Mathematical Intelligence
The real nature of intelligence as it appears in the development of mathematics is something quite other than that of sensory analysis. Intelligence is fundamentally skill, and although skill may be acquired in connection with some sort of sensory contact of an organism and environment, it is only determined by that contact in the sense that if the sensory conditions were different the needs of the organism might be different, and the kind and degree of skill it could attain would be other than under the conditions at first assumed. Whenever the beginnings of mathematics appear with primitive people, we find a stage of development that calls for the exercise of skill in dealing with certain practical situations. Hence we found early in our investigations that it was impossible to affirm a weak intelligence from limited achievements in counting, just as it would be absurd to assume the feeble intelligence of a philosopher from his inability to manipulate a boomerang. The instance merely suggests a kind of skill that he has never been led to acquire.
Yet it is possible to distinguish intellectual skill, or better skills, from physical or athletic prowess. Primarily, it is directed at the formation and use of concepts, and the concept is only a symbol that can be substituted for experiences. A well-built concept is a part of a system of concepts where relations have taken the place of real connections in such a fashion that, forgetting the actuality, it is possible to present situations that have never occurred or at least are not immediately given at the time and place of the presentation, and to substitute them for actual situations in such a fashion that these may be expediently met, if or when such situations present themselves. An isolated concept, that is, one not a part of any system, is as mythical an entity as any savage ever dreamed. Indeed, it would add much to the clearness of our thinking if we could limit the use of "intelligence" to skill in constructing and using different systems of concepts, and speak concretely of mathematical intelligence, philosophical intelligence, economic intelligence, historic intelligence, and the like. The problem of creative intelligence is, after all, the problem of the acquisition of certain forms of skill, and while the general lines are the same for all knowledge (because the instruments are everywhere symbolic presentations, or concepts), in each field the situation studied makes different types of difficulties to be overcome and suggest different methods of attaining the object.
In mathematics, the formal impulse to reduce the content of fundamental concepts to a minimum, and to stress merely relations has been most successful. We saw its results in such geometries as Hilbert's and Peano's, where the empty name "entity" supplants the more concrete "point," and the "1" of arithmetic has the same character. In the social sciences, however, such examples as the "political" and the "economic" man are signal failures, while, perhaps, the "atom" and the "electron" approach the ideal in physics and chemistry. In mathematics, all further concepts can be defined by collections of these fundamental entities constituted in certain specified ways. And it is worth noting that both factually and logically a collection of entities so defined is not a mere aggregate, but possesses a differentiated character of its own which, although the resultant of its constitution, is not a property of any of its elements. A whole number is thus a collection of 1s, but the properties of the whole number are something quite different from that of the elements through which it is constituted, just as an atom may be composed of electrons and yet, in valency, possess a property that is not the direct analogue of any property possessed by electrons not so organized.
Natural science, however, considers such building up of its fundamental entities into new entities as a process taking place in time rather than as consequent upon change of form of the whole rendering new analytic forms expedient. Hence it points to the occurrence of genuine novelties in the realm of objective reality. Mathematics, on the other hand, has generalized its concepts beyond the facts implied in spatial and temporal observations, so that while significant in both fields by virtue of the nature of its abstractions, its novelties are the novelties of new conceptual formations, a distinguishing of previously unnoted generalizations of relations existent in the realm of facts. But the fact that time has thus passed beyond its empirical meaning in the mathematical realm is no ground for giving mathematics an elevated position as a science of eternal realities, of subsistent beings, or the like. The generalization of concepts to cover both spatial and temporal facts does not create new entities for which a home must be provided in the partition of realities. Metaphysicians should not be the "needy knife grinders" of M. Anatole France (cf. Garden of Epicurus, Ch. "The Language of the Metaphysicians"). Nevertheless, the success of abstraction for mathematical intelligence has been immense.
No significant thinking is wholly the work of an individual man. Ideas are a product of social coöperation in which some have wrested crude concepts from nature, others have refined them through usage, and still others have built them into an effective system. The first steps were undoubtedly taken in an effort to communicate, and progress has been in part the progress of language. The original nature of man may have as a part those reactions which we call curiosity, but, as Auguste Comte long ago pointed out (Lévy-Bruhl, A. Comte, p. 67), these reactions are among the feeblest of our nature and without the pressure of practical affairs could hardly have advanced the race beyond barbarism. Science was the plaything of the Greek, the consolation of the Middle Ages, and only for the modern has it become an instrument in such fashion as to mark an epoch in the still dawning discovery of mind.
Man is, after all, rational only because through his nervous system he can hold his immediate responses in check and finally react as a being that has had experiences and profited by them. Concepts are the medium through which these experiences are in effect preserved; they express not merely a fact recorded but also the significance of a fact, not merely a contact with the world but also an attitude toward the future. It may be that the mere judgment of fact, a citation of resemblances and differences, is the basis of scientific knowledge, but before knowledge is worthy of the name, these facts have undergone an ideal transformation controlled by the needs of successful prediction and motivated by that self-conscious realization of the value of control which has raised man above the beasts of the field.
The realm of mathematics, which we have been examining, is but one aspect of the growth of intelligence. But in theory, at least, it is among the most interesting, since in it are reached the highest abstractions of science, while its empirical beginnings are not lost. But its processes and their significance are in no way different in essence from those of the other sciences. It marks one road of specialization in the discovery of mind. And in these terms we may read all history. To quote Professor Woodbridge (Columbia University Quarterly, Dec., 1912, p. 10): "We may see man rising from the ground, startled by the first dim intimation that the things and forces about him are convertible and controllable. Curiosity excites him, but he is subdued by an untrained imagination. The things that frighten him, he tries to frighten in return. The things that bless him, he blesses. He would scare the earth's shadow from the moon and sacrifice his dearest to a propitious sky. It avails not. But the little things teach him and discipline his imagination. He has kicked the stone that bruised him only to be bruised again. So he converts the stone into a weapon and begins the subjugation of the world, singing a song of triumph by the way. Such is his history in epitome—a blunder, a conversion, a conquest, and a song. That sequence he will repeat in greater things. He will repeat it yet and rejoice where he now despairs, converting the chaos of his social, political, industrial, and emotional life into wholesome force. He will sing again. But the discovery of mind comes first, and then, the song."