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math and language

I was talking to a friend of mine the other day (okay, okay, it was Catherine). She had been to a party at the French Embassy in Washington, and got talking with a French gentleman there about mathematics.

"Mathematics is a language," he told her.

"It is?" she said.

"Furthermore, it is a dead language," he went on (I bet he says this sort of thing to all the women he meets at parties).

Later, Catherine asked me if I think that mathematics is a language. I've been hearing this sort of thing for years, and had never really thought too deeply about it. I don't even know exactly what qualifies something to be a language. But Catherine asked me, and Catherine doesn't ask questions lightly, so I gave it some thought.

What is a language, exactly? I, for one, certainly don't know. But whatever a language is, I thought, it ought to be able to stand on its own. The jargon of a specialized field shouldn't count as a language by itself; it's just jargon. If you take away the English or German or French or Chinese words that support the jargon, the jargon doesn't stand on its own. And that's how it is with mathematics.

You can write out some simple proofs, of course, without using any words at all. But you can only write out the simplest arguments that way, basically those that follow directly from manipulating expressions and equations. I think a language should be able to support all sorts of complicated ideas from all walks of life; if mathematics is a language, then it's a pretty limited one.

"No", I said, "I don't think it is."

When I mentioned this later to Bernie, he pointed this article out to me. These research results suggest that mathematics and language are pretty much independent functions, as far as our brain functioning is concerned.

I would guess that the intersection of language and math occurs at word problems. Word problems are very hard, perhaps because we do have to integrate totally different functions in our brains in order to understand them. But turning word problems into algebraic expressions isn't translation; it's distillation. A lot of the meaning in the original word problem is left behind in the process; the names of the kids who exchanged marbles, the fact that it was marbles that were exchanged, and so forth. You can't work backward from the algebraic expression and uniquely reconstruct the word problem as it was originally.

So I think Catherine's French acquaintance was wrong. If anyone knows any more on the topic of what language is than I do, and if mathematics actually qualifies as a language, I'd like to hear about it.

What Counts: How Every Brain is Hardwired for Math, by Brian Butterworth
The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene
Children's Mathematical Development: Research and Practical Applications by David C. Geary
(fyi: It is possible to buy Geary's book for far less than the $124 Amazon wants for it, or the $55 I paid for a used & extensively highlighted copy...)

Carolyn on math and language 7-2-05
Carolyn on math and language again 7-3-05
"the language of numbers is not language" 7-3-05

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I was gobsmacked when our acquaintance told me math was a dead language--just like Latin, he said.

Later on he cautioned me against investing too much time and energy in thinking about math, or learning math, or studying math books, because, he said, 'math isn't the future.'

Then he said, 'Your husband is the future.'

Neither of us had a clue what this meant.

Ed is a historian, and I would hate to think historians would ever disappear. Since September 11 I've become aware, for the first time, that what historians do is different from what the rest of us think they do. Ed was telling me that history is a 'discipline,' and I'm starting to see what that means. Historians have rules for evidence, etc.....they're not just writing stuff down in sequence!

OK, that was a tangent.

The point is, math has been around for a very long time, and history has been around for a very long time, and thus far no one's noticed any big conflict.

Why would history be the future any more than math?

He may not have been referring to Ed's profession, but to Ed's involvement with transnational....transnational something-or-others.

The man's wife has been the French liaison to U.S. schools; she promotes the study of French language, history, and culture. Essentially, her job is public diplomacy.

Ed, because he is the Director of the Institute for French Studies at NYU, has been part of that. It's his job to teach college & graduate students French history.

So.....I guess the man may have been seeing Ed as some kind of transnational vanguard.

Though how anyone's going to earn a living from that, I don't know.

Scratch that.

How more than a tiny few are going to earn a living from that, I don't know.

I think there are going to be many more jobs for software designers & economists down the line than for directors of institutes whose mission is to study other countries.

I'm betting on math.

-- CatherineJohnson - 02 Jul 2005

The other strange thing was that there are 3 separate very elite colleges in France: one in science, engineering and math; one in finance and economics; one in literature and the arts.

(They have a system I don't understand, in which everyone can go to college (?), but this has resulted in the colleges being terrible. There are 3 elite schools the elite attend in order to bypass the colleges everyone else goes to. The students who make it into the elite schools don't necessarily go into the field they study. What they want is the elite degree and position in society that the school gives them. I'm probably garbling this, but it's not wildly wrong.)

His son had, I think, just been accepted into one of the three.

Science & engineering.

-- CatherineJohnson - 02 Jul 2005

I love this topic, which shouldn't surprise anyone who knows me. And to the question, Is math a language? I always answer 'yes.'

Now, if you ask me whether or not I think mathematics is a science or a philosophy, I would answer, without a doubt, 'philosophy.'

But anyway, back to language. The first idea here is that a language, to be a language, need not be comprehensive or robustly expressive. The communications of whales, bees, and some very primitive tribal cultures come to mind as possible examples of very limited, ultra-practical languages. But even if we would not accept these as languages, we can imagine, for argument's sake, an object that we would refer to as a language that is extremely limited in its expressiveness.

Should mathematics be discounted as a language because it derives most of its "language" from other languages? Two thoughts on this: (1) If we took everything French out of English, it would be impossible to even say . . . "impossible." (2) An important question needed to decide this is, Is it possible to understand mathematics without language? If so--if mathematics is truly extra-linguistic--than the criterion does not apply.

Anyway, I'm certainly not an expert on the ins and outs of these questions. My sense that mathematics is a language comes from my reflections on the topic in light of my experience.

In my youth I spent some time abroad--East Germany--and picked up the language. If you talk to anyone who knows two or more languages, I think they will agree: Many things are simply untranslatable. I often find myself, to this day, substituting German words into my English sentences (especially the word erklären), because they serve better the sense I'm trying to communicate.

Mathematics is like another language for me, because, while it is very limited in what it discusses, it sees the world in a way that is almost impossible to see if you don't know the language.

-- JdFisher - 03 Jul 2005

Well, it sounds as though we'll need an expert to decide this issue. There is a definite, worked-out definition of language, and I just don't know whether mathematics qualifies.

Actually, I believe that almost everything expressible in math is also expressible in English. It's just not as easy to manipulate symbolically.

But even if math is a language -- I don't believe for a minute that it's a dead one!

-- CarolynJohnston - 03 Jul 2005

I would say that algebra is the language of (most) math. (See, I'm hedging already.) Algebra, like language, is of little use all by itself, although some like to talk to themselves. You can use algebra, like a language, to transmit ideas and come to conclusions. Without algebra, much of math would be impossible - well, there is always guess and check. Oh, never mind. Where are my sparklers. I'm going out to chase fireflys.

-- SteveH - 03 Jul 2005

Why does my watch say July 2 and the blog say 03 Jul 2005? Is it in a European time zone?

-- SteveH - 03 Jul 2005

I've been wondering that very thing myself for quite a while. My best guess is that TWiki (the program this blikiwikiblog runs on) is putting everything in Greenwich Mean Time.

-- CarolynJohnston - 03 Jul 2005

Why does my watch say July 2 and the blog say 03 Jul 2005? Is it in a European time zone?

OK, now I'm freaking out.

-- CatherineJohnson - 03 Jul 2005

Title: math and language
TopicType: WebLog
SubjectArea: EducationResearch
LogDate: 200507012334