[1576]. Where there is nothing to measure there is nothing to calculate, hence it is impossible to employ mathematics in psychological investigations. Thus runs the syllogism compounded of an adherence to usage and an apparent truth. As to the latter, it is wholly untrue that we may calculate only where we have measured. Exactly the opposite is true. Every hypothetically assumed law of quantitative combination, even such as is recognized as invalid, is subject to calculation; and in case of deeply hidden but important matters it is imperative to try on hypotheses and to subject the consequences which flow from them to precise computation until it is found which one of the various hypotheses coincides with experience. Thus the ancient astronomers tried eccentric circles, and Kepler tried the ellipse to account for the motion of the planets, the latter also compared the squares of the times of revolution with the cubes of the mean distances before he discovered their agreement. In like manner Newton tried whether a gravitation, varying inversely as the square of the distance, sufficed to keep the moon in its orbit about the earth; if this supposition had failed him, he would have tried some other power of the distance, as the fourth or fifth, and deduced the corresponding consequences to compare them with the observations. Just this is the greatest benefit of mathematics, that it enables us to survey the possibilities whose range includes the actual, long before we have adequate definite experience; this makes it possible to employ very incomplete indications of experience to avoid at least the crudest errors. Long before the transit of Venus was employed in the determination of the sun’s parallax, it was attempted to determine the instant at which the sun illumines exactly one-half of the moon’s disk, in order to compute the sun’s distance from the known distance of the moon from the earth. This was not possible, for, owing to psychological reasons, our method of measuring time is too crude to give us the desired instant with sufficient accuracy; yet the attempt gave us the knowledge that the sun’s distance from us is at least several hundred times as great as that of the moon. This illustration shows clearly that even a very imperfect estimate of a magnitude in a case where no precise observation is possible, may become very instructive, if we know how to exploit it. Was it necessary to know the scale of our solar system in order to learn of its order in general? Or, taking an illustration from another field, was it impossible to investigate the laws of motion until it was known exactly how far a body falls in a second at some definite place? Not at all. Such determinations of fundamental measures are in themselves exceedingly difficult, but fortunately, such investigations form a class of their own; our knowledge of fundamental laws does not need to wait on these. To be sure, computation invites measurement, and every easily observed regularity of certain magnitudes is an incentive to mathematical investigation.—Herbart, J. F.
Werke [Kehrbach], (Langensalza, 1890), Bd. 5, p. 97.
[1577]. Those who pass for naturalists, have, for the most part, been very little, or not at all, versed in mathematicks, if not also jealous of them.—Boyle, Robert.
Works (London, 1772), Vol. 3, p. 426.
[1578]. However hurtful may have been the incursions of the geometers, direct and indirect, into a domain which it is not for them to cultivate, the physiologists are not the less wrong in turning away from mathematics altogether. It is not only that without mathematics they could not receive their due preliminary training in the intervening sciences: it is further necessary for them to have geometrical and mechanical knowledge, to understand the structure and the play of the complex apparatus of the living, and especially the animal organism. Animal mechanics, statical and dynamical, must be unintelligible to those who are ignorant of the general laws of rational mechanics. The laws of equilibrium and motion are ... absolutely universal in their action, depending wholly on the energy, and not at all on the nature of the forces considered: and the only difficulty is in their numerical application in cases of complexity. Thus, discarding all idea of a numerical application in biology, we perceive that the general theorems of statics and dynamics must be steadily verified in the mechanism of living bodies, on the rational study of which they cast an indispensable light. The highest orders of animals act in repose and motion, like any other mechanical apparatus of a similar complexity, with the one difference of the mover, which has no power to alter the laws of motion and equilibrium. The participation of rational mechanics in positive biology is thus evident. Mechanics cannot dispense with geometry; and beside, we see how anatomical and physiological speculations involve considerations of form and position, and require a familiar knowledge of the principal geometrical laws which may cast light upon these complex relations.—Comte,A.
Positive Philosophy [Martineau], Bk. 5, chap. 1.
[1579]. In mathematics we find the primitive source of rationality; and to mathematics must the biologists resort for means to carry on their researches.—Comte, A.
Positive Philosophy [Martineau], Bk. 5, chap. 1.