12 - Betwixt and Between: Plato and the Objects of Mathematics

Summary

Plato’s dialoguesespecially the Republiclead us to wonder what the objects of mathematics are. For Plato, no perceptible three is unqualifiedly three, a necessary condition for being an object of knowledge. Aristotle controversially ascribes to Plato the view that mathematical objects are “intermediates,” between perceptibles and Forms: multiple but also eternal, lacking change, and separate from perceptibles. The hunt for or against intermediates in Plato’s dialogues has depended on two ways of understanding Plato on scientific claims, a Form-centric approach and a subject-centric (semantic) approach. Although Socrates does not present intermediates in the Republic, it is difficult to see how the units of the expert arithmetician or motions of the real astronomer could be simply Forms or perceptibles. The standard over-reading of the Divided Line, where the middle sections are equal, further obscures our understanding. The Phaedo and the Timaeus provide candidates for mathematical objects, although these have only some of the attributes ascribed to intermediates. We are left with no clear answer, but exploring options may be exactly what Plato wants.