The mathematical sciences Plato has expressly excluded from philosophy. He considers them as helps to philosophical thinking (Rep. VII. 526), as necessary steps of knowledge, without which no one can come to philosophy (Ib. VI. 510); but mathematics with him is not philosophy, for it assumes its principles or axioms, without at all accounting for them, as though they were manifest to all, a procedure which is not permitted to pure science; it also serves itself for its demonstrations, with illustrative figures, although it does not treat of these, but of that which they represent to the understanding (Ib.). Plato thus places mathematics midway between a correct opinion and science, clearer than the one, but more obscure than the other. (Ib. VII. 533.)
IV. The Platonic Dialectics. 1. Conception of Dialectics.—The conception of dialectics or of logic, is used by the ancients for the most part in a very wide sense, while Plato employs it in repeated instances interchangeably with philosophy, though on the other hand he treats it also as a separate branch of philosophy. He divides it from physics as the science of the eternal and unchangeable from the science of the changeable, which never is, but is only ever becoming; he distinguishes also between it and ethics, so far as the latter treats of the good not absolutely, but in its concrete exhibition in morals and in the state; so that dialectics may be termed philosophy in a higher sense, while physics and ethics follow it as two less exact sciences, or as a not yet perfected philosophy. Plato himself defines dialectics, according to the ordinary signification of the word, as the art of developing knowledge by way of dialogue in questions and answers. (Rep. VII. 534). But since the art of communicating correctly in dialogue is according to Plato, at the same time the art of thinking correctly, and as thus thinking and speaking could not be separated by the ancients, but every process of thought was a living dialogue, so Plato would more accurately define dialectics as the science which brings speech to a correct issue, and which combines or separates the species, i. e. the conceptions of things correctly with one another. (Soph. p. 253. Phædr. p. 266). Dialectics with him has two divisions, to know what can and what cannot be connected, and to know how division or combination can be. But as with Plato these conceptions of species or ideas are the only actual and true existence, so have we, in entire conformity with this, a third definition of dialectics (Philebus p. 57), as the science of being, the science of that which is true and unchangeable, the science of all other sciences. We may therefore briefly characterize it as the science of absolute being or of ideas.
2. What Is Science? (1.) As opposed to sensation and the sensuous representation.—The Theatætus is devoted to the discussion of this question in opposition to the Protagorean sensualism. That all knowledge consists in perception, and that the two are one and the same thing, was the Protagorean proposition. From this it followed, as Protagoras himself had inferred, that things are, as they appear to me, that the perception or sensation is infallible. But since perception and sensation are infinitely diversified with different individuals, and even greatly vary in the same individual, it follows farther, that there are no objective determinations and predicates, that we can never affirm what a thing is in itself, that all conceptions, great, small, light, heavy, to increase, to diminish, &c., have only a relative significance, and consequently, also, the conceptions of species, as combinations of the changeful many, are wholly wanting in constancy and stability. In opposition to this Protagorean thesis, Plato urges the following objections and contradictions. First. The Protagorean doctrine leads to the most startling consequences. If being and appearance, knowledge and perception are one and the same thing, then is the irrational brute, which is capable of perception, as fully entitled to be called the measure of all things, as man, and if the representation is infallible, as the expression of my subjective character at a given time, then need there be no more instruction, no more scientific conclusion, no more strife, and no more refutation. Second. The Protagorean doctrine is a logical contradiction; for according to it Protagoras must yield the question to every one who disputes with him, since, as he himself affirms, no one is incorrect, but every one judges only according to truth; the pretended truth of Protagoras is therefore true for no man, not even for himself. Third. Protagoras destroys the knowledge of future events. That which I may regard as profitable may not therefore certainly prove itself as such in the result. To determine that which is really profitable implies a calculation of the future, but since the ability of men to form such a calculation is very diverse, it follows from this that not man as such, but only the wise man can be the measure of things. Fourth. The theory of Protagoras destroys perception. Perception, according to him, rests upon a distinction of the perceived object and the perceiving subject, and is the common product of the two. But in his view the objects are in such an uninterrupted flow, that they can neither become fixed in seeing nor in hearing. This condition of constant change renders all knowledge from sense, and hence (the identity of the two being assumed), all knowledge impossible. Fifth. Protagoras overlooks the apriori element in knowledge. It is seen in an analysis of the sense-perception itself, that all knowledge cannot be traced to the activity of the senses, but that there must also be presupposed besides these, intellectual functions, and hence an independent province of supersensible knowledge. We see with the eyes, and hear with the ears, but to group together the perceptions attained through these different organs, and to hold them fast in the unity of self-consciousness, is beyond the power of the activity of the senses. Again, we compare the different sense-perceptions with one another, a function which cannot belong to the senses, since each sense can only furnish its own distinctive perception. Still farther, we bring forward determinations respecting the perceptions which we manifestly cannot owe to the senses, in that we predicate of these perceptions, being and not-being, likeness and unlikeness, &c. These determinations, to which also belong the beautiful and the odious, good and evil, constitute a peculiar province of knowledge, which the soul, independently of every sense-perception, brings forward through its own independent activity. The ethical element of this Plato exhibits in his attack upon sensualism, and also in other dialogues. He maintains (in the Sophist), that men holding such opinions must be improved before they can be instructed, and that when made morally better, they will readily recognize the truth of the soul and its moral and rational capacities, and affirm that these are real things, though objects of neither sight nor of feeling.
(2.) The Relation of Knowing to Opinion.—Opinion is just as little identical with knowing as is the sense-perception. An incorrect opinion is certainly different from knowing, and a correct one is not the same, for it can be engendered by the art of speech without therefore attaining the validity of true knowledge. The correct opinion, so far as it is true in matter though imperfect in form, stands rather midway between knowing and not-knowing, and participates in both.
(3.) The Relation of Science to Thinking.—In opposition to the Protagorean sensualism, we have already referred to an energy of the soul independent of the sensuous perception and sensation, competent in itself to examine the universal, and grasp true being in thought. There is, therefore, a double source of knowledge, sensation and rational thinking. Sensation refers to that which is conceived in the constant becoming and perpetual change, to the pure momentary, which is in an incessant transition from the was, through the now, into the shall be (Parm. p. 152); it is, therefore, the source of dim, impure, and uncertain knowledge; thinking on the other hand refers to the abiding, which neither becomes nor departs, but remains ever the same. (Tim. p. 51.) Existence, says the Timæus (p. 27) is of two kinds, “that which ever is but has no becoming, and that which ever becomes but never is. The one kind, which is always in the same state, is comprehended through reflection by the reason, the other, which becomes and departs, but never properly is, may be apprehended by the sensuous perception without the reason.” True science, therefore, flows alone from that pure and thoroughly internal activity of the soul which is free from all corporeal qualities and every sensuous disturbance. (Phæd. p. 65.) In this state the soul looks upon things purely as they are (Phæd. p. 66) in their eternal being and their unchangeable condition. Hence the true state of the philosopher is announced in the Phædon (p. 64) to be a willingness to die, a longing to fly from the body, as from a hinderance to true knowledge, and become pure spirit. According to all this, science is the thinking of true being or of ideas; the means to discover and to know these ideas, or the organ for their apprehension is the dialectic, as the art of separating and combining conceptions; the true objects of dialectics are ideas.
3. The Doctrine of Ideas in its Genesis.—The Platonic doctrine of ideas is the common product of the Socratic method of forming conceptions, the Heraclitic doctrine of absolute becoming, and the Eleatic doctrine of absolute being. To the first of these Plato owes the idea of a knowing through conceptions, to the second the recognition of the becoming in the field of the sensuous, to the third the position of a field of absolute reality. Elsewhere (in the Philebus) Plato connects the doctrine of ideas with the Pythagorean thought that every thing may be formed from unity and multiplicity, from the limit and the unlimited. The aim of the Theatætus, the Sophist, and the Parmenides is to refute the principles of the Eleatics and Heraclitics: this refutation is effected in the Theatætus by combating directly the principle of an absolute becoming, in the Sophist by combating directly the principle of abstract being, and in the Parmenides by taking up the Eleatic one and showing its true relations. We have already spoken of the Theatætus; we will now look for the development of the doctrine of ideas in the Sophist and Parmenides.
The ostensible end of the former of these dialogues is to show that the Sophist is really but a caricature of the philosopher, but its true end is to fix the reality of the appearance, i. e. of the not-being, and to discuss speculatively the relation of being and not-being. The doctrine of the Eleatics ended with the rejection of all sensuous knowledge, declaring that what we receive as the perception of a multiplicity of things or of a becoming is only an appearance. In this the contradiction was clear, the not-being was absolutely denied, and yet its existence was admitted in the notion of men. Plato at once draws attention to this contradiction, showing that a delusive opinion, which gives rise to a false image or representation, is not possible, since the whole theory rests upon the assumption that the false, the not-true, i. e. not-being cannot even be thought. This, Plato continues, is the great difficulty in thinking of not-being, that both he who denies and he who affirms its reality is driven to contradict himself. For though it is inexpressible and inconceivable either as one or as many, still, when speaking of it, we must attribute to it both being and multiplicity. If we admit that there is such a thing as a false opinion, we assume in this very fact the notion of not-being, for only that opinion can be said to be false which supposes either the not-being to be, or makes that, which is not, to be. In short, if there actually exists a false notion, so does there actually and truly exist a not-being. After Plato had thus fixed the reality of not-being, he discusses the relation of being and not-being, i. e. the relation of conceptions generally in their combinations and differences. If not-being has no less reality than being, and being no more than not-being, if, therefore, e. g. the not-great is as truly real as the great, then every conception may be apprehended according to its opposite sides as being and not-being at the same time: it is a being in reference to itself, as something identical with itself, but it is not-being in reference to every one of the numberless other conceptions which can be referred to it, and with which, on account of its difference from them, it can have nothing in common. The conception of the same ταὐτὸν and the different θάτερον represent the general form of an antithesis. These are the universal formulæ of combination for all conceptions. This reciprocal relation of conceptions as at the same time being and not-being, by virtue of which they can be arranged among themselves, forms now the basis for the art of dialectics, which has to judge what conceptions can and what cannot be joined together. Plato illustrates here by taking the conceptions of being, motion (becoming), and rest (existence), and showing what are the results of the combinations of these ideas. The conceptions of motion and rest cannot well be joined together, though both of them may be joined with that of being, since both are; the conception of rest is therefore in reference to itself a being, but in reference to the conception of motion a not-being or different. Thus the Platonic doctrine of ideas, after having in the Theatætus attained its general foundation in fixing the objective reality of conceptions, becomes now still farther developed in the Sophist to a doctrine of the agreement and disagreement of conceptions. The category which conditions these reciprocal relations is that of not-being or difference. This fundamental thought of the Sophist, that being is not without not-being and not-being is not without being, may be expressed in modern phraseology thus: negation is not not-being but determinateness, and on the other hand all determinateness and concreteness of conceptions, or every thing affirmative can be only through negation; in other words the conception of contradiction is the soul of a philosophical method.
The doctrine of ideas appears in the Parmenides as the positive consequence and progressive development of the Eleatic principle. Indeed in this dialogue, in that Plato makes Parmenides the chief speaker, he seems willing to allow that his doctrine is in substance that of the Eleatic sage. True, the fundamental thought of the dialogue—that the one is not conceivable in its complete singleness without the many, nor the many without the one, that each necessarily presupposes and reciprocally conditions the other—stands in the most direct contradiction to Eleaticism. Yet Parmenides himself, by dividing his poem into two parts, and treating in the first of the one and in the second of the many, postulates an inner mediation between these two externally so disjointed parts of his philosophy, and in this respect the Platonic theory of ideas might give itself out as the farther elimination, and the true sense of the Parmenidean philosophizing. This dialectical mediation between the one and the not-one or the many Plato now attempts in four antinomies, which have ostensibly only a negative result in so far as they show that contradictions arise both whether the one be adopted or rejected. The positive sense of these antinomies, though it can be gained only through inferences which Plato himself does not expressly utter, but leaves to be drawn by the reader—is as follows. The first antinomy shows that the one is inconceivable as such since it is only apprehended in its abstract opposition to the many; the second, that in this case also the reality of the many is inconceivable; the third, that the one or the idea cannot be conceived as not-being, since there can be neither conception nor predicate of the absolute not-being, and since, if not-being is excluded from all fellowship with being, all becoming and departing, all similarity and difference, every representation and explanation concerning it must also be denied; and lastly, the fourth affirms that the not-one or the many cannot be conceived without the one or the idea. What now is Plato’s aim in this discussion of the dialectic relations between the conceptions of the one and the many? Would he use the conception of the one only as an example to explain his dialectic method with conceptions, or is the discussion of this conception itself the very object before him? Manifestly the latter, or the dialogue ends without result and without any inner connection of its two parts. But how came Plato to make such a special investigation of this conception of the one? If we bear in mind that the Eleatics had already perceived the antithesis of the actual and the phenomenal world in the antithesis of the one and the many, and that Plato himself had also regarded his ideas as the unity of the manifold, as the one and the same in the many—since he repeatedly uses “idea” and “the one” in the same sense, and places (Rep. VII. 537) dialectics in the same rank with the faculty of bringing many to unity—then is it clear that the one which is made an object of investigation in the Parmenides is the idea in its general sense, i. e. in its logical form, and that Plato consequently in the dialectic of the one and the many would represent the dialectic of the idea and the phenomenal world, or in other words would dialectically determine and establish the correct view of the idea as the unity in the manifoldness of the phenomenal. In that it is shown in the Parmenides, on the one side, that the many cannot be conceived without the one, and on the other side, that the one must be something which embraces in itself manifoldness, so have we the ready inference on the one side, that the phenomenal world, or the many, has a true being only in so far as it has the one or the conception within it, and on the other side, that since the conception is not an abstract one but manifoldness in unity, it must actually have manifoldness in unity in order to be able to be in the phenomenal world. The indirect result of the Parmenides is that matter as the infinitely divisible and undetermined mass has no actuality, but is in relation to the ideal world a not-being, and though the ideas as the true being gain their appearance in it, yet the idea itself is all that is actual in the appearance or phenomenon; the phenomenal world derives its whole existence from the ideal world which appears in it, and has a being only so far as it has a conception or idea for its content.
4. Positive Exposition of the Doctrine of Ideas.—Ideas may be defined according to the different sides of their historical connection, as the common in the manifold, the universal in the particular, the one in the many, or the constant and abiding in the changing. Subjectively they are principles of knowing which cannot be derived from experience they are the intuitively certain and innate regulators of our knowledge. Objectively they are the immutable principles of being and of the phenomenal world, incorporeal and simple unities which have no relation to space, and which may be predicated of every independent thing. The doctrine of ideas grew originally out of the desire to give a definite conception to the inner essence of things, and make the real world conceivable as a harmoniously connected intellectual world. This desire of scientific knowledge Aristotle cites expressly as the motive to the Platonic doctrine of ideas. “Plato,” he says (Metaph. XIII. 4), “came to the doctrine of ideas because he was convinced of the truth of the Heraclitic view which regarded the sensible world as a ceaseless flowing and changing. His conclusion from this was, that if there be a science of any thing there must be, besides the sensible, other substances which have a permanence, for there can be no science of the fleeting.” It is, therefore, the idea of science which demands the reality of ideas, a demand which cannot be granted unless an idea or conception is also the ground of all being. This is the case with Plato. According to him there can be neither a true knowing nor a true being without ideas and conceptions which have an independent reality.
What now does Plato mean by idea? From what has already been said it is clear that he means something more than ideal conceptions of the beautiful and the good. An idea is found, as the name itself (εἰδος) indicates, wherever a universal conception of a species or kind is found. Hence Plato speaks of the idea of a bed, table, strength, health, voice, color, ideas of simple relations and properties, ideas of mathematical figures, and even ideas of not-being, and of that, which in its essence only contradicts the idea, baseness and vice. In a word, we may put an idea wherever many things may be characterized by a common name (Rep. X. 596): or as Aristotle expresses it (Met. XII. 3). Plato places an idea to every class of being. In this sense Plato himself speaks in the beginning of the Parmenides. Parmenides asks the young Socrates what he calls ideas. Socrates answers by naming unconditionally the moral ideas, the ideas of the true, the beautiful, the good, and then after a little delay he mentions some physical ones, as the ideas of man, of fire, of water; he will not allow ideas to be predicated of that which is only a formless mass, or which is a part of something else, as hair, mud and clay, but in this he is answered by Parmenides, that if he would be fully imbued with philosophy, he must not consider such things as these to be wholly despicable, but should look upon them as truly though remotely participating in the idea. Here at least the claim is asserted that no province of being is excluded from the idea, that even that which appears most accidental and irrational is yet a part of rational knowledge, in fact that every thing existing may be brought within a rational conception.