The Life of Reason: The Phases of Human Progress

by George Santayana

Confusion comes of imperfect abstraction, or ambiguous intent.

When dialectic is employed, as in ethics and metaphysics, upon highly complex ideas—concretions in discourse which cover large blocks of existence—the dialectician in defining and in deducing often reaches notions which cease to apply in some important respect to the object originally intended. Thus Socrates, taking “courage” for his theme, treats it dialectically and expresses the intent of the word by saying that courage must be good, and then develops the meaning of good, showing that it means the choice 01 the greater benefit; and finally turns about and ends by saying that courage is consequently the choice of the greater benefit and identical with wisdom. Here we have a process of thought ending in a paradox which, frankly, misrepresents the original meaning. For “courage” meant not merely something desirable but something having a certain animal and psychological aspect. The emotion and gesture of it had not been excluded from the idea. So that while the argument proves to perfection that unwise courage is a bad thing, it does not end with an affirmation really true of the original concept. The instinct which we call courage, with an eye to its psychic and bodily quality, is not always virtuous or wise. Dialectic, when it starts with confused and deep-dyed feelings, like those which ethical and metaphysical terms generally stand for, is thus in great danger of proving unsatisfactory and being or seeming sophistical.

The mathematical dialectician has no such serious dangers to face. When, having observed the sun and sundry other objects, he frames the idea of a circle and tracing out its intent shows that the circle meant cannot be squared, there is no difficulty in reverting to nature and saying that the sun’s circle cannot be squared. For there is no difference in intent between the circularity noted in the sun and that which is the subject of the demonstration. The geometer has made in his first reflection so clear and violent an abstraction from the sun’s actual bulk and qualities that he will never imagine himself to be speaking of anything but a concretion in discourse. The concretion in nature is never legislated about nor so much as thought of except possibly when, under warrant of sense, it is chosen to illustrate the concept investigated dialectically. It does not even occur to a man to ask if the sun’s circle can be squared, for every one understands that the sun is circular only in so far as it conforms to the circle’s ideal nature; which is as if Socrates and his interlocutors had clearly understood that the virtue of courage in an intemperate villain meant only whatever in his mood or action was rational and truly desirable, and had then said that courage, so understood, was identical with wisdom or with the truly rational and desirable rule of life.

The fact that mathematics applies to existence is empirical.

The applicability of mathematics is not vouched for by mathematics but by sense, and its application in some distant part of nature is not vouched for by mathematics but by inductive arguments about nature’s uniformity, or by the character which the notion, “a distant part of nature,” already possesses. Inapplicable mathematics, we are told, is perfectly thinkable, and systematic deductions, in themselves valid, may be made from concepts which contravene the facts of perception. We may suspect, perhaps, that even these concepts are framed by analogy out of suggestions found in sense, so that some symbolic relevance or proportion is kept, even in these dislocated speculations, to the matter of experience. It is like a new mythology; the purely fictitious idea has a certain parallelism and affinity to nature and moves in a human and familiar way. Both data and method are drawn from applicable science, elements of which even myth, whether poetic or mathematical, may illustrate by a sort of variant or fantastic reduplication.

The great glory of mathematics, like that of virtue, is to be useful while remaining free. Number and measure furnish an inexhaustible subject-matter which the mind can dominate and develop dialectically as it is the mind’s inherent office to develop ideas. At the same time number and measure are the grammar of sense; and the more this inner logic is cultivated and refined the greater subtlety and sweep can be given to human perception. Astronomy on the one hand and mechanical arts on the other are fruits of mathematics by which its worth is made known even to the layman, although the born mathematician would not need the sanction of such an extraneous utility to attach him to a subject that has an inherent cogency and charm. Ideas, like other things, have pleasure in propagation, and even when allowance is made for birth-pangs and an occasional miscarriage, their native fertility will always continue to assert itself. The more ideal and frictionless the movement of thought is, the more perfect must be the physiological engine that sustains it. The momentum of that silent and secluded growth carries the mind, with a sense of pure disembodied vision, through the logical labyrinth; but the momentum is vital, for the truth itself does not move.

Its moral value is therefore contingent.

Whether the airy phantoms thus brought into being are valued and preserved by the world is an ulterior point of policy which the pregnant mathematician does not need to consider in bringing to light the legitimate burden of his thoughts. But were mathematics incapable of application, did nature and experience, for instance, illustrate nothing but Parmenides’ Being or Hegel’s Logic, the dialectical cogency which mathematics would of course retain would not give this science a very high place in the Life of Reason. Mathematics would be an amusement, and though apparently innocent, like a game of patience, it might even turn out to be a wasteful and foolish exercise for the mind; because to deepen habits and cultivate pleasures irrelevant to other interests is a way of alienating ourselves from our general happiness. Distinction and a curious charm there may well be in such a pursuit, but this quality is perhaps traceable to affinities and associations with other more substantial interests, or is due to the ingenious temper it denotes, which touches that of the wit or magician. Mathematics, if it were nothing more than a pleasure, might conceivably become a vice. Those addicted to it might be indulging an atavistic taste at the expense of their humanity. It would then be in the position now occupied by mythology and mysticism. Even as it is, mathematicians share with musicians a certain partiality in their characters and mental development. Masters in one abstract subject, they may remain children in the world; exquisite manipulators of the ideal, they may be erratic and clumsy in their earthly ways. Immense as are the uses and wide the applications of mathematics, its texture is too thin and inhuman to employ the whole mind or render it harmonious. It is a science which Socrates rejected for its supposed want of utility; but perhaps he had another ground in reserve to justify his humorous prejudice. He may have felt that such a science, if admitted, would endanger his thesis about the identity of virtue and knowledge.

Quantity submits easily to dialectical treatment.

Mathematical method has been the envy of philosophers, perplexed and encumbered as they are with the whole mystery of existence, and they have attempted at times to emulate mathematical cogency. Now the lucidity and certainty found in mathematics are not inherent in its specific character as the science of number or dimension; they belong to dialectic as a whole which is essentially elucidation. The effort to explain meanings is in most cases abortive because these meanings melt in our hands—a defeat which Hegel would fain have consecrated, together with all other evils, into necessity and law. But the merit of mathematics is that it is so much less Hegelian than life; that it holds its own while it advances, and never allows itself to misrepresent its original intent. In all it finds to say about the triangle it never comes to maintain that the triangle is really a square. The privilege of mathematics is simply to have offered the mind, for dialectical treatment, a material to which dialectical treatment could be honestly applied. This material consists in certain general aspects of sensation—its extensity, its pulsation, its distribution into related parts. The wakefulness that originally makes these abstractions is able to keep them clear, and to elaborate them infinitely without contradicting their essence.