Chapter XIII.[25] On 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 [pg 322] 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, “Αγεωμετρητος μηδεις εισιτω,” 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 [pg 323] 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.
Chapter XIV. On The Association Of Ideas.
The presence of ideas and thoughts in our consciousness is as strictly subordinated to the principle of sufficient reason in its different forms as the movement of bodies to the law of causality. It is just as little possible that a thought can appear in the mind without an occasion as that a body can be set in motion without a cause. Now this occasion is either external, thus an impression of the senses, or internal, thus itself also a thought which introduces another thought by means of association. This again depends either upon a relation of reason and consequent between the two; or upon similarity, even mere analogy; or lastly upon the circumstance that they were both first apprehended at the same time, which again may have its ground in the proximity in space of their objects. The last two cases are denoted by the word à propos. The predominance of one of these three bonds of association of thoughts over the others is characteristic of the intellectual worth of the man. The first named will predominate in thoughtful and profound minds, the second in witty, ingenious, and poetical minds, and the third in minds of limited capacity. Not less characteristic is the degree of facility with which one thought recalls others that stand in any kind of relation to it: this constitutes the activeness of the mind. But the impossibility of the appearance of a thought without its sufficient occasion, even when there is the strongest desire to call it up, is proved by all the cases in which we weary [pg 325] ourselves in vain to recollect something, and go through the whole store of our thoughts in order to find any one that may be associated with the one we seek; if we find the former, the latter is also found. Whoever wishes to call up something in his memory first seeks for a thread with which it is connected by the association of thoughts. Upon this depends mnemonics: it aims at providing us with easily found occasioners or causes for all the conceptions, thoughts, or words which are to be preserved. But the worst of it is that these occasioners themselves have first to be recalled, and this again requires an occasioner. How much the occasion accomplishes in memory may be shown in this way. If we have read in a book of anecdotes say fifty anecdotes, and then have laid it aside, immediately afterwards we will sometimes be unable to recollect a single one of them. But if the occasion comes, or if a thought occurs to us which has any analogy with one of those anecdotes, it immediately comes back to us; and so with the whole fifty as opportunity offers. The same thing holds good of all that we read. Our immediate remembrance of words, that is, our remembrance of them without the assistance of mnemonic contrivances, and with it our whole faculty of speech, ultimately depends upon the direct association of thoughts. For the learning of language consists in this, that once for all we so connect a conception with a word that this word will always occur to us along with this conception, and this conception will always occur to us along with this word. We have afterwards to repeat the same process in learning every new language; yet if we learn a language for passive and not for active use—that is, to read, but not to speak, as, for example, most of us learn Greek—then the connection is one-sided, for the conception occurs to us along with the word, but the word does not always occur to us along with the conception. The same procedure as in language becomes apparent in the particular case, in the learning of [pg 326] every new proper name. But sometimes we do not trust ourselves to connect directly the name of this person, or town, river, mountain, plant, animal, &c., with the thought of each so firmly that it will call each of them up of itself; and then we assist ourselves mnemonically, and connect the image of the person or thing with any perceptible quality the name of which occurs in that of the person or thing. Yet this is only a temporary prop to lean on; later we let it drop, for the association of thoughts becomes an immediate support.