when applied to describing our world, is in fact to make the least amount of assumptions and to make the assumptions as uncontestable as possible (but we can never escape assumptions). It would make quite a lot of sense that there'd be a lot of similarities in that and very good reasons to create such things. Just like I would expect every alien civilization to invent knives, scissors, wheels, and other such things. I mean these are rather consequences of interacting in our universe and dealing with the least energy rather than the wheel existing as a thing that is to be discovered. But if so, we come back to language, and I again cannot differentiate how the wheel or math can be existing in reality (by this abstract definition of minima in solution spaces) but language is not.