NB. I was sent this book as a review copy.

The last book of Stillwell’s that I reviewed was Reverse Mathematics which was utterly fascinating, and truly mind-bending. I was very much looking forward to another of his books, and this one did not disappoint. It is a much less alternative perspective on mathematics than the previous, but no less beautifully written or compelling.

I teach pure mathematics to first year undergraduates (amongst others), and so often find that the very concept of a mathematical proof is something that is so hard to grasp. What is sufficient to concretely prove something? What can be assumed? What sort of proof is appropriate within a given context? High school maths generally sets students up very badly in this realm.

Stillwell’s book on the Story of Proof is perhaps a little beyond what could be grasped easily by most first year students, though very keen ones, with patience could certainly make their way through it, and would benefit enormously from doing so.

The book starts with the origins of proof from Chinese, Indian, Islamic, Greek and Roman roots, taking us on the journey through our understanding of proof within geometry (mostly) from pre-Euclid to post-Euclid. It builds things up beautifully, going into detail into the real meanings of definitions, theorems and proofs which students often fail to distinguish at a deep intuitive level. These first chapters are filled with how axiomatic systems are built up, and what can be drawn from them, going through mostly easy to understand theorems in geometry.

From geometry the book moves naturally into algebra, where the transition to the abstract is dealt with very beautifully. With this approach, a keen student with a supply of pen and paper at hand who hasn’t come across Fields, Rings and Groups would still be able to follow these topics.

One of the highlights of the book was the historical building of proofs over time, showing how a proof would be written, which may have had holes in it, which would be filled by a later proof coming at the theorem from a different angle. To any first year studying calculus, the discussion around the Fundamental Theorem of Algebra should be easy to follow, seeing the gaps allows one to see the subtlety inherent in what they may have found either seemingly obvious, or baffling when they first came across it.

The role of logical rules with which to formalise the proving of concepts, and the computational tools with which to do so is brought together, and that, Stillwell points out, is one of the aims of the book – to show how logic, computation and abstraction come together in the world of proof. This in particular comes together as Stillwell moves into the latter parts of the book on Set Theory and then Propositional Logic

The book is structured through mathematical topics, including proofs in Calculus, Topology, Algebraic Number Theory, Set Theory and more. The aim of this is to highlight the different methods of proof which are used in each area. I appreciated this approach, however, while one picks up these differences through the examples, I think that having a chapter really highlighting these differences in one place may have also been insightful. While there are different toolkits needed for different areas of mathematics, a student often struggles with really understanding the ontology of such tools. The keen student mentioned before may like to build up this ontology as they are reading through the book. The more we can understand the tools at our disposal, the more comfortable we can feel within the landscape of mathematical proof.  To a large extent, the chapter comparing the different axiomatic systems for numbers, geometry and sets gives a nice starting point at least within a confined domain (in the colloquial sense).

Overall, I hugely enjoyed this book and learned a massive amount from it, both from topics that I thought that I knew well, to topics that I had less familiarity with. This book would be perfect for any keen undergraduate, keen amateur, or indeed a teacher of mathematics, who wants a book to dip into to use for the classroom.

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