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The Ancient Tradition of Geometric Problems

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This is a history of Greek mathematics from the point of view of problems and problem solving, not least the three classical problems: cube duplication, circle quadrature, angle trisection. Knorr argues that this restriction is not arbitrary: "One seems typically to assume that metamathematical concerns were the effective motivating force underlying efforts of geometers ... In general, I will find most convincing the 'internalist' position that technical research is directed toward the solution of problems arising from previous and current research efforts", mathematics thus being "on the whole autonomous in setting the directions of research". So the focus on problems is a way to understand the development of Greek mathematics by exposing the mindsets of the mathematicians. This aspect is naturally very interesting, and the mere presence of such perspectives sets this book apart from the bare bones histories of Heath et al. However, grand programs and pretty pots on the cover is not everything. Our fragmented knowledge of Greek mathematics does not allow Knorr's program to be carried out in a completely satisfactory manner; neither is the big picture one of Knorr's main concerns. Instead much of the book amounts to quite specialised scholarly analysis of sources and critique of other scholarly interpretations. In the end, one is not entirely convinced that Knorr has unveiled the key to understanding Greek mathematics. His criticism of other interpretative schemes is convincing but sometimes suspiciously convenient. So, for instance, Knorr rather enjoys arguing that philosophy never had a major influence on mathematics (thus supporting his point of view) while the more interesting and relevant questions of the influences of astronomy, mechanics and optics are largely silenced.