Timaeus 31b4-32c4再考 : 宇宙の身体の事実上の不滅性のテーゼと立体幾何学

抄録

One of the crucial claims made in the Timaeus appears to be that the body of the universe is indestructible by any agent other than the Demiurge who had bound it together (31b8-32c4) This claim is striking, for this kind of indestructibility of the universe, which I shall call de facto indestructibility, contradicts the fundamental proposition at Timaeus 31b4 that the universe is bodily, visible and tangible It should be remembered that in the Platonic corpus 'bodily nature' usually connotes evil, mortality and destructibihty, this motif is observed clearly in, for example, the Phaedo and the Pohticus myth This state of affairs needs to be explained I shall be claiming in this paper that the notion of de facto indestructibility of the body of the universe is founded in the Timaeus on the achievements and ideas of stereometry Mathematical entities in themselves, of course, are not subject to coming-to-be and passing-away By describing the fundamental constitution of the body of the universe in stereometrical terms, Plato facilitates the transition to the claim that the only mode of destruction to which the universe would be liable would be if the Demiurge himself wished to undo his work-which of course he never would The key to this transition is the move from σωματοειδεζ(31b4)to στερεοειδη(32b1), and it is this move that will be the chief focus of attention in this paper Two steps are to be observed in the argument which connects the fundamental statement that the universe must be bodily, visible and tangible (31b4) with the framework of fire, earth, air and water The first step consists in linking 'visibility and tangibility' to 'fire and earth', where the term 'στερεον' is deliberately introduced at 31b6 The word 'στερεον' slides into the text initially to signify merely solidity, but carrying overtone of mathematical extension, which, it might be said, allows Plato to introduce the concept of στερεοειδη (a three-dimensional mathematical solid) The second step is composed of the introduction of the theory of proportion (αναλογια) which requires 'air' and 'water' as the middle terms uniting 'fire' and 'earth' The move from the first step to the second step, which the term 'στερεον' plays a crucial role in facilitating, is followed by the striking introduction of the basic rule of proportional equation, the mathematical formulation of proportional equation is used to provide a reason why the Demiurge chooses, in particular, the principle of proportion as the cosmic bond to unite fire, earth, air and water The theory of proportion is derived from the achievements of stereometry in the fifth century B C and incorporated in the Timaeus The stereometrical ideas relating to the de facto indestructibility of the universe are, the theory of the two mean proportionals, and the method of constructing the regular solids The theory of the two mean proportionals introduces an original idea concerning the nature of the bond that unites the primary bodies While in Empedocles the cosmic bond is 'love', what we have in the Timaeus is 'geometrical proportion' (αναλοια), which is described as the fairest of bonds(δεσμων καλλιστοζ) (Timaeus 31c2) This secures the de facto indestructibility of the universe at one level Stereometry gives an assurance of eternal bodily existence, which is a necessary attribute in the Greek tradition of mathematical entities bound by dissoluble numerical proportions The ideas derived from stereometry thus form a crucial part of the metaphysical foundation of the Timaeus