math and language, again

Now that my opinion of math-as-language is challenged (I say math is not a language, and JdFisher says it is, but neither of us really knows), I've had to go out on the internet to see what kind of stuff people have said about it.

One of the first papers I came across was on this very topic: this one by Tony Brown. Brown is perhaps a student of Jacques Derrida, the well-known and recently deceased deconstructionist philosopher. I include a snippet from this paper for your amusement:

Whilst Mason's distinction might offer a valuable rhetorical device in initiating or analysing mathematical performance, such a distinction suppresses the historicity endemic in anything commonly recognised as mathematical performance, or even mathematics itself, and thus obscures the values associated with this (cf. Derrida, 1989). In particular, the linguistic forces driving (and being driven by) mathematical constructing get squeezed out of the picture. Mathematical constructing, I would suggest, is always linguistic to a degree, oscillating in a hermeneutic circle, between more or less sturdy linguistic frames.

OK, so there you have the philosopher's perspective (in an entirely new language, I think).

Here's a column by Edward Willett, and he perfectly expresses where people are coming from when they claim math is a language.

In fact, elements of these arts can sometimes be set out in mathematical terms--no one who has listened to Bach can doubt the mathematical underpinnings of music, and Leonardo da Vinci even wrote a mathematical treatise on the depiction of perspective in paintings.

The fact that he could do so demonstrates that mathematics is also a language, which, like other languages, uses agreed-upon symbols and grammar to describe objects and relationships. Using nouns (constants), pronouns (variables) and verbs (operations), you construct sentences (equations), which build upon each other to create whole paragraphs and even books.

But I'm not quite convinced, though I agree with each of his items in the second paragraph. I've always thought Bach had a style that was reminiscent of math -- but his music itself isn't math, it's music. They're not the same. The presence of nouns and verbs alone doesn't qualify mathematics as a full-blown language, either.

One of the most useful items I came across in my search was this article, "Mathematics as a Language". This guy (and I couldn't figure out who the author was -- I think it might have been Alex Bogomolny, but I'm not sure) has a different perspective on the math-as-language argument:

When I think of the development of Mathematics over the last 2500 years, I am less surprised that early mathematicians left lasting results than that, given the tools they possessed, they achieved anything at all that could have lived through centuries.

The really great thing about his article is a table that he presents about 4 or 5 paragraphs down, comparing the descriptions of the great mathematical results (expressed by their inventors in their native languages, and translated for the purposes of the table) with their expressions in mathematical notation.

For example, contrast this statement by Euclid:

If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments. (Euclid, Elements, II.4, 300B.C.)

with the same thing in mathematical notation: (a + b)^2 = a^2 + b^2 + 2ab.

(Note: ^2 means 'squared').

Several other examples are given, and they're impressive: I suggest you have a look. He also concludes that mathematics is a language, but his examples (it seems to me) can actually be used to disprove it! They suggest that mathematics is actually a shorthand because, after all, the ancients were able to express their ideas in their own language.

OK, I'm going out on a limb to try to prove a point that a. I'm not sure about myself, and b. I'm not really equipped to prove. I still don't think math is a language, but I think mathematical notation might be a sort of a simple or pidgin language.

But the amazing thing about mathematical notation is that, once you've expressed your idea (that is, once you've translated the word problem), you can take off in a completely different direction and solve the equation by manipulating its components. There's really no comparable way to take the building blocks of language and reorganize them until the answer pops out. If mathematical notation is a language, then the addition of that symbol-manipulation capability makes math a lot more than just a language.

And now I would like Steven Pinker or Noam Chomsky to wade in and sort this out for us.


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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