An argument for “meta-mathematics”

by Luke Wolcott

Definition: meta-mathematics – the study of the human context of research mathematics. Includes the sociology of the math community, the psychology and cognitive science of mathematics, and implicitly the history and philosophy of math. (Yes, I’m co-opting the term.)

Have you ever wanted to get inside the head of one of those “brilliant” mathematicians? Did you know that the mathematician Jacques Hadamard conducted a survey of well-known mathematicians – asking them specific questions about how they do math – and carefully analyzed the responses?

Have you ever stopped to think about beauty in mathematics? Did you know Thomas Tymoczko wrote an article exploring the possibility of a theory of mathematical aesthetics?

Have you ever contemplated the mathematician’s place in society – for example, the communication barrier that exists between ‘us’ and ‘them’? Did you know about the harsh critique of mathematical elitism that the poet H.M. Enzensberger wrote and delivered to the International Congress of Mathematicians in 1998?

I think we all have moments when we step back and reflect on the mathematical experience. But not everyone knows about the (granted, quite rare) books and articles written by sociologists, cognitive scientists, etc, who study the math culture and the minds of mathematicians. Or about the thoughtfully-written personal accounts of self-reflecting masters of math.

I’ve started compiling a meta-mathematics reference list on my blog, including short descriptions and a nifty classification table. Everything there has had an impact on how I think about and do math. Check it out, and please tell me what I should add.

I know, I know: you don’t have time to read these kinds of things. They’re interesting, but not relevant to getting a PhD.

But the above examples are all articles that could be skimmed in the time it takes to proctor a calculus exam. As for the fascinating books, that take much more commitment, I think it’s still a worthwhile undertaking. It’s not a zero-sum game. Reading meta-mathematics will make you a better, more effective, and possibly happier, researcher and teacher.

How? The more you are aware of and understand your mathematical process, the more productive and efficient you’ll be, I think. For example, go ahead and read Bill Thurston describing his process. Then ask yourself, “How do I do math? How do I do my best math?” Such contemplation, I think, on the individual level and in our community as a whole, will produce a higher yield of our best, and most beautiful, mathematics.

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