I’m a lapsed mathematician but I spent the evening of new year’s eve 2009 with a group of practising mathematicians. I was struck, yet again, by their varied interests and, more than anything else, by their readiness to engage with ideas, to explore concepts, turn them over and, even in casual conversation, to explore a range of possibilities. They have the habit of disagreeing without rancour. (Disagreeing like this is an effective way to exchange ideas and loosen prejudices.)

Mathematicians seem to share ideas and engage with ideas more often and more routinely than many scientists. (Pure mathematicians are not scientists in the normal sense because their central concern isn’t experimentally disprovable hypotheses about the physical world.) Mathematicians certainly have far broader interests than most engineers and other technical people. Social conversations among mathematicians may seem like those among traditional academics in the liberal arts until you realise that hard sciences are included on the list of topics not to gloss over, disparage or trivialise. In fact, mathematicians sometimes have a quite shocking ability to engage intellectually with diverse topics, including topics that seem at first to be utterly banal!

Mathematicians are more like composers (musicians) than scientists. Professionally, they are engaged in making structures in a cultural context—not structures of sounds in time, but structures of abstract ideas—with an emphasis on extended chains of closely-connected ideas where the linear dominates because of the underlying logical rules, just as the linear dominates music because of its underlying temporal base. This much is clear enough. What is often less clear is the way that the underlying rules of logic (the rules of argument, what is accepted as proof) are, in part, culturally dependant in the same way as the culturally-dependant, unwritten rules about what it takes for a sequence of sounds to be accepted as music.