The Cave and the Light: Plato Versus Aristotle, and the Struggle for the Soul of Western Civilization

by Arthur Herman

Taken together, they laid the future cornerstone of what comes to be called calculus, or the mathematics of infinity (Archimedes was the first mathematician to use the concept of infinity in his work). Without it, modern math and science as we know it would not exist. In one sense, the impetus to Archimedes’s math research was proving that Plato had been wrong: you can’t form a sphere out of a series of triangles or pentagons, as Plato claimed in the Timaeus, any more than you can form a circle out of a square. Archimedes showed this by demonstrating that you can’t measure the area of a circle or sphere or cylinder using any kind of straight line. However, you can measure it by slicing it into pieces that can be bounded by straight lines, then slicing that part left over into measurable pieces again and again, until finally there is nothing left unmeasured.28 This is what calculus, and the concept of infinity, enabled Archimedes to do. He discovered, for example, that the area of a sphere is four times the area of its greatest circle; and that the area of a parabola (like the one described by a stone flung from a catapult) is four-thirds the area of the triangle enclosed within it. Archimedes’s work was also establishing in passing another scientific landmark for the future. If you can’t measure something, then it probably doesn’t exist. It’s a notion that Aristotle would have rejected at once. It was, however, another logical outgrowth of his own scientific method.b Archi...