Hamkins packs an immense amount in to Lectures on the Philosophy of Mathematics, but manages to provide a kind of structure to it all, not with an overarching theme or narrative but with many smaller stories that cohere. There is effective chunking into subsections, sections and chapters, with an order to the latter which keeps the more accessible material to the front: "Numbers, "Rigor", "Infinity", "Geometry", "Proof", "Computability", "Incompleteness", and "Set Theory". And smaller parts are often usable in isolation — the section on "Complexity", for example, contains a reasonably self-contained half page on "Feasibility as polynomial-time computation" — and a proper index (which includes notation) makes it easy to find particular topics.

Hamkins writes exceptionally clearly and manages to give robust but succinct explanations for often complex ideas. There are fairly extensive exercises accompanying each chapter, which are not rigorous but not completely informal either: just understanding them is useful reinforcement, but thinking through them forces a real engagement with the ideas involved.

One proviso to all of this is that Lectures is very much written for mathematicians. It doesn't assume much specific knowledge — and those unfamiliar with topics such as category theory can skip over references to those easily enough — but anyone without a general familiarity and comfort with mathematics is going to be floundering. (It is based on lectures for an Oxford University honours course.) The focus is also on material of current interest to mathematicians, with context provided to understand that but without any attempt to give a general history of the philosophy of mathematics or to explore the broader influence of mathematics on philosophy.

One notable feature of Lectures in the Philosophy of Mathematics is its outstanding design and layout. It is both easy to use and attractive, with occasional colour diagrams used to good purpose. And there are no typographical or other errors, at least that I picked up.

September 2021

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