Mathematics is not (only) a language

Posted by: Gary Ernest Davis on: January 9, 2010

In: Uncategorized
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Every so often someone will assert that mathematics is a language.

What Â concerns me about the statement “mathematics is a a language” Â is what I am not hearing directly: “mathematics is ONLY a language” Â which I worry is implicit.

For example in an interview, Paul Samuelson, Â said:

Well, my belief is that nothing that can be expressed by mathematics cannot be expressed by careful use of literary words. All the mathematics itself originally grew out of axioms — assumptions stated in words — and [it reminds me of] the only speech that the great Willard Gibbs, at Yale, ever gave at a faculty meeting, when there was a debate between studying languages and mathematics. He said four words: “mathematics IS a language.” And that’s on the frontispiece of my Foundations of Economic Analysis, but I’ve written somewhere that he should have cut it down to three words: “Mathematics is language.” It’s just a very compact form of language.

So there we have it from the winner of the 1970Â Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel himself: Mathematics is just a very compact from of language.

This view of mathematics – that it is just a language for expressing something about another topic of interest, such as physics, economics, chemistry, sociology – Â is common among users of applied mathematics For such people mathematics is not their major concern. Rightly so – their focus of attention is on physics, or chemistry, or economics. If they could avoid mathematics altogether they probably would.

But what about people who are mathematicians: are they really just a bunch of linguists, getting together to discuss the language of mathematics? Wouldn’t you think that the people whose life is mathematics might know if mathematics is just a language?

My own view is that mathematics is a science – it is Â a science of number, space, pattern and arrangement.

If this view has any validity then mathematics ought to have well established facts. Does it? You bet it does, and one of the great grand-daddy facts of mathematics is the prime number theorem.

The prime number theorem is not “just language”. It is a deep fact about the way prime numbers are spaced out in the collection of whole numbers. It was discovered empirically by Gauss and Legendre, and proved to be true by Jacques Â Hadamard.

The prime number theorem asserts that the $n^{th}$ prime number is approximately equal to $n \log (n)$ , with the relative error of this approximation approaching 0 as $n$ Â increases without bound. Â Here $\log (n) =\int_1^n\frac{1}{x}dx$ is the natural logarithm of $n$ : Â the area under the curve $y=\frac{1}{x}$ from $1 \textrm{ to } n$ .

Alain Connes has asserted that we know such facts about mathematical reality better than we know any fact about physical reality. He writes:

“The scientific life of mathematicians can be pictured as a trip inside the geography of the mathematical reality which they unveil gradually in their own private mental frame.” [A_view_of_mathematics]

Connes has even gone so far as to assert:

Is this an isolated example, essentially the only real fact in mathematics. No, to the contrary, mathematics is full of mathematical facts.

Here’s another simple, but beautiful fact discovered by Euler and overlooked by the ancient Greeks in their study of polyhedra. A polyhedron has a number of vertices (v), edges (e) and faces (f). Euler discovered, and proved, that v-e+f=2. Â Is this just a language statement. No: it is a provable, but non-obvious, fact about plane graphs which applies to the graphs of Â polyhedra.

Dave Richerson has a beautiful account of this lovely finding of Euler in his book Euler’s Gem. If you still think that mathematics is (only) a language, do yourself, and me, a favor by getting hold of this book and reading it with pleasure. It is a beautifully written book, that tells what the science of topology is about.

So mathematics is a subject with deep facts, and is not just a language for expressing facts about some other topic.

Where, then does the idea that mathematics is just a language come from? I have already said that users of mathematics in other fields often see mathematics this way. But where did they learn that? My view is that school mathematics by and large promotes this idea that mathematics is just a language, because school mathematics is, by and large, devoid of mathematical facts. School mathematics is, in the main, a collection of recipes and methods for doing … who knows what.

It’s time, in my view, for an infusion of facts and mathematical science into school mathematics. Â Then we might get kids loving mathematics, as we might if we read Dave Richerson’s lovely book.