Timothy Gowers

This paper considers the reasons mathematicians give for making probabilistic judgments about unproved mathematical statements, and discusses how one might interpret and justify such judgments more formally. Following Pólya, I argue that we update our probabilistic judgments in a broadly Bayesian way, while to explain what they mean in the first place, I argue that they are referring not so much to the truth of the statements as to the likely existence or otherwise of reasons for them. The link between the two is provided by a "no-miracle" principle, which says that a surprising mathematical statement will not be true unless it is true for a reason. This principle applies only to statements that are sufficiently natural, so the paper also sets out criteria for a statement to be more or less natural.

Mathematics is a subject we are all exposed to in our daily lives, but one which many of us fear. In this introduction, Timothy Gowers elucidates the most fundamental differences, which are primarily philosophical, between advanced mathematics and what we learn at school, so that one emerges with a clearer understanding of such paradoxical-sounding concepts as 'infinity', 'curved space', and 'imaginary numbers'. From basic ideas, through to philosophical queries, to common sociological questions about the mathematical community, this book unravels some of the mysteries of space and numbers.