I. In his book on Darwinism (Ch. XV.) Wallace, in contesting the theory that applies the law of conservation of variations, useful in the struggle for existence, to the mental faculties, insists at length upon the mathematical faculty; he maintains that it is an inexplicable exception, a case that cannot be reduced to law. The inaptitude of inferior races for even the simplest calculations is well known; how—from such a rudimentary origin—could it develop into the genius of a Newton, a Laplace, or a Gauss? What motive power accounts for this development? The author establishes by a host of sufficiently useless historical details, that mathematical superiority played no part in the struggle of tribe with tribe, and later on of people with people (Greeks against Persians), and that the victory resulted from other causes, moral and social. For this there is abundant evidence. But since mathematical aptitude is only a particular instance of abstraction, albeit one of the most perfect, the question ought to be proposed under a more general form. Had the aptitude for abstraction, ab initio, any practical value? Yes, “the motive power that caused its development, that Wallace claims without specifying it, is utility.”
To avoid possibility of equivocation, let us remark that the development of the attitude for abstracting and generalising may be explained in a two-fold manner: by acknowledging the influence of heredity, and by omitting it.
In the former case, it is supposed that this aptitude appears as a “spontaneous variation” in the individual or race, that it fixes itself, is reciprocal, grows by slow accumulation in the course of generations. This theory postulates the heredity of acquired characters, which is accepted by some, rejected by others, more especially since the advent of Weismann. I refrain from invoking it, by reason of its hypothetical and disputed nature. The probability of any transmission would moreover be far harder to establish here than in other psychical directions, such as imagination, or feeling.
In the second case, with elimination of the hereditary factor, progress must be attributed to social causes, utility and imitation. From all time there have been minds which when face to face with practical problems knew better than others how to extract the essential, and neglect the accessory, in the complex of facts. The utility of abstraction is identical with that of attention, which does not require demonstration; it may be summed up in a single word: to simplify. As the process succeeds, it finds imitators. There is no need to admit at the outset any reflected and fully conscious abstraction: a happy instinct, guided by the needs of life, is sufficient at the commencement. Races that are poorly gifted in this respect, or little apt at imitating their betters, have never got beyond a low level. In effect, abstraction and generalisation are the nerve of all knowledge that transcends sensation. Is this mode of cognition useful? There can be no possible doubt as to the answer.
2. The rôle of inventors corresponds to the fact which, in transformist terminology, is known as spontaneous variation. By inventors, we mean those who are born with the talent or the genius for abstraction. It is superfluous to prove that such have been found, in considerable numbers. They are abstract thinkers by instinct, as others are musicians, mechanicians, designers. The biography of the great mathematicians abounds in examples: Pascal inventing geometry out of a few vague indications from his father; Newton divining Euclid’s demonstrations from the simple enunciation of the theorems; Ampère, before he could read or understand the use of figures, making long calculations by means of a few pebbles; Gauss, at five years old, rectifying the arithmetic of a workman, etc. If fewer analogous facts can be quoted from the other sciences, it is because mathematical precocity is frequent, and is more surprising. All that is the effect of innate disposition: this word serving only to recapitulate our ignorance of the causes which produce such minds. In the development of knowledge by abstraction and generalisation, the first cause—utility—may be likened to the part played by slow actions in geology; whether in the case of practical inventions, or of the constitution of an idiom, it is continuous, collective, and anonymous. The rôle of the great abstract thinkers, on the contrary, resembles the rapid and epoch-making actions.
II. If we now consider the progress of abstraction from a more general point of view (instead of following it step by step, from its lowest to its highest degree, as in the preceding chapters), i. e., according to its orientation towards a given end, we find that it has followed three principal directions during its history: practical, speculative, scientific. These are, indeed, inseparable, inasmuch as practical abstraction leads to science, scientific abstraction is profitable to practice, and speculation cannot entirely forego the other two.[137]
Abstraction and practical generalisation are necessarily the first in order, as we found in studying their first appearance in the lower animals, in children, and in savages. They serve to distinguish the qualities of things by some word, or sign; they subserve the simple adaptations of daily life. Later on, at a higher stage, they note the appearance of mixed processes, which, while more especially directed to utility, are already the prelude to scientific knowledge. Disinterested curiosity has awakened, and timidly makes for daylight. A minimum acquaintance with the history of the sciences teaches us that all were at their origin processes of applied research, and that often, in their uncertain efforts, our forbears found what they were not looking for. The numerative systems issued from the need of counting objects, and later on, from rude commercial exchanges. Elementary geometry was required, in order to measure the fields, to determine a right angle, to fix relative positions, and to furnish the indispensable parts of primitive architecture. The invention of the lever, of the balance, of rudimentary engines for the lifting of heavy masses, gave the first foundations of mechanics. Astronomy arose in the desire to regulate civil life and the religious festivals, and the wish (e. g., among the Peruvians and Mexicans) not to irritate the gods by delaying the sacrifices due to them. Metallurgy, and later on the search for the philosopher’s stone, and the elixir of life, were the prelude to scientific chemistry. The historical outset of each science would furnish a profusion of similar facts.
The two other operations issued by an internal division of labor from this—at first the only—tendency of the mind.
First come purely speculative, i. e., philosophical or metaphysical abstraction and generalisation. This new trend has clean and well defined characteristics; and it was, in antiquity, the privilege of two peoples alone, the Greeks and the Hindus. Abstraction leads immediately to the highest generalisations; from the crude and direct simplification of a few facts, the mind leaps at a bound to the final causes of things; it skips the intermediate stages: it ignores the sequence of slow and progressive evolution. This procedure where, in point of fact, abstraction and generalisation are only the servants of a particular form of imagination, found its first complete expression in Plato, and the Theory of Ideas. With Plato, the human intellect tasted for the first time the supreme pleasures of playing with the highest abstractions, and believing firmly that the universe can be summed up, constructed and explained by the help of some few entities. In this direction, notwithstanding its manifold changes of aspect, the generalising process has remained fundamentally the same, and has done no more than repeat itself. We are here concerned with statement, not with criticism. Psychologists must needs admit that this tendency to construct the world (whether or no it be illusory) is a fact inherent in the nature of the human intellect. Stallo, in the book already quoted,[138] gives an incisive critique of the fundamental concepts of the physical sciences, and their unconscious trend towards metaphysics. His appreciation of the characteristics proper to the purely speculative process of abstraction and generalisation is so apt, that we cannot do better than transcribe it: