Near the beginning of this relaxed and illuminating set of reflections on the life of a professional mathematician, Michael Harris recalls an early encounter with an academic adviser at Princeton. The deep but distant voice of the professor cut through swathes of information about courses and textbooks and confronted Harris with an awkward home truth that ‘You want to be the world’s greatest mathematician.’ Harris was sixteen at the time and was astonished by the professor’s X-ray insight into the secrets of his soul. As he grew older, however, he realised that his hectic ambition was not so much a private adolescent fantasy as an essential element of mathematics as a field of modern scientific research. 

Once upon a time, mathematics was like any other discipline: you began with a few elements that might appeal to any moderately alert child – counting everyday objects, calculating with whole numbers and fractions, constructing circles and triangles with rulers and compasses – and even when you moved on to