youth in the fancies of the poets, and in the laws and customs of the State;—then there is the training of the body to be a warrior athlete, and a good servant of the mind;—and thirdly, after an interval follows the education of later life, which begins with mathematics and proceeds to philosophy in general.
There seem to be two great aims in the philosophy of Plato,—first, to realize abstractions; secondly, to connect them. According to him, the true education is that which draws men from becoming to being, and to a comprehensive survey of all being. He desires to develop in the human mind the faculty of seeing the universal in all things; until at last the particulars of sense drop away and the universal alone remains. He then seeks to combine the universals which he has disengaged from sense, not perceiving that the correlation of them has no other basis but the common use of language. He never understands that abstractions, as Hegel says, are 'mere abstractions'—of use when employed in the arrangement of facts, but adding nothing to the sum of knowledge when pursued apart from them, or with reference to an imaginary idea of good. Still the exercise of the faculty of abstraction apart from facts has enlarged the mind, and played a great part in the education of the human race. Plato appreciated the value of this faculty, and saw that it might be quickened by the study of number and relation. All things in which there is opposition or proportion are suggestive of reflection. The mere impression of sense evokes no power of thought or of mind, but when sensible objects ask to be compared and distinguished, then philosophy begins. The science of arithmetic first suggests such distinctions. The follow in order the other sciences of plain and solid geometry, and of solids in motion, one branch of which is astronomy or the harmony of the spheres,—to this is appended the sister science of the harmony of sounds. Plato seems also to hint at the possibility of other applications of arithmetical or mathematical proportions, such as we employ in chemistry and natural philosophy, such as the Pythagoreans and even Aristotle make use of in Ethics and Politics, e.g. his distinction between arithmetical and geometrical proportion in the Ethics (Book V), or between numerical and proportional equality in the Politics.
The modern mathematician will readily sympathise with Plato's delight in the properties of pure mathematics. He will not be disinclined to say with him:—Let alone the heavens, and study the beauties of number and figure in themselves. He too will be apt to depreciate their application to the arts. He will observe that Plato has a conception of geometry, in which figures are to be dispensed with; thus in a distant and shadowy way seeming to anticipate the possibility of working geometrical problems by a more general mode of analysis. He will remark with interest on the backward state of solid geometry, which, alas! was not encouraged by the aid of the State in the age of Plato; and he will recognize the grasp of Plato's mind in his ability to conceive of one science of solids in motion including the earth as well as the heavens,—not forgetting to notice the intimation to which allusion has been already made, that besides astronomy and harmonics the science of solids in motion may have other applications. Still more will he be struck with the comprehensiveness of view which led Plato, at a time when these sciences hardly existed, to say that they must be studied in relation to one another, and to the idea of good, or common principle of truth and being. But he will also see (and perhaps without surprise) that in that stage of physical and mathematical knowledge, Plato has fallen into the error of supposing that he can construct the heavens a priori by mathematical problems, and determine the principles of harmony irrespective of the adaptation of sounds to the human ear. The illusion was a natural one in that age and country. The simplicity and certainty of astronomy and harmonics seemed to contrast with the variation and complexity of the world of sense; hence the circumstance that there was some elementary basis of fact, some measurement of distance or time or vibrations on which they must ultimately rest, was overlooked by him. The modern predecessors of Newton fell into errors equally great; and Plato can hardly be said to have been very far wrong, or may even claim a sort of prophetic insight into the subject, when we consider that the greater part of astronomy at the present day consists of abstract dynamics, by the help of which most astronomical discoveries have been made.
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 connexion 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; 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.' 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 (Philebus).
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), 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
