This is a wide-ranging collection of essays on the intersection of mathematics with other fields of knowledge, especially philosophy. The content is primarily not mathematical, although it includes a lot of mathematics. The back-cover blurb says “The book is directed towards the math buffs of the world and, more generally, towards the literate and interested public.” It’s not a math book, or a popular-math book; it’s really about “how do we know things?”, where some of the things are mathematical and some are in other fields where math can help. It’s a very erudite book and not for the general public.

The author, still active today at 92 years old, is well known for his work in numerical analysis and approximation theory, and for his popular-math works such as *The Mathematical Experience* (with Reuben Hersh). The essays were all written in the last few years and many have been published previously, in a variety of markets, so they are in all different formats and organizations.

My favorite essays:

- “Evidence in Mathematics”: Proof is certainly the gold standard in mathematics, but Davis points out that in practice mathematicians believe and use many claims for which they have never examined the proofs; he covers many reasons why mathematicians believe things, without suggesting an “end of proof”.
- “Spengler’s Mathematics Considered and a Phoenix Reborn?”: The reference is to the chapter “The Meaning of Numbers” in Oswald Spengler’s famous and controversial 1918 work
*The Decline of the West*. Davis points out that, despite the book’s fame, no professional mathematician has ever critiqued Spengler’s view, and he proceeds to do so here. James R. Newman (a lawyer, not a mathematician) did reprint the chapter in his anthology *The World of Mathematics*, along with a commentary. Davis’s treatment is very thorough and he approaches Spengler’s ideas from many viewpoints.
- “The Media and Mathematics Look at Each Other”: To some extent this rehashes familiar complaints that the media don’t cover mathematics, and when they do it is as human-interest stories casting mathematicians as mad scientists. But it ends with some positive suggestions for improvement, and correctly points out that (for example) newspapers do present a large amount of educational and intricate material, such as cooking recipes and detailed weather reports. Couldn’t they do something for mathematics?
- “Can the Mathematical/Physical Notions of Entropy be Usefully Imported into the Social Sphere?”: This one is an interesting failure: it points out correctly the widespread use of the term entropy with a correspondingly widespread lack of consistent definition. But after pointing out all these examples the essay doesn’t really do anything to bring order to the situation. (Not mentioned here, but there was even a 1972 science-fiction novel by George Alec Effinger titled
*What Entropy Means to Me*.)

Bottom line: an anthology full of interesting things, but not everything will interest everyone.

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Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.