Alan J. Cain

This ebook is an annotated edition of G. H. Hardy’s _A Mathematician’s Apology_. It aims to supply sources for all quotations and to clarify allusions to works, people, or events, as well as to give background information. Hardy made a number of minor misquotations, suggesting that he quoted from memory or used paraphrased notes of his own; the annotations point these out. This edition also includes an annotated version of Hardy’s essay ‘Mathematics in war-time’, which formed the kernel around which he shaped the _Apology_. The annotations point out how parts of this essay were incorporated into the _Apology_. Also included is a list of editions, reprintings, and excerpts of the _Apology_ and ‘Mathematics in war-time’, and three essays by the annotator: the first sets the _Apology_ in context in the debate about the justification for mathematics, particularly as an aesthetic pursuit; the second attempts to survey comprehensively contemporary reviews of the _Apology_; the third examines the legacy and ongoing influence of the _Apology_.

This paper studies how spatial thinking interacts with simplicity in [informal] proof, by analysing a set of example proofs mainly concerned with Ferrers diagrams (visual representations of partitions of integers) and comparing them to proofs that do not use spatial thinking. The analysis shows that using diagrams and spatial thinking can contribute to simplicity by (for example) avoiding technical calculations, division into cases, and induction, and creating a more surveyable and explanatory proof (both of which are connected to simplicity). In response to one part of Hilbert's 24th problem, the area between two proofs is explored in one example, showing that between a proof that uses spatial reasoning and one that does not, there is a proof that is less simple yet more impure than either. This has implications for the supposed simplicity of impure proofs. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.