I read and reread proofs in much the same way as I listen to a piece of music: understanding how themes are established, mutated, interwoven and transformed. It is the journey that excites the mathematician. Just as music is not about reaching the final chord, mathematics is about more than just the result. It was a realisation that linked the seemingly separate worlds of primes and squares. Take the discovery by Fermat, the 17th-century French mathematician, that a prime number that has a remainder of 1 after division by 4 (like 41) can always be written as the sum of two square numbers (41=16+25). As we've explored the universe of numbers, we've discovered strange connections and stories about numbers that excite and surprise us. Mathematics is about structure and pattern. Just as notes and rhythms are not all there is to music, so arithmetic and counting are not all there is to mathematics. But, as a professor of mathematics, I think this argument misunderstands what mathematics is truly about. Many people react angrily to such a claim, believing music to be so much richer and more emotional than mathematics, and that to make such a comparison is to misunderstand what music is truly about. And it is in putting the notes together to create, say, the Goldberg Variations or Don Giovanni that I believe the true connection between mathematics and music reveals itself.
Yet, while the combinations of notes we have been drawn to over the centuries can all be explained through numbers, music is more than just notes and beats – just as Shakespeare is more than just words from a dictionary. (This is because the frequencies of the two notes are in an exact 1:2 ratio.) When we hear two notes an octave apart, we feel we're hearing the same note, so much so that we give them the same name. So is there really a link? Or is it crazy to try to connect the creative art of music with the steely logic of mathematics? Certainly the grammar of music – rhythm and pitch – has mathematical foundations.
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