[This is an extract from the chapter 'The Platonic Theory of Being', published in The Sacred History of Being on November 2, 2015]
.....The context of this passage is an attempt to explain water and air as intermediary between fire and earth, but what Plato is giving us is more general: a theory of participation which has been the root of the western tradition in art and architecture ever since:
For whenever the middle term of any three numbers, cubic or square, is such that as the first term is to it, so is it to the last term - and again, conversely, as the last term is to the middle, so is the middle to the first.
- then the middle term becomes in turn the first and the last, while the first and last become in turn middle terms, and the necessary consequence will be that all the terms are interchangeable, and being interchangeable they all form a unity. [i]
Why is it a necessary consequence that all the terms are interchangeable? Each of the terms bears a relation to the others, a proportionate similitude, and each can become first, middle and last terms in an extended sequence, but they are not the same as each other. They are conjoined with one another, but in sequence. They bear likeness to each other in the proper sequential order but not otherwise. It depends on the arrangement. Given the proper arrangement, one may pass through the sequence and establish degrees of similitude between all the different terms. But they are still not the same. They participate in each other, but their proportionate similitude is not identity, and that is essentially what Plato is claiming here.
I think that there is no doubt that this is what Plato means, and it is up to us to explain it. Clearly it underpins the description of the activity of the philosopher in the Republic, where it is said that the process of argument:
...treats assumptions not as principles, but as assumptions in the true sense, that is, as starting points and steps in the ascent to something which involves no assumption and is the first principle of everything; when it has grasped that principle it can again descend, by keeping to the consequences that follow from it, to a conclusion. The whole procedure involves nothing in the sensible world but moves solely through forms to forms, and finishes with forms. [ii]
What Plato is arguing is that, by systematic dialectical enquiry, we can rise from the realms of likelihood and opinion, where we encounter only similitudes, to the realm in which certain knowledge is possible. This is to be achieved by passing through the similitudes, on account of their similitude, to their ultimate origin, the Form of the Good.
In a sense the Platonic dialogues can be understood in this way - as excursions from selected assumptions - and the Republic and the Timaeus illustrate this very well - both employ series of images, of likelihoods which are to be understood as metaphors or similitudes of the general notion which is to be grasped. We are to reject the accounts of the making of the universe in the Timaeus, on the grounds that these accounts possess merely likelihood, and no more. The general notion or the ultimate reality behind the accounts is not to be grasped by collation of the versions, but by an altogether more mysterious process of induction....
[End of chapter extract]