Morning! I thought you would answer along these lines ;-)
> for computable functions.
No, I believe Busy Beaver numbers are real, too. Because TMs either halt, or they don't, independently of what you can prove in some axiom system about them.
> 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?
No, the definition would not change: Yes, we can have an infinitely long tape, as a mathematical object, that is part of the overall mathematical object that is the TM. Axiom systems describe mathematical objects, so your choice of axioms to describe a TM is not arbitrary, but must make a TM a model of these axioms. So if you are describing a TM with an infinitely long tape with axioms that demand finiteness, your description is wrong.
> You said math is real (claim, no proof)
Yes, I have no proof for this, and there might never be, because, as Gödel showed, you cannot prove everything that is true (note that Gödel also thought that math is real). But it is the only thing that makes sense, and actually allows you to move forward doing proper mathematics, instead of saying things such as "there is no infinitely long tape", although you know perfectly well how such an infinitely long tape looks like.
> Why would math be any lesser were it a made up thing?
It is not a moral judgement on my part. I don't think language is bad, I really like language. And you can make up descriptions in math of course, using language and axioms, I do it all the time. And some of these descriptions will describe real mathematical objects that exist, and some of your descriptions will be inconsistent, and so the things you described won't exist. In general, it is not possible to prove which of your descriptions are consistent, you will need faith here, although you can make relative judgements. I have faith in the natural numbers, for example.
> 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.
I cannot prove you wrong, so you can keep your opinion. But I find it more convincing simply to accept that math is real, than that there is a "convergent solution space", whatever that is (and is it real?).
This would be a good thing to provide a citation for.
> you will need faith here,
Sorry, I don't do that.
> than that there is a "convergent solution space", whatever that is (and is it real?).
Well yes. Physics does optimize for the least energy. For example, if you intend to build a device that you hold in your hand and can smack things you're going to probably come up with a hammer. You're probably also going to come up with a hammer who's head is a cylinder too. But that's because we live in a universe and that the thing we're doing is bound by its rules and influences.
Idk man, it seems like your argument is just consistent with "everything is real." I've asked and you can't define to me what is real and what isn't real, with you clearly using a nonstandard definition. That's not a proof, that's just definition.
There is no proof for this kind of thing, just faith. And there is a proof that faith must be enough. So I know if you don't even have faith in your foundational position, you have actually nothing at all. You can call it "conviction", if you want. If you want to know more about Gödel, there is a thing called the Internet. Look it up [1].
Well, I think it is just a clear sign of someone who is not interested in a proper discussion, and it is also somewhat insulting: To ask for a reference, when you can just look it up yourself, in about the same amount of time it takes to ask for a reference. Especially if you name your account after Gödel (it's Gödel, not Godel, by the way), then it just shows that you have no clue what you are actually talking about.
> for computable functions.
No, I believe Busy Beaver numbers are real, too. Because TMs either halt, or they don't, independently of what you can prove in some axiom system about them.
> 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?
No, the definition would not change: Yes, we can have an infinitely long tape, as a mathematical object, that is part of the overall mathematical object that is the TM. Axiom systems describe mathematical objects, so your choice of axioms to describe a TM is not arbitrary, but must make a TM a model of these axioms. So if you are describing a TM with an infinitely long tape with axioms that demand finiteness, your description is wrong.
> You said math is real (claim, no proof)
Yes, I have no proof for this, and there might never be, because, as Gödel showed, you cannot prove everything that is true (note that Gödel also thought that math is real). But it is the only thing that makes sense, and actually allows you to move forward doing proper mathematics, instead of saying things such as "there is no infinitely long tape", although you know perfectly well how such an infinitely long tape looks like.
> Why would math be any lesser were it a made up thing?
It is not a moral judgement on my part. I don't think language is bad, I really like language. And you can make up descriptions in math of course, using language and axioms, I do it all the time. And some of these descriptions will describe real mathematical objects that exist, and some of your descriptions will be inconsistent, and so the things you described won't exist. In general, it is not possible to prove which of your descriptions are consistent, you will need faith here, although you can make relative judgements. I have faith in the natural numbers, for example.
> 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.
I cannot prove you wrong, so you can keep your opinion. But I find it more convincing simply to accept that math is real, than that there is a "convergent solution space", whatever that is (and is it real?).