Workshop in Ancient Philosophy (Thursday - Week 6, HT25)

Abstract: There is a curiosity of Socrates’ analysis of the mathematical sciences in Republic VI and VII. His criticisms of the sciences seem to centre around necessary features: mathematicians treat hypotheses as first principles; they speak of ‘changing’ what they admit to be changeless entities; and they make use of ‘visible kinds’. Intuitively, Socrates is not here criticising mathematicians themselves for proceeding in this way, but rather commenting on the intellectual limitations of mathematics, given the kind of science that it is.

The idea is that for the mathematicians to stop doing any of what Socrates complains about would be to stop doing mathematics itself, and to take up dialectic. Yet it seems at least questionable whether mathematics evaporates at the removal of e.g. visible kinds. More, Socrates explicitly suggests that astronomy and harmonics can be upgraded, analogising them to geometry. I argue that the analogy between geometry on the one hand and astronomy/harmonics on the other implies that the use of ‘visible kinds’ is – at least in principle – dispensable. Further, I argue that Socrates envisions a version of mathematics which avoids his criticisms while still retaining its status as mathematics, and conclude by suggesting some consequences for the relationship of mathematics to dianoia, the second-best condition of the soul.