History of Western Philosophy - Bertrand Russell [88]
This leads to a somewhat curious theory of space, as something intermediate between the world of essence and the world of transient sensible things.
There is one kind of being which is always the same, uncreated and indestructible, never receiving anything into itself from without, nor itself going out to any other, but invisible and imperceptible by any sense, and of which the contemplation is granted to intelligence only. And there is another nature of the same name with it, and like to it, perceived by sense, created, always in motion, becoming in place and again vanishing out of place, which is apprehended by opinion and sense. And there is a third nature, which is space, and is eternal, and admits not of destruction and provides a home for all created things, and is apprehended without the help of sense, by a kind of spurious reason, and is hardly real; which we beholding as in a dream, say of all existence that it must of necessity be in some place and occupy a space, but that what is neither in heaven nor on earth has no existence.
This is a very difficult passage, which I do not pretend to understand at all fully. The theory expressed must, I think, have arisen from reflection on geometry, which appeared to be a matter of pure reason, like arithmetic, and yet had to do with space, which was an aspect of the sensible world. In general it is fanciful to find analogies with later philosophers, but I cannot help thinking that Kant must have liked this view of space, as one having an affinity with his own.
The true elements of the material world, Timaeus says, are not earth, air, fire, and water, but two sorts of right-angled triangles, the one which is half a square and the one which is half an equilateral triangle. Originally everything was in confusion, and 'the various elements had different places before they were arranged so as to form the universe'. But then God fashioned them by form and number, and 'made them as far as possible the fairest and best, out of things which were not fair and good'. The above two sorts of triangles, we are told, are the most beautiful forms, and therefore God used them in constructing matter. By means of these two triangles, it is possible to construct four of the five regular solids, and each atom of one of the four elements is a regular solid. Atoms of earth are cubes; of fire, tetrahedra; of air, octahedra; and of water, icosahedra. (I shall come to the dodecahedron presently.)
The theory of the regular solids, which is set forth in the thirteenth book of Euclid, was, in Plato's day, a recent discovery; it was completed by Theaetetus, who appears as a very young man in the dialogue that bears his name. It was, according to tradition, he who first proved that there are only five kinds of regular solids, and discovered the octahedron and the icosahedron.4 The regular tetrahedron, octahedron, and icosahedron, have equilateral
triangles for their faces; the dodecahedron has regular pentagons, and cannot therefore be constructed out of Plato's two triangles. For this reason he does not use it in connection with the four elements.
As for the dodecahedron, Plato says only 'there was yet a fifth combination which God used in the delineation of the universe'. This is obscure, and suggests that the universe is a dodecahedron; but elsewhere it is said to be a sphere.5 The pentagram has always been prominent in magic, and apparently owes this position to the Pythagoreans, who called it 'Health' and used it as a symbol of recognition of members of the brotherhood:6 It seems that it owed its properties to the fact that the dodecahedron has pentagons for its faces, and is, in some sense, a symbol