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 cooeperation 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 (Levy-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."