What’s location got to do with it? Place, space, and the infinite in classical greek mathematics

Abstract

Here is a basic question: how much philosophy of mathematics can one pull out of Greek mathematical texts? Obviously, this depends on what we are taking as philosophy of mathematics. We can describe easily enough what goes on in a typical Greek mathematical treatise, but even here, ‘typical’ is a loaded word. We define the scope to our liking and to some 7 or 8 extant authors, or maybe to more, but the more we extend our list, the less does the result fit, for example, the tidy and austere world that Netz (1999) portrays in his ground breaking study. Even within this group, we should expect difference and variation. Austerity aside, unless they tell us, we have not a clue how Greek mathematicians thought about their work. In discussions of the philosophy of Greek mathematics, it is very common, following in the path of Proclus, for moderns to find a philosophy of Greek mathematics and then to trace the philosophy to Aristotle or Plato. If we distinguish, however, issues that are intrinsic to a mathematical exposition from external questions such as the ontology of mathematics, we can observe that there is very little evidence of any views about ontology expressed in any Hellenistic mathematicians, while later mathematicians tend to come out of neo-Platonism. I shall illustrate this point by exploring two related issues in Greek philosophies and mathematical practices before the emergence of neo-Platonism, those of place and the infinite.