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There are several books available on the history of mathematics. Some give an account on the development of a certain area, others focus on a group of persons and some do hardly more than story telling. I was looking for one that tells the story of the development of the main ideas and the understanding of what mathematics and science in general is (or what people thought it is and should be). Howard Eves' book is the first book I bought that gives me the answers I was looking for. Starting with pre-Euclidean fragments, going on with Euclid, Aristotle and the Pythagoreans, straight to non-Euclidean geometry it focuses on the axiomatic method of geometry. What pleased me most here is that the author really takes each epoch for serious. He quotes longer (and well chosen) passages from Euclid, Aristotle and Proclus to demonstrate their approaches. Each chapter ends with a Problems section. I was surprised to see how much these problems reveal of the epoch, its problems and thinking. The book goes on with chapters on Hilbert's Grundlagen, Algebraic Structure etc, always showing not only the substance of these periods but also the shift in the way of thinking and the development towards rigor. The last chapter is titled Logic and Philosophy. Eves divides "contemporary" philosophies of mathematics into three schools: logistic (Russel/Whitehead), intuitionist (Brouwer) and the formalist (Hilbert). The book ends with some interesting appendices on specific problems like the first propositions of Euclid, nonstandard analysis and even Gödel's incompleteness theorem. Bibliography, solutions to selected problems and an index are carefully prepared to round up an excellent book. Should you buy this book ? Yes. What kind of mistake can you make in spending US$ 12.95 on a book that has withstood the test of time through three editions (each with a different publisher). I havent completed reading the book yet, but I dont regret having bought it.
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