diagonal, we arrive at two contradictory requirements: the requirement of the ratio to be reducible to lowest terms and the requirement of the side and diagonal to both be divisible by two. If in our thinking, we attempted to enforce the first requirement, we would deprive the square of its shape. If we attempted to enforce the second, number would lose its character as number. In either case, we reach an absurdity: a requirement that deprives a thing of what it is. We may as well be attempting to make the square circular. The ratio of a square’s diagonal and its side is not the only case where two magnitudes cannot be related as a number to a number. Whenever this impossibility occurs, the two magnitudes are said to incommensurable. The significance of incommensurability lies in the fact that it undermines a claim to know the world. So long as there are incommensurables, the beauty and order of nature cannot have its ultimate source in number: “The impossibility of there being a common measure of length for the side and diagonal of a square was, in effect, a crisis of intelligibility: it meant that there was a ratio or logos in the world that was unspeakable or alogos for the being and determinateness of things resided, for the Pythagoreans, in our ability to count them.” 16 The depth of the crisis is perhaps best illustrated in the attempt by the Pythagoreans to hide the existence of incommensurables. The failure of number to be ultimate put their very way of life at risk. Nonetheless, the incommensurables were always there to be discovered. “It is said that when the discovery of incommensurability was first revealed to outsiders, thus making public the insufficiency of number, the man who thus had undermined the Pythagorean enterprise was murdered.” 17
Kalkavage, Peter. Plato’s Timaeus. (Newburyport, MA: Focus Publishing, 2001), 157. We should note that because magnitudes describe geometrical figures, incommensurability represents a division between ratios that can be grasped only by sight and those that can be grasped through speech. 16
Flaumenhaft, Harvey. “Why We Won’t Let You Speak of the Square Root of Two.” St. John’s Review 48, no. 1: 25 17
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