Appendix 11.

(507) [Plato's ideas and Pythagoras' numbers]

To confirm my explanation of Plato's ideas and throw some light on an important point of ancient philosophical history, let me add an observation. Modern thinkers have discussed at length whether Pythagoras' numbers were the same as Plato's ideas. For our purpose we should note that numbers are abstract ideas, whereas Plato's ideas, which must serve as exemplars, can be only specific, not abstract ideas - Cicero writes, nos recte speciem possumus dicere [we can correctly say species] (Academ., bk. 1). Pure numbers can never be exemplars of things, just as something totally abstract cannot be used by a sculptor as an exemplar for a statue. Plato, by substituting ideas for numbers, perfected the teaching glimpsed by Pythagoras, at least in the forms of its expression. The Pythagoreans themselves seem to have made some progress in perfecting the teaching according to certain passages in the Timaeus. It is extraordinary that modern authors (particularly Brucker) did not sense the difference which Plato himself had to some extent made clear. The fine Tuscan philosopher, Marsilio Ficino, mentions Plato's opinion: 'He (Plato) posits a first and a second intelligible. He places ideas, that is, the species and motions of the divine mind, other minds and souls in the first, and numbers and shapes in the second intelligible (Thaeatetus. Cf. also the end of the sixth dialogue of the Republic). We see here that exemplar ideas are separate from numbers. Ideas precede numbers because numbers are obtained from ideas by abstraction; numbers are a part of ideas, like all abstracts.