A concise history of mathematics, but not for philosophers
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A concise history of mathematics, but not for philosophers John Stillwell: A concise history of mathematics for philosophers. Cambridge: Cambridge University Press, 2019. 69pp, £15. Fenner Stanley Tanswell1
© Springer Nature B.V. 2020
This short contribution to the Cambridge Elements series on the philosophy of mathematics offers a 65-page summary of the history of mathematics. The nine chapters cover Ancient Greek mathematics (Chapters 1 and 2), imaginary numbers (Chapter 3), the development of analysis (Chapters 4 and 5), non-Euclidean geometry (Chapter 6), set theory (Chapter 7), and formal systems and their limits (Chapters 8 and 9). In such a short space, these chapters do not and cannot go into any great depth, but do cover the major points on these topics. Each chapter begins with a preview of the upcoming pages and ends with a section on philosophical issues that are raised by the history. The title proclaims that this book is aimed at philosophers, and the author suggests that he is giving an account of “philosophically instructive mathematics” (1). The main sign of this is that every chapter ends with these philosophical issues, which are divided into the three themes of “Intuition and Logic”, “Meaning and Existence”, and “Continuous and Discrete”. These themes act to weave three main threads of philosophy into the historical story that is told. Stillwell has many previous books on the history of mathematics, in particular the general volumes (Stillwell 1989, 2016). The current work acts as a dramatically shortened version of these longer books. In particular, Stillwell uses many of the same examples, even with the same diagrams to illustrate them, as in the 2016 book. The advantage of distilling the book down into only major milestones in the history of mathematics is that this leads to shorter readings that students are more likely to actually do. The disadvantage is that many important things are left out, including some of the more pedagogical material that would make the topics more accessible for students that do not already know them. Let me give a couple of examples of important things directly relevant to the book’s contents that are glossed over too briefly. First of all, higher levels of infinity are alluded to and even have a section, Section 7.3, dedicated to them. In this * Fenner Stanley Tanswell [email protected] 1
Mathematics Education Centre, Loughborough University, Loughborough, UK
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section, the author discusses how the diagonal argument can be generalised and that it also shows the impossibility of a set of all sets. However, the book never quite makes the idea of constructing bigger and bigger cardinals explicit, nor does it contain any discussion of large cardinals, something which is a major topic in the overlap of mathematics and philosophy. Secondly, the Church-Turing Thesis is discussed in Chapter 9, but is never stated or given in a way that would allow someone to come to learn what it means. The odd claim that it “can in princip
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