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Morris Kline
Mathematics, The loss of certainty
Mathematics has an air of being the most secure form of knowledge. In Mathematics, The Loss of Certainty, however, Morris Kline shows that this is not necessarily deserved. He shows how, rather than mathematics being an obvious progression of knowledge, in fact many ideas in the subject were strongly resisted when first introduced, and even when accepted often rested on insecure foundations. He explains how Euclidean geometry turned out not be as 'obviously true' as people thought, how calculus was based on the shaky ground of infinitesimals, and how grudgingly imaginary numbers came to be accepted as a valid way to do calculations.
I felt that Kline tends to overdo the uncertainty of mathematics, in particular at the start of the book, making it look like each surprising result implied that the subject was in total disarray. The book improves as it goes on though. The later parts look at the development of Cantor's set theory and the axiomatisation of mathematics leading up to Göel's incompleteness theorem and beyond. This is certainly worth reading for those with an interest in the philosophy of mathematics, especially since the book requires very little previous mathematical experience on the part of the reader.