Proving Math is a Language: A Mathematical Argument

[Mickey]

In the old days (i.e. just a few decades ago) candidates had to know a foreign language in order to become a PhD. Very sly students, like my macroeconomics professor, were able to maneuver through this requirement by convincing their superiors that mathematics is a language.

Until I heard that story, I didn't know the idea was taken that seriously. Actually, I have never heard a mathematician or scientist argue otherwise. A quick Google search for the phrase "mathematics is not a language" as of this date returns a paltry 26 results, and even some of those are conditional. The opposite and more familiar statement returns 14,500 results.

Saying that mathematics is a language sounds all nice and poetic, and it's possible for one to wax philosophically on the subject, but I'm not looking for philosophy or poetry here. Please no philosophizing in this thread. This is the math section. I'm looking for a mathematical argument showing that mathematics is a language.

After all, if the proposition is true, then shouldn't it be possible to use mathematics to prove it? If it's not true, can we use mathematics to disprove it?

Even if one could show that it couldn't be proved or disproved, one should still be able to state the proposition mathematically. Has that ever been done?

I have never seen that little. Until the statement is formalized, I'm afraid the idea is pseudo-mathematics.

Remember, please no philosophizing. Conjecture or postulate instead, if you must. I just request that you make it a mathematical argument.

 

[neurocomp2003]

i guess then the question someone would pose to u first is what your definition of a language is...since There is a math field called Langauge THeory(or cs field to some)

 

[HallsofIvy]

Every now and again, someone will propose allowing "computer languages" as a "language" option, bringing the wrath of the liberal arts faculty down on their heads!

(What do you mean "in the old days". As far as I know, any decent Ph.D. program requires two foreign languages today.)

 

[J77]

(What do you mean "in the old days". As far as I know, any decent Ph.D. program requires two foreign languages today.)

Why?

 

[shmoe]

Why?

Not all mathematics is done in English (or whatever your native tongue is). Some departments language requirements are based solely on the ability to translate mathematical documents, no conversational skills required.

These requriements are definitely vanishing at some schools, or more and more phd programs are falling from HallsofIvy's "decent" category (like mine is, which requires no foreign language).

 

[Dragonfall]

Well, define 'language', and we'll see if mathematics fits.

 

[Mickey]

Well, define 'language', and we'll see if mathematics fits.

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?

 

[desA]

Definitions of language on the Web:

* a systematic means of communicating by the use of sounds or conventional symbols; "he taught foreign languages"; "the language introduced is standard throughout the text"; "the speed with which a program can be executed depends on the language in which it is written"
* speech: (language) communication by word of mouth; "his speech was garbled"; "he uttered harsh language"; "he recorded the spoken language of the streets"
* terminology: a system of words used to name things in a particular discipline; "legal terminology"; "biological nomenclature"; "the language of sociology"
* linguistic process: the cognitive processes involved in producing and understanding linguistic communication; "he didn't have the language to express his feelings"
* the mental faculty or power of vocal communication; "language sets homo sapiens apart from all other animals"
* lyric: the text of a popular song or musical-comedy number; "his compositions always started with the lyrics"; "he wrote both words and music"; "the song uses colloquial language"
wordnet.princeton.edu/perl/webwn

* Langauge is a system of finite arbitrary symbols combined according to rules of grammar for the purpose of communication. Individual languages use sounds, gestures and other symbols to represent objects, concepts, emotions, ideas, and thoughts.
en.wikipedia.org/wiki/Language

* In mathematics, logic and computer science, a formal language is a set of finite-length words (i.e. character strings) drawn from some finite alphabet, and the scientific theory that deals with these entities is known as formal language theory. Note that we can talk about formal language in many contexts (scientific, legal, linguistic and so on), meaning a mode of expression more careful and accurate, or more mannered than everyday speech. ...
en.wikipedia.org/wiki/Language_(computability)

Results of a Google search: define:language.

The 2nd definition seems reasonable in the context of a relationship to Mathematics.

 

[neurocomp2003]

MIckey: your question is still unclear to me because you must define what your defintion of language is?

Unless the question you ask is whether Mathematics considers its symbolic/formalistic Structure a Langauge in mathematics. The answer would be yes because of the field of Langauge theory(sometimes a subtopic in Computability books).

 

[J77]

Not all mathematics is done in English (or whatever your native tongue is). Some departments language requirements are based solely on the ability to translate mathematical documents, no conversational skills required.

These requriements are definitely vanishing at some schools, or more and more phd programs are falling from HallsofIvy's "decent" category (like mine is, which requires no foreign language).

The only research which I've come across that's needed translation is that done in Russian.

People can spend careers translating Russian texts to English - after all, they did do everything first!

Also helps that a lot of former Soviets work in the West now.

I see no reason why a 'decent' PhD 'program' would include learning another language.

 

[Mickey]

MIckey: your question is still unclear to me because you must define what your defintion of language is?

Unless the question you ask is whether Mathematics considers its symbolic/formalistic Structure a Langauge in mathematics. The answer would be yes because of the field of Langauge theory(sometimes a subtopic in Computability books).

Okay, that sounds reasonable. But, as far as I can tell, you said it in English.

Can you please say it again in the language of mathematics?

 

[shmoe]

The only research which I've come across that's needed translation is that done in Russian.

It depends on your field, where the main contributors publish and so on. I've had to stumble through a few French articles myself.

I don't think it's a bad thing to include to give some practice in not being afraid of foreign papers. The exams I've heard of are usually written translation exams given a dictionary, so it is pratical stuff that many mathematicians will find handy at some point. It's not asking directions to the cinema or grocery lists.

 

[mathwonk]

if you think you never need another language other than english in math, try reading galois' original work, or riemanns complete works (partially available now in english after 150 years).

not all interesting research is that being done now. and what is being done now should ideally not duplicate what has already been done. that's why we have research libraries. i myself submitted a solved problem for a thesis until it was pointed out hurwitz had already done it, in german, in the 1800's.

and my ultimate thesis problem was an answer to a question posed in german by wirtinger in 1895, but not answered then.

since the best way to understand any work is to read the authors version, you are dependent on either translations or ability to read the original language. besides, when you go there for a conference, it is rude and less effective to order dinner and wine in english. one always needs the local language to feel comfortable and make ones hosts feel so.

but my department too is weakening the language standard regularly until now it has almost disappeared. some foreign students who get credit for their own native tongue as a foreign language, have found it difficult to find time from their math studies even to learn fluent english.

 

[mathwonk]

but mathematics is clearly not a language, even if it has its own vocabulary, anymore than plumbing is a language. every subject has its own vocabulary and special definitions.

 

[oldelectrician]

from a shop floor perspective Mathwonk and others of similar
opinion rings all too true. Try working with Japanese, French
Germans and British, along with Texas engineers who don't
understand PID drawings they submitted which are supposed
to drive a UV water purification system for a state of the art
IBM chip plant. Japanese are upset, French are perplexed and
German dude asks "what is wrong with [US]? - you used to set the
standard." Two electricians, one American, one Canadian
somehow muddled through thanks to what languages we could
remember from school/traveling/military service. The Canadian's
language skills better than the American's (oldelectrician), and to
the amazement of many turned more than a few disasters into
more work for our employer - Both Japanese and French companies'
engineers were appreciative of our efforts to communicate - a
Berlitz book in the tool box. Learn a language - the math/science vocab,
which indeed is a universal, only goes so far --and one may be suprised
at one has been missing once even a little bit of skill has been obtained -
After all, someone may be saying to the person next to him that you
are going off a cliff and there you stand oblivious with a smile on your
face. The engineering company reps from Texas, replete with attitude, an example - ended up being asked to leave the project.

 

[DeadWolfe]

Out of curiosity mathwonk, were those thesis problems?

 

[cogito²]

Learning foreign languages should most definitely be a requirement of a Ph.D. Mathematics is now done on international levels and we shouldn't expect everyone else to learn our language. Why not take some time and learn the language (and culture) of your peers?

 

[Mickey]

It seems like mathematicians who refer to mathematics as a language aren't thinking like mathematicians.

I find it very confusing and a little unsettling to hear academic authority figures, the people teaching mathematics and languages, say that mathematics is a language to people who initially don't know much about mathematics and language, when it looks so patently untrue, and not back it up.

I await those who espouse the notion to use mathematics to simply state the notion. ;)

 

[0rthodontist]

Sure mathematics is a language, a specialized one. It is an effective method for conveying certain messages, which is all a language needs to be. But this is only a quibble over semantics, because your real question is whether it is a reasonable substitute for a foreign natural language like German, which it is not.

 

[mathwonk]

when i shop i ask each cashier how to say thank you in her/his language. each has his own, and it is very tricky to match them, but when they do it is amgic. amharic, somali, vietnamese, polish, german, russian, spanish, french, italian..all have a different verbal button that makes them smile.

once walking down the stret with long hair and beard, some italian construction workers made fun of me so i shouted "buon giorno!" and they all smiled and shouted back friendly greetings.

 

[Mickey]

Sure mathematics is a language, a specialized one. It is an effective method for conveying certain messages, which is all a language needs to be. But this is only a quibble over semantics, because your real question is whether it is a reasonable substitute for a foreign natural language like German, which it is not.

No, I've made no assumption that they must be substituted.

Actually, I am very open to the idea that a mathematical proposition that mathematics is a language could not be meaningfully "translated" to English at all.

 

[mathwonk]

deadwolfe if you asked what was my thesis problem, i first worked on orders of groups of automorphisms of riemann surfaces of genus 5. it is well known that hurwirtz proved that the maximum order of the group of complex analytic automorphisms of a riemaNN SURFACE OF GENUS G IS 84(G-1). SO I PROVED THAT FOR GENUS 5 THE MAXIMUM IS IN FACT 192.

then i wrote the expert macbeath in britain who referred me to hurqitz original paper where i learned he actually proved more. namely if 84(g-1) is not achieved, then the next possibility is 48(g-1), etc,...

so my work was fruitless since i had not read the original work in german.

then i proved that mumfords vanishing theorem on abelian varieties followed from an extension of kodairas vanishing theorem. namely kodairaproved that if a line bundle has a positive difeinite metric, then all its cohomology above degree one vanishes. but the proof shows that the vanishing in degree d depends only on the positiveity of the curvature operator in degree d.

now the eigenvalues of that curvature operator are d fold sums of eigenvaklues of the first order operator, so if any sum of d eigenvaklues is positive then the dth cohomology vanishes.

of course if all the eigenvalues are positive then any sum of d of them is too, but this can happen otherwise. e.g. if exactly r of them are negative but not very large, and all the others are positive abd quite large then the sum of any r+1 eigenvalues is positive and you get vanishing in certain high dimensions.this turned out to ahve already been proved by deligne, so i scrAPPED THAT THESIS TOO.

then i learne3d that wirtinger had shown that the process of assigning to a smooth curve of genus g, an abelian variety of dimension g, the jacobian, could be generalized to assign to an unramified double cover of a curves of genus g, an abelian variety of dimension g-1, its prym variety.

curves of genus 6 have a moduli space of dimension 3g-3 = 15, and abelian varieties (principaLLY POLARIZED) of dimension 5 are parametrized by a modul;i space of dimension 15 also. wirtingewr asserted that this asignment was generically finite to one, but did not determine the degree of the map. i showed it had degree 27.

my method was to find a point, namely A JACOBIAN OF DIMENSION 5, over which the fiber could be computed by beauville's results, to be a union of three connected components of dimensions 0, 1, and 2.

then by a generalization of the implicit function theorem, i was able to determine the degree of the map near each component, as 1,10, and 16, for a total of 27.

some of my work was a bit rough and some of the refined arguments were obtained or provided by other people or jointly with them, and the projects and approaches to them were suggested by my advisor.

but i am very proud of what i did. in particular the geometric ideas that unlocked the solution, e.g. the generalized implicit function theorem, were mine, and they subsequently found other important applications in the hands of other people.

 

[neurocomp2003]

what makes a language a language? can we include pictographic languages? Can i let "A" represent the action of "punching"? infact aren't all languages mathematical languages? after all isn't math a "tool" about symbols and when you concatenate sets of symbols(alphabetic or pictorial) you create strings then you define certain strings/2D images to create "words" and sequence words to create sentences and define rules upon which only certain combination of words are coherent or allowed. And then define more symbols to separate sentences. And define more structures to organize what you have written.

Note: Every word I used above can be defined by a symbol(but I'm not very good with LaTeX).

 

[mathwonk]

mathematics employs its own speciaL language, but to say it is a language seems wrong to me. i cannot imagine a mathematician, i.e. anyone who has ever discovered any mathematics, saying that.

such a universal statement as mine is always incorrect, but it would still surprize me.

ideas can be expressed in many languages. i once read a book of linguistics where the author asserted that thioguht is impossible without words. i thought, this is a person who has never had an abstract thought.

later perusing the book Psychology iof invention in the mathematical field, by the great mathematician jacques hadamard, confirmed my opinion that abstract creative thought is often done independently of words, and it was confirmed in the cases and testimony of mozart, einstein, poincare, and others,..i.e. in that haze of shapes and forms in the brain from which mathematical discoveries crystallize, there is no discernible language with which to communicate the ideas to others. But there is mathematics being done there.

of course one cannot know this without having done some mathematical thinking.

 

[mathwonk]

hadamard made it a rule for compiling his testimonies that only real mathematicians, and outstanding ones at that, should be consulted for their views. which of course let's me out.

 

[mathwonk]

by the way deadwolfe, didi ans wer your question aBOUT THESIS RPOBLEMS, or did i misunderstand it?

 

[physicsmasta]

wow that sux

 

[mathwonk]

deadwolfe i assumed you meant to ask "what were those thesis problems?". but maybe not.

 

[neurocomp2003]

but mathwonk...from what i remmeber in psychology mathematics/pictographic languages/spatial manipulations all reside on the same side of the brain or so the experiments say.

What is a language if not a form of communication? with less rules and with more sounds

 

[Office_Shredder]

Ok, here we go:

|-|3110

It looks like leet-speak, but it's really all math

| probability of A given B

- minus sign

and the rest are numbers

 

[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?