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Mathematics certainly serves as a language. It is also full of nice games. But it is not just that. Mathematics is real. As I said, a Turing Machine is real, an

Mathematics certainly serves as a language. It is also full of nice games. But it is not just that. Mathematics is real. As I said, a Turing Machine is real, and it either halts, or it doesn't. And what turns a Turing Machine into something real is not that you and I agree on what it is, communicating with each other, 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've seen people having trouble with my use of "real" before. When I say real, I mean that these mathematical objects exist in our reality. There is only one reality, and we share it. That's why you and I can use these mathematical objects to communicate with each other, and that's why mathematics also serves as a language. 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.

godelski 3y ago
> 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.
- auggierose 3y ago
  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?).
  - godelski 3y ago
    > note that Gödel also thought that math is 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.

godelski 3y ago

> 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.

auggierose 3y ago
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?).
- godelski 3y ago
  > note that Gödel also thought that math is 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.