The Methods Of Mathematics

Euclid’s method of demonstration has brought forth from its own womb its most striking parody and caricature in the famous controversy on the theory of parallels, and the attempts, which are repeated every year, to prove the eleventh axiom. This axiom asserts, and indeed supports its assertion by the indirect evidence of a third intersecting line, that two lines inclining towards each other (for that is just the meaning of “less than two right angles”) if produced far enough must meet—a truth which is supposed to be too complicated to pass as self-evident, and therefore requires a demonstration. Such a demonstration, however, cannot be produced, just because there is nothing that is not immediate.

This scruple of conscience reminds me of Schiller’s question of law:—

“For years I have used my nose for smelling. Have I, then, actually a right to it that can be proved?” Indeed it seems to me that the logical method is hereby reduced to absurdity. Yet it is just through the controversies about this, together with the vain attempts to prove what is directly certain as merely indirectly certain, that the self-sufficingness and clearness of intuitive evidence appears in contrast with the uselessness and difficulty of logical proof—a contrast which is no less instructive than amusing. The direct certainty is not allowed to be valid here, because it is no mere logical certainty following from the conceptions, thus resting only upon the relation of the predicate to the subject, according to the principle of contradiction. That axiom, however, is a synthetical proposition a priori, and as such has the guarantee of pure, not empirical, perception, which is just as immediate and certain as the principle of contradiction itself, from which all demonstrations first derive their certainty. Ultimately this holds good of every geometrical theorem, and it is quite arbitrary where we draw the line between what is directly certain and what has first to be demonstrated. It surprises me that the eighth axiom is not rather attacked. “Figures which coincide with each other are equal to each other.” For “coinciding with each other” is either a mere tautology or something purely empirical which does not belong to pure perception but to external sensuous experience. It presupposes that the figures may be moved; but only matter is movable in space. Therefore this appeal to coincidence leaves pure space—the one element of geometry—in order to pass over to what is material and empirical.

The reputed motto of the Platonic lecture-room, “‘3μÉ1⁄4μÄÁ·Ä¿Â 1⁄4· ́μ1 μ1Ã1ÄÉ,” of which mathematicians are so proud, was no doubt inspired by the fact that Plato regarded the geometrical figures as intermediate existences between the eternal Ideas and particular things, as Aristotle frequently mentions in his “Metaphysics” (especially i. c. 6, p. 887, 998, et Scholia, p. 827, ed. Berol.) Moreover, the opposition between those self-existent eternal forms, or Ideas, and the transitory individual things, was most easily made comprehensible in geometrical figures, and thereby laid the foundation of the doctrine of Ideas, which is the central point of the philosophy of Plato, and indeed his only serious and decided theoretical dogma. In expounding it, therefore, he started from geometry. In the same sense we are told that he regarded geometry as a preliminary exercise through which the mind of the pupil accustomed itself to deal with incorporeal objects, having hitherto in practical life had only to do with corporeal things (Schol. in Aristot., p. 12, 15).

This, then, is the sense in which Plato recommended geometry to the philosopher; and therefore one is not justified in extending it further. I rather recommend, as an investigation of the influence of mathematics upon our mental powers, and their value for scientific culture in general, a very thorough and learned discussion, in the form of a review of a book by Whewell in the Edinburgh Review of January 1836. Its author, who afterwards published it with some other discussions, with his name, is Sir W. Hamilton, Professor of Logic and Metaphysics in Scotland. This work has also found a German translator, and has appeared by itself under the title, “Ueber den Werth und Unwerth der Mathematik” aus dem Englishen, 1836. The conclusion the author arrives at is that the value of mathematics is only indirect, and lies in the application to ends which are only attainable through them; but in themselves mathematics leave the mind where they find it, and are by no means conducive to its general culture and development, nay, even a decided hindrance. This conclusion is not only proved by thorough dianoiological investigation of the mathematical activity of the mind, but is also confirmed by a very learned accumulation of examples and authorities. The only direct use which is left to mathematics is that it can accustom restless and unsteady minds to fix their attention. Even Descartes, who was yet himself famous as a mathematician, held the same opinion with regard to mathematics. In the “Vie de Descartes par Baillet,” 1693, it is said, Liv. ii. c. 6, p. 54: “Sa propre expérience l’avait convaincu du peu d’utilité des mathématiques, surtout lorsqu’on ne les cultive que pour elles mêmes…. Il ne voyait rien de moins solide, que de s’occuper de nombres tout simples et de figures imaginaires,” &c.