The metaphysical philosopher from his point of view recognizes mathematics as an instrument of education—which strengthens the power of attention, developes the sense of order and the faculty of construction, and enables the mind to grasp under simple formulae the quantitative differences of physical phenomena. But while acknowledging their value in education, he sees also that they have no connection with our higher moral and intellectual ideas. In the attempt which Plato makes to connect them, we easily trace the influences of ancient Pythagorean notions. There is no reason to suppose that he is speaking of the ideal numbers at 525 E; but he is describing numbers which are pure abstractions, to which he assigns a real and separate existence, which, as “the teachers of the art” (meaning probably the Pythagoreans) would have affirmed, repel all attempts at subdivision, and in which unity and every other number are conceived of as absolute. The truth and certainty of numbers, when thus disengaged from phenomena, gave them a kind of sacredness in the eyes of an ancient philosopher. Nor is it easy to say how far ideas of order and fixedness may have had a moral and elevating influence on the minds of men, “who,” in the words of the “Timaeus,” “might learn to regulate their erring lives according to them” (47 C). It is worthy of remark that the old Pythagorean ethical symbols still exist as figures of speech among ourselves. And those who in modern times see the world pervaded by universal law, may also see an anticipation of this last word of modern philosophy in the Platonic idea of good, which is the source and measure of all things, and yet only an abstraction. (Compare “Philebus” sub fin.).
Two passages seem to require more particular explanations. First, that which relates to the analysis of vision. The difficulty in this passage may be explained, like many others, from differences in the modes of conception prevailing among ancient and modern thinkers. To us, the perceptions of sense are inseparable from the act of the mind which accompanies them. The consciousness of form, colour, distance, is indistinguishable from the simple sensation, which is the medium of them. Whereas to Plato sense is the Heraclitean flux of sense, not the vision of objects in the order in which they actually present themselves to the experienced sight, but as they may be imagined to appear confused and blurred to the half-awakened eye of the infant. The first action of the mind is aroused by the attempt to set in order this chaos, and the reason is required to frame distinct conceptions under which the confused impressions of sense may be arranged. Hence arises the question, “What is great, what is small?” and thus begins the distinction of the visible and the intelligible.
The second difficulty relates to Plato’s conception of harmonics. Three classes of harmonists are distinguished by him:—first, the Pythagoreans, whom he proposes to consult as in the previous discussion on music he was to consult Damon—they are acknowledged to be masters in the art, but are altogether deficient in the knowledge of its higher import and relation to the good; secondly, the mere empirics, whom Glaucon appears to confuse with them, and whom both he and Socrates ludicrously describe as experimenting by mere auscultation on the intervals of sounds. Both of these fall short in different degrees of the Platonic idea of harmony, which must be studied in a purely abstract way, first by the method of problems, and secondly as a part of universal knowledge in relation to the idea of good.
The allegory has a political as well as a philosophical meaning. The den or cave represents the narrow sphere of politics or law (compare the description of the philosopher and lawyer in the “Theaetetus,” 172–176), and the light of the eternal ideas is supposed to exercise a disturbing influence on the minds of those who return to this lower world. In other words, their principles are too wide for practical application; they are looking far away into the past and future, when their business is with the present. The ideal is not easily reduced to the conditions of actual life, and may often be at variance with them. And at first, those who return are unable to compete with the inhabitants of the den in the measurement of the shadows, and are derided and persecuted by them; but after a while they see the things below in far truer proportions than those who have never ascended into the upper world. The difference between the politician turned into a philosopher and the philosopher turned into a politician, is symbolized by the two kinds of disordered eyesight, the one which is experienced by the captive who is transferred from darkness to day, the other, of the heavenly messenger who voluntarily for the good of his fellow-men descends into the den. In what way the brighter light is to dawn on the inhabitants of the lower world, or how the idea of good is to become the guiding principle of politics, is left
