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previous: Chapter 18 This webpage reproduces a chapter of A History of Philosophy by Frederick Copleston, S. J. as reprinted by Image Books — Doubleday New York • London • Toronto • Sydney • Auckland 1993 The text, and illustrations except as noted, are in the public domain. next: Chapter 20

Part III: Plato

 p142  Chapter XIX
Theory of Knowledge

Plato's theory of knowledge cannot be found systematically expressed and completely elaborated in any one dialogue. The Theaetetus is indeed devoted to the consideration of problems of knowledge, but its conclusion is negative, since Plato is therein concerned to refute false theories of knowledge, especially the theory that knowledge is sense-perception. Moreover, Plato had already, by the time he came to write the Theaetetus, elaborated his theory of degrees of "knowledge," corresponding to the hierarchy of being in the Republic. We may say, then, that the positive treatment preceded the negative and critical, or that Plato, having made up his mind what knowledge is, turned later to the consideration of difficulties and to the systematic refutation of theories which he believed to be false.​1 In a book like the present one, however, it seems best to treat first of the negative and critical side of the Platonic epistemology, before proceeding to consider his positive doctrine. Accordingly, we propose first of all to summarise the argument of the Theaetetus, before going on to examine the doctrine of the Republic in regard to knowledge. This procedure would seem to be justified by the fact that the Republic is not primarily an epistemological work at all. Positive epistemological doctrine is certainly contained in the Republic, but some of the logically prior presuppositions of that doctrine are contained in the later dialogue, the Theaetetus.

The task of summarising the Platonic epistemology and giving it in systematic form is complicated by the fact that it is difficult to separate Plato's epistemology from his ontology. Plato was not a critical thinker in the sense of imanuel Kant, and though it is possible to read into his thoughts an anticipation of the Critical Philosophy (at least, this is what some writers have endeavoured to do), he is inclined to assume that we can have knowledge and to be primarily interested in the question what  p143 is the true object of knowledge. This means that ontological and epistemological themes are frequently intermingled or treated pari passu, as in the Republic. We will make an attempt to separate the epistemology from the ontology, but the attempt cannot be wholly successful, owing to the very character of the Platonic epistemology.

I. Knowledge is not Sense-perception

Socrates, interested like the Sophists in practical conduct, refused to acquiesce in the idea that truth is relative, that there is no stable norm, no abiding object of knowledge. He was convinced that ethical conduct must be founded on knowledge, and that that knowledge must be knowledge of eternal values which are not subject to the shifting and changing impressions of sense or of subjective opinion, but are there are for all men and for all peoples and all ages. Plato inherited from his Master this conviction that there can be knowledge in the sense of objective and universally valid knowledge; but he wished to demonstrate this fact theoretically, and so he came to probe deeply into the problems of knowledge, asking what knowledge is and of what.

In the Theaetetus Plato's first object is the refutation of false theories. Accordingly he sets himself the task of challenging the theory of Protagoras that knowledge is perception, that what appears to an individual to be true is true for that individual. His method is to elicit dialectically a clear statement of the theory of knowledge implied by the Heraclitean ontology and the epistemology of Protagoras, to exhibit its consequences and to show that the conception of "knowledge" thus attained does not fulfil the requirements of true knowledge at all, since knowledge must be, Plato assumes, (i) infallible, and (ii) of what is. Sense-perception is neither the one nor the other.

The young mathematical student Theaetetus enters into conversation with Socrates and the latter asks him what he thinks knowledge to be. Theaetetus replies by mentioned geometry, the sciences and crafts, but Socrates points out that this is no answer to his question, for he had asked, not of what knowledge is, but what knowledge is. The discussion is thus meant to be epistemological in character, though, as has been already pointed out, ontological considerations cannot be excluded, owing to the very character of the Platonic epistemology.  p144 Moreover, it is hard to see how in any case ontological questions can be avoided in an epistemological discussion, since there is no knowledge in vacuo: knowledge, if it is knowledge at all, must necessarily be knowledge of something, and it may well be that knowledge is necessarily related to so particular type of object.

Theaetetus, encouraged by Socrates, make another attempt to answer the question proposed, and suggests that "knowledge is nothing but perception"​2 thinking no doubt primarily of vision, though in itself perception has, of course, a wider connotation. Socrates proposes to examine this idea of knowledge, and in the course of conversation elicits from Theaetetus an admission of Protagoras' view that perception means appearance, and that appearances vary with different subjects. At the same time he gets Theaetetus to agree that knowledge is always of something that is, and that, as being knowledge, it must be infallible.​3 This having been established, Socrates next tries to show that the objects of perception are, as Heraclitus taught, always in a state of flux: they never are, they are always becoming. (Plato does not, of course, accept Heraclitus' doctrine that all is becoming, though he adds the doctrine in regard to the objects of sense-perception, drawing the conclusion that sense-perception cannot be the same as knowledge.) Since an object may appear white to one at one moment, grey at another, sometimes hot and sometimes cold, etc., "appearing to" must mean "becoming for," so that perception is always of that which is in process of becoming. My perception is true for me, and if I know what appears to me, as I obviously do, then my knowledge is infallible. So Theaetetus has said well that perception is knowledge.

This point having been reached, Socrates proposes to examine the idea more closely. He raises the objection that if knowledge is perception, then no man can be wiser than any other man, for I am the beside the judge of my own sense-perception as such. What, then, is Protagoras' justification for setting himself up to teach others and to take a handsome fee for doing so? And where is our ignorance that makes us sit at his feet? For is not each one of us tape measure of his own wisdom? Moreover, if knowledge and perception are the same, if there is no difference between seeing and knowing, it follows that a man who has come to know (i.e. see) a thing in the past and still remembers it, does not know it — although he remembers it — since he does not see it.  p145 Conversely, granted that a man can remember something he has formerly perceived and can know it, even while no longer perceiving it, it follows that knowledge and perception cannot be equated (even if perception were a kind of knowledge).

Socrates then attacks Protagoras' doctrine on a broader basis, understanding "Man is the measure of all things," not merely in reference to sense-perception, but also to all truth. He points out that the majority of mankind believe in knowledge and ignorance, and believe that they themselves or others can hold something to be true which in point of fact is not true. Accordingly, anyone who holds prog' doctrine to be false is, according to Protagoras himself, holding the truth (i.e. if the man who is the measure of all things is the individual man).

After these criticisms Socrates finishes the claims of perception to be knowledge by showing (i) that perception is not the whole of knowledge, and (ii) that even within its own sphere perception is not knowledge.

(i) Perception is not the whole of knowledge, for a great part of what is generally recognised to be knowledge consists of truths involving terms which are not objects of perception at all. There is much we know about sensible objects, which is known by intellectual reflection and not immediately by perception. Plato gives existence or non‑existence as examples.​4 Suppose that man sees a mirage. It is not immediate sense-perception that can inform him as to the objective existence or non‑existence of the mirage perceived: it is only rational reflection that can tell him this. Again, the conclusions and arguments of mathematics are not apprehended by sense. One might add that our knowledge of a person's character is something more than can be explained by the definition, "Knowledge is perception," for our knowledge of a person's character is certainly not given in bare sensation.

(ii) Sense-perception, even within its own sphere, is not knowledge. We cannot really be said to know anything if we have not attained truth about it, e.g. concerning its existence or non‑existence, its likeness to another thing or its unlikeness. But turn is given in reflection, in the judgment, not in bare sensation. The bare sensation may give, e.g. one white surface and a second white surface, but in order to judge the similarity between the two, the mind's activity is necessary. Similarly, the  p146 railway lines appear to converge: it is in intellectual reflection that we know that they are really parallel.

Sense-perception is not, therefore, worthy of the name of knowledge. It should be noted how much Plato is influenced by the conviction that sense-objects are not proper objects of knowledge and cannot be so, since knowledge is of what is, of the stable and abiding, whereas objects of sense cannot really be said to be — qua perceived, at least — but only to become. Sense-objects are objects of apprehension in some sort, of course, but they elude the mind too much to be objects of real knowledge, which must be, as we have said, (i) infallible, (ii) of what is.

(It is noteworthy that Plato, in disposing of the claim of perception to be the whole of knowledge, contrasts the private or peculiar objects of the special senses — e.g. colour, which is the object of vision all — with the "common terms that apply to everything," and which are the objects of the mind, not of the senses. These "common terms" correspond to the Forms or Ideas which are, ontologically, the stable and abiding objects, as contrasted with the particulars or sensibilia.)

II. Knowledge is not simply "True Judgment"

Theaetetus sees that he cannot say that judgment tout simple is knowledge, for the reason that false judgments are possible. He therefore suggests that knowledge is true judgment, at least as a provisional definition, until examination of it shows whether it is true or false. (At this point a digression occurs in which Socrates tries to find out how false judgments are possible and come to be made at all. Into this discussion at any length, but I will mention one or two suggestions that are made in its course. For example, it is suggested that one class of false judgments arises through the confusion of two objects of different sorts, one a present object of sense-perception, the other a memory-image. A man may judge — mistakenly — that he sees his friend so way off. There is someone there, but it is not his friend. The man has a memory-image of his friend, and something in the figure he sees recalls to him this memory-image: he then judges falsely that it is his friend who is over there. But, obviously, not all cases of false judgment are instances of the confusion of a memory-image with a present object of sense-perception: a mistake in mathematical calculation can hardly be reduced to this. The famous simile of the "aviary" is introduced,  p147 in an attempt to show how other kinds of false judgment may arise, but it is found to be unsatisfactory; and Plato concludes that the problem of false judgment cannot be advantageously treated until the nature of knowledge has been determined. The discussion of false judgment was resumed in the Sophistes.)

In the discussion of Theaetetus' suggestion that knowledge is true judgment, it is pointed out that a judgment may be true without weight fact of its truth involving knowledge on the part of the man who makes the judgment. The relevance of this observation is easily grasped. If I were to make at this moment the judgment, "Mr. Churchill is talking to President Truman over the telephone," it might be true; but it would not involve knowledge on my part. It would be a guess or random shot, as far as I am concerned, even though the judgment were objectively true. Similarly, a man might be tried on a charge of which he was actually not guilty, although the circumstantial evidence was very strong against him and he could not prove his innocence. If, now, a skilful lawyer defending the innocent man were able, for the sake of argument, so to manipulate the evidence or to play on the feelings of the jury, that they gave the verdict "Not guilty," their judgment would actually be a true judgment; but they could hardly be said to know the innocence of the prisoner, since ex hypothesi the evidence is against him. Their verdict would be a true judgment, but it would be based on persuasion rather than on knowledge. It follows, then, that knowledge is not simply true judgment, and Theaetetus is called on to make another suggestion as to the right definition of knowledge.

III. Knowledge is not True Judgment plus an "Account"

True judgment, as has been seen, may mean no more than true belief, and true belief is not the same thing as knowledge. Theaetetus, therefore, suggests that the addition of an "account" or explanation (λόγος) would convert true belief into knowledge. Socrates begins by pointing out that if giving an account or explanation means the enumeration of elementary parts, than these parts must be known or knowable: otherwise the absurd conclusion would follow that knowledge means adding to true belief the reduction of the complex to unknown or unknowable elements. But what does giving an account mean?

1. It cannot mean merely that a correct judgment, in the  p148 sense of true belief, is expressed in words, since, if that were the meaning, there would be no difference between true belief and knowledge. And we have seen that there is a difference between making a judgment that happens to be correct and making a judgment that one knows to be correct.

2. If "giving an account" means analysis into elementary parts (i.e. knowable parts), will addition of an account in this sense suffice to convert true belief into knowledge? No, the mere process of analyzing into elements does not convert true belief into knowledge, then a man who could enumerate the parts with go to make up a wagon (wheels, axle, etc) would have a scientific knowledge of a wagon, nature man who could tell you what letters of the alphabet go to compose a certain word would have a grammarian's scientific knowledge of the word. (N. B. We must realize that Plato is speaking of the mere enumeration of parts. For instance, the man who could recount the various steps that lead to a conclusion in geometry, simply because he had seen them in a book and had learnt them by heart, without having really grasped the necessity of the premisses and the necessity and logical sequence of the deduction, would be able to enumerate the "parts" of the theorem; but he would not have the scientific knowledge of the mathematician.

3. Socrates suggests a third interpretation of "plus account." It may mean "being able to name some mark by which the thing one is asked about differs from everything else."​5 If this is correct, then to know something means the ability to give the distinguishing characteristics of that thing. But this interpretation too is disposed of, as being inadequate to define knowledge.

(a) Socrates points out that if knowledge of a thing means the addition of its distinguishing characteristics to a correct notion of that thing, we are involved in an absurd position. Suppose that I have a correct notion of Theaetetus. To convert this correct notion into knowledge I have to add some distinguishing characteristic. But unless this distinguishing characteristic were already contained within my correct notion, how could the latter be called a correct notion I cannot be said to have a correct notion of Theaetetus, unless this correct notion includes Theaetetus' distinguishing characteristics: if these distinguishing characteristics are not included, then my "correct notion" of Theaetetus  p149 would equally well apply to all other men; in which case it would not be a correct notion of Theaetetus.

(b) If, on the other hand, my "correct notion" of Theaetetus includes his distinguishing characteristics, then it would also be absurd to say that I convert this correct notion into knowledge by adding the differentia, since this would be equivalent to saying that I convert my correct notion of Theaetetus into knowledge by adding to Theaetetus, as already apprehended in distinction from others, that which distinguishes him from others.

N. B. It is to be noted that Plato is not speaking here of specific differences, he is speaking of individual, sensible objects, as is clearly shown by the examples that he takes — the sun and a particular man, Theaetetus.​6 The conclusion to be drawn is not that no knowledge is attained through definition by means of a difference, but rather that the individual, sensible object is indefinable and is not really the proper object of knowledge at all. This is the real conclusion of the dialogue, namely, that true knowledge of sensible objects is unattainable, and — by implication — that true knowledge must be knowledge of the universal and abiding.

IV. True Knowledge

1. Plato has assumed from the outset that knowledge is attainable, and that knowledge must be (i) infallible and (ii) of the real. True knowledge must possess both these characteristics, and any state of mind that cannot vindicate its claim to both these characteristics cannot be true knowledge. In the Theaetetus he shows that neither sense-perception nor true belief are possessed of both these marks; neither, then, can be equated with true knowledge. Plato accepts from Protagoras the belief in the relativity of sense and sense-perception, but he will not accept a universal relativism: on the contrary, knowledge, absolute and infallible knowledge, is attainable, but it cannot be the same as sense-perception, which is relative, elusive and subject to the influence of all sorts of temporary influences on the part of both subject and object. Plato accepts, too, from Heraclitus the view that the objects of sense-perception, individual and sensible particular objects, are always in a state of becoming, of flux, and so are uniformity unfit to be the objects of true knowledge. They come into being and pass away, they are indefinite in number, cannot be clearly grasped in  p150 definition and cannot become the objects of scientific knowledge. But Plato does not draw the conclusion that there are no objects that are fitted to be the objects of true knowledge, but only that sensible particulars cannot be the objects sought. The object of true knowledge must be stable and abiding, fixed, capable of being grasped in clear and scientific definition, which is of the universal, as Socrates saw. The consideration of different states of mind is thus indissolubly bound up with the consideration of the different objects of those states of mind.

If we examine those judgments in which we think we attain knowledge of the essentially stable and abiding, we find that they are judgments concerning universals. If, for example, we examine the judgment "The Athenian Constitution is good," we shall find that the essentially stable element in this judgment is the concept of goodness. After all, the Athenian Constitution might be so changed that we would no longer qualify it as good, but as bad. This implies that the concept of goodness remains the same, for if we term the changed Constitution "bad," that can only be because we judge it in reference to a fixed concept of goodness. Moreover, if it is objected that even though the Athenian Constitution may change as an empirical and historical fact, we can still say "The Athenian Constitution is good," if we mean the particular form of the Constitution that we once called good (even though it may in point of fact have since been changed), we can point out in answer that in this case our judgment has reference, not so much to the Athenian Constitution as a given empirical fact, as to a certain type of constitution. That this type of constitution happens at any given historical moment to be embodied in the Athenian Constitution is more or less irrelevant: what we really mean is that this universal type of constitution (whether found at Athens or elsewhere) carries with it the universal quality of goodness. Our judgment, as far as it attains the abiding and stable, really concerns a universal.

Again, scientific knowledge, as Socrates saw (predominantly in connection with ethical valuations), aims at the definition, at crystallising and fixing knowledge in the clear and unambiguous definition. A scientific knowledge of goodness, for instance, must be enshrined in the definition "Goodness is . . .," whereby the mind expresses the essence of goodness. But definition concerns the universal. Hence true knowledge is knowledge of the universal. Particular constitutions change, but the concept of goodness  p151 remains the same, and it is in reference to this stable concept that we judge of particular constitutions in respect of goodness. It follows, then, that it is the universal that fulfils the requirements for being an object of knowledge. Knowledge of the highest universal will be the highest kind of knowledge, while "knowledge" of the particular will be the lowest kind of "knowledge."

But does not this view imply an impassable gulf between true knowledge on the one hand and the "real" world on the other — a world that consists of particulars? And if true knowledge is knowledge of universals, does it not follow that true knowledge is knowledge of the abstract and "unreal"? In regard to second question, I would point out that the essence of Plato's doctrine of Forms or Ideas is simply this: that the universal concept is not an abstract form devoid of objective content or references, but that to each true universal concept there corresponds an objective reality. How far Aristotle's criticism of Plato (that the latter hypostasised the objective reality of the concepts, imagining a transcendent world of "separate" universals) is justified, is a matter for discussion by itself: whether justified or unjustified, it remains true that the essence of the Platonic theory of Ideas is not to be sought in the notion of the "separate" existence of universal realities, but in the belief that universal concepts have objective reference, and that the corresponding reality is of a higher order than sense-perception as such. In regard to the first question (that of the gulf between true knowledge and the "real" world), we must admit that it was one of Plato's standing difficulties to determinate the precise relation between the particular and the universal; but to this question we must return when treating of the theory of Ideas from the ontological viewpoint: at the moment one can afford to pass it over.

2. Plato's positive doctrine of knowledge, in which degrees or levels of knowledge are distinguished according to objects, is set out in the famous passage of the Republic that gives us the simile of the Line.​7 I give here the usual schematic diagram, which I will endeavour to explain. It must be admitted that there are several important points that remain very obscure, but doubtless Plato was feeling his way towards what he regarded as the truth; and, as far as we know, he never cleared up his precise meaning  p152 in unambiguous terms. We cannot, therefore, altogether avoid conjecture. IMAGE8

The development of the human mind on its way from ignorance to knowledge, lies over two main fields, that of δόξα (opinion) and that of ἐπιστήμη (knowledge). It is only the latter that can properly be termed knowledge. How are these two functions of the mind differentiated? It seems clear that the differentiation is based on a differentiation of object. δόξα (opinion), is said to be concerned with "images," while ἐπιστήμη, a least in the form of νόησις, is concerned with originals or archetypes, ἀρχαί. If a man is asked what justice is, and he points to imperfect embodiments of justice, particular instances which fall short of the universal ideal, e.g. the action of a particular man, a particular constitution or set of laws, having no inkling that there exists a principle of absolute justice, a norm and standard, then that man's state of mind is a state of δόξα: he sees the images or copies and mistakes them for the originals. But if a man has an apprehension of justice in itself, if he can rise above the images to the Form, to the Idea, to the universal, whereby all the particular instances must be judged, then his state of mind is a state of knowledge, of ἐπιστήμη or γνῶσις. Moreover, it is possible to progress from one state of mind to the other, to be "converted," as it were; and when a man comes to realise that what he formerly took to be originals are in reality only images or copies, i.e. imperfect embodiments of the  p153 ideal, imperfect realisations of the norm or standard, when he comes to apprehend in some way the original itself, then his state of mind is no longer that of δόξα, he has been converted to ἐπιστήμη.

The line, however, is not simply divided into two sections; each section is subdivided. Thus there are two degrees of ἐπιστήμη and two degrees of δόξα. How are they to be interpreted? Plato tells us that the lower degree, that of εἰκασία, has as its object, in the first place, "images" or "shadows," and in the second place "reflections in water and in solid, smooth, bright substances, and everything of the kind."​9 This certainly sounds rather peculiar, at least if one takes Plato to opium that any man mistakes shadow and reflections in water for the original. But one can legitimately extend the thought of Plato to cover in general images of images, imitations at second hand. Thus we said that a man whose only idea of justice is the embodied and imperfect justice of the Athenian Constitution or of some particular man, is in a state of δόξα in general. If, however, a rhetorician comes along, and with specious words and reasonings persuades him that things are just and right, which in reality are not even in accord with the empirical justice of the Athenian Constitution and its laws, then his state of mind is that of εἰκασία. What he takes for justice is but a shadow or caricature of what is itself only an image, if compared to the universal Form. The state of mind, on the other hand, of the man who takes as justice the justice of the law of Athens or the justice of a particular just man is that of πίστις.

Plato tells us that the objects of the πίστις section are the real objects corresponding to the images of the εἰκασία section of the line, and he mentions "the animals about us, and the whole world of nature and of art."​10 This implies, for instance, that the man whose only idea of a horse is that of particular real horses, and who does not see that particular horses are imperfect "imitations" of the ideal horse, i.e. of the specific type, the universal, is in a state of πίστις. He has not got knowledge of the horse, but only opinion. (Spinoza might say that he is in a state of imagination, of inadequate knowledge.) Similarly, the man who judges that external nature is true reality, and who does not see that it is a more or less "unreal" copy of the invisible world (i.e. who does not see that sensible objects are imperfect realisations of the specific type) has only πίστις. He is not so badly off as the  p154 dreamer who thinks that the ideas that he sees are the real world (εἰκασία), but he has not got ἐπιστήμη: he is devoid of real scientific knowledge.

The mention of art in the above quotation helps us to understand the matter a little more clearly. In the tenth book of the Republic, Plato says that artists are at the third remove from truth. For example, there is the specific form of man, the ideal type that all individuals of the species strive to realise, and there are particular men who are copies or imitations or imperfect realisations of the specific types. The artist now comes and paints a man, the painted man being an imitation of an imitation. Anyone who took the painted man to be a real man (one might say anyone who took the wax policeman at the entrance of Madame Tussaud's to be a real policeman) would be in a state of εἰκασία, while anyone whose idea of a man is limited to the particular men he has seen, heard of or read about, and who has no real grasp of the specific type, is in a state of πίστις. But the man who apprehends the ideal man, i.e. the ideal type, the specific form of which particular men are imperfect realisations, has νόησις.​11 Again, a just man may imitate or embody in his actions, although imperfectly, the idea of justice. The tragedian then proceeds to imitate this juman on the stage, but without knowing anything of justice in itself. He merely imitates an imitation.

Now, what of the higher division of the line, which corresponds in respect of object to νοητά, and in respect of state of mind to ἐπιστήμη? In general it is connected, not with ὁρατά or sensible objects (lower part of the line), but with ἀορατά, the invisible world, νοητά. But what of the subdivision? How does νόησις in the restricted sense differ from διάνοια? Plato says that the object of διάνοια is what the soul is compelled to investigate by the aid of the imitations of the former segments, which it employs as images, starting from hypothesis and proceeding, not to a first principle, but to a conclusion.​12 Plato is here speaking of mathematics. In geometry, for instance, the mind proceeds from hypotheses, by the use of a visible diagram, to a conclusion. The geometer, says Plato, assumes the triangle, etc., as known, adopts these "materials" as hypotheses, and then, employing a visible diagram, argues to a conclusion, being interested, however, not in the diagram itself (i.e. in this or that particular triangle or particular square or particular diameter). Geometers thus employ  p155 figures and diagrams, but "they are really endeavouring to behold those objects which a person can only see with the eye of thought."13

One might have thought that the mathematical objects of this kind would be numbered among the Forms or ἀρχαί, and that Plato would have equated the scientific knowledge of the geometer with νόησις proper; but he expressly declined to do, and it is impossible to suppose (as some have done) that Plato was fitting his epistemological doctrines to the exigencies of his simile of the line with its divisions. Rather must we suppose that Plato really meant to assert the existence of a class of "intermediaries," i.e. of objects which are the object of ἐπιστήμη, but which are all the same inferior to ἀρχαί, and so are the objects of διάνοια and not of νόησις.​14 It becomes quite clear from the close of the sixth book of the Republic​15 that the geometers have not got νοῦς or νόησις in regard to their objects; and that because they do not mount up above their hypothetical premisses, "although taken in connection with a first principle these objects come within the domain of the pure reason."​16 These last words show that the distinction between the two segments of the upper part of the line is to be referred to a distinction of state of mind and not only to a distinction of object. And it is expressly stated that understanding or διάνοια is intermediate between opinion (δόξα) and pure reason (νόησις).

This is supported by the mention of hypotheses. Nettleship thought that Plato's meaning is that the mathematician accepts his postulates and axioms as if they were self-contained truth: he does not question them himself, and if anyone else questions them, he can only say that he cannot argue the matter. Plato does not use the word "hypothesis" in the sense of a judgment which is taken as true while it might be untrue, but in the sense of a judgment which is treated as if it were self-conditioned, not being seen in its ground and in its necessary connection with being.​17 Against this it might be pointed out that the examples of "hypotheses" given in510c are all examples of entities and not of judgments, and that Plato speaks of destroying hypotheses rather than of reducing them to self-conditioned or self-evident propositions. A further suggestion on this matter is given at the close of this section.

 p156  In the Metaphysics,​18 Aristotle tells us that Plato held that mathematical entities are "between forms and sensible things." "Further, besides sensible things and forms, he says there are the objects of mathematics, which occupy an intermediate position, differing from sensible things in being eternal and unchangeable, from Forms in that there are many alike, while the Form itself is in each case unique." In view of this statement by Aristotle, we can hardly refer the distinction between the two segments of the upper part of the line to the state of mind alone. There must be a difference of object as well. (The distinction would be drawn between the states of mind exclusively, if, while τὰ μαθηματικά belonged in their own right to the same segment as αἱ ἀρχαί, the mathematician, acting precisely as such, accepted his "materials" hypothetically and then argued to conclusions. He would be in the state of mind that Plato calls διάνοια, for he treats his postulates as self-conditioned, without asking further questions, and argues to a conclusion by means of visible diagrams; but his reasoning would concern, not the diagrams as such but ideal mathematical objects, so that, if he were to take his hypotheses "in connection with a first principle," he would be in a state of νόησις instead of διάνοια, although the true object of his reasoning, the ideal mathematical objects, would remain the same. This interpretation, i.e. the interpretation that would confine the distinction between the two segments of the upper part of the line to states of mind, might well seem to be favoured by the statement of Plato that mathematical questions, when "taken in connection with a first principle, come within the domain of the pure reason"; but Aristotle's remarks on the subject, if they are a correct statement of the thought of Plato, evidently forbid this interpretation, since he clearly thought that Plato's mathematical entities were supposed to occupy a position between αἱ ἀρχαί and τὰ ὁρατά.)

If Aristotle is correct and Plato really meant τὰ μαθηματικά to constitute a class of objects on their own, distinct from other classes, in what does this distinction consist? There is no need to dwell on the distinction between τὰ μαθηματικά and the objects of the lower part of the line, τὰ ὁρατά, since it is clear enough that the geometrician is concerned with ideal and perfect objects of thought, and not with empirical circles or lines, e.g. cart-wheels or hoops or fishing-rods, or even with geometrical diagrams as such, i.e. as sensible particulars. The question, therefore, resolves  p157 itself into this: in what does the distinction between τὰ μαθηματικά, as objects of διάνοια, and αἱ ἀρχαί, as objects of νόησις, really consist?

A natural interpretation of Aristotle's remarks in the Metaphysics is that, according to Plato, the mathematician is speaking of intelligible particulars, and not of sensible particulars, nor of universals. For example, if the geometer speaks of two circles intersecting, he is not speaking of the sensible circles drawn nor yet of circularity as such — for how could circularity intersect circularity? He is speaking of intelligible circles, of which there are many alike, as Aristotle would say. Again, to say that "two and two make four" is not the same as to say what will happen if twoness be added to itself — a meaningless phrase. This view iss uttered by Aristotle's remark that for Plato "there must be a first 2 and 3, and the numbers must not be addable to one another."​19 For Plato, the integers, including 1, form a series in such a way that 2 is not made up of two 1's, but is a unique numeral form. This comes more or less to saying that the integer 2 is twoness, which is not composed Ottawa "onenesses." These integer numbers Plato seems to have identified with the Forms. But though it cannot be said of the integer 2 that there are many alike (any more than we can speak of many circularities), it is clear that the mathematician who does not ascend to the ultimate formal principles, does in fact deal with a plurality of 2's and a plurality of circles. Now, when the geometer speaks of intersecting circles, he is not treating of sensible particulars, but of intelligible objects. Yet of these intelligible objects there are many alike, hence they are not real universals but constitute a class of intelligible particulars, "above" sensible particulars, but "below" true universals. It is reasonable, therefore, to conclude that Plato's τὰ μαθηματικά are a class of intelligible particulars.

Now, Professor A. E. Taylor,​20 if I understand him correctly, would like to confine the sphere of τὰ μαθηματικά to ideal spatial magnitudes. As he points out, the properties of e.g. curves can be studied by means of numeral equations, but they are not themselves numbers; so that they would not belong to the highest section of the line, that of αἱ ἀρχαί or Forms, which Plato identified with Numbers. On the other hand, the ideal spatial magnitudes, the objects which the geometrician studies, are not sensible  p158 objects, so that they cannot belong to the sphere of τὰ ὁρατά. They therefore occupy an intermediate position between Number-Forms and Sensible Things. That this is true of the objects with which the geometer deals (intersecting circles, etc.) I willingly admit; but is one justified in excluding from τὰ μαθηματικά the objects with which the arithmetician deals? After all, Plato, when treating of those whose state of mind is that of διάνοια, speaks not only of students of geometry, but also of students of arithmetic and the kindred sciences.​21 It would certainly not appear from this that we are justified in asserting that Plato confined τὰ μαθηματικά to ideal spatial magnitudes. Whether or not we think that Plato ought to have so confined the sphere of mathematical entities, we have to consider, not only what Plato ought to have said, but also what he did say. Most probably, therefore, he understood, as comprised in the class of τὰ μαθηματικά, the objects of the arithmetician as well as those of the geometer (and not only of these two, as can be inferred from the remark about "kindred sciences"). What, then, becomes of Aristotle's statement that for Plato numbers are not addable (ἀσυμβλητοί)? I think that it is certainly to be accepted, and that Plato saw clearly that numbers as such are unique. On the other hand, it is equally clear that we add groups or classes of objects together, and speak of the characteristic of a class as a number. These we add but they stand for the classes of individual objects, though they are themselves the objects, not of sense but of intelligence. They may, therefore, be spoken of as intelligible particulars, and they belong to the sphere of τὰ μαθηματικά, as well as the ideal spatial magnitudes of the geometer. Aristotle's own theory of number may have been erroneous, and he may thus have misrepresented Plato's theory in some respects; but if he definitely stated, as he did, that Plato posited an intermediate class of mathematical entities, it is hard to suppose that he was mistaken, especially as Plato's own writings would seem to leave no reasonable doubt, not only that he actually posited such a class, but also that he did not mean to confine this class to ideal spatial magnitudes.

(Plato's statement that the hypotheses of the mathematicians — he mentions "the odd and the even not figures and three kinds of angles and the cognates of these in their several branches of science"​22 — when taken in connection with a first principle, are  p159 cognisable by the higher reason, and his statement that the her reason is concerned with first principles, which are self-evident, suggest that he would welcome the modern attempts to reduce pure mathematics to their logical foundations.)

It remains to consider briefly the highest segment of the line. The state of mind in question, that of νόησις, is the state of mind of the man who uses the hypotheses of the διάνοια segment as starting-points, but passes beyond them and ascends to first principles. Moreover, in this process (which is the process of Dialectic) he makes no use of "images," such as are employed in the διάνοια segment, but proceeds in and by the ideas themselves,​23 i.e. you strictly abstract reasoning. Having clearly grasped the first principles, the mind then descends to the conclusions that follow from them, again making use only of abstract reasoning and not of sensible images.​24 The objects corresponding to νόησις are αἱ ἀρχαί, the first principles or Forms. They are not merely epistemological principles, but also ontological principles, and I will consider them more in detail later; but it is as well to point out the following fact. If it were merely a question of seeing the ultimate principles of the hypotheses of the διάνοια section (as e.g. in the modern reduction of pure mathematics to their logical foundations), there might be no very great difficulty in seeing what Plato was driving at; but he speaks expressly of dialectic as "destroying the hypotheses," ἀναιροῦσα τὰς ὑποθήσεις,​25 which is a hard saying, since, though dialectic may well show that the postulates of the mathematician need revision, it is not so easy, at first sight at least, to see how it can be said to destroy the hypotheses. As a matter of fact, Plato's meaning becomes clearer if we consider one Petrarch hypothesis he mentions — the odd and the even. It would appear that Plato recognised that there are numbers which are neither even nor odd, i.e. irrational numbers, and that in the Epinomis​26 he demands the recognition of quadratic and cubic "surds" as numbers.​27 If this is so, then it would be the task of dialectic to show that the traditional hypotheses of the mathematician, that there are no irrational numbers, but that all numbers are integers and are either even or odd, is not strictly true. Again, Plato roughed to accept the Pythagorean idea of the point-unit and speak of the point as "the beginning of a line,"​28 so that the point-unit, i.e. the point as having magnitude of its  p160 own, would be a fiction of the geometer, "a geometrical fiction,"​29 an hypothesis that needs to be "destroyed."

3. Plato further illustrated his epistemological doctrine by the famous allegory of the Cave in the seventh book of the Republic.​30 I will briefly sketch the allegory, since it is valuable as showing clearly, if any further proof be needed, that the ascent of the mind from the lower sections of the line to the higher is an epistemological process, and that Plato regarded this process, not so much as a continuous process of evolution as a series of "conversions" from a less adequate to a more adequate cognitive state. IMAGE

Plato asks us to imagine an under­ground cave which has an opening towards the light. In this cave are living human beings, with their legs and necks chained from childhood in such a way that they face the inside wall of the cavalry and have never seen the light of the sun. Above and behind them, i.e. between the prisoners and the mouth of the cave, is a fire, and between them and the fire is a raised way and a low wall, like a shrine. Along this raised way there pass men carrying statues and figures of animals and other objects, in such a manner that the objects they city appear over the top of the low wall or screen. The  p161 prisoners, facing the inside wall of the cavalry, cannot see one another nor the objects carried behind them, but they see the shadows of themselves and of these objects thrown on to the walls they are facing. They see only shadows.

These prisoners represent the majority of mankind, that multitude of people who remain all their lives in a state of εἰκασία, beholding only shadows of reality and hearing only echoes of the truth. Their view of the world is most inadequate, distorted by "their own passions and prejudices, and by the passions and prejudices of other people as conveyed to them by language and rhetoric."​31 And though they are in no better case than children, they cling to their distorted views with all the tenacity of adults, and have no wish to escape from their prison-house. Moreover, if they were suddenly freed and told to look at the realities of which they had formerly seen the shadows, they would be blinded by the glare of the light, and would imagine that the shadows were far more real than the realities.

However, if one of the prisoners who has escaped grows accustomed to the light, he will after a time be able to look at the concrete sensible objects, of which he had formerly seen but the shadows. this man beholds his fellows in the light of the fire (which represents the visible sun) and is in a state of πίστις, having been "converted" from the shadow-world of εἰκόνες, prejudices and passions and sophistries, to the real world of ζῷα, though he has not yet add to the world of intelligible, non‑sensible realities. He sees the prisoners for what they are, namely prisoners, prisoners in the bonds of passion and sophistry. Moreover, if he perseveres and comes out of the cave into the sunlight, he will see the world of sun‑illumined and clear objects (which represent intelligible realities), and lastly, though only by an effort, he will be able to see the sun itself, which represents the Idea of the Good, the highest Form, "the universal cause of all things right and beautiful — the source of truth and of reason."​32 He will then be in a state of νόησις. (To this Idea of the Good, and also to the political considerations that concerned Plato in the Republic, I shall return in later chapters.)

Plato remarks that if someone, after ascending to the sunshine, went back into the cave, he would be unable to see properly because of the darkness, and so would make himself "ridiculous"; while if he tried to free another and lead him up to the light, the  p162 prisoners, who love the darkness and consider the shadows to be true reality, would put the offender to death, if they could but catch him. Here we may understand a reference to Socrates, who endeavoured to enlighten all those who would listen and make them apprehend truth and reason, instead of letting themselves be misled by prejudice and sophistry.

This allegory makes it clear that the "ascent" of the line was regarded by Plato as a progress, though this progress is not a continuous and automatic process: it needs effort and mental discipline. Hence his insistence on the great importance of education, whereby the young may be brought gradually to behold eaten and absolute truths and values, and so saved from passing their lives in the shadow-world of error, falsehood, prejudice, sophistical persuasion, blindness to true values, etc. This education is of primary importance in the case of those who are to be statesmen. Statesmen and rulers will be blind leaders of the blind, if they dwell in the spheres of εἰκασία or πίστις, and the wrecking of the ship of State is a more terrible thing than the wreck of anyone's individual barque. Plato's interest in the epistemological ascent is thus no mere academic or narrowly critical interest: he is concerned with the conduct of life, tendance of the soul and with the good of the State. The man who does not realise the true good of man will not, and cannot, lead the truly good human life, and the statesman who does not realise the true good of the State, who does not view political life in the light of eternal principles, will bring ruin on his people.

The question might be raised, whether or not there are religious implications in the epistemology of Plato, as illustrated by the simile of the Line and the allegory of the Cave. That the conceptions of Plato were given a religious colouring and application by the Neo‑Platonists is beyond dispute: moreover, when a Christian writer, such as the Pseudo-Dionysius, traces the mystic's ascent to God by the via negativa, beyond visible creatures to their invisible Source, the light of which blinds by excess of light, so that the soul is in a state of, so to speak, luminous obscurity, he certainly utilises themes which came from Plato via the Neo‑Platonists. But it does not necessarily follow that Plato himself understood the ascent from a religious viewpoint. In any case this difficult question cannot be profitably touched on until one has considered the ontological nature and status of Plato's Idea of the Good; and even then one can scarcely reach definitive certainty.

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The Author's Notes:

1 We do not thereby mean to imply that Plato had not made up his mind as to the status of sense-perception long before he wrote the Theaetetus (we have only to read the Republic, for instance, or consider the genesis and implications of the Ideal Theory): we refer to systematic consideration in published writings.

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2 151E 2‑13.

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3 152C 5‑7.

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4 185C 4‑e2.

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5 208C 7‑8.

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6 208C 7‑e4.

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7 Rep., 509D 6‑511e5.

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8 On the left side of the line are states of mind: on the right side are corresponding objects. In both cases the "highest" are at the top. The very close connection between the Platonic epistemology and the Platonic ontology is at once apparent.

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9 Rep., 509e1–510a3.

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10 Rep., 510a5‑6.

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11 Plato's theory of art is discussed in a later chapter.

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12 Rep., 510b4‑6.

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13 Rep., 510e2–511a1.

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14 cf. W. R. F. Hardie, A Study in Plato, p52 (O. U. P., 1936).

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15 Rep., 510.

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16 Rep., 511c8‑d2.

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17 Lectures on the Republic of Plato (1898), pp252 f.

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18 987b14 ff. Cf. 1059b2 ff.

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19 Metaph., 1083a33‑5.

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20 Cf. Forms and Numbers, Mind, Oct. 1926 and Jan. 1927. (Reprinted in Philosophical Studies.)

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21 Rep., 510c2 ff.

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22 Rep., 510c4‑5.

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23 Rep., 510b6‑9.

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24 Rep., 511b3‑c2.

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25 Rep., 533c8.

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26 Epin., 990c5–991b4.

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27 Cf. Taylor, Plato, p501.

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28 Metaph., 992a20 ff.

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29 Metaph., 992a20‑1.

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30 Rep., 514a1–518d1.

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31 Nettleship, Lectures on the Republic of Plato, p260.

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32 Rep., 517b8‑c4.

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