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.

The main aim of Part 2 is to explain how the form of the good gives rise to knowledge of forms, the forms in question being of virtues and virtue-related things. This ramifies into discussions of dialectic and mathematics, the ambiguous property 'clearness' (saphēneia), hypotheses, and the non-hypothetical principle. It is proposed that the form of the good is interrogative. This position is defended against philosophical and textual objections, and argued to be preferable to alternatives. There is discussion of why Plato excludes the use of diagrams from dialectic and whether he can allow input from experience. The role of context in the rulers' dialectic is explained, and becomes the basis for explaining why Plato's treatment of dialectic in the Republic remains at the level of a sketch. There is an exploration of the difference between true philosophers and sight-lovers, and of the criteria and scope of 'good' in dialectic. This last discussion encounters the classic problem of the connection between Plato's 'justice in the soul' and just conduct as ordinarily recognized, and a solution to this problem is proposed.