Like the circle, [the right triangle is] one of the sturdiest things in the cosmos and, at the same time, utterly immaterial. Physical circles are never really perfectly circular; physical right triangles always fall short of perfection. We grasp such geometrical figures only with our minds, never with our hands. When we do grasp the demonstration, even though we may do it through particular drawings, we lay hold of why it must be so in any particular case from the nature of the right triangle as such - but there is no "right triangle as such" out there floating in the cosmos that we can either see with our eyes or grasp with our hands. The struggle to understand the demonstration about right triangles is the struggle to grasp something that is immaterial.
But if we are successful in this struggle, and we glimpse the necessity of the geometric relationships as such, then and only then can we see why it must be so in every possible case and not just in this or that particular drawn triangle. Even more profound, if we reflect on our own reflection, we receive a more beautiful proof, a demonstration that we have, in our reason, a power to grasp immaterial truths - a power that somehow exceeds the particular, physically defined powers of our senses and imagination and is capable of grasping universal truth. Could this be a proof of the immateriality of the soul?
Wednesday, November 07, 2007
Is Platonism a means, not an end?
In "A Meaningful World", Wiker and Witt write, on Euclid's demonstration of Pythagoras' theorem: