Don’t worry, this post is not about factors irrelevant to mathematics such as gender, ethnicity and sexual orientation.
Nor is it about important factors, highlighted in the 2011 International Review of UK Mathematics although largely ignored by the research council that commissioned it: research area, size of research group, and size of institution.
But I discovered recently a diversity in the way we do mathematics, which I found surprising and potentially significant.
A few weeks ago, I was at dinner with a visiting colloquium speaker. The conversation turned to whether mathematical thought is done in words, or is “pre-linguistic”.
This is a topic about which Jacques Hadamard, in his book originally called The Psychology of Invention in the Mathematical Field but re-published as The Mathematician’s Mind, had a lot to say. Some linguists and linguistic philosophers, notably Max Müller, insist that language is essential to thought, and that no thoughts can be pre-linguistic. Hadamard, from his own intuition and from the writings of others from Poincaré to Einstein, is convinved that this is not the case, and is bewildered that Müller can hold this view with such vehemence. In a footnote, he says,
I have also seen the following topic (a deplorable subject, as far as I can judge) proposed for an examination—an elementary one, the “baccalauréat”—in philosophy in Paris: “To show that language is as necessary for us to think as it is to communicate our thoughts.”
For me, I know for sure that my best insights (those which are not just routine calculations) are pre-linguistic, and I struggle to put them into words: similarly, if the insight is a conjecture, I struggle to see how the conjecture might be proved. I assumed that most mathematicians would be like me, and would agree with Hadamard rather than Müller.
So it was a bit of a surprise when, of the five research mathematicians at the table, we were split 3 to 2 in Hadamard’s favour.
This is of course an anecdote, and not survey data. But we noticed a curious thing. The two who said they did mathematics in words had something probably significant in common: their cradle tongue (in both cases, a Slavic language) was not the language in which they do mathematics (in both cases, English); and both of them had learnt English at a comparatively advanced age. The other three of us were all native English speakers.
Not sure what to make of this. But I am glad that it drove me back to Hadamard’s book. I had completely forgotten that, at a certain point, he admits to his failure to be able to think creatively about group theory!