Josiah Willard Gibbs is a name likely to be unfamiliar to many people today. I must be honest and tell you that I myself wasn’t too familiar with it, and needed a couple of searches on Google to help me out.
I was familiar, though, with a quote that he is responsible for. It is a quote that appears in a book that I am not particularly fond of, and we will get into my reasons for my dislike in a bit. Here is the quote though, in all of its four word glory:
“Mathematics is a language”
I was reminded of the quote when I read Dilip D’Souza’s lovely little rumination on just this question: whether mathematics ought to be considered a language or not. Dilip, in his essay, focuses on whether mathematics was invented or discovered, and says it is perhaps a bit of both. As an amateur student of mathematics, I cannot answer the question definitively one way or the other, and I’m happy to go along with his best guess.
But to go back to Dilip’s original question: is mathematics a language? Well, if it is one, it certainly isn’t one of the easier ones. Back in college, if you had given me the choice to study irregular verb conjugations in French or learn mathematics better, I would undoubtedly have chosen the former.
But over time, my attitude towards mathematics has changed, and I think for the better. The older I get, the more I am inclined to agree with Gibbs: it really is a language, and a beautiful one at that.
It is a language with beauty, as Euler’s identity makes abundantly clear. It is a language pregnant with mystery. And not just the kind of mystery that one associates with school-time tribulations! Try finding out what the sum of all natural numbers is, for example. And then wonder how Ramanujan thought the answer to be self-evident. If that doesn’t strike you as a mystery, nothing will.
And as with all things mysterious and beautiful, the more time you spend with mathematics, the more mysterious and beautiful it becomes. Dilip hints at the wonder that is Bolyai and Reimann geometry in his essay, and both are worth learning more about. But why stop there? Once you find yourself going down the rabbit-hole of discovering mathematics for its own sake, you find yourself in a wonderland that makes Alice’s look positively quotidian.
But as with all languages, so also with mathematics. Your love and affinity for it is very much a function of the way you were introduced to it. If your introduction to it was through dull and dreary classroom exercises, carefully designed to suck every single trace of fun out of the experience, then you are unlikely to have fallen in love with it.
And that, I suspect, is the case with most of us. Myself included, to be clear. But I was lucky. I was reintroduced to the subject by a professor of mine, who introduced to me the beauty that lies in wait beneath the seemingly impenetrable surface of the subject. And over the years, I have fallen in love with the language.
Noam Chomsky is famous for hypothesizing that we are all born with an innate ability to understand language – any language. Now, some languages may well be more difficult than others, but the more time I’ve spent with the subject, the more I have come to believe that Chomsky may well have a point. Mathematicians such as Steven Strogatz, Edward Frenkel and Grant Sanderson have helped me appreciate the language more.
And while I may never be able to compose even a limerick in this language, let alone author a magnum opus, the more I learn about it, the more I am able to appreciate the works of those who have gained some mastery over it.
A brief coda:
Gibbs’s quote, with which I started this essay, comes as an epigraph in a book that I don’t particularly like. As any economist of a certain age will tell you, that book is The Foundations of Economic Analysis, written by Paul Samuelson.
The reason I do not like it is because I think the book helped push the study of economics a little bit too far in the direction of mathematical analysis for its own sake. Mathematics helps make economic analysis more tractable, and more amenable to logical analysis, but it is the dose that makes the poison. The study of economics may have become more tractable because of mathematics, but in my opinion, the mathematical formalism has been taken too far.
As the wise Kenneth Boulding put it: “”Mathematics brought rigor to economics. Unfortunately, it also brought mortis.”
But make no mistake, this is at best a criticism of how economics as a field has developed over the years, if that. Samuelson himself admitted as much, and it is a sign of the greatness of the man that he did it in 1952(!). 1
For mathematics, today, I have nothing but gentle love and affection, and above all, a sense of awe and wonder.
And I wish and hope for more of the same for you!
Economic Theory and Mathematics–An Appraisal Paul A. Samuelson The American Economic Review Vol. 42, No. 2, Papers and Proceedings of the Sixty-fourth Annual Meeting of the American Economic Association (May, 1952), pp. 56-66 (11 pages).