I'm not sure what your argument proves. No one said it is just games either. I think you're turning what I said into something else. Languages are incredibly impressive and powerful things. I feel like you're making languages appear simple and of low utility. But language has been one of the, if not the most, powerful tool humans have ever created, especially when you consider written language. The latter allowing us to pass along information asynchronously through time and space. Do not miss the power of this tool just because it is something we use so frequently. There's a reason we use it so frequently.
The reason we use the language of math is because it is so precise. But you just described a turing machine without using math. Keep in mind that physics is also a language. Physics is an ever evolving model, a description of our reality. But it is not more than that, a model. But all models being wrong does not equate to models being useless. Physicses (yes, there are multiple ones), are clearly quite powerful and useful models that have enabled us to do quite amazing things as well. But I'm also not going to start programming in Brainfuck or with Magic cards. But I do have a deep passion for mathematics and it is something I use every day and spend quite a lot of time learning even outside my work.
Your argument demonstrates the usefulness of mathematics, but does not demonstrate that it isn't a language. I also would like to remind you that you can create pretty useless algebras. Maybe also checkout Wolfram's section on recreational mathematics. Maybe also dig into my name sake's multiple theorems. Most people misunderstand the results, but I don't think it diminishes their worth, they are quite important and amazing. And the incompleteness theorem is deeply related to turing completeness as well as superposition and a famous logic paradox.
I am having trouble following your opinion through this thread.
> Your argument demonstrates the usefulness of mathematics, but does not demonstrate that it isn't a language.
What is “a language” to you? What is an “isn’t a language” to you?
I can grok you referring to axioms and lemmas as a “mathematical language”, but I see such as just the way we communicate something more essential and wholly independent of any need to have been communicated.
A lot of contemporary research mathematics is layered and wrought of “useful” complexities for its desired domain, but how do you dismiss the essential and seemingly unrealness of its abstraction from our perceived reality?
Counting is an example.
Subjective boundaries illuminate the essentialism of distinctness. 2 apples describe the same abstract phenomena as 2 atoms, or 2 galaxies, or 2 orientations of stereoisomers.
What is the “language” here? The word/symbol 2? The subjective boundary that separates something more continuous into discrete forms?
Transcendentals and irrationals alight my meditation on what the hell all this is that we’re experiencing.
You have a triangle with edges that terminate at each vertex, but if two of those edges have equal length than you can interpret their length as unit 1 where the third edge then has a length of (sqrt 2) which is a number without a finite decimal expansion.
What language can be used to defend an infinitesimal equating to a finite value?
This points at an essentialism to me.
Any amount of “language” is incapable of both explaining this completely or explaining it away.
Similar with pi and its relation to a circle which has a well defined circumference that somehow expresses itself with a number that is itself incapable of being expressed or defined.
As you brought up the incompleteness theorems, they too have a similar “infinite in finite” quality.
I am unsure how you can understand godel but argue against the essentialism of the sur-real abstractions he brings attention to.
> "Infinitesimal" is just an idea, as far as I know. Nothing real is infinitesimal.
The unreal (re: abstracted) aspect is what places it outside the confines of “language” for me.
Are black holes real? Do they have singularities? If yes, that can be an example of your “real” infinitesimal.
My opinion is that infinitesimals are more than real they are essential. They are the building blocks of all that is “real”.
Ultimately, what we’re talking about is a philosophical debate that would require one to step “outside” reality to confirm or deny outright so we are just providing our opinions on an unknowable concept.
What is “real” in this context?
Is pi “real”? Is the plank constant? The former was my path to the essentialism of infinitesimals. The latter my path to the essentialism of discrete counting.
The math leads us there but I don't think anyone is particularly happy about it.
> Is pi “real”?
¯\_(ツ)_/¯
> Is the plank constant?
¯\_(ツ)_/¯
Fuck man, I can't tell you if a quark is real. I'm also not aware of anyone who can. The best we got is our interpretation that the model being indistinguishable from the real thing might as well be the real thing. Metaphysics and metamathematics are mind bending areas that require a deep understanding of the non-meta concepts first.
But given all you've said, I highly suggest looking into the various set theories I mentioned previously. Specifically start with Finite ZF set theory and Peano Arithmetic, where you'll find you can indeed operate on such concepts as pi without infinities.
> What is “a language” to you? What is an “isn’t a language” to you?
A language is an abstract concept that describes a method of communication. It need not be spoken (such as English), written (such as what we're doing now). We frequently use body language to communicate, and so do many animals. We have braille, smoke signals, maritime flags, we communicate with knots on a string, and so many more things. You're right that language is quite a broad and vague thing. But recognize that all these things are also not of the universe, but of us humans (or similar of other animals). Something like English is something we may better refer to as a social construct, as it is a collective agreement, though body language may be a bit more ingrained but I still do not think you would refer to it as something other than language or something of the universe (distinct from us being of the universe in the trivial sense).
> What is the “language” here? The word/symbol 2? The subjective boundary that separates something more continuous into discrete forms?
(This is HN, so I'm going to assume you're familiar with programming languages.) If I give you these 14 characters (p, t, k, s, m, n, l, j, w, a, e, i, o, u) are we able to communicate? Maybe after some trial and error, but certainly not something we could throw into a translation machine. It'd be hard to call these even tokens since we have not distinguished consonants from vowels or if that even is a thing here, so we can't really lex. We need words, phrases, and context before you can even from syntax. Then we need to build our syntax, which is equally non trivial despite looking so (build a PL, it is a great exercise for any computer scientist or mathematician. For the latter, build your own group, ring, field, ideal, and algebra. You'd do this in an abstract algebra course). We need all this to really start making a real means to communicate. These are things we take for granted but are far more complex when we actually have to do them from scratch, forcing our hands.
Do you have a problem calling a programming language a language? I'd assume not because we collectively do so tautologically. Great, you agree that math is a language. Thank you lambda calculus. We can have an isomorphic relationship between programming languages and various mathematical systems. I'll point out here that there are different algebras and calculus with different rules and forms, though many that are not deep in mathematics may not be exactly familiar with these. I think this is often where the confusion arises, since we most often are using our descriptions that are most useful, just like how no one programs in brainfuck and just how most drawings are communicative visualizations rather than abstract art. I again remind you of Poincare who says that mathematics is not the study of numbers, but the study of relationships. He does not specify numbers in the latter part, on purpose. Category theory may be something you wish to take up in this case, as it takes the abstraction to the extreme. Speaking of which
> but how do you dismiss the essential and seemingly unrealness of its abstraction from our perceived reality?
I could ask you the same about English. Why is this any different? Is that because you are aware of other languages that people speak? Or is it because you recognize that these languages are a schema of encoding and decoding mechanisms which result in a lossy communication of information between different entities?
You discuss counting, but are not recognizing that you can not place an apple into text, nor atoms, galaxies, or stereoisomers. It is because mathematics is the map, the language, not the thing itself. We can duplicate these at will or modify them in any way. Math is not bound to physical laws like an apple is. Its bound is the same of the apple that exists in my mind, not in my hand. (If you want to make this argument in the future, a stronger one might revolve around discussion of primes and their invariances)
> What language can be used to defend an infinitesimal equating to a finite value?
If this is the essential part, I think this is probably the best point to focus on. Specifically because infinities are not real. Nor are they even numbers. If you disagree then you disagree with physics. Rather infinities are a conceptual tool that is extremely useful. But if we were able to count and use infinities then we'd have the capacity for magic via the Banach-Tarski Pardaox, and completely violate the no cloning theorem. But our universe does not appear to actually have arbitrary precision, rather our tool does due to its semantics. Maybe finite ZF is a better choice than ZF or ZFC set theory. Why not NBG which has a finite number of axioms or why not MK which isn't?
Infinities, singularities, and such are not things in our universe. You may point to a black hole but this would represent a misunderstanding of our understandings of them. We cannot peer in beyond the event horizon, which certainly is not a singularity and has real measurable and finite volume. It is what is inside that is the singularity. But can you say that this is not in fact just an error in the math? It wouldn't be the first time such a thing has happened. Maybe it is at the limits of our math and so thus is a result of the inconsistency of axiomatic systems? There are many people working on this problem, and I do not want to undermine their hard work, and neither should you.
You're biased because you're looking at how we use the tool rather than what the tool is itself. We use mathematics as the main descriptive language for science because of its precision. But we've also had to do a lot of work to ensure its consistency and make it more precise along the way. But I think you may have not been exposed to the levels of abstraction that math has, as this is not seen by most people until well beyond a calculus class.
> I am unsure how you can understand godel but argue against the essentialism of the sur-real abstractions he brings attention to.
And I cannot see the reverse. Are you saying that the universe is incomplete? Are you saying that the universe is not consistent? This sounds like a better argument for the idea that the universe is a simulation (as in we are being simulated, not as in you can represent and draw parallels between the universe and simulation. The former begs the question "on what" and we get turtles all the way down). Rather, as the old saying goes, I do not believe that the map is the territory. Just like how our brain creates an incomplete model of the world we live in, mathematics too is used to create an incomplete model to help describe not only what we can see but what we don't. But do not trivialize or diminish the notion of a model, as I certainly would not claim our brains and senses are useless. Models are quite powerful things, there is a reason we use them. But a model is not the thing itself.
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.
> 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.
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.
The reason we use the language of math is because it is so precise. But you just described a turing machine without using math. Keep in mind that physics is also a language. Physics is an ever evolving model, a description of our reality. But it is not more than that, a model. But all models being wrong does not equate to models being useless. Physicses (yes, there are multiple ones), are clearly quite powerful and useful models that have enabled us to do quite amazing things as well. But I'm also not going to start programming in Brainfuck or with Magic cards. But I do have a deep passion for mathematics and it is something I use every day and spend quite a lot of time learning even outside my work.
Your argument demonstrates the usefulness of mathematics, but does not demonstrate that it isn't a language. I also would like to remind you that you can create pretty useless algebras. Maybe also checkout Wolfram's section on recreational mathematics. Maybe also dig into my name sake's multiple theorems. Most people misunderstand the results, but I don't think it diminishes their worth, they are quite important and amazing. And the incompleteness theorem is deeply related to turing completeness as well as superposition and a famous logic paradox.