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Everyone knows that ancient Greek geometers treated circles and many people know that that they treated conic sections. (Indeed, they treated conic sections as sections of cones.) It is perhaps less well known that ancient geometers also discussed a variety of other curves, including spirals, and the quadratrix, cissoid, and conchoid. Their discussion of these curves is interesting from the perspective of the history of mathematics--they explode a simplistic, but still wide-spread, picture of Greek mathematics as limited by the straight-edge and compass. But we will focus on the questions these curves raise in the philosophy of mathematics (including the philosophical views held by the ancients). We will read the original texts (in translation) in which these curves were presented and defined, with a focus on the philosophical issues that they raise. We are especially interested in the way in which these curves are defined and constructed. In various ways, their definition and construction differ from the definition and construction of other mathematical objects. For instance, motion is involved in their definition and construction, and sometimes machine-like tools are invoked, which go far beyond the familiar resources of the straight-edge and compass. To what extent did that cast doubt on the mathematical and ontological acceptability of these curves? To what extent did the definitions of these curves make important innovations in the practice of defining? To what extent did the theories of these curves extend or change the standards (explicit or implicit) for mathematical knowledge and understanding?
Prerequisites: Language of instruction is English. Papers (Hausarbeiten) may be written in German or English. Knowledge of Greek welcome but not required. Advanced knowledge of mathematics welcome but not required; nothing more than standard Gymnasium-level mathematics is presupposed. No special knowledge of Greek mathematics is required. |
