here is a concept which corrupts and upsets all others,” Jorge Luis Borges wrote; “I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite.” The infinite and its avatars—the incommensurable, the unbounded, the immeasurable—have always occasioned intellectual horror. The Pythagoreans dreamed that all things have an articulable logos (reason, proportion, ratio). The ancients claimed that the philosopher Hippasus drowned at sea, a vestige of the Chaos from which Hesiod taught that all things spring, following his discovery of irrational magnitudes. While this story may be apocryphal, the discovery of the unspeakable (alogon) √2, which no ratio of whole numbers can express (and whose decimal representation is interminable), was inevitably memorialized as tragedy. In 1874, Georg Cantor proved that the infinite set of real numbers is non-denumerable—that it exceeds the infinite set of positive integers, and so could not be “counted” (matched one-to-one with the natural numbers 1, 2, 3, . . . ) even by God himself. He was thereafter tormented by mental illness. What would Euripides, whose maddening Dionysus explodes the most fundamental law of logic, the principle of non-contradiction, have made of that coincidence?
The Greek philosophers struggled to find measures of world and soul that would not dissolve on close inspection. Heraclitus declared that “you could not discover the limits of the soul, even if you journeyed the whole way, so deep is its logos.” Plato plays with similar thoughts in his Statesman, where he compares untamed human nature to √2. In the same dialogue, he has Socrates suggest to his companion, a mathematician who demands threefold recompense for speeches about the sophist, statesman, and philosopher, that the lover of wisdom is radically incommensurable with the first two types. Socrates, who claimed not to know whether he was a multiform monster or a tame and simple animal, sought sunlit uplands of truth beyond the Cave of human existence because he understood that it is not man who is the measure of all things, as the sophist Protagoras declared, but the cryptic god of Delphi, or the mysterious Good that transcends all beings.
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