Proving Math is a Language: A Mathematical Argument

[loseyourname]

Are you asking for its English definition? If so, that's unacceptable. There must be a precise mathematical definition of language so that a proposition containing it as a mathematical object can have some rigor.

Maybe it is a set or class of objects? Or could it be that there is only one language and mathematics is in a set of its own?

What about Loglan, which uses the predicate calculus as its grammar? As such, the entire language can be translated into a non-verbal, symbolic form. I know it isn't mathematics, but the point being, it can be derived from first-order logic, and is a language, so why can't other formalisms derived from first-order logic, like mathematics, be a language?

I can understand why you want the statement made using mathematical symbols rather than English, as you are implicitly making the claim that the relevant statement should be translatable into any language, including mathematics if it is a language, but I have two slight quibbles. The first arises directly from this implicit claim: I'd simply like to bring up the fact that plenty of languages are limited in what can be expressed using them. Sumerian cannot be used to talk about the space shuttle program; heck, to use a more banal example, during the Hebrew revival that has been taking place since the founding of the modern state of Israel, new words have continually needed to be invented to communicate about things that did not exist when Hebrew first died off in native use. It is entirely possible that mathematics is a language, but does not possesses the vocabulary necessary to formulate the statement 'mathematics is a language' using nothing but mathematical symbols. You couldn't do it in Khudzul, either, but that is still a language.

The second quibble is that I cannot understand why you are asking for proof. Prove that English is a language using English. There is no formalized, rigorous process necessary. After all, 'language' is simply a word, and any entity or set of entities that fits the meaning of the word qualifies as a language. To give a property-laden definition, any set of symbols used to communicate ideas in accordance with a socially agreed-upon syntax and semantics is a language. Heck, I suppose it doesn't even need to be socially agreed-upon, as an artificial language that only one person knows and no one uses is still a language. The point being, objects don't fit the definitions of words a priori in a manner provable from first principles. They fit the definitions of words because they have properties that put them within the extension of the word based on the way it is used by people. If people use the word 'language' in such a way that its extension includes symbolic formalisms like mathematics as a referrent, then mathematics is a language, de facto rather than de jure.

Getting back to the original point about expressability using mathematical symbols, my definition is a truth-functional statement, so I could translate it into purely symbolic, computable form while still making the statement that mathematics is a language, but is it really necessary? You could easily do it as well.

 

[buddyholly9999]

Wow...that was deep. I think loseyourname makes a good point.

I would have just given you an eloquent math proof.

Proof by "because I said so":
Mathematics is a language...because I said so. QED.


We could also have done it as
proof by "loseyourname is da man":
loseyourname is da man and loseyourname says that math
is a language all of its own...thus mathematics is a language. IYF.

I protest that we should all use IYF at the end of proofs now.

 

[honestrosewater]

If you say that a language is a set of strings, a theory is a set of formulas, and formulas are strings, as are common definitions of these things in math and logic, then the theories of math are languages. Math could qualify as a language by many other reasonable definitons as well. But you never said what languages or mathematics are.

 

[Mickey]

I wrote two rather lengthy replies but each time my browser quit. It's my own fault, since I already know that my computer is dying. :(

Anyway, I'll be pithy. I'm not asking for notation or symbols. I'm asking for mathematics!

Appreciating the distinction is part of the problem. Mathematics is a consistent system, but mathematical notation is not. The restrictions of typesetting have forced us to use the same arrangements of symbols for different topics, and this has caused some needless irritation and confusion over the years.

Saying that there is a unique language we can use to communicate mathematics is not the same thing as saying that mathematics is itself a language. If it was a language, it would be a consistent language and much easier to learn, especially if we can count its axioms on two hands. loseyourname, since you made the comparison to loglan, I think you'll agree that loglan is easy to learn precisely because of its consistency.

Consistency is so much of what mathematics is all about! Although notation may be consistent within mathematical topics, making it easier for specialists, it is unfortunate that we don't have a consistent language for learning the whole of mathematics.

Perhaps one day in the future, students will learn mathematics by first learning a consistent language, such as loglan, and then use it to learn mathematics. It would likely mean that they would also be learning a computer language at the same time. It would be even more interesting because, if mathematics describes nature, then we could potentially communicate with a more phenomenal language, rather than a purely symbolic one.

I'd absolutely love to see the day when mathematical language is as consistent as mathematics. Maybe then there would be no real distinction. Then, we might be able to communicate in a true language of nature.

 

[neurocomp2003]

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?

 

[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}
--------------------------------------------
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.