The Republic
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,