All these sciences are the prelude of the strain, and are profitable if they are regarded in their natural relations to one another. 'I dare say, Socrates,' said Glaucon; 'but such a study will be an endless business.' What study do you mean – of the prelude, or what? For all these things are only the prelude, and you surely do not suppose that a mere mathematician is also a dialectician? 'Certainly not. I have hardly ever known a mathematician who could reason.' And yet, Glaucon, is not true reasoning that hymn of dialectic which is the music of the intellectual world, and which was by us compared to the effort of sight, when from beholding the shadows on the wall we arrived at last at the images which gave the shadows? Even so the dialectical faculty withdrawing from sense arrives by the pure intellect at the contemplation of the idea of good, and never rests but at the very end of the intellectual world. And the royal road out of the cave into the light, and the blinking of the eyes at the sun and turning to contemplate the shadows of reality, not the shadows of an image only – this progress and gradual acquisition of a new faculty of sight by the help of the mathematical sciences, is the elevation of the soul to the contemplation of the highest ideal of being.

'So far, I agree with you. But now, leaving the prelude, let us proceed to the hymn. What, then, is the nature of dialectic, and what are the paths which lead thither?' Dear Glaucon, you cannot follow me here. There can be no revelation of the absolute truth to one who has not been disciplined in the previous sciences. But that there is a science of absolute truth, which is attained in some way very different from those now practised, I am confident. For all other arts or sciences are relative to human needs and opinions; and the mathematical sciences are but a dream or hypothesis of true being, and never analyse their own principles. Dialectic alone rises to the principle which is above hypotheses, converting and gently leading the eye of the soul out of the barbarous slough of ignorance into the light of the upper world, with the help of the sciences which we have been describing – sciences, as they are often termed, although they require some other name, implying greater clearness than opinion and less clearness than science, and this in our previous sketch was understanding. And so we get four names – two for intellect, and two for opinion, – reason or mind, understanding, faith, perception of shadows – which make a proportion – being: becoming::intellect: opinion – and science: belief::understanding: perception of shadows. Dialectic may be further described as that science which defines and explains the essence or being of each nature, which distinguishes and abstracts the good, and is ready to do battle against all opponents in the cause of good. To him who is not a dialectician life is but a sleepy dream; and many a man is in his grave before his is well waked up. And would you have the future rulers of your ideal State intelligent beings, or stupid as posts? 'Certainly not the latter.' Then you must train them in dialectic, which will teach them to ask and answer questions, and is the coping-stone of the sciences.

I dare say that you have not forgotten how our rulers were chosen; and the process of selection may be carried a step further: – As before, they must be constant and valiant, good-looking, and of noble manners, but now they must also have natural ability which education will improve; that is to say, they must be quick at learning, capable of mental toil, retentive, solid, diligent natures, who combine intellectual with moral virtues; not lame and one-sided, diligent in bodily exercise and indolent in mind, or conversely; not a maimed soul, which hates falsehood and yet unintentionally is always wallowing in the mire of ignorance; not a bastard or feeble person, but sound in wind and limb, and in perfect condition for the great gymnastic trial of the mind. Justice herself can find no fault with natures such as these; and they will be the saviours of our State; disciples of another sort would only make philosophy more ridiculous than she is at present. Forgive my enthusiasm; I am becoming excited; but when I see her trampled underfoot, I am angry at the authors of her disgrace. 'I did not notice that you were more excited than you ought to have been.' But I felt that I was. Now do not let us forget another point in the selection of our disciples – that they must be young and not old. For Solon is mistaken in saying that an old man can be always learning; youth is the time of study, and here we must remember that the mind is free and dainty, and, unlike the body, must not be made to work against the grain. Learning should be at first a sort of play, in which the natural bent is detected. As in training them for war, the young dogs should at first only taste blood; but when the necessary gymnastics are over which during two or three years divide life between sleep and bodily exercise, then the education of the soul will become a more serious matter. At twenty years of age, a selection must be made of the more promising disciples, with whom a new epoch of education will begin. The sciences which they have hitherto learned in fragments will now be brought into relation with each other and with true being; for the power of combining them is the test of speculative and dialectical ability. And afterwards at thirty a further selection shall be made of those who are able to withdraw from the world of sense into the abstraction of ideas. But at this point, judging from present experience, there is a danger that dialectic may be the source of many evils. The danger may be illustrated by a parallel case: – Imagine a person who has been brought up in wealth and luxury amid a crowd of flatterers, and who is suddenly informed that he is a supposititious son. He has hitherto honoured his reputed parents and disregarded the flatterers, and now he does the reverse. This is just what happens with a man's principles. There are certain doctrines which he learnt at home and which exercised a parental authority over him. Presently he finds that imputations are cast upon them; a troublesome querist comes and asks, 'What is the just and good?' or proves that virtue is vice and vice virtue, and his mind becomes unsettled, and he ceases to love, honour, and obey them as he has hitherto done. He is seduced into the life of pleasure, and becomes a lawless person and a rogue. The case of such speculators is very pitiable, and, in order that our thirty years' old pupils may not require this pity, let us take every possible care that young persons do not study philosophy too early. For a young man is a sort of puppy who only plays with an argument; and is reasoned into and out of his opinions every day; he soon begins to believe nothing, and brings himself and philosophy into discredit. A man of thirty does not run on in this way; he will argue and not merely contradict, and adds new honour to philosophy by the sobriety of his conduct. What time shall we allow for this second gymnastic training of the soul? – say, twice the time required for the gymnastics of the body; six, or perhaps five years, to commence at thirty, and then for fifteen years let the student go down into the den, and command armies, and gain experience of life. At fifty let him return to the end of all things, and have his eyes uplifted to the idea of good, and order his life after that pattern; if necessary, taking his turn at the helm of State, and training up others to be his successors. When his time comes he shall depart in peace to the islands of the blest. He shall be honoured with sacrifices, and receive such worship as the Pythian oracle approves.

'You are a statuary, Socrates, and have made a perfect image of our governors.' Yes, and of our governesses, for the women will share in all things with the men. And you will admit that our State is not a mere aspiration, but may really come into being when there shall arise philosopher-kings, one or more, who will despise earthly vanities, and will be the servants of justice only. 'And how will they begin their work?' Their first act will be to send away into the country all those who are more than ten years of age, and to proceed with those who are left…

At the commencement of the sixth book, Plato anticipated his explanation of the relation of the philosopher to the world in an allegory, in this, as in other passages, following the order which he prescribes in education, and proceeding from the concrete to the abstract. At the commencement of Book VII, under the figure of a cave having an opening towards a fire and a way upwards to the true light, he returns to view the divisions of knowledge, exhibiting familiarly, as in a picture, the result which had been hardly won by a great effort of thought in the previous discussion; at the same time casting a glance onward at the dialectical process, which is represented by the way leading from darkness to light. The shadows, the images, the reflection of the sun and stars in the water, the stars and sun themselves, severally correspond, – the first, to the realm of fancy and poetry, – the second, to the world of sense, – the third, to the abstractions or universals of sense, of which the mathematical sciences furnish the type, – the fourth and last to the same abstractions, when seen in the unity of the idea, from which they derive a new meaning and power. The true dialectical process begins with the contemplation of the real stars, and not mere reflections of them, and ends with the recognition of the sun, or idea of good, as the parent not only of light but of warmth and growth. To the divisions of knowledge the stages of education partly answer: – first, there is the early education of childhood and 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 the inhabitants of the lower world, or how the idea of good is to become the guiding principle of politics, is left unexplained by Plato. Like the nature and divisions of dialectic, of which Glaucon impatiently demands to be informed, perhaps he would have said that the explanation could not be given except to a disciple of the previous sciences. (Symposium.)

Many illustrations of this part of the Republic may be found in modern Politics and in daily life. For among ourselves, too, there have been two sorts of Politicians or Statesmen, whose eyesight has become disordered in two different ways. First, there have been great men who, in the language of Burke, 'have been too much given to general maxims,' who, like J.S. Mill or Burke himself, have been theorists or philosophers before they were politicians, or who, having been students of history, have allowed some great historical parallel, such as the English Revolution of 1688, or possibly Athenian democracy or Roman Imperialism, to be the medium through which they viewed contemporary events. Or perhaps the long projecting shadow of some existing institution may have darkened their vision. The Church of the future, the Commonwealth of the future, the Society of the future, have so absorbed their minds, that they are unable to see in their true proportions the Politics of to-day. They have been intoxicated with great ideas, such as liberty, or equality, or the greatest happiness of the greatest number, or the brotherhood of humanity, and they no longer care to consider how these ideas must be limited in practice or harmonized with the conditions of human life. They are full of light, but the light to them has become only a sort of luminous mist or blindness. Almost every one has known some enthusiastic half-educated person, who sees everything at false distances, and in erroneous proportions.

With this disorder of eyesight may be contrasted another – of those who see not far into the distance, but what is near only; who have been engaged all their lives in a trade or a profession; who are limited to a set or sect of their own. Men of this kind have no universal except their own interests or the interests of their class, no principle but the opinion of persons like themselves, no knowledge of affairs beyond what they pick up in the streets or at their club. Suppose them to be sent into a larger world, to undertake some higher calling, from being tradesmen to turn generals or politicians, from being schoolmasters to become philosophers: – or imagine them on a sudden to receive an inward light which reveals to them for the first time in their lives a higher idea of God and the existence of a spiritual world, by this sudden conversion or change is not their daily life likely to be upset; and on the other hand will not many of their old prejudices and narrownesses still adhere to them long after they have begun to take a more comprehensive view of human things? From familiar examples like these we may learn what Plato meant by the eyesight which is liable to two kinds of disorders.

Nor have we any difficulty in drawing a parallel between the young Athenian in the fifth century before Christ who became unsettled by new ideas, and the student of a modern University who has been the subject of a similar 'aufklarung.' We too observe that when young men begin to criticise customary beliefs, or to analyse the constitution of human nature, they are apt to lose hold of solid principle (Greek). They are like trees which have been frequently transplanted. The earth about them is loose, and they have no roots reaching far into the soil. They 'light upon every flower,' following their own wayward wills, or because the wind blows them. They catch opinions, as diseases are caught – when they are in the air. Borne hither and thither, 'they speedily fall into beliefs' the opposite of those in which they were brought up. They hardly retain the distinction of right and wrong; they seem to think one thing as good as another. They suppose themselves to be searching after truth when they are playing the game of 'follow my leader.' They fall in love 'at first sight' with paradoxes respecting morality, some fancy about art, some novelty or eccentricity in religion, and like lovers they are so absorbed for a time in their new notion that they can think of nothing else. The resolution of some philosophical or theological question seems to them more interesting and important than any substantial knowledge of literature or science or even than a good life. Like the youth in the Philebus, they are ready to discourse to any one about a new philosophy. They are generally the disciples of some eminent professor or sophist, whom they rather imitate than understand. They may be counted happy if in later years they retain some of the simple truths which they acquired in early education, and which they may, perhaps, find to be worth all the rest. Such is the picture which Plato draws and which we only reproduce, partly in his own words, of the dangers which beset youth in times of transition, when old opinions are fading away and the new are not yet firmly established. Their condition is ingeniously compared by him to that of a supposititious son, who has made the discovery that his reputed parents are not his real ones, and, in consequence, they have lost their authority over him.

The distinction between the mathematician and the dialectician is also noticeable. Plato is very well aware that the faculty of the mathematician is quite distinct from the higher philosophical sense which recognizes and combines first principles. The contempt which he expresses for distinctions of words, the danger of involuntary falsehood, the apology which Socrates makes for his earnestness of speech, are highly characteristic of the Platonic style and mode of thought. The quaint notion that if Palamedes was the inventor of number Agamemnon could not have counted his feet; the art by which we are made to believe that this State of ours is not a dream only; the gravity with which the first step is taken in the actual creation of the State, namely, the sending out of the city all who had arrived at ten years of age, in order to expedite the business of education by a generation, are also truly Platonic. (For the last, compare the passage at the end of the third book, in which he expects the lie about the earthborn men to be believed in the second generation.)

BOOK VIII. And so we have arrived at the conclusion, that in the perfect State wives and children are to be in common; and the education and pursuits of men and women, both in war and peace, are to be common, and kings are to be philosophers and warriors, and the soldiers of the State are to live together, having all things in common; and they are to be warrior athletes, receiving no pay but only their food, from the other citizens. Now let us return to the point at which we digressed. 'That is easily done,' he replied: 'You were speaking of the State which you had constructed, and of the individual who answered to this, both of whom you affirmed to be good; and you said that of inferior States there were four forms and four individuals corresponding to them, which although deficient in various degrees, were all of them worth inspecting with a view to determining the relative happiness or misery of the best or worst man. Then Polemarchus and Adeimantus interrupted you, and this led to another argument, – and so here we are.' Suppose that we put ourselves again in the same position, and do you repeat your question. 'I should like to know of what constitutions you were speaking?' Besides the perfect State there are only four of any note in Hellas: – first, the famous Lacedaemonian or Cretan commonwealth; secondly, oligarchy, a State full of evils; thirdly, democracy, which follows next in order; fourthly, tyranny, which is the disease or death of all government. Now, States are not made of 'oak and rock,' but of flesh and blood; and therefore as there are five States there must be five human natures in individuals, which correspond to them. And first, there is the ambitious nature, which answers to the Lacedaemonian State; secondly, the oligarchical nature; thirdly, the democratical; and fourthly, the tyrannical. This last will have to be compared with the perfectly just, which is the fifth, that we may know which is the happier, and then we shall be able to determine whether the argument of Thrasymachus or our own is the more convincing. And as before we began with the State and went on to the individual, so now, beginning with timocracy, let us go on to the timocratical man, and then proceed to the other forms of government, and the individuals who answer to them.

But how did timocracy arise out of the perfect State? Plainly, like all changes of government, from division in the rulers. But whence came division? 'Sing, heavenly Muses,' as Homer says; – let them condescend to answer us, as if we were children, to whom they put on a solemn face in jest. 'And what will they say?' They will say that human things are fated to decay, and even the perfect State will not escape from this law of destiny, when 'the wheel comes full circle' in a period short or long. Plants or animals have times of fertility and sterility, which the intelligence of rulers because alloyed by sense will not enable them to ascertain, and children will be born out of season. For whereas divine creations are in a perfect cycle or number, the human creation is in a number which declines from perfection, and has four terms and three intervals of numbers, increasing, waning, assimilating, dissimilating, and yet perfectly commensurate with each other. The base of the number with a fourth added (or which is 3:4), multiplied by five and cubed, gives two harmonies: – the first a square number, which is a hundred times the base (or a hundred times a hundred); the second, an oblong, being a hundred squares of the rational diameter of a figure the side of which is five, subtracting one from each square or two perfect squares from all, and adding a hundred cubes of three. This entire number is geometrical and contains the rule or law of generation. When this law is neglected marriages will be unpropitious; the inferior offspring who are then born will in time become the rulers; the State will decline, and education fall into decay; gymnastic will be preferred to music, and the gold and silver and brass and iron will form a chaotic mass – thus division will arise. Such is the Muses' answer to our question. 'And a true answer, of course: – but what more have they to say?' They say that the two races, the iron and brass, and the silver and gold, will draw the State different ways; – the one will take to trade and moneymaking, and the others, having the true riches and not caring for money, will resist them: the contest will end in a compromise; they will agree to have private property, and will enslave their fellow-citizens who were once their friends and nurturers. But they will retain their warlike character, and will be chiefly occupied in fighting and exercising rule. Thus arises timocracy, which is intermediate between aristocracy and oligarchy.