At different occasions in his dialogues, Plato had drawn a comparison between the Ideas and
view sometimes expressed by scholars, both ancient and modem, that the Ideas are simply thoughts, in the divine or in the human mind." Therefore, the possibility that the artist-craftsman would have first conceived and Idea in his mind and then transferred this to the work of art should be considered alien to the spirit of Platonic theory because, even though Plato's carpenter might have an eye on the eidos of the bed when he produces a particular bed, he is not imitating the abstract Form-bed but actually making an individual bed that participates in the nature of the universal Form.
particulars would be imperfect exemplifications of the Ideas. However, when expressions like particulars 'imitating' the universals or Ideas are taken literally the true meaning of Plato's theory is falsified. As Ross has contended, "the expressions 'share' and 'imitate'[...]are alike metaphors inadequate to express the relation of particulars to an Idea, because they both treat the Idea as if it were a thing, instead of being a characteristic of things."
It seems to be established then that Plato's Ideas should be excluded from being considered as objects of imitation, although some authors have speculated on such possibility. In Plato's theory, abstract Ideas are absolute essences separated
from the human world of mind and action. As Panofsky has contended, the Platonic Ideas, unlike the Ideas of philosophers and art theorists in the sixteenth century, are not "notions or concepts residing in the mind of man" that "reveal themselves in artistic creativity" but rather they are "metaphysical substances existing outside the world of sensory appearances as well as outside the human intellect." In the same line, Ross contends that "there is nothing in Plato to justify the
geometric figures and mathematics. At some point, Plato seems to have thought that mathematics and geometry are the expression of those abstract Forms or Ideas that only exist in an intelligible realm. For example, in the Republic527b, geometrical knowledge is equated with the sort of absolute, eternal truth that Ideas stand for: 'Tor geometrical knowledge is of that which always is." Later, in the position, differing from sensible things in being eternal and unchangeable, from Forms in that there are many alike, while the Form is in each case unique." Quoted in Ross, op. cit, p. 177.
Timaeus, the form of the universe is assimilated to a sphere, while the smallest components from which all matter is composed are associated with the Pythagorean solids.
The question that arises then is to which extent Plato's Ideas should be considered the same as geometric figures or mathematics. At this junction, we are confronting one of the most intricate problems in Plato's theory of Ideas: the relationship between the abstract universals and the sensible particulars. In spite of Plato's equation of Ideas with mathematics and geometry, it cannot be concluded that he thought that these were the same as the Ideas. David Melling makes this point clear when he writes that "knowing the importance of the Pythagorean influence on Plato, and noting the importance of mathematics in the programme of education designed to lead to knowledge of the Forms, we may be led to suspect they are mathematical realities of some kind. There is, however, nothing in the text of the Phaedo or the Republic to force us to such a conclusion, and we might come to quite a different view, for example that the Forms are eternal archetypes or paradigms of sense-perceptible things." Moreover, Plato consistently maintained during the later development of the theory of Ideas that there are three kinds of entities: Ideas, mathematical objects and sensible objects. Even though he explored the connections that exist between Ideas and mathematical objects, he was far from conceding that they were the same. Plato only admitted to the identity of Ideas with 'ideal numbers' (e.g. the notion of oneness, twoness) but did not contend that Ideas were expressed through particular numbers (e.g. 1, 2, 3, 4). Something similar could be said with regard to an hypothetical identity of Ideas with geometric figures: the Idea of a circle is not so much a perfect circle as the abstract notion of circularity.
A second possibility, for the Ideas to be considered as objects of imitation, is that Ideas are taken as sensible models of paradigms after which other individuals are produced. At different moments in the development of his theory, Plato spoke of the particulars 'sharing' and 'imitating' the Ideas. In the Parmenides 132d, for example, Socrates says that the Ideas are "as it were paradigms" and that "other things are like them and are copies of them." According to this, the