Proving Math is a Language: A Mathematical Argument

[Mickey]

mickey: I'm still confused about what you define as language and what you define as mathematics? Have you ever taken compiler or language theory?

No. I don't have definitions for language or mathematics. I'm asking for mathematicians to provide them, if they are going to insist that mathematics is a language.

I ask for "proof" just because I want it to be consistent and rigourous. So, if they are able to show that they are unable to provide a proof, that's good too.

Until they make it clear, I'm as confused as you are, and I really don't appreciate mathematicians confusing us more than they already do. ;)

 

[honestrosewater]

You say you want math, but all you seem to be talking about is how real people use physical symbols in the real world.

If you're asking for someone to come up with a proof that math is a language, and you're also going to let us define math and language, that's easy.

Defintion: Math is a language.
Proof: Math is a language (by definition).

Voila!

 

[neurocomp2003]

honestrosewater: aw no clapping emoticons...i'll edit to give you the rolling happy face...(was going to ask what your name meant...hone strose water hehe then i realized the st was on the wrong word)...

Mickey: if honestrosewater definition isn't sufficient

Combining the fields of Set Theory, Logic Theory, and Langauge Theory( and perhaps compiler theory) I'm sure you'll get one thus get cracking and have fun...however I don't think you know what language theory is, or for some reason you refuse to answer whether you know it or not.

Also Would you believe that math will prove itself as a language if it can be mapped or translated to all other languages in our language space? Btw do you consider languages that come from other languages as languages? My anthro and linguistics is horrible but didn't the english language emerge from other language and constantly takes words from other languages as its own? Lastly like honestrosewater implied we don't need a rigourous proof if your going to let us define mathematics...if you require a proof then you have already partially defined mathematics...and thus i think you'd need to define it completely.

However Like any other language you would have to communicate by a certain fundamental set(of sounds or actions) to start off. I don't know how to use LaTeX. so a fundamental set of sounds or symbols would be something like {thereexists} or {thereexists,x,y,=,implies or such that{},(),,negate,and,or}. Then you can define strings from language theory and formulas from set theory

--------------------------------------------
Fundamental(each symbol requires a sound)
thereexists (i'll gesture it to you,you can pick what sound you want)
thereexists x (make a symbol,make a sound,,,over a period of a couple of months we'll agree that sound is symbol)
thereexists y (make a symbol,make a sound,,,over a period of a couple of months we'll agree that sound is t symbol)
thereexists 1 (we start chucking stuff at each other)
thereexists {} (we'll go foraging)
thereexists () (we'll go foraging)
thereexists +1 (you'll steal some from my pile)
thereexists -1 (I'll steal some from your pile)

thereexists ->(we'll punch either till we get the meaning "implies")
thereexists symbol if (we'll punch either till we get the meaning "if")
thereexists symbol then (we'll punch either till we get the meaning "then")

thereexists yes/no true/false 1/0
thereexists neg(x) (i wave my figure at you and "hit" you a couple of times)
thereexists = ->x=x
thereexists isin -> x isin{x}
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Only if you've defined a certain fundamental set of symbols and formula with meaning can you have your rigorous proof. But then again only when you have a certain set of actions and sounds can you define a language. And languages arise out of other languages.
--------------------------------------------
thereexists Define A -> thereexists A
Define Symbol
Define Alphabet
Define Word
Define String
Define Function/Map/Transform
Define Graph
Define 2D Bitmaps(pictographs)
Define 3D Bitmaps(pictographs)
Define 2D Bmp-2D Bmp maps/transforms
Define 2D Bmp-3D Bmp maps/transforms
Define 3D Bmp-2D Bmp maps/transforms
Define 3D Bmp-3D Bmp maps/transforms

 

[Mickey]

That's interesting, neurocomp. I'm not a language theorist, but I don't need to be one to demand rigourous arguments, because then I may learn them without hesitation of their validity.

If you could show that mathematics is a language via language theory, that would be a significant achievement.

 

[neurocomp2003]

i meant the mathematical field Langauge Theory(as seen as part of Computability) not linguistics. Also there is always the definition that Mathematics is the "Universal Language"(or rather the Turing machine,which is a mathematical concept). Though again to have some sort of rigorous proof you mostlikely need a fundamental set of symbols and sounds to portray any language. But yet again if you need a rigourous proof you have defined partially what mathematics is.

I hope this isn't a homework/take home exam problem.

 

[honestrosewater]

I'm not a language theorist,

Well, mathematical linguistics is my speciality, so perhaps you are in luck.

but I don't need to be one to demand rigourous arguments, because then I may learn them without hesitation of their validity.

Or you could learn to construct rigorous arguments and recognize valid ones yourself. If you are going to be a mathematician, you will need those abilities anyway. And by the bye, rigor is no guarantee of validity, in case you were thinking so.

How was my proof not rigorous? There aren't even any steps. I really don't understand how anyone could complain about rigor in a proof by definition.

Do you want a formal proof? You're letting us assume as an axiom what you want us to prove, and in all formal systems, an axiom is a proof of itself. Since you didn't specify a language, just let a denote that axiom in whatever language you want your proof written in.

Proof: a.

By not telling us what math and language are supposed to mean, your demand sounds a lot like 'prove that ______ is a ______'. I am just trying to point out why it is silly and self-defeating to refuse to tell us what you expect math and language to be. Your objections to people's answers show that you do indeed have some expectations.

If taken literally, you seem to be asking for a proof that there exist some x and y such that x is in y, whatever x and y are. That's easy.

I think you're asking for a proof that mathematics is a member of the set of all languages. But mathematics, as a field of study, includes things that wouldn't reasonably count as being part of a language.

If you are instead asking whether mathematicians speak something special that should count as a language on its own, separate from their natural languages, or that has some other special properties, I think that's an interesting question. But it's not a mathematical question since it concerns physical objects rather than mathematical objects. It's a question for linguistics.

If you want the set of all mathematical theories -- which are the special things that mathematicians are saying -- to be a subset of the set of all languages, I already told you how this can be so. Every set of strings is a language. A mathematical theory is a set of strings. Again, every set of strings is a language. What more do you want anyone to say? I was just talking about theories and such in another thread, in case you want to know a little more about what a mathematical theory is by my definition.

This isn't restrictied to only formal languages either. Every language can be characterized as a set of strings. Natural languages, or, more accurately, certain aspects of them, are studied as sets of strings. The interesting thing is the formal grammar, which tells you which strings are in the set, or generates those strings. You might get a more satisfying answer by asking about the properties of the grammars, if any exist, that generate some mathematical theory (or class of mathematical theories) and then comparing those grammars with grammars for other languages (or classes of languages).

If you could show that mathematics is a language via language theory, that would be a significant achievement.

How so? What is language theory? Formal language theory? Formal language theory defines a language as a set of strings.

My anthro and linguistics is horrible but didn't the english language emerge from other language and constantly takes words from other languages as its own?

Modern English evolved from Middle English, which evolved from Old English, which is a descendant of Proto-Germanic*, which is a descendant of Proto-Indo-European*. That leaves you at about 4500 BC and is as far back as I know how to go. Most languages are like English in that they are descendants of other languages and do or can borrow from other languages, though I wouldn't say that any language borrows 'constantly'. I talked a little about these things here.

*these are unattested, hypothetical languages reconstructed from their hypothetical descendant languages using the comparative method.