27 February 2010
The Greeks believed that men could in heroic moments live as gods, with the sole exception being that they must some day die: immortality is the one thing denied them. Heroes are god-like in their moment of triumph, and gods can be as foolish as men in their weaknesses, but the gods are the immortals, and this exempts them from a particular human experience of finitude: mortality. (I previously discussed this in Reason in Moderation.)
We might similarly characterize the Greek attitude to reason: in his lucid intervals the mind of man is like unto the gods, embodying a heroism of the intellect. But man cannot sustain his reason beyond its proper span, any more than life can be preserved past mortal limits. Thus the life appropriate to man is that of the cultivation of proper limits and a prudent respect for the boundaries that he ought not to pass into the unlimited. Moderation is the watchword here: “All things in moderation” and “Nothing in excess” were famous proverbs in the ancient world that are still with us today. Human virtue, then, is a function of finitude, and folly the lure of the apeiron, that is to say, the undetermined, the formless, the unstructured, the arbitrary, and the unbounded.
How shall one set the limits for oneself proper to human finitude? The above extrapolation of ancient Greek heroism to the realm of the mind has a perfect exemplification in a surviving fragment from one of the many shadowy presocratic philosophers, Epicharmus of Syracuse: A mortal should think mortal thoughts, not immortal thoughts. (Fragment 19 in the Ancilla to the Pre-Socratic Philosophers by Kathleen Freeman) Does this represent an aspiration to achievement, i.e., to achieve that which a human being can achieve in actuality, or an admonition to humility, i.e., to avoid the hubris of that which is denied to human achievement? Is there a difference between the two, or is it a matter of the glass being half-empty or half-full?
The idea contained in Ephicharmus’ aphorism may have had a certain currency in classical antiquity. The injunction, “O mortal man, think mortal thoughts” has been attributed to Euripides, though it is not to be found in his plays or fragments. However, in The Bacchae Euripides wrote, “…not to think mortal thoughts is to see few days.” This clearly implies a parallel line of thought. For a man to think immortal thoughts, that is to say, thoughts appropriate to the gods, he is courting doom and disaster, and, of course, Greek tragedy is filled with doom and disaster. The Furies visit doom and disaster upon men who exhibit hubris.
Ever since the Greeks, to whom we owe our mathematics and philosophy, the actual infinite has been rejected. The famous Pythagorean table of opposites, which was headed by apeiron and peras, associated the apeiron with the ugly, the crooked, and the bad. Thus the infinite was not only theoretically rejected, but also made the object of moral disapproval. Thinking the infinite represents the hubris of the intellect. Descartes said that we shouldn’t call things infinite, but rather indefinite. Fear of the infinite is almost a theme in Pascal’s Pensées. Gauss in a letter to Schumacher explicitly rejected infinite totalities. Kant’s antinomies not only questioned metaphysical ideas, but in showing the unsolvability of the finitude or infinitude of space and time also casts doubt on the concepts of the infinite employed in the demonstration.
This much of Kant is well known. Less known is the analysis of the infinite in the Critique of Judgement (Kritik der Urteilskraft), § 26, in the consideration of the mathematical sublime (“sublime” is the English term for Kant’s “Erhabene”). Kant makes a distinction between apprehensio and comprehensio, and holds that the former can go on ad infinitum, but the latter becomes more difficult and is eventually overwhelmed. It is the sublime which overwhelms comprehensio, and it is the sublime which is great beyond all comparison. If the infinite has overwhelmed philosophers one suspects it must be the terrifying sublime (in § 1 of Beobachtungen über das Gefühl des Schönen und Erhabenen, Kant distinguishes between the terrifying sublime, the noble sublime, and the splendid sublime), which suggests the ancient horror infinitum.
Like the Greek hero Prometheus who stole fire from the gods, there was one man who was not humbled by the horror infinitum, and that was Georg Cantor. The power of Cantor’s ideas are precisely his rejection of this tradition, sanctioned by history, and the introduction of a method to compare the sizes of infinite sets. The infinite in Cantor does not overwhelm and it is not incomparable. Indeed, Cantor’s great technical innovations — one-to-one correspondence and diagonalization — allow us to take the measure of the infinite and soberly assess its significance.
We have not only learned to think immortal thoughts, but we have learned to think them systematically and rigorously. The intuitive breakthrough of Cantor to set theory and transfinite numbers was a salto mortale, a death-defying leap of the intellect. In this, it is like Darwin’s intuitive breakthrough to natural selection or Einstein’s intuitive breakthrough to relativity. Such moments in the history of science are difficult to reconcile with sober theorizing. They represent the mind’s singular function, and it is only through such singular accomplishments that scientific progress is possible.
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