Meta-theoretical Passage

Abstract

This passage is a locus classicus for methodology in ancient mathematics. It is perhaps the best-known passage in Coll. IV. A doublet can be found in Coll. III, and a shorter version in Coll. VII. There are to be three kinds of mathematics: plane, solid, linear, corresponding to three kinds of basic curves. In addition, a homogeneity requirement holds: only arguments that use means from the mathematical kind to which the problem belongs are fully valid mathematically. The passage is referred to in Descartes’ Géométrie (Descartes 1637, p. 315, pp. 40/41 Smith/Latham). Newton also quotes it with approval, and employs it against the Cartesian program in geometry. Up until relatively recently, it was taken to be the communis opinio for mathematics throughout antiquity, and quoted or referred to in secondary literature in this way. In fact, it is, at least in this generality, only to be found in Pappus. For him, it is obviously important. He is committed to this view in the following sense: he uses it to structure his material to give a representative survey of ancient mathematics, to give a coherent methodologically oriented picture of the geometrical tradition. It is not certain, and in fact not all that relevant for the understanding of Coll. IV itself, whether this meta-theoretical position was shared, in this full generality, by the mathematicians. Pappus may very well be generalizing a feature to be found in Apollonius’ analytical works on locus problems: separate plane problems from solid ones. Still, he is well-informed, competent, and manages to tell a reasonably coherent story. It should be appreciated as a whole. An extensive discussion will not be given here (for the full text, see the translation in Part I). In the present edition, I have taken this passage quite literally, and propose a reading of the whole of Coll. IV in light of it. In what follows, I will comment on the two main items in the passage: the mathematical kinds, and the homogeneity criterion, and briefly indicate how the different parts of Coll. IV relate to remarks in the passage.