Philosophical Studies, Part I.

By A. E. Taylor


The papers here collected, with the exception of the first, have all appeared in print at different dates; the order of their present arrangement is very different from that of their composition. Naturally, things are said in some of them which I should now say rather differently, if at all. But I have felt bound to leave the essays substantially as they were written, except for the deletion of a few mere errors and the insertion of an occasional qualifying footnote. I could indeed have wished, in connection with the third and longest of them, to take account of the friendly criticisms contained in the second edition (1933) of Dr. Stenzel's Zahl und Gestalt bei Platon und Aristoteles, had not that work appeared too late for such a purpose. It must be understood that all comments on Dr. Stenzel's views are based on the first edition of his essay (1924). In connection with the fourth paper I must at once call attention to the facts that there is now, as there was not when the essay was written, an adequate critical text of the Στοιχείωσις Θεολογική of Proclus, that of Professor E. R. Dodds (Oxford, 1933), and that Professor Dodds has convinced me both that the doctrine of the "divine henads" is really due to Syrianus, not to Proclus himself, and that the attempt made in the essay to "rationalize" this doctrine is unsatisfactory, though I have not thought it right to disguise my short-comings by alteration of the text. Three of the essays |vi| were originally delivered as public lectures, the sixth at Manchester in connection with the celebrations of the six hundredth anniversary of the canonisation of St. Thomas Aquinas in 1924, the seventh before the British Academy in 1926, in commemoration of the three hundredth anniversary of the death of Francis Bacon, the ninth before the University of Cambridge as the Leslie Stephen lecture of 1927.

I have to express my thanks for the permission to republish, in the case of the second, fourth, tenth, and eleventh essays, to the Aristotelian Society and the publishers of its Proceedings, Messrs. Williams & Norgate; in that of the third, fifth, and eighth, to the editor of Mind and the publishers, Messrs. Macmillan; in that of the sixth, to Mr. Basil Blackwell; in that of the seventh, to the President and Council of the British Academy; in that of the ninth, to the Syndics of the Cambridge University Press. My best thanks are also due to Messrs. R. & R. Clark for the pains they have taken with the printing of the volume.

A. E. Taylor
Edinburgh, March 1934

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I. Aeschines of Sphettus

[This paper was originally presented in 1928.]

When the philosopher Socrates was on his trial, he urged that his accusers, if they believed their own charges, ought to have produced evidence of his bad influence on the young from the fathers and elder brothers of his junior companions. Among other persons, actually present in court, he named in this connection Lysanias of the deme Sphettus, the father of one Aeschines (Apology 33 e). Aeschines, we see, was at this time a young man, presumably at most not older than Plato, whose eldest brother, Adimantus, is named by Socrates in the same way. From Phaedo 59 b we learn that Aeschines was one of the group of devoted friends who were present at the philosopher's death. Plato tells us no more of him, and Xenophon never mentions his name. What was known or surmised about him in Alexandrian times we learn from later writers. Most of it is collected in the brief biographical sketch of Diogenes Laertius (ii. 60-64), and seems in the main trustworthy, if we discount one or two bits of manifest tittle-tattle. According to this narrative, Aeschines was always in needy circumstances, and this one fact disposes of a tale put about by Idomeneus of Lampsacus, a member of the clique of Epicurus, that it was really he who made the plan for getting Socrates out of prison ascribed to Crito by Plato; out of spite, according to this story, because Plato resented the friendship of |2| Aeschines with Aristippus (Diogenes Laertius ii. 60, iii. 36). More than thirty years later, Aeschines, like other Socratic men, tried to profit by the temporary zeal of Dionysius II. for philosophy at the beginning of his reign. He came to Syracuse, we are told, with some of his dialogues, but was at first neglected, until he had been preferred to the young king by the good offices, of Plato according to Plutarch (Moralia 67 d-e), according to the detractors of Plato, of Aristippus (Diogenes Laertius ii. 61, iii. 36). Apparently he only returned to Athens some years later, after the capture of Syracuse by Dion in 357 B.C. (Diogenes Laertius ii. 63). For the rest of his life he took private pupils and composed speeches for the law-courts (Diogenes Laertius ii. 62). Athenaeus (611 e) has preserved the proem of an interesting discourse against him, ascribed to Lysias, on behalf of a creditor who claims to have lent him the means to start in a perfumery business, on the strength of a simple faith that a disciple of Socrates could be trusted to pay his debts, but to have been rudely disillusioned. Like other creditors of Aeschines, he found he could recover neither interest nor principal. If the speech was really composed by Lysias, the lawsuit must be dated long before the voyage to Syracuse. The tales of sharp practice need mean nothing worse than that Aeschines, like borrowers all the world over, wanted to “renew” when his creditors were not too willing.

The one point of capital importance in the Alexandrian stories is that Aeschines founded no school, and had no philosophy of his own to recommend. We may fairly suppose, then, that he, at least, had no motive for mystification in his picture of Socrates. His dialogues were highly esteemed for their pure and simple Attic style. According to the most careful writers, there were seven which were certainly genuine, Alcibiades, |3| Aspasia, Axiochus, Callias, Miltiades, Rhinon, Telauges. There was a foolish story that these were really the work of Socrates, and had been given to the nominal author after the philosopher's death by Xanthippe (Diogenes Laertius ii. 60, Athenaeus 611 d), a story which at least implies that they were supposed to reproduce the personality and manner of Socrates with remarkable fidelity. It is this which makes him of special interest to the student of Plato as a possible “control” of the greater writer's methods of work. Unfortunately, there is no certain means of knowing which writer was first in the field, though I shall try immediately to give probable reasons for assigning the priority to Plato. However that may be, since Aeschines had no doctrine of his own to “push,” representations of Socrates in which we find him in accord with Plato have a fair claim to be accepted as substantially true to fact.

One interesting consideration is at once suggested by the very titles of the seven dialogues. It is a striking feature of Plato's account of Socrates that he is depicted as closely connected from an early date with the personal circle of Pericles, a representation hardly in keeping with some modern views of the philosopher as a sort of "working-man of genius" outside the influences potent in the "upper circles" of the day. Aspasia, Alcibiades, his uncle Axiochus, Callias the "millionaire," all of them closely connected in one way or another with Pericles, figure in Plato among the habitual associates of Socrates; other personages introduced into the dialogues who illustrate the same point are Zeno, Protagoras, Damon the musical theorist, Pythodorus, Cephalus, Pyrilampes (Plato's own stepfather), and his family. All of them are in one way or another specially connected with Pericles. Now four out of the seven dialogues of Aeschines (Alcibiades, Aspasia, |4| Axiochus, Callias) are shown by their titles to be concerned with members of this group, and the titles of two of the other three confirm the suggestion thus made about the philosopher's “social standing.” The Miltiades after whom a dialogue is named is not the famous son of Cimon, but is certainly a near relative, cousin or nephew. This reminds us that Archelaus, the teacher of Socrates, had belonged to the circle of Cimon, the son of the hero of Marathon. Rhinon, who gives his name to a dialogue of which nothing but the title is known, is pretty certainly the man of the name who was the leading figure of the board of Ten which took temporary control of affairs at Athens after the fall of the Thirty, and who is said by Aristotle to have shown himself a καλὸς κἀγαθός in that trying position. Socrates is thus shown by Aeschines, as well as by Plato, as standing in close relations with the most famous of ancient Athenian houses, the Alcmaeonidae (Pericles), Philaidae (Miltiades, Alcibiades), Eteobutadae (Callias). This should, I believe, go far to make us suspicious of the later story, which first appears in extant literature in a satirical allusion of Timon of Phlius, according to which Socrates was the son of a working stone-cutter or statuary. We may reasonably suspect this of being no more than an Alexandrian misunderstanding of the playful allusion of Plato's Socrates, in the Euthyphro, to Daedalus as his ancestor, an allusion still rightly understood by the fourth-century author of Alcibiades I (121 a), and correctly explained by the scholium, on that passage. If this is so, though either comparison is misleading, it would be truer to speak of Socrates as belonging to a family of ci-devants than to call him a man of the bas peuple.1

|5| It is a curious fact about the literary methods of Aeschines that he seems specially to have affected for his dialogues the form of narrated drama (with Socrates, apparently, for the narrator). We can prove this for the Alcibiades, where, as the extant fragments show, Socrates was made to describe ex post facto the unbounded ambition and arrogance of Alcibiades as a lad, and his own attempts to convict him of ignorance of self. The Aspasia must have been of the same type, since it contained a narrative by Socrates of a conversation between Aspasia, Xenophon, and Xenophon's young wife (Cicero, De invent, i. 31. 51), and apparently also a report (by Socrates) of the relations of Aspasia with Lysicles, her protector after the death of Pericles. Probably, though not demonstrably, H. Dittmar is right in supposing that in both dialogues Socrates was represented as joining in a discussion of the character of a notable person recently deceased. The case is not so clear with the Callias, of which we know little but that it described the διαφορά between Callias and his father (Athenaeus, 220 6), whether διαφορά here means “quarrel” or “difference in character,” and that it was Plutarch's source for an anecdote about the neglect of the just Aristides by his wealthy relative (Plutarch Aristides 25). For chronological reasons the Callias of this anecdote cannot be the associate of Socrates who figures in Plato's Protagoras and Xenophon's Symposium, and at a later date as a person of considerable importance in Xenophon's Hellenica, the butt of Aristophanes and Eupolis, and the “villain” of Andocides' speech on the Mysteries, but must be his grandfather, the second of the name. But whether the διαφορά of the dialogue means a “difference,” in whatever sense, between this Callias and his father, or between Callias No. 3, the Callias of Plato and Xenophon, and his father |6| does not seem to me so clear, though Athenaeus appears to me to suppose the latter to be intended. According to him there was also in the dialogue an attack on Protagoras and Prodicus in which Prodicus was reproached with having been the teacher of Theramenes. This shows that Callias III was a character in the dialogue, since it was he who had notoriously spent his money lavishly on “sophists” and had been depicted as the patron of Prodicus and the rest in the Parasites (Κόλακες) of Eupolis. It follows, I think, that the dramatic date of the conversation must have been at some time when Theramenes had brought himself into general odium. This could hardly be before 411, and is most naturally taken to be some years later, e.g. in 405, when the Athenian people were in the mood of self-condemnation which followed the unconstitutional execution of the Arginusae generals, and anxious to make Theramenes the scapegoat for the proceedings. (Aristophanes, in the same way, in this very year makes a malicious point against Euripides by mentioning Theramenes as the kind of man produced by his theatre.) I suggest, then, as probable that in the dialogue Callias III — the Callias of Plato — was the chief personage, as we have seen that he was also a character in the Aspasia, and that he was made to tell the story of a διαφορά between his grandfather Callias II and his father, the anecdote about Aristides being part of the story.

All we know of the Axiochus — not, of course, to be confused with the “spurious” work of that name fathered on Plato — is what Athenaeus tells us (220 c), that it contained a bitter invective against Alcibiades as a hard drinker and seducer of women. (In another place Athenaeus quotes from Lysias the scandal that Alcibiades and his uncle Axiochus kept one wife between them for their common use at Abydos, and that |7| a daughter of this woman, whom each of the pair fathered on the other, was afterwards the mistress of both.) It is clear that the two men, who were both exiled over the famous scandal about the “profanation of the Mysteries,” were alleged to have spent their years of exile in very discreditable courses, much like Byron at Venice. This explains how Aeschines comes, in a dialogue Axiochus, to be attacking Alcibiades as a lecher. From the nature of its contents, this also must have been a reported dialogue. The parties incriminated must have been discussed in their absence (and presumably, after their deaths?).

From the Miltiades nothing remains but a eulogistic description of the model youth and boyhood of Miltiades, son of Stesagoras. I should suppose the Stesagoras in question to be the cousin of the Miltiades of Marathon of whom we read in the sixth book of Herodotus, but for the express statement by Herodotus (vi. 38) that this person left no son. In any case the combination of the names Miltiades and Stesagoras is reasonable proof that the person meant belonged to the same famous family, the great house of the Philaidae. We are told that he “gave such attention to his body” in early life “that he is still in better condition than any of his contemporaries,” a remark which shows that he is supposed to be of an advanced age, and would be in place if he is supposed to be a nephew of the more famous Miltiades and a cousin of Cimon. As a pure conjecture, I would suggest that the context of the passage may probably have been a general encomium on the old-fashioned education given to Attic boys before the rise of the "sophists," much in the spirit of the “Righteous Argument” of the Clouds. Presumably the speaker was defending this type of education against that made popular by Protagoras, and in that case it is hardly |8| likely that the words are spoken by Socrates. If they are, I should suppose that they are a reproduction of an alleged earlier speech of someone else on the subject, so that we are again dealing with a reported dialogue.

The Telauges is shown by the words “said I” (ἔφην ἐγώ) in one of its two surviving sentences to have belonged to the same type. Thus we can be fairly sure that four, if not five, of the seven dialogues had this form. For the rest we have no direct evidence, though there is perhaps a piece of indirect evidence which should count. I have mentioned the curious later tale that the real author of the dialogues was Socrates himself. This is a mere idle tale, to be sure, but it occurs to me that it could not have arisen, even as a bit of Alexandrian literary gossip, if Socrates had not spoken all through the seven dialogues as the narrator, as he does in Plato's Republic. If this reasoning is sound, it follows that all seven were of the same simple type of narrated conversation as Plato's Charmides and Lysis. And if this is so, I think a further consequence follows with high probability. Since the prose dialogue pretty clearly arose by direct imitation of the dramatic sketch, or mime, it is natural to suppose that the earliest dialogues to be written were of the “directly dramatic” type, and that the narrated conversation is a subsequent improvement, intended to provide fuller openings for characterisation of the speakers. Now among the Platonic dialogues, those which usually impress readers with a sense that they are first attempts, like the two called after Hippias, are of this directly dramatic type; the early reported conversations, such as the Charmides, seem to reveal a more practised hand, and this conclusion is borne out by the more special evidence of “stylometry.” This is intelligible enough if Plato had |9| to discover for himself the form most suitable to his unusual gifts of humour and dramatic characterisation. If there had already been a model in this kind before him, it is hard to think he could ever have been blind to its suitability to his genius. It seems to me therefore most probable, though not demonstrable, that Plato was first in the field as the inventor of the Socratic dis-course, and that the form of the dialogues of Aeschines was imitated from that of the Charmides and its fellows, and not vice versa.

It is of more interest to consider how far the remains of Aeschines confirm the portrait of Socrates with which Plato has made us familiar. Such scanty quotations, naturally enough, contain little which throws light on the personal appearance, manners, or habits of the hero, though the extant opening words of an unnamed dialogue, perhaps the Alcibiades, “we were sitting on the benches in the Lyceum from which the athlothetae direct the competition,” at least bear out one of Plato's statements about the favourite haunts of Socrates. It is also worth notice that an Aeschines who can only be the Socratic made the statement that Aristippus of Cyrene was first attracted to Athens by the reputation of Socrates (κατὰ κλέος Σωκράτους, Diogenes Laertius ii. 65), exactly as Xenophon relates the same thing about Simmias, Cebes, and Phaedondas. This is in keeping with the representation of Plato that Socrates had very early attracted the attention of men like Parmenides and Protagoras, and enjoyed, as he is made to say himself in the Apology, the universal reputation of being an exceptional man; it is inconsistent with the view which has sometimes been mooted that no one outside his own little circle knew anything much about him, even as late as the date of the Clouds. Aeschines, like Plato, clearly assumed that an “intelligent foreigner” would hear of |10| Socrates as one of the notabilities of Athens, much as an inquiring Frenchman or German in the eighteenth century would hear of Samuel Johnson as one of the notabilities of London, and that such a man might quite naturally visit Athens expressly to gratify his curiosity about Socrates. One later writer at least seems to have kept this representation in mind. There still exists a Syriac version, dating from the sixth century of our era, of a Socratic dialogue in which a stranger, anxious to know the real fate of the soul after death, explains to Socrates that he has come to him for an answer to the question on the strength of his widespread reputation. The name of the stranger is given in the version as Herostrophos, which appears to be a mere miswriting of Aristippus (the second r being the only difference between the two words written in the Syriac character).

For evidence of the thought and manner of Socrates as represented by Aeschines we have to go to the three dialogues of which we can still make out something of the main argument, Alcibiades, Aspasia, Telauges. We may begin with the Alcibiades, from which we possess, besides briefer citations, one fairly continuous passage preserved pretty fully by Aelius Aristides in his 46th oration, and now in some respects capable of fuller reconstruction from a papyrus (No. 1608) published in vol. xiii. of the Oxyrhynchus papyri. This long fragment, in which Socrates is describing his attempts to influence the youthful Alcibiades, runs as follows (Fr. 1, Krauss, amplified from Oxyrhynchus papyrus 1608):

“…to have treated your parents as Themistocles is said to have treated his? Why, God-a-mercy, Socrates! said he. And do you think men must necessarily be incompetent musicians (ἀμούσους) before they are competent (μουσικούς), incapable horsemen (ἀφίππους) before |11| they are capable (ἱππικούς)? I think they must be incapable first in both cases, said he. [Here follows a mutilated passage in the papyrus, in which Alcibiades apparently spoke severely about the youthful undutifulness of Themistocles.] … “and that Apollodorus made a good apology for the common-place man (ὑπὲρ τοῦ φαύλου). But, said he, there is one thing I could hardly have believed — that Themistocles was disowned by his father. A man must be a very common fellow indeed, and far gone in insanity to involve himself in that kind of quarrel and enmity with his own parents; even a small boy would know how to avoid such a thing. Why, Alcibiades, said I, do you think it so shocking a thing to quarrel with one's parents that anyone and everyone must ≪ avoid ≫ it” … [Here follow a few broken lines, and then we come to the passage previously known from Aristides.] “So, as I could see that he was emulous of Themistocles, I went on thus, Since you presume to find fault with the life of Themistocles, reflect that” … [here five lines are lost] “to know such things as that, Socrates. Well then, has it ever struck you that all this vast territory which is called Asia, and stretches from one end of the sun's journey to the other, is subject to one man? Of course, he said, to the Great King. You know then, I said, that this monarch led a force against ourselves and the Lacedaemonians, in the confidence that if he could subdue our two cities, the rest of the Greeks would be very ready to become his subjects. He threw the Athenians into such dismay that they deserted their country and fled to Salamis, first electing Themistocles their commander, and authorising him to deal with the situation at his discretion. In fact, the supreme hope of deliverance for Athens lay upon his plans for the defence. Themistocles was not moved to despair of the situation by the inferiority of |12| the Greek cities, and the superiority of the Persian in naval and military equipment and financial resources; nay, he understood that if his opponent should not surpass him in counsel, his other advantages, great as they were, would avail him little, and his conviction was that success falls as a rule to the side whose affairs are directed by the better man. As, in fact, in this very case, the King felt his situation the weaker from the very day that he encountered a better man than himself. Themistocles disposed of his forces, great as they were, with such confidence that when once he had defeated him on the water he urged the Athenians to break up the bridge the King had built. Failing in this, he sent a report to the King contradicting the resolutions adopted by the city, to the effect that he was himself doing his best to save the King and his force by opposing the Athenian demand for the destruction of the bridge. Hence it is not merely ourselves and the Greek world who regard Themistocles as the author of our deliverance; the very King whom he defeated believed that he owed his preservation to him, and to him alone, such was his preeminence in intelligence. Consequently, when he was exiled from the city, the King rewarded him for his supposed preservation by the gift of the whole government of Magnesia, and many other presents. Thus even in exile he was more flourishing than many Athenians who were at home, unmolested and in the highest repute for merit. Who has a better claim, then, to be considered the most powerful man of his age than Themistocles, the generalissimo of the Greeks, and conqueror of a king whose dominions reach from sunrise to sunset? And yet, Alcibiades, said I, you must remember that though he was all I have said, all his service was not enough to save him from exile and disfranchisement at the hands of the city, but proved too little. What then, |13| think you, must be the case with mere ordinary men who bestow no tendance on themselves? Is it not a miracle if they succeed, even in trifles? Nor must you charge me, Alcibiades, said I, with thinking irreligiously and oddly of Fortune and the supernatural, because I credit the man with science of all he accomplished, and hold that none of these exploits were due to Fortune. I should find it much easier to prove to you that the irreligion is on the side of those who disagree with me than they could to prove as much of me, since they believe that Fortune comes indifferently to the evil and the good, that Heaven does not favour the more deserving as the more pious party.”

The two remaining shorter fragments come from the framework in which this narrative was set. They are as follows: “Had I imagined I could benefit him by any rule of art (τέχνῃ), I should certainly plead guilty to gross folly. But in point of fact, I supposed this advantage over Alcibiades had been given me by Providence (θείᾳ μοίρᾳ) and there is nothing in this to be surprised at” (Fr. 3, Krauss). “My passion (ἔρως) for Alcibiades was like the experience of the Bacchanals: when the god takes possession of them, they can draw honey and milk in places where others cannot even get water from the wells. It was just so with me; I knew no doctrine (μάθημα) which I could benefit a man by teaching him; still I imagined I might make him the better by my companionship because of my passion for him” (διὰ τὸ ἐρᾶν. Krauss, Fr. 4).

To deal first with the two briefer fragments. We see from them that Aeschines concurs with Plato on two capital points connected with the personality of Socrates. He, like Plato but unlike Xenophon, ascribed to him a very special relation to Alcibiades, going back to the boyhood of the latter, and, in the earlier years |14| of the headstrong youth, at any rate, a very marked influence over him, an influence which Socrates hoped to use for the moral betterment of a youth of such brilliant promise. The strength of this influence was indicated in the dialogue, as we learn from Aristides, by the effect of Socrates' account of Themistocles on the lad. He drove him to lay his head on his knees and shed tears of despair at the contrast between his own “preparation” for public life and that of his prototype (Cf. Plutarch Moralia 69 f). Also, Socrates was made to describe his own feelings for the wayward and brilliant lad in the language of exalted passion. He calls it ἔρως and compares it with the condition of a Bacchic votary in the hour of actual “possession” by the god. Aeschines thus agrees with the representation of the early relations between the two men familiar to us from Plato's Symposium, and moreover agrees with Plato in crediting Socrates with the temperament of the “lover,” the fourth species of μανία described in the Phaedrus. This is entirely unlike the colourless version of the matter given, for apologetic reasons, in Xenophon's Memorabilia, where it is vital to the writer's purposes to minimise the connection between the two men, because, as his own language shows, the gravest of all the charges against the philosopher which he has to meet is that Alcibiades and Critias had been his “pupils.” Since Xenophon has thus an obvious motive for understatement, whereas it is not conceivable what common motive Plato and Aeschines could have had for exaggeration, it should be obvious which account is likely to be the truer to fact. It is possible, no doubt, that the dialogue of Aeschines may be later than the Symposium and is influenced by it, but even if that should be so, it would be significant that Aeschines should evidently accept Plato's representation as in |15| keeping with his own recollections of his old friend and teacher.

The second point seems to me still more important. Not only does Aeschines agree with Plato, and disagree profoundly with Xenophon, in making Socrates disclaim the possession of any ready-made ἐπιστήμη or τέχνῃ but he also definitely connects the “fine frenzy” of the lover, as Plato does, with the personality of Socrates. This is, to my mind, one among the other reasons for holding that in the famous “erotic discourses,” as elsewhere, Plato has artistically suppressed his own personality. Whether Plato had the temperament he and Aeschines agree in ascribing to Socrates, or had not, is, it seems to me, a question we have no means of answering. Like others, I may have my own personal conviction on the point, one way or the other. But naturally I cannot treat my personal conviction as evidence of its own truth. It should, however, be clear that if we trust the agreement between Aeschines and Plato as proof that the temperament ascribed to Socrates in the Symposium is that of the historical man, a whole large literature which has grown up around the name of Plato will have to be relegated to the limbo of unverifiable hypotheses. Not a few deserving students, particularly in Germany, continue to find in a certain form of “erotics” the key to both Plato's personality and his philosophy. If it should prove to be the fact that what has been taken for profound self-disclosure is really a masterpiece of inspired portrait-painting, the whole of this literature loses its supposed value for the biographer of Plato.

As minor points of agreement with Plato we may observe, as I have already said, Socrates' disclaimer of the possession of any ἐπιστήμη or τέχνῃ which could be imparted to a pupil, with the implication that, as the |16| Apology puts it, he had never had a real μαθητής his young friends did not stand to him in the relation of pupils to a preceptor, though, at the same time, it is suggested that association with him might be expected to make them “better men,” a phrase which really presupposes the doctrine of the "tendance of the soul" as the great business of life expounded by Plato in the Apology and elsewhere. We note, too, certain familiar tricks of speech, like the phrase θείᾳ μοίρᾳ.

When we come to the long continuous passage about Themistocles, we see again that its purport is a double one, and agrees with that of so many of the discourses of Plato's Socrates. The underlying thought is that the secret of success in life and in statesmanship is something which is called by two names, both familiar to us from Plato, ἐπιστήμη and ἐπιμέλεια αὑτοῦ, “care for the self.” And the point to be made appears to be that even Themistocles, though he tried to “prepare” himself, and was better qualified for public life than the self-confident youth who dreamed of surpassing him, did not really possess the true ἐπιστήμη, did not “care for himself” in the right way. He rightly judged that the victory would be decided, not by numbers nor wealth, but by the personality of the commanders on either side; but with all his knowledge, he did not make a true success of his life. He ended in disgrace and exile, — not such an exile as a Socrates might have accepted from loyalty to high principle, but an involuntary exile which meant the ruin of his whole scheme of life. The main thought of the dialogue would thus appear to have been this. Themistocles, the model whom Alcibiades is anxious to surpass, had an immense advantage over his rivals in sagacity, invention, knowledge of affairs; yet he had not the knowledge a man must have to make his life a moral success, the knowledge how |17|
  …to keep the law
  In calmness made, and see what he foresaw.

He did not “care for himself” in the proper way.

Behind all this lies the set of ideas worked out later by some pupil of Plato in a dialogue obviously based on that of Aeschines, the Alcibiades I, where it is argued that “tendance of the self” is the secret of moral success, this tendance must rest on a true knowledge of the thing tended, and this thing, the “self,” as distinguished from the instruments it wields, is the soul; so that the true statesman must, before all things, be a man who makes it the business of his life to see that his own soul, and those of those for whom he acts, are "as good as possible. “We thus recover the familiar thesis that the virtues are all one and the same thing, knowledge of good, and the fragments of our dialogue are an important link in the chain of evidence by which the theory is vindicated for the "historical Socrates.”

We note, also, that the appeal to the careers of Athenian public men is used in much the same way in which it is employed by the Socrates of Plato. The comments on the final collapse of the career of Themistocles reminds us at once of the thesis of the Gorgias, that the famous leaders, from Miltiades to Pericles, may have been good “body-servants” of the people, but cannot be pronounced true statesmen, since they regularly ended by being disgraced by the society they had “tended,” and thus must have been deficient in the knowledge on which all “tendance” of a creature should be based; they did not really know what good is, and therefore, whatever they may have done for the subjects of their “tendance,” they could not make them good. Here too, I believe, we can detect at least one echo of the actual phraseology of Socrates, when the words are put into his mouth that Themistocles was the superior |18| of Xerxes τῷ φρονεῖν. As Burnet has observed, the use of φρονεῖν in the sense of “to be wise,” “to be intelligent” is definitely non-Attic, being, in fact, a loan from the vocabulary of Ionian science. The word is peculiarly in point on the lips of the old disciple of Archelaus, the man against whom Aristophanes raised a laugh by calling him a φροντιστής, his “brave notions"” φροντίδες, and his house a φροντιστήριον.

Possibly we should attend to a further point, though this is more conjectural. We see that the description of Themistocles' amazing career was preceded in the dialogue by some observations, now lost to us, about an ugly incident in his early life, his alleged repudiation by his father Neocles (Aeschines is likely to be the source from which this incident has passed into Plutarch's Life of Themistocles). We cannot, of course, be sure what use was made of this anecdote. But it looks as though Socrates were saying, in the immediate context, that just as a man begins by being a bad horseman before he is a good one, Themistocles must have begun, as we all begin, without the knowledge which enables a man to “care for himself.” (His quarrel with his father will be proof of the fact.) Socrates is thus openly or covertly attacking the theory that “goodness” comes by nature. I therefore suspect that he must have been made also to dwell on another point, duly made in the Alcibiades I, as it is by Plato in the Protagoras and Meno, that, if "goodness" is not simply born in us, neither do we pick it up from our parents, as we do our native speech. I suspect that the story of Neocles' quarrel with his son was used to show that the son cannot be supposed to have “learned goodness” from a father who disowned him.

(The meaning of the allusion to a defence of the “vulgar man” (the φαῦλος) by Apollodorus can, I believe, |19| no longer be explained. It should, at any rate, be clear for chronological reasons apparent to any reader of the Symposium, the apologist for the φαῦλος cannot, as Burnet suggested to the editors of vol. xiii. of the Oxyrhynchus papyri, be Apollodorus of Phalerum, son of Aeantodorus, known to us from the Symposium, Apology, and Phaedo. He would be still unborn in the days when Socrates was conversing with the youthful Alcibiades.) The Aspasia also presents fresh and interesting points of contact with Plato. The precise structure of the dialogue is unknown, but it looks as though Socrates, speaking presumably after the death of Aspasia, dwelt on her remarkable abilities. We know that this was proved by two considerations. Someone, probably Socrates, told the story of her connection with Lysicles in the years immediately after the death of Pericles, and insisted on the point that Lysicles owed his temporary success as a δημήγορος to her instructions. Apparently the thought was that Aspasia was able to train him in the political ideals and methods of Pericles. (Plutarch, Pericles 24, and the valuable scholium on Plato Menexenus, 235 e.) Her practical wisdom was also illustrated by the story referred to by Cicero (De invent, i. 31. 51 ff.), Quintilian (v. 11. 27 ff.) and Victorinus, of the good advice she gave to Xenophon and his young wife about the avoidance of conjugal disagreements. There is a curious problem here to which I can only refer in passing. Though the name Xenophon is not uncommon, it is difficult to think of any known Xenophon, with a marked taste for farming and horsemanship (referred to by Aspasia), who could well have been introduced into the dialogue as a young married man, except the well-known writer. And Cicero, who had the full text before him, clearly made the identification, since |20| Xenophon, in his mouth, without further specification, can hardly mean anyone else. But this Xenophon left Athens in 401, and never saw Socrates alive again. He was a very young man at the time, and not yet married to the only wife he is certainly known to have had. And we cannot well suppose that Aeschines committed a gross error about the personal affairs of a contemporary whom he must have known well: this kind of “anachronism” is hardly in nature. It seems to me easier to believe that the well-known Xenophon should, in early manhood, have married and lost a wife who happens not to be mentioned elsewhere in extant literature. After all, young married women are exposed to certain perils by the very fact that they are young married women, and if Xenophon lost a wife early in life, there is no place in his writings where a reference to the fact would be particularly relevant.

The real interest of the dialogue is independent of these personal issues. From the notices preserved to us, it is clear it contained a discussion of the possibility that any woman could exhibit the capacity for affairs presupposed by the story of the help given by Aspasia to Lysicles. It must have been to meet a doubt on this point that someone (most likely Socrates) was made to appeal to two earlier precedents, that of a certain Thargelia, like Aspasia an ἐταίρα of Miletus, who was said to have used her fascinations freely to enlist prominent persons in Thessaly and elsewhere in the cause of Xerxes against Hellas, and that of a real or imaginary Persian princess Rhodogyne, who had the reputation of an “Amazon,” and was honoured by a statue in which she was figured with half-braided hair, the tale being that, receiving the news of a rebellion as she was at her toilet, she hurried into the field “as she was.” We are not actually told by the later writers who allude to |21| the stories into whose mouth Aeschines put them, but since Maximus of Tyre (xxxviii. 4) relates an incident which must come from a dialogue about Aspasia, that Socrates advised Callias (III) to put his son under Aspasia's tuition, and Athenaeus (220cb) that Callias was spoken of, and called by someone a κοάλεμος in our dialogue (where he was therefore presumably a personage), it seems to me that the story about Thargelia, and presumably therefore also about Rhodogyne, was told by Socrates to prove that the qualities he ascribed to Aspasia were not without precedent. The two stories together are intended to support the familiar Socratic tenet on which Plato has based the developments of Republic v., that “the goodness of a man and that of a woman are the same.” Since the two commonly recognised chief forms of the goodness of a man are valour in the field and prudence in counsel, the two stories are meant to show that neither is confined to one sex. We have thus evidence from Aeschines that however much in the details of Republic v. may be “development” due to Plato, the central thought that both sexes should be expected to take their share in the work of administration and military life is strictly Socratic.

There is one further remark suggested by what we know of the Aspasia. H. Dittmar, in his valuable edition of the remains of Aeschines, rightly dwells on the point that it was the writers of Socratic dialogues, particularly Aeschines and Antisthenes (and to a lesser degree Plato in the Menexenus), who, so far as we can see, created the romantic Pericles-Aspasia motiv. The treatment of the subject by the contemporary comedians, Cratinus, Aristophanes, and the rest, was not romantic. In their view, Aspasia was simply a “whore” and “bawd,” and Pericles a “lecher.” Dittmar further contends that the Socratics so far agreed with the |22| comedians as to regard the devotion of Pericles to his lady as a mere sensual weakness, and a proof that he was no true philosophic statesman. Now though Antisthenes, the immediate source of the well-known story that Pericles never left his house without kissing Aspasia good-bye, is likely enough, in view of his known opinions, to have held such a view, I confess I find it hard to think Aeschines had set him the example. If we take this view, we must suppose that all that was said in the dialogue about the capacities of women, including the advice of Socrates to Callias, was meant for irony pure and simple. But our existing fragments do not enable us to judge whether the “irony” which is so characteristic of Socrates in Plato was reproduced by Aeschines or not. And we can hardly conceive the great asserter of the “unity of virtue” treating the thesis that the “masculine” virtues are not confined to the male sex in a mere spirit of persiflage. Is it not more probable that Aeschines meant his account of the ability of Aspasia to be taken seriously? The conclusion is, I believe, borne out by the parallel with Plato's Menexenus, where Socrates professes to have learned his patriotic speech from Aspasia, who is also said to have inspired the famous “funeral oration” of Pericles. The speech in the Menexenus is only half ironical; it is a subtle mixture of sound patriotism with superficial “chauvinism.” It could not have been put into the mouth of a πολιτικός of the kind in whom Plato believed, a statesman inspired by genuine knowledge of good, but it is exactly the sort of compound of nobility and baseness, wisdom and prejudice, to be expected from one whose “goodness” rests on uncriticised “opinion,” and this, according to the Platonic view, was the case with Pericles himself.

The remaining dialogue of which it is possible to say |23| something, the Telauges, again throws light on a side of Socrates duly represented in Plato, but often forgotten, his relations with Orphic and Pythagorean ascetics and “Salvationists,” and so helps to illustrate the Euthyphro and Phaedo. Like Plato, Aeschines seems to have represented the philosopher's attitude towards such persons as a curious blend of appreciation and detached criticism. (Demetrius Περὶ ἑρμηνείας remarks that a reader might be puzzled to know whether the account of Telauges was meant by Aeschines as θαυμασμός or as χλευασμός.) Little is known of the structure of the dialogue (which, as we have seen, was a reported one), except that it introduced, besides Telauges, two personages familiar to us in members of the Socratic circle, Critobulus, the fashionable and dissipated son of Crito, who figures in Xenophon and in the Euthydemus, and Hermogenes, the half-brother of Callias III, known to us from the Cratylus as interested in Orphicism, and from Xenophon's Apologia as the professed authority for Xenophon's account of the last days of Socrates. Whether Cratylus, who appears in Plato as a friend of Hermogenes, was also a character in Aeschines, as Burnet has maintained, is not so clear, though Aristotle's remark that Aeschines (presumably our author) had used the expression τοῖν χεροῖν διασείων of Cratylus points in that direction (Aristotle Rhetoric 1417 b 1).

Telauges is not otherwise known to us. The name belongs to Pythagorean legend, in which it appears as that of a son of Pythagoras, but the person meant by Aeschines was clearly a man of the same time as Hermogenes himself, since one of the very few later notices of the dialogue (Proclus in Cratylus 21) tells us that Hermogenes was reproached for not caring for the necessities of his ἐταῖρος Telauges. Presumably then, Telauges is a real person of the time of the Archidamian |24| War, a practitioner, as we shall see directly, of extreme asceticism and simplification of life, and probably a Pythagorist of the “…strict observance.” (Only a man belonging to such a circle is likely to have received such a name, just as no one in our own society is likely to be called Ignatius or Aloysius, unless he comes from a family with Roman Catholic connections.) The little we know of the argument of the dialogue comes partly from Athenaeus (220 a), partly from an allusion in the notebook of Marcus Aurelius (vii. 66). Athenaeus, or rather the speaker in Athenaeus, uses the dialogue to support the charge that the philosophers are even more satirical rascals than the comedians. Aeschines, he says, in his Telauges, satirises Critobulus for his stupidity and foulness of life, and raises immoderate laughter against Telauges on the score of his slovenly habits. He depicts him as hiring a ἱμάτιον from a fuller for half an obol a day, going about in a κώδιον — apparently a sort of leather apron — lacing his sandals with thongs of esparto-grass. (The text of the rest of the sentence seems too corrupt for certain restitution, so that we only hope Dittmar is mistaken about the very unsavoury trait with which he completes the picture.)

The reflections of the Stoic emperor really tell us little beyond the fact that he had read Aeschines. How can we be sure, he asks, that Telauges may not have been a better man than Socrates? It is no sufficient answer to tell of the splendid figure made by Socrates before his judges or in the prison. The question is not what Socrates seemed to human observers to be, but what he was in the inner man, whether his attitude was dictated by pure love for virtue for her own sake.

It seems to me foolish to see, as some critics have seen, in these remarks an attempt, in the interests of the “Cynic” ideal, to set up Telauges as a rival to Socrates. |25| The Emperor means only what Kant meant when he said that apparently heroic discharge of duty may be prompted by a disguised vanity which destroys its moral worth, and all we can really infer about the dialogue of Aeschines from this passage is that in the dialogue Socrates figured as the type of the truly virtuous man, Telauges as the caricature. Marcus is saying that you cannot be sure that Socrates really has this superiority, unless you can read to the bottom of his heart.

Putting the two notices together, it seems safe to say that the main point with Aeschines must have been to contrast Socrates at once with Critobulus, an exquisite and dandy who is “foul within,” and with Telauges, who tries to make external raggedness and dirt a proof of real goodness. The point was worth making, since the association of Socrates with persons of this type, the extreme simplicity of his life, and the poverty of his later years might lead to misapprehension. We all know what play Aristophanes in the Clouds makes with the suggestions of rags, dirt and beggary. We remember again, that in the curious story preserved by Xenophon of the philosopher's relations with Antiphon, his lack of a shirt and of changes of raiment to meet the changes of the seasons are brought up against him. The suspicion that he was something not very different from the kind of fanatic who despises cleanliness, neatness, and decency had to be refuted by those who cherished his memory. It would be the more resented that the leading Socratic men — if we except the eccentric Antisthenes — were not of the type which finds slovenliness attractive. The “spruceness” of the young men of the Academy in particular was made by comic poets ground for an accusation of foppishness. Aeschines contrasts the decent poverty of Socrates with the |26| deliberate filthiness of Telauges in the same spirit which prompted the recollection or invention of the rebuke to Antisthenes, who had paraded the rent in his cloak, “the tear shows your — vanity.” It is presumably for the same purpose that Plato is careful to let us know that, for all his usual severe simplicity in such things, Socrates could and did “dress,” and apparently dress well, for such an occasion as Agathon's party. Even in the most “unworldly” of the Platonic dialogues, when Socrates is enlarging on the theme that the cult of the body is something that must be despised by the man who means to “make his soul,” we are only told that such a man will condemn “fine clothes and shoes and such adornments of the body, — so far as it is not a downright necessity to have them” (Phaedo, 64 d). The qualifying clause marks the difference between a Socrates and a Telauges. It is meant to show that the simplicity of the Socratic life is not based on superstition, and has nothing to do either with veneration of dirt or with disregard of the decencies of civilised living. Socrates, to put the thing in modern language, has no “evening” coat; it is not that he thinks the wearing of one sinful, but that he cannot afford one, and the reason why he cannot is that all through a protracted “world-war,” he has been too busy with his mission to the souls of men to give a thought to his property.

We may conjecture that Socrates was made to condemn at once the elegant of the type of Critobulus, who is still neat and dressed without but filthy within, and the Pythagorist who lodges a clean soul in a “pigsty.” Perhaps it was to anticipate a possible rejoinder from Telauges that his squalor was the inevitable effect of poverty that Socrates made the remark, whatever it was, which Proclus takes as censure on Hermogenes for neglect of his friend. The only actual words preserved |27| from the dialogue are quoted by Priscian: “let us reap some benefit from the excellence of your intelligence” (or? “purpose” διανοίας); “again, Solon the legislator is dead, but to this day we owe great benefits to him.” Nothing can be built on such scraps, but they read like part of an argument against “fugitive and cloistered virtue.” The point seems to be that
          If our virtues
  Did not go forth of us, 'twere all alike
  As if we had them not.

Perhaps it is not fanciful to see in this a remonstrance from Socrates, the active missionary, addressed to Telauges, a “contemplative,” who not only “lights his torch for himself,” but by cultivating habits which make his company insupportable, actually smothers it “under a bushel.” But this is, of course, avowedly pure conjecture.


1^ Grote had long ago seized on the point that Socrates belonged to “the ancient gens” of the Daedalidae. (History, ed. 1888, vii. 82.)

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II. Parmenides, Zeno, and Socrates

{This paper was originally presented before the Aristotelian Society in 1916. Allusions to Professor Burnet's views, unless otherwise stated, are from the analysis of the Parmenides in his volume, Greek Philosophy: Thales to Plato.}

There is a laudable unwritten custom of well-bred society by which metaphysical discussions are carefully excluded from polite conversation. The reason of the rule is probably, as Mr. Jourdain has lately explained, that such discussions commonly involve the perception of jokes of the fourth order, and jokes of a higher degree than the second, or at best the third, are imperceptible by all but an insignificant minority of mankind. Hence the prohibition of their perpetration in general conversation is an easy and obvious deduction from the principle of the Categorical Imperative. History, however, presents us with two brilliant exceptions to the general rule: the conversation held at a memorable tea-party between Alice, the Mad Hatter, and the March Hare, and that which, if we may believe Plato, took place at Athens at a certain celebration of the Panathenaic festival, some time about 451 or 450 B.C., in the house of the well-known admiral and politician, Pythodorus, the son of Isolochus, between Parmenides, Zeno, and the youthful but already distinguished Socrates. Mr. Jourdain has already published an entertaining and illuminating commentary on one of these singular conversations: I propose this evening to invite your attention to some points of interest connected with the other. |29| I cannot, of course, undertake to deal here with so wide a subject as the purpose and argument of the Parmenides considered as a whole. All that I intend is to offer a slight contribution to the history of early Greek logical theory by attempting to throw some light on one or two lines of reasoning which are made prominent in the dialogue, and I shall select for special consideration two topics, the use made by Parmenides of the appeal to an infinite regress, and his attempted Refutation of Idealism. Before I can deal with either point in detail it will be necessary to say something in general about the dramatic setting which Plato has provided for the discussion, a subject on which the commentators, so far as I am acquainted with them, have been unduly silent.

If we examine the Parmenides, as we have the right to examine any dialogue of Plato, simply as a work of dramatic art, we shall see at once that it has certain peculiarities which give it a unique place among the Platonic "discourses of Socrates."Its form, to begin with, is unusually complicated; it is a narration by an otherwise unknown speaker of a narration of a narration of a conversation. Hence its "formula," as Professor Burnet calls it, is "Antiphon said that Pythodorus said that Parmenides, Zeno, or Socrates said such and such a thing."The scheme is, of course, far too cumbrous to be kept up at all rigidly, and Plato repeatedly allows himself to drop for convenience into direct reproduction of the conversation. So far as this scheme goes, however, the Parmenides does not stand alone; we have an almost exact counterpart in the Symposium, with the exception that there the speaker who relates what he had heard about the famous dinner in honour of Agathon's victory is himself a known person, and that his story has come to him at only one |30| remove, so that the formula reduces to "Aristodemus told me that Socrates, or Aristophanes, spoke as follows."The full singularity of the scheme adopted for the Parmenides only becomes manifest from a rather fuller examination of the imaginary circumstances of the recitation. The speaker who relates Antiphon's account of Pythodorus' account of the interview between the three famous philosophers is indeed named, but beyond his mere name we learn no more of him than that he belongs to a group of citizens of Clazomenae who take a keen interest in philosophy (μάλα φιλόσοφοι, 126 b). Where, or to whom, he is speaking we are not told. The scene is certainly not in or near Athens, and to judge from the way in which the word οἴκοθεν, in his opening sentence, is explained by the addition ἐκ Κλαζομενῶν, it is not in Clazomenae. We are really entitled to say no more than that the story of the meeting of Socrates with the Eleatic philosophers is related somewhere by a person interested in philosophy to a like-minded audience. This complete silence about the place and the personnel is a thing unparalleled in the rest of Plato's dialogues. In the case of directly dramatic dialogues, the mere presence of Socrates himself provides sufficient indication of place. Even in a work which avoids all more specific references, like the Philebus, we are at least sure that we are to imagine ourselves in Athens or its immediate neighbourhood. The Theaetetus is supposed to be read aloud at Megara long after the conversation which it records, but the opening discourse between Euclides and Terpsion is intended to make it quite clear when and where and in what circumstances the reading takes place. So in the Phaedo Plato is quite careful to direct our attention to the point that Phaedo's narrative of the master's last day on earth is delivered some little while after the |31| event before the Pythagorean community of Phlius. With reported dialogues the case is much the same. Apollodorus in the Symposium, for example, expressly explains that his recollection of Aristodemus' narrative is just and vigorous because he had rehearsed the whole only a day or two before (πρῴην) in conversation with a friend as he walked from his home in Phalerum to the city. In the Republic, Socrates repeats a conversation in which he had been the central figure only the day before, and we are told just where it had been held, in the house of Polemarchus in the Peiraeus; in the Protagoras he has only just left the circle in the house of Callias when he meets the friend to whom he relates the events of the day. Even in the Laws what we may call the stage directions are perfectly clear and distinct. The Sophistes and Politicus, indeed, so far as their contents go, have nothing to indicate time and place, but both are carefully attached to the Theaetetus in such a way as to date them immediately after the filing of the accusation against Socrates in the year 400-399. That the immediate speaker in the Parmenides should be, as he is, quite uncharacterised, and should be speaking no one knows where and to no one knows whom, is quite against Plato's usual practice, and the departure from custom has, therefore, presumably a reason.

Still, if we learn little about Cephalus, the one definite thing that we do learn is significant enough. We are expressly told that he and his friends made the journey from Clazomenae to Athens for no other purpose than to learn from Plato's younger half-brother, Antiphon, the details of the conversation between Socrates and the Eleatics (πάρειμί γ ͗ ἐπ ͗ αὐτὸ τοῦτο δεησόμενος ὑμῶν, 126 a, and the more express statement of 126 b just below). This conversation, we must remember, is supposed to have been held when Socrates, who |32| was born in or shortly before 470 B.C., was still, "exceedingly young" (σφόδρα νέον, 127 c), i.e. not later than about 450. It is assumed that, at the time when Cephalus is speaking, all the persons who had actually been present on this memorable occasion were already dead, and a correct account of what happened could only be obtained from Antiphon, who, we are told, had heard the tale from Pythodorus, in whose house the meeting took place, so often that he had got it by heart (εὖ μάλα διεμελέτηεν, 126 c). That Proclus is right in pointing out that the death of Socrates is presumed in this narrative is obvious. So long as one of the parties to the original conversation was alive, it would have been ridiculous to make Cephalus go to a secondhand source for his information. How long after 399 Cephalus is supposed to be speaking cannot perhaps be decided. Antiphon is now no longer a μειράκιον but a young man, but, in the absence of any positive knowledge about the date of his birth, we can draw no inferences from this. The important point is simply that the journey of Cephalus to Athens must be supposed to happen not less than half a century after the meeting of the three philosophers, and quite possibly a number of years later. What may we reasonably infer from Plato's assumption of this story as the basis for his dialogue? First of all, I think it is clear that Professor Burnet is right (Phaedo, p. xxiii) in calling attention to Plato's habit of calling attention to the fact that he could not have been personally present at some of the scenes which he describes. Thus the device of making Apollodorus repeat at second-hand from Aristodemus the incidents of the Symposium serves to remind us that Plato, who was a mere boy at the time of Agathon's tragic victory, could not have been present at its celebration, and is not proposing to speak as an eye-witness. |33| Similarly, the insistence in the Parmenides on the point that there is now only one person living who can satisfy the curiosity of Cephalus, and that he himself had got his knowledge, when a mere lad, from a much older man who is now dead, is an effective device for warning us that the scene to be described belongs to a very remote past, of which Plato could himself have no direct knowledge. And we may suspect that one reason for the pains which he has taken to explain how the narrative was passed on by Pythodorus to Antiphon and by Antiphon to Cephalus is to make it clear that it has been derived from sources entirely independent of himself.

To the reader this means, of course, that Plato declines to pledge his personal credit for the historical accuracy of all the details. If we find the Eleatic philosophers apparently conducting their dialectic with a special view to fourth-century controversies between Plato and his contemporaries — well, Plato has as good as told us that he is not responsible for the accuracy of the narrative. He was not there to hear what Parmenides and Zeno actually said, and the version he puts before us makes no profession to come in any way from Socrates. It is what Antiphon professed to have learned from Pythodorus; we might be interested to know whether Socrates would have confirmed it on all points, but … Socrates is unfortunately no more, and, even for what Pythodorus said, we have only the recollection of one much younger man whose testimony cannot be subjected to any process of control, and must be taken for what it is worth. In no other dialogue has Plato been at such elaborate pains to make it quite clear that he has left himself free to colour his account of a conversation in the distant past with an eye to the philosophical situation in the present.

But there is another and even more important |34| inference suggested by the opening narrative of the dialogue which, so far as I know, has never yet been pointed out with sufficient plainness. The initial assumption of the story about Cephalus and his visit to Athens is that the meeting of Socrates with the famous Eleatics was not merely an historical fact — that it was so seems to be now the current view of most writers on the history of early Greek philosophy — but that it was an event of absolutely first-rate importance. It is taken for granted that the conversation of the three philosophers was so notable that half a century or more afterwards it was remembered as something of remarkable interest by the friends of Cephalus at Clazomenae, who, indeed, sent to Athens for the express purpose of getting the true account of what had passed from the one person on earth who could supply it.

Of course, I do not suggest here that it is necessary to suppose that the mission of Cephalus to Athens is an historical fact. It is most likely to be no more than an artistic fiction on the part of Plato. The really important point is that Plato should have thought the story, true or false, sufficiently plausible to make use of it as he does. It implies at the very least that the philosophers of Clazomenae took the same sort of interest in Socrates and his doings which the Phaedo attests for the Pythagoreans of Thebes and Phlius and the Theaetetus for the philosophers of Megara. Nor would it be hard to account for the existence of this interest. When all the available evidence for the dates in the life of Anaxagoras are carefully compared, it seems almost certain that the prosecution which terminated that philosopher's thirty years of residence in Athens must have occurred somewhere about 450 B.C.,1 in spite of the general agreement of modern historians in favour of placing the |35| event nearly twenty years later. This explains among other things why in the Phaedo the influence of Anaxagoras on Socrates is represented as exerted partly at secondhand, partly through his book, and nothing is said of any personal intercourse between the two men, why again in the Greater Hippias Socrates is made to contrast Anaxagoras as one of the "ancients" with the men of his own time, why the doxographic tradition, which goes back to Theophrastus, always mentions not Anaxagoras himself but his successor, Archelaus, as the teacher of Socrates, and finally how Anaxagoras was able, between his disappearance from Athens and his death in the opening years of the Peloponnesian War, to organise a philosophical school in Ionia, which appears to have been still in existence in the time of Epicurus. It also explains the interest of the philosophers at Clazomenae in Socrates. For Clazomenae was the native city of Anaxagoras himself, and though all the accounts agree in naming Lampsacus as the actual centre from which he propagated his philosophy after his enforced retirement from Athens, we may be sure, even if the history of Epicureanism did not prove the point, that science continued to be studied in Ionia generally, and that the fame of a brilliant pupil of Anaxagoras' successor, Archelaus, would be sure to spread to the birthplace of Anaxagoras himself. A meeting of Socrates with the great Eleatics would be memorable as marking the beginning of the process by which the science of the Ionian East and the Italian West were for the first time brought together at the only place which, for historical reasons, was adapted to serve as a general clearinghouse for Greek speculations — the Athens which was already becoming the great political and commercial centre of the civilisation of the Mediterranean basin. Hence the naturalness of the |36| fiction that even after more than half a century the event should have been so vividly recollected that the scientific men of Clazomenae sent a special deputation to recover a detailed account of it from the only living man who was in a position to supply one.

It should also be borne in mind that there are special reasons why it is humanly certain that a young man of philosophical genius living in the middle of the fifth century, and already feeling dissatisfied, as we are told in the Phaedo Socrates was dissatisfied, with the current Ionian views about science, would make a point of being introduced to the most famous representatives of Western ideas. In the Parmenides itself all that we are told by way of explanation of the presence of Parmenides and Zeno in Athens is that they had come to visit the Panathenaic festival. To understand the full meaning of this we need to recur to information supplied partly by the poem of Parmenides himself, partly by statements made in the Platonic Corpus and elsewhere about Zeno. Our thanks for the preservation of the eschatological proem to the poem of Parmenides are due to Sextus Empiricus, who inserted the whole of it in the first of his treatises "against the dogmatists" (Sextus Empiricus, Advanced Mathematicos, vii. iii). The recent reexamination of the manuscripts by Mutschmann for his still incompleted edition of Sextus, shows that according to the best text Parmenides began his verses with an invocation to the divinity ἣ κατὰ πάντ ͗ ἄστη φέρει εἰδότα φῶτα, "who guides the man who knows through all cities." This means, of course, that Parmenides himself was in the habit of traveling from city to city and giving epideictic displays of his philosophy. Like the evidence, which goes back to Isocrates, for the actual education of Pericles by Anaxagoras, the allusion shows how far it is from being true that there |37| was in the middle of the fifth century any hard-and-fast line of distinction between the man of science and the so-called "sophist" who undertook the "education of men" as a profession. We need not, of course, suppose that Parmenides made nothing by his epideixeis any more than that Anaxagoras derived no personal advantages from his position as instructor to Pericles. If Protagoras came to be popularly regarded as the inventor of the sophistic profession, we must remember both that according to the account of Plato he must have been in the field at least twenty years at the date of the visit of Parmenides and Zeno to Athens, and that the special novelty of his programme was not that he was paid for his services but that he substituted the art of political success for science as the subject of his instructions.

About Zeno the case is even clearer. It is quite beyond reasonable doubt that Zeno not only taught for pay but that he must have settled in Athens and practised his calling there for some considerable time. This is explicitly stated in a dialogue, the First Alcibiades (119 a), which though probably not Platonic, is at any rate shown by its style and contents to be a fourth-century Academic work little if anything later in composition than the latter years of Plato's life. We are there expressly told that two well-known public men of the fifth century, Pythodorus son of Isolochus — and it is manifestly he, as Proclus saw, who is the Pythodorus of our dialogue — and Callias the son of Calliades, the commander who fell honourably before the walls of Potidaea, paid him 100 minae each for his instructions. Zeno's permanent residence in Athens is equally implied by Plutarch's story that Pericles had been one of his hearers, and by the well-known allusions of |38| figured as discussing problems connected with the notion of the infinitesimal. In fact, it is precisely this professional activity from which Zeno derived the name of "the sophist. "Writers who wish to distinguish Zeno of Elea from Zeno of Cittium and other persons of the same not unusual name call him ὁ σοφιστής, not, as Earl Russell has imagined, by way of disparagement of his mathematical paradoxes, but simply because he did, as a matter of fact, follow the calling of a paid instructor of young men, just as we might speak today of "So-and-so the Army coach" or "Such-a-one the journalist."And if we will believe Plato, as there is no reason why we should not, acquaintance with Zeno's works had already had a great influence on the mind of Socrates himself in early youth. According to the famous autobiographical passage of the Phaedo, prominent among the difficulties which had led Socrates to the formulation of his doctrine of Forms were not only the problem raised by Anaxagoras about growth and nutrition (Phaedo 96 c), but Zeno's puzzle about unity and plurality (96 e), and the method of "hypothesis" finally adopted by Socrates as the only proper instrument of philosophical inquiry is just that method of Zeno in which the Parmenides represents him as receiving a lesson from the two earlier philosophers.

The situation, in fact, as imagined by Plato, and as likely to have occurred in fact, is that Socrates has just thought out for himself as a theory which will solve both sets of difficulties, the doctrine of Forms. (That this solution is genuinely his own is stated with the utmost distinctness. Parmenides' very first question, on hearing it (130 a), is αὐτὸς σὺ οὕτω διῄρησαι ὡς λέγεις, "Have you made this distinction of which you speak by yourself and for yourself?"; αὐτὸς here |39| means just what it does in such a phrase as αὐτοί ἐσμεν, "we are by ourselves," and it is implied in the whole passage that the answer is affirmative. The theory of Socrates is plainly something of which Parmenides is hearing for the first time, though it is so far constructed on lines familiar to him that he only requires to hear it stated once before showing himself an acute and formidable critic of it.) Plato's assumption then is that the meeting between the Eleatics and Socrates was a memorable event in the history of Greek philosophy for very obvious reasons. Socrates had already been interested in the work of Zeno, but, according to Plato's account, had hitherto not been under the personal influence of Zeno. Zeno is, it appears, in Athens for the first time, since it is implied that all the copies of his book which have got abroad there are reproductions of a surreptitious copy: the true text has now been brought to Athens for the first time by the author (127 c). Such a first meeting between the greatest thinkers of an earlier generation and Socrates in the very flush of his first eager speculation must necessarily be of moment, and hence Plato can readily ask us to believe that men might take much pains to secure an authentic account of the interview even fifty or sixty years later. In fact it would be just the death of the last survivors of the party that would make persons with an interest in the history of ideas feel the necessity to obtain a narrative of the kind without further delay.

I have dwelt so long on the character of Plato's piece of introductory narrative because, as it seems to me, if I have divined its purpose correctly, an inference of some importance may be made about the reasoning to which it preludes. We shall naturally expect, if the whole work is to be of a piece, that the proper historical illusion will be kept up throughout it. However many |40| covert shafts Plato may be aiming at contemporaries of his own living towards the middle of the fourth century, we shall expect that his drama will respect the unities sufficiently to be in its main lines true to the spirit of the fifth century. The chief lines of reasoning, however they may be worked out into detail, should be such as might naturally have been followed in a discussion of the age of Zeno. And we can see that Plato really felt this too. The whole form of the dialogue with its ingenious antinomies has, as Professor Burnet has said, been adapted to the pretence that it is just one of those exhibitions of the Zenonian dialectic to which Aristotle refers. It pretends to be just such a dialogue as that quoted by Aristotle in which Zeno was represented as posing Protagoras with the notion of petites perceptions which are "beneath the threshold."It may, in my own opinion, be fairly said that, so far as the main lines of discussion are concerned, there is little if anything in Plato's Parmenides which might not have been said at an actual joyous passage of arms between dialecticians in the middle of the fifth century. To prove this statement completely it would be necessary to subjoin an elaborate critical commentary on the whole dialogue, taken clause by clause. But I propose to do something towards establishing the point in the present paper by singling out for consideration two arguments put forward in the early part of the dialogue which have always attracted a great deal of attention, — that which turns upon the logical objection to an "infinite regress," and that in which Plato's Parmenides anticipates Kant's attempt to make a formal refutation of "Idealism."

To appreciate these pieces of dialectic it is necessary to begin by understanding exactly what is the point in the doctrine of Forms, as propounded by the youthful Socrates, against which the Eleatics are directing their |41| attack. Unfortunately modern writers on Plato have often approached the Parmenides with a complete misunderstanding of its main purpose. They have supposed Socrates to be asserting, Parmenides and Zeno to be disputing, the existence of "Forms which are only to be apprehended by thought." This is a misconception which is fatal to any real insight into the dialogue. Parmenides and Zeno nowhere raise any difficulty about the existence of such Forms as the proper objects of knowledge; in fact, since the very "One" of which their own philosophy speaks is just such a Form, and is, in fact, called by the equivalent name μορφή in the poem of Parmenides himself, they could not well make a difficulty on the point. From their neglect to ask for any explanation of the matter, we must assume that they are supposed already to know quite well what sort of thing a Form is, and to have met before persons who believe in the reality of such Forms. Indeed, if Proclus is right in taking it as familiarly known that the extreme "friends of Forms" criticised in the Sophistes are Italian Pythagoreans, the Eleatics must have known all about the matter. What does strike them as unfamiliar in the theory expounded by Socrates is that he believes in Forms "of the things we perceive," and it is about this very assertion of a precise correspondence between Forms and "things we perceive" that they ask the question whether he had really hit upon the doctrine for himself. The whole object of the dialectical difficulties which they go on to raise is to suggest to Socrates that it is impossible to give a coherent account of the relation asserted in his theory to subsist between a Form and the sensible things of which it is the Form, and Parmenides ends his dialectical examination of the doctrine by the express declaration that though, as can now be seen, Socrates |42| has formidable difficulties to face before he can claim to have justified the assumption of Forms of things, philosophical thought is impossible unless there are Forms (135 a-c).

We have, therefore, to bear in mind that the object of the argumentation is not to throw a doubt on the existence of Forms, but to urge the need for a plain and explicit account of the relation which Socrates commonly called that of participation, by which a thing is connected with what he calls the Form of that thing. As Professor Burnet says, expressing the point with perfect exactness in the terminology of a later generation, it is not the existence of the intelligible but the existence of the sensible which is, according to Parmenides and Zeno, the crux in Socrates' theory. And, in fact, it was precisely the crux. In the account given in the Phaedo sensible things figure as mere temporary vehicles of a number of Forms; they are, apparently, what they have sometimes been called by later thinkers, meeting-places of universals, terms which sustain complexes of relations, but what more than this a thing is does not appear. It would seem that Socrates himself felt that this could not be the last word on the matter; at any rate, he is careful in the Phaedo to suggest a plurality of names for the relation between thing and Form, and appears not to be wholly satisfied with his account of it. That Plato himself felt the necessity of giving a different doctrine on the point is manifest not only from the Timaeus and Philebus but from the hints furnished by Aristotle's criticisms of him. The impression left by the Parmenides is that Plato at least wishes us to think that Socrates had quite early in life struck into the right line of thought, but to the day of his death had never been able to follow it up with complete success. Indeed our dialogue even professes |43| to give the reason for his partial failure; he had never in his early life had a thoroughly adequate training in hard and dry dialectic. He was trying to define "beautiful" and "right" and "good" and the other Forms before going through a sufficient "discipline" in hard logic, or, in other words, his interests were too exclusively ethical and not logical enough. To myself this passage (Parmenides 135 c-d) has all the appearance of being intended to suggest a serious criticism of what Plato felt to be a weak spot in the Socratic philosophy. I find it quite incredible that such a direct criticism should be leveled either at a purely imaginary person, who had no existence outside his creator's imagination, or at some unnamed person, Plato himself in an earlier stage of his development, or some disciple of Plato, or some Socratic man, under the disguise of the Master. With so much in the way of preliminary orientation we may turn to the treatment of the two specific arguments I have selected for consideration. And I will begin with the argument from the illegitimacy of the infinite regress, which occurs twice over, at Parmenides 132 a-b, and Parmenides 132 d-e. I will begin by a fairly literal rendering of the relevant passages:

"I suppose your reason for thinking that there is in each case such a one Form is this. When you judge that several things are big, perhaps when you consider them all you hold that there is one and the same Form, and hence you think that 'the big' is one." "You are right."

"But if you consider together in the same way the big and the other big things, will there not again appear one big something in virtue of which they all appear big?" "So it would seem."

"Then there will appear a second Form of bigness, over and above the big and the things that partake of the big, and there will be a third on the top of all these |44| in virtue of which they will all be big. Thus each of your Forms will no longer be one but indefinitely numerous."

[Socrates hereupon makes the suggestion that the difficulty might be evaded by supposing that a Form is only a "thought in a mind," and this leads to what I have called the Refutation of Idealism. He then tries the alternative explanation that the relation of a thing to a Form is simply that the Form is a type, and the thing is like the type. This is met by recurrence to the argument from the regress as follows.]

"Then if anything is like the Form can the Form be other than like that which has been stamped with its likeness, in so far as it was modelled on it? Can the like by any artifice be prevented from being like what is like it?" "Certainly not" [i.e. the relation of likeness is symmetrical]. "And must not the like and its like both partake of one identical something?" "They must." "And that by partaking of which likes are like — will not it be just the Form?" "To be sure it will." "Then it follows that nothing else is like the Form, nor the Form like anything else. Otherwise besides the Form there will always appear another Form, and if it is like anything, still another, and there will be an unending series of fresh Forms, if the Form proves like that which partakes of it."

The argument from the "infinite regress" is thus employed first against the general theory of the "participation" of things in forms and then, in a specialised form, against the suggested identification of the relation of "participation" with the relation between a copy of an original and the original.

The questions which naturally occur to us on reading the two passages are two, whether the reasoning ascribed to Parmenides in the dialogue is sound, and what, so far as we can still discover, was the history of |45| this type of argument in antiquity before Plato composed the Parmenides? Has Plato invented the difficulty which Zeno is made to raise for himself? Or was it invented by some contemporary and unfriendly thinker as a criticism on the type of doctrine expounded in the Phaedo? Or is it possible to hold that it is at least historically possible that the real Zeno may have argued in this fashion against theories which were actually current in his own times? The answer usually given to these questions is, I think, that the reasoning is valid, or at least that Plato has not given any reason to think it invalid, and further that, as it is said we know from Alexander of Aphrodisias, the argument was invented by the Megarian logician Polyxenus and is identical with that often alluded to by Aristotle as the τρίτος ἄνθρωπος or "third man."What I propose to show is that the appeal to the regress, though valid against certain ways in which the doctrine of Forms might be understood, is not valid against anything which Plato has advanced anywhere in his writings, that there is no ground for supposing it to be identical with the argument of Polyxenus and that it is not what Aristotle usually has in mind when he speaks of a certain type of argument as the "third man."I will consider first the more general form of the objection.

The argument as formulated by Parmenides at 132 amounts to this. The reason, and the only reason, why we should believe in Forms is that when many particulars have a common predicate — e.g. when it is true to say of several men that each of them is tall — this must mean that they have a common character, a common objective determination to which the common predicate of speech answers, and this common character is one and the same definite determination. That is why we say that though the particulars are many |46| there is one Form in which they all "partake."But, Parmenides contends, we may once more ascribe this common predicate not only to each of the several "particulars," but also to their "common nature" itself. We can say not only that A1 is great or good or beautiful, A2 great or good or beautiful, A3 great or good or beautiful, but that greatness is great, goodness good, beauty beautiful, and so on. Thus if the resemblance between A1, A2, A3 requires to be accounted for by saying that each of them is an "instance" of A, by parity of reasoning we must say that since A itself has a predicate in common with A1, A2, A3, there is a second Form — call it A1 of which A, A1, A2, A3 all "partake," and the same considerations will avail to establish in the place of every Form A postulated by the theory of Socrates, a simply infinite series of Forms A, A1, A2, … An, … Aw. And this, it is assumed, is an absurdity.

Now, in the first place, I should like to observe that this argument, whatever it is worth, is not directed against the reality of Forms or universals, but against the possibility of appealing to that reality as a ground for believing in the revelations of sense-perception and an explanation of what we mean when we make a perceptual judgement. It is not the doctrine that there are νοητὰ εἴδη and that we can be acquainted with them, but the doctrine that what we perceive by our senses — "sense-data" — "partake in" them and thus acquire a secondary reality which furnishes the starting-point of the argument, and, as I shall try to show immediately, the conclusion that there is not one form of "good," "beautiful," etc., but a whole hierarchy of orders of good, beauty, etc., is not per se absurd. The real difficulty is that if this is so the theory of Forms becomes useless as a device for "saving the appearances" |47| of the world as perceived by sense. The argument is exactly in the right place when put into the mouth of an Eleatic who wishes not to "save" these appearances, but to "give them a fall" (καταβάλλειν τὰς αἰσθήσεις), and if valid against Socrates it is only valid because Socrates is the champion of perception against ultra-Rationalism.

First, then, as to the validity of the general argument from the "regress," which has always been much affected by metaphysicians as an instrument for the discomfiture of their rivals. It is still often assumed that a theory which leads to an "indefinite regress" in any form is thereby logically discredited. I cannot myself agree with this view. It seems to be no better than a prejudice based on that confusion between infinity and indeterminateness which has been finally exploded by the researches of modern mathematicians into the character of infinite collections. The doctrine that only the finite has determinate structure or order is one which a few hours' study of any elementary work on the Theory of Assemblages is sufficient to explode. Hence I think Russell is plainly right in distinguishing between a harmless and a logically vicious type of the "regress." There can be no logical objection to the "regress" so long as it is constituted merely by implications between propositions. It is no objection either to the significancy or the truth of a proposition p0 to say that p0 implies p1, which again implies p2, and so on interminably. For there is no reason why each of an endless series of propositions {p} should not be true. In fact, on the hypothesis of "Idealists" of the kind who usually make the most frequent employment of the "regress" against their opponents, every true proposition p must imply an infinite series of true propositions. For they commonly hold that a proposition cannot be |48| true without being actually known by some mind and that this is part of what we mean by calling p true. Hence the true proposition p implies, on their theory, the true proposition, "x knows p," and this, being itself a true proposition, again implies "y — who may of course be identical with x — knows that x knows p," and so on in indefinitum.2

Professor Royce has correctly drawn this conclusion, and since it is a fundamental article of his philosophical belief that to be known by someone is part of what we mean by being true, he rightly accepts the view that this particular "regress" must be accepted. But he seems also to make the further assumption, which is not warranted by his premises, that it is never an objection to a philosophical doctrine that it leads to the "regress." Here, again, I think Russell clearly right in holding that there is one kind of "regress" which is always fatal to any hypothesis which implies it. No intelligible proposition can be such that an infinite "regress" arises in the very attempt to state its meaning. An apparent proposition p0 which turns out to be such that we can- not state its meaning without first stating as parts of that meaning the infinite series of propositions p1, p2, … pn, … pw must be no proposition at all but a mere unmeaning noise. For, as we can never exhaust an infinite series by enumeration of its terms, we could never know definitely what such a p0 means, and every proposition must have a fully determinate meaning. Hence for us, at any rate, p0 is no proposition at all. The importance of this distinction will be seen when it is remembered that all the attempts made by philosophers, |49| and notably by Kant, to discover contradictions in our notions of space and time involve only a "regress" of the harmless kind; they only show that certain propositions, if true, involve the truth of an infinity of other propositions, as there is no reason why they should not. So, again, if Zeno's well-known argument from indefinite divisibility were alleged as a reason for denying that a line can be divided at all, there would be an open fallacy. It was only valid ad homines because part of the case of his opponents was that a point is a minimum length. As an argument for Spinoza's thesis of the indivisibility of real extension it is no more cogent than it would be to argue that there can be no such number as 1 because there is an infinity of rational fractions less than 1. We cannot, however, meet the argument of Parmenides against Socrates by urging that the "regress" of which he speaks is of the harmless kind. If he is right in finding that "regress" in the theory of μέθεξις, the "regress" is vicious and shows that the theory of Socrates is indefensible. For the reasoning is as follows: Two things A1 and A2 are both A (e.g. Socrates and Zeno are both men), because they have a common "nature" (humanity), and it is only because they possess this common nature that we can truly predicate the same term of them. But we can predicate A of A itself in precisely the same sense in which we predicate A of A1 and of A2 (i.e. we can say that Humanity is human, or Man is a man, exactly as one can say Zeno is human or Socrates is a man). Hence A, A1 A2 must, on our own theory, have a still more ultimate common nature, and so on indefinitely. Hence you will never be able to say exactly what it is that Zeno and Socrates have in common; you do not know what the predicate you assert about both of them is.

|50| Thus the solution of the puzzle, if there is one, cannot lie in admitting the "regress" but pronouncing it harmless; if the theory of Socrates is to be defended at all, it must be shown that the alleged "regress" does not really arise. That is, we must deny the tacit premise of Parmenides that a universal can be predicated of itself as it is predicated of its "instances." A1 and A2, we must say, have the common nature A, or are "instances of" A, but A and A1 are not two "instances of" A; A has not to itself the relation it has to A1 or to A2." We may say of two white things that each of them is white, but we must not say in the same way that whiteness, or white, is white. Or, to use Plato's language, which makes the point clearer, though we may say that a white surface has whiteness, or white colour, we must not say that white colour, or whiteness, has white colour or whiteness. We must say that a concept, or meaning, or intension can be predicated of each constituent of the corresponding extension, but can never be predicated of itself, — in fact that the subject-predicate relation is an alio-relative. This seems to me an obvious truth which is only concealed from us by the linguistic fact that we commonly use the same word "is" to symbolise both predication and identity. "White is white," "goodness is goodness," and the like, if they are significant expressions at all, are not predications but assertions of identity. They mean that "white is the same thing as white," etc., but "Socrates is a man" does not mean that Socrates is identical with Man. To say that snow is white means that snow has the colour white. What that means I must not discuss here, but, whatever it means, it would be nonsense to say that whiteness, or white colour, has a white colour, as snow has. White has not itself any colour at all; it is a colour.

The solution of Parmenides' puzzle, then, is that |51| identity and the relation of predicate to subject are different and disparate, and this is why every system of logical symbolism has always found it necessary to avoid the trap laid for thought by the inexactitude of ordinary speech in this matter. Hence the alleged "regress" does not really arise from the original statement about the "participation" of things in Forms or universals. It arises not from the doctrine of Socrates himself but from Parmenides' skillful combination of what Socrates had said with the further premise that the Form "participates in" itself, (τί δ᾽ αὐτὸ τὸ μέγα καὶ τἆλλα τὰ μεγάλα, ἐὰν ὡσαύτως τῇ ψυχῇ ἐπὶ πάντα ἴδῃς, οὐχὶ ἕν τι αὖ μέγα φανεῖται, ᾧ ταῦτα πάντα μεγάλα φαίνεσθαι; 132 a.) So, to recur once more to my example, the "common nature" of all white things is just their white colour, but the "common nature" of a thing and a colour cannot itself be a colour, and you do not need to know what it is in order to know what the white colour which is the "common nature" of all white things is. You can know what "white" is without requiring to have any view on the question what colours and things other than colours have in common. Hence no "regress" is involved in the meaning of the assertion that such and such a particular "partakes" in such and such a Form. It is perhaps important to note that the source of the apparent fallacy, the ambiguity of "is," is also, as Plato was to show in the Sophistes, the source of all the old "eristic" difficulties about negative propositions. Since, as everyone admits, Plato saw and explained the ambiguity so far as it affects the possibility of significant denial, it is only reasonable to suppose he was aware of the presence of the same ambiguity in the argument we have just analysed. But it would have been bad art, and probably also bad history, to allow the youthful Socrates of the dialogue to see through and expose the fallacy. Consequently Plato does not let him |52| discuss it at all. He is made to turn without discussion to a fresh point. Historically, I take it, this means that the appeal to the "regress" had been used against the doctrine of μέθεξις, but presumably after the death of Socrates himself. The persons who used it must have meant primarily not so much to discredit the doctrine that the proper objects of knowledge are intelligible Forms as to deny that these Forms are in any way connected with the things and events of the perceived world. That is, they must have been logicians with an ultra-intellectualistic bias like the "friends of Forms" mentioned in the Sophistes whom Proclus identifies with Italian Pythagoreans.

The detailed examination of the history of the argument has to begin with the consideration of certain passages in the Metaphysics of Aristotle and the explanation given of these passages by Alexander of Aphrodisias. Aristotle more than once makes the statement that the "most finished" versions of the theory of Forms lead to a difficulty which he speaks of as the "third man."Thus at Metaphysics A, 990 b 15, where he is arguing that the reasons currently given in the Platonic school for believing in the Forms and their relation to the world of sense-data are not above criticism, he says ἔτι δὲ οἱ ἀκριβέστεροι τῶν λόγων οἱ μὲν τῶν πρός τι ποιοῦσιν ἰδέας, … οἱ δὲ τὸν τρίτον ἄνθρωπον λέγουσι, "of the more accurate (but the meaning is rather 'more finished,' 'more subtle') arguments, some lead to Ideas of relations … others involve the difficulty of the 'third man'" (tr. Ross). The same remark occurs,and, so far as the words I have cited go, in identical language, except that ἀκριβέστατοι is substituted for ἀκριβέστεροι at Metaphysics 1079 a 11. From the facts that Alexander in his commentary refers to the argument with which we have just dealt as an example of a "third man" argument, |53| and also mentions the Megarian "sophist" — i.e. formal logician — Polyxenus as the inventor of a form of the "third man," it has become customary to say that τρίτος ἄνθρωπος is a name for what we call the appeal to the "indefinite regress," that the controversial use of this appeal was invented by Polyxenus, and that it is to this that both the Parmenides and the Aristotelian references allude. According to this nowgenerally accepted view, Plato is here recalling and dramatically ascribing to Parmenides a criticism directed — so it is assumed — against Plato himself by Polyxenus. The dramatic justification of this is that Polyxenus belonged to a school whose founder Euclides was originally a disciple of Parmenides. If, however, we read the passage of Alexander with proper care, we shall see that we must not assume without discussion either that Polyxenus invented the particular argument rehearsed in the Parmenides, or that it is the Parmenides argument to which Aristotle is alluding in the Metaphysics. We must therefore consider the whole question for ourselves in a little detail.

[It may be as well to begin with a brief statement of what is known about the personality of Polyxenus. Our one contemporary reference to him comes from the correspondence between Plato and Dionysius II." In the last letter of the correspondence as arranged in our texts, which is also the earliest in order of time and belongs to the year 366-5, Plato mentions that he is sending to Dionysius a person whose society will, he hopes, be agreeable to him and to Archytas. This person is Helicon of Cyzicus, a member of the astronomical school of Eudoxus, and, Plato adds, one who has enjoyed the society of a certain unnamed pupil of Isocrates and Polyxenus, one of the disciples of Bryson (τῶν Βρύσωνός τινι ἑταίρων, 360 c). Thus Helicon was apparently selected by Plato on the ground that he |54| would be able to represent at once the mathematics and astronomy of Eudoxus, the political ideals of Isocrates, and the formal logic of the Megarians; from the context it is clear that, though Plato thought well of his man, his feelings towards persons of Megarian antecedents were not at this date over-cordial. The important point for my purposes is, however, that the reference gives us, in a rough way, the date of Polyxenus. He is a disciple of Bryson, whose interesting contributions to the problem of the quadrature of the circle are discussed by Aristotle in the Posterior Analytics and Sophistical Refutations in an unappreciative and pedantic way, which suggests that the maestro di color che sanno did not really understand the nature of the problem. Now, Bryson was one of the original members of the Megarian school, and had been a personal associate of Socrates, as we see from the fact that Theopompus, the historian, a pupil of Isocrates, in an attempt (fr. 247) to depreciate Plato, charged him with borrowing his ideas from Aristippus, Antisthenes, and Bryson. Polyxenus thus belongs to the second generation of the school of Euclides, and must be, roughly speaking, contemporary with Plato, so that it is quite credible in itself that he may be the author of criticisms referred to by Aristotle and by Plato himself in a work as late as the Parmenides.

I can hardly carry the discussion further without actually quoting almost in full what Alexander said about the "third man" in his comment on Met. A, 990 b 15, as there are several points in his statement to which I would direct attention. This, then, is what he says:

"The argument which brings in the 'third man' is as follows: They [i.e. the believers in Forms] say that the substances which are predicated generally are the |55| true and proper substances κυρίως εἶναι τοιαῦτα I.e. the Academy, unlike Aristotle, who regards individual things like 'this horse', 'this man', as the primary substances, regard universals or kinds, 'man', 'horse', etc., as the 'true and proper' substances, whereas Aristotle will only allow them to be called substances in a secondary and derivative sense], and that these are the Forms. Further, things which are like one another are so in virtue of participation [μετονςία, a word never used by Plato in this connection, as Professor Burnet has noted] in one and the same something which properly is that [i.e. is 'horse' or 'man', or whatever each of two like things is said to be], and this is the Form. But if this is so, and if that which is predicated in like manner of several things, when not identical with any one of those things, is another thing over and above them — and it is just because the Form of Man, though predicated of particular men, is not identical with any of them that it is a kind — there must be a third man besides man particular — as, e.g. Socrates or Plato — and the Form, which last is also itself numerically one.

"Now, there was an argument used by the sophists and introducing the 'third man' to this effect. If we say 'there is a man walking' we do not mean that Man, in the sense of the Form, is walking — for the Form is unmoving — nor yet a determinate particular man — and how can we mean this if we do not recognise the man? We are aware that a man is walking, but not who the particular man is of whom we assert this; we are saying that another third man different from these [i.e. different from both Man and from this or that man whom we know] is walking. Ergo, there is a third man of whom we have predicated that he is walking. To be sure, this argument is sophistical, but an opening is made for it |56| by those who postulate the Forms. And Phanias says, in his reply to Diodorus, that the sophist Polyxenus introduces the 'third man' in these words: 'If man is man in virtue of partaking and participation [κατὰ μετοχήν τε καὶ μετουσίαν, both words non-Platonic in this sense] in the Form, or αὐτοάνθρωπος, there must be a man who will have his being correlatively to the Form. This cannot be the αὐτοάνθρωπος, who is the Form, nor yet the particular man who is by participation (μετοχή) in the Form. The only alternative is that there is yet a third man who is relatively to the Form.

"The 'third man' is also demonstrated thus: If what is truly predicated of a plurality of subjects is a reality alongside those of which it is predicated and distinct from them — and those who postulate the Forms believe they can prove this …," if so, I say there will be a 'third man'. For if Man as predicate is other than the men of whom the term is predicated, and has a substantial being of its own, and if Man is predicated in like manner, both of particular men and of the Form, then there must be a third man distinct both from the particular men and from the Form. And in the same way a fourth, predicable in like manner of this third man, of the Form and of the particular man, and again a fifth, and so on in indefinitum. This argument is identical with the first, since it was assumed that like things are like in virtue of their participation (μετουσία) in one and the same thing."

Now, it is to be observed that we are here offered three distinct arguments, each of which brings in a "third man"; one is ascribed to the "sophists," that is, according to the Aristotelian use of that word, the Megarian logicians generally, the second by name to the Megarian Polyxenus, and the third is identical with the argument put by Plato into the mouth of Parmenides. |57| It is only in this last version, which Alexander gives in two forms, that any question of a "regress" arises, and this argument is not that attributed to Polyxenus. The other two will easily be seen on analysis to be of a quite different type. The first, that of the "sophists" generally, is based on the ambiguity of the indefinite article, or, in Greek, of the common noun without any article. If I say, as I quite well may, "a man is walking down the street" without knowing what man it is whom I see at the far end of the street, though I am saying what is true and significant, I plainly do not mean that "humanity" is going down the road. Particular men may be met in the Strand, but you would hardly expect to encounter Man, "the substance of men which is Man," there. And I do not mean that this or that known man, Lord Kitchener or Russell, is going along the Strand, since by hypothesis I do not know who the man in question is. Hence besides "Man" with the capital and Lord Kitchener or Russell, the words "a man" or "man" must have some third sense. This is, of course, simply true. When I say "Man is fallible," I mean by Man what Socrates would have called "just man," the Form of man. When I say "a man wrote Hamlet," if I have any knowledge of English literature I mean that Hamlet was written by a particular man whose name, birthplace, and so forth I could mention if I chose. But when I say "a man wrote Junius," I — who am not convinced by any hypothesis yet put forward on the identity of Junius — mean neither that the Form of man, Man with the capital, wrote Junius nor that, e.g., Philip Francis or Edmund Burke wrote Junius. I really mean to assert the disjunctive proposition "either a wrote Junius, or b wrote Junius. or c wrote Junius or …" and so forth, where a, b, c … stand for the different individuals of English speech |58| who were alive and adult during the whole period in which the Letters of Junius were appearing. I mean that some one of this set, though I do not know which, was the author. That this observation was well worth making is shown by the fact that Russell has had to make it again at some' length in his Principles of Mathematics. But it is not in any way inconsistent with the theory of Forms; It is no objection to a doctrine of universalia in rebus or even of universalia ante res to say that it cannot tell me which of the many beings who "partake of" humanity wrote the Letters of Junius. If, as would seem, the argument is Megarian, it shows no trace of being directed against Socrates or Plato; it is merely a correct reflection on the ambiguity of the article such as would naturally occur to anyone interested in the formal development of logic.

The argument of Polyxenus is rather different and distinctly more subtle. Our view of its exact purport must depend on a point of textual criticism. In my rendering I have followed, with just a shade of doubt, a transposition of the words of one clause suggested by Clemens Bäumker. Professor Burnet, in his recent work on Greek Philosophy from Thales to Plato, does the same thing, but oddly enough subjoins an interpretation which seems only possible if the transposition is not made. As I understand the passage, the argument is this: According to the theory of Forms, man means (1) the Form of Man, (2) each of the particular men who, on this theory, have an inferior kind of reality due to their "participation" in the Form. Thus man in sense (1) is identical with the Form, in sense (2) depends on and derives his being from the Form. Polyxenus maintains that there must be a third sense intermediate between the two. There should be a "man" who is not identical with the Form and yet is "on the same |59| footing" with it, not derivative like the "particular" man.

The point of this can, I think, be illustrated by what we know to have been Plato's doctrine about the objects studied in geometry. As we know from Aristotle, he held that these "mathematicals" are "intermediate" between Forms and sensibles. Thus the Form of circularity is one and only one; the circle as a type of plane curve is one determinate type, and all circles belong to this same type. The round figures we draw with ink or chalk are not really true to the type; they are only approximations, or, in the language used by Socrates in the Phaedo, "they are not circles, but would like to be circles if they could." But the circles of which Euclid reasons stand in an intermediate position. There are many of them. We talk, e.g., of two circles which cut or touch, or of a nest of concentric circles, or of the three circles each of which touches one side of a triangle and the two other sides "produced."Yet each of the geometer's many circles is an exact, and not, like the visible round figure, a merely approximate realisation of the one type. Now, as I understand Polyxenus, he was arguing that on the theory of Forms there ought always to be something which mediates between the Form of Man and the imperfect embodiments of it which figure in actual life as the geometer's circles mediate between "the circle" of Analysis and the things we draw on paper or on the blackboard. But there seems to be no such thing in the case of Man. This reasoning is most naturally understood as a criticism directed against that very extension of the doctrine of Forms from mathematics to cover the realm of organisms about which Socrates himself is made to express a doubt in our dialogue at p. 130 d. If this was really the point Polyxenus intended to make, his criticism appears |60| to me to speak very highly for his philosophical acumen. Plato himself indicates in the passage to which I have just referred that the recognition of Forms of organisms is one of the most ticklish points in the whole theory. How he himself in the end escaped from the difficulty cannot be considered here, as any serious discussion of the matter would require an elaborate investigation of what Aristotle has told us about the Platonic reduction of philosophy to arithmetic. But it may at least be said that the Platonic doctrine, as known to Aristotle, only preserves the conceptions ascribed in the Phaedo to Socrates by a transformation which makes them at first sight almost unrecognisable. For my present purpose it is enough to note that if I have rightly discerned the real point of Polyxenus, his criticism must have been specially directed against Plato himself and no other, and this would explain why it is, as a matter of fact, never answered in the Parmenides, where it would be an anachronism to put Platonism, as distinct from the cruder doctrine expounded in the Phaedo, into the mouth of the youthful Socrates. For the conception of this gradation from Forms or numbers through mathematicals down to sensibles is always connected by Aristotle with what he represents as the personal theories of Plato. He ascribes it to Plato as peculiar to himself, as an ἴδιον Πλάτωνος, in a way in which he never ascribes the general theory of Forms to him. Professor Burnet, who takes a different view, remarks, indeed, that the words used in the account given by Alexander on the authority of Phanias of Eresus, an original member of the Lyceum, of what Polyxenus had said, to represent the relation between a Form and a sensible, μετουςία, μετοχή, are not technical terms of Plato's vocabulary, and infers that the argument of Polyxenus was not specially directed against Plato. I |61| do not myself think the inference of much weight. If it proves anything, it should surely prove that the criticism of Polyxenus was not directed even against Socrates, for it is Socrates who, in Plato's writings, habitually talks of as the relation between sensibles and Forms. The only other person in the dialogues who ever says much about the matter is the Pythagorean Timaeus, and he avoids the use of the words μετέχειν and μέθεξις in a very remarkable manner, for which I shall directly give the true reason. But if the argument is meant neither to tell against Socrates nor against Plato, against whom is it directed? Do we know of any other "friends of Forms" who held the view that sensible things are what they are by "participating" in Forms at all, except just Socrates and his associates? Professor Burnet who, like myself, regards this account of sensible things as the distinctive contribution of Socrates to the theory of Forms is, I think, under a special obligation to face this question.

With regard to his one definite argument, that from the un-Platonic character of the words μετέχειν and μετοχή, I might remark (a) that even if it were absolutely certain that the actual words of Polyxenus have undergone no modification in reaching us at two removes, I see no reason why his preference for μετοχή as the verbal noun to μετέχειν should be regarded as proof that he is not thinking of Plato, μετοχή is, at any rate, as old as the fourth century as a verbal noun to μετέχειν. Thus in Met. Z, 1030 a 11 ff., we are told that "nothing which is not a species of a genus will have an essence (τὸ τί ἦν εἶναι), only species will have one, for in these the subject is not held to participate in the attribute" (ταῦτα γὰρ δοκεῖ οὐ κατὰ μετοχὴν λέγεσθαι), where the so-called "Alexander" sees a direct allusion to the Socratic-Platonic doctrine, δύναται τὸ οὐ |62| κατὰ μετοχὴν νοεῖσθαι ἀντὶ τοῦ οὐ κεχωρισμένα ἐστὶ τὰ εἴδη καὶ οὐ καθάπερ φησὶ Πλάτων κατὰ μετοχὴν αὐτῶν τὰ καθ ͗ ἕκαστά ἐστιν "the words is not held to participate may be understood to mean that the species are not separable, and individuals do not exist in virtue of participation in them, as Plato asserts" ("Alexander" in loc.). Aristotle again uses μετοχή as the noun of μετέχειν at Ethica Eudemia, 1217 a 29, though without reference to the theory of Forms. The word indeed is used by Plato himself in one of his latest writings, Epinomis vii. 344 e, though not in a technical sense, ὡς παιδείς δὴ. Merovala, again, though not a Platonic or Aristotelian word, is no coinage of a later age but belongs to the Greek of Aristophanes and Demosthenes, and I have observed the use of it in later Platonists as an equivalent for the Platonic μέθεξις. (b) And, as an illustration to show that inferences from verbal expressions must not be pushed too far, I would remind Professor Burnet that he himself expresses a doubt whether the name "indeterminate duality" given by Aristotle to the continuum called by Plato the "great- and-small" is Platonic, though he has, of course, no doubt that the concept is characteristically Platonic. Similarly, it is notorious that Aristotle expresses the Platonic theory of matter by the statement that "Plato says in the Timaeus that space (χώρα) and matter (ὕλη) are the same," though Aristotle must have known that as a matter of language the Timaeus does not use the word ὕλη in the sense of "matter" at all. I am, therefore, not convinced by the linguistic argument that the reasoning of Polyxenus is aimed at someone other than Plato.

Now let us see how the argument will be affected if we refuse to make the transposition of words introduced by Bäumker into the passage about Polyxenus. In the |63| MSS. of Alexander the text runs thus: "If man is man by partaking or participation in the Form or αὐτοάνθρωπος, there must be a man who has his being relatively to the Form. But neither the αὐτοάνθρωπος who is the Form, nor the particular man, is in virtue of participation in the Form. The remaining possibility is that there should be a third man who has his being relative to the Form." If this is what Polyxenus said, he must mean one and the same thing by "having one's being relative to" the Form and "partaking in" the Form. The sense then is: What do you mean by the man who is said to "partake in" the Form of man? You cannot mean Man, because Man does not "participate in" but is the Form. And you do not mean this or that actual man; therefore you must mean "man" in some unintelligible third sense. Thus understood, Polyxenus simply assumes it as conceded by those against whom he is reasoning that this and that man do not "participate in" the Form, that is, as Professor Burnet says, the actual men stand in no relation to the Form. He is not attempting to prove this but making it one of the premises of his syllogism. But once more we have to ask ourselves against whom such a polemic can be directed. Can we point to any "friends of Forms" who admitted that things of some kinds "partake of" Forms but held that none of these things are sensibles? Such a theory is, no doubt, an abstract possibility. We can imagine a philosopher holding that all the "things" which "partake in" Forms are what Plato called "mathematicals," — the many circles, triangles, etc., of the geometer, not "sensibles."And something like this may — nay, almost must — have been the doctrine of the "friends of Forms" criticised in the Sophistes. But Aristotle is explicit and emphatic on the point that the phrase about "participation" was never Pythagorean. |64| The Pythagoreans, he says at Met. A, 987 b11, "said that things are by imitation of the numbers, whereas Plato said it was by participation." (This, I may observe in passing, is the simple explanation of the fact that the Pythagorean speaker in the Timaeus talks throughout of μιμησις not of μέθεξις.) And he is equally clear in the same context that it was sensibles which were said to "have their being by participation."Thus, whether we follow Bäumker in his transposition or not, it still seems to me plain that the argument of Polyxenus is aimed against either Plato or Socrates as he is represented in Plato, and more probably than not against Plato himself.

As I understand Alexander's account of the matter, he means that this argument is a special application of the more general one to which he refers simply as an "argument of the sophists," and of which he says that it was provoked by those who "separate the common (nature) from the particulars. "This seems to mean that even the more general form of the argument was devised for the purposes of the polemic against Plato. I agree with Professor Burnet that Alexander does not say that Polyxenus invented the "third man," but only that he "brought it on the stage" (for this seems to be the metaphor underlying the expression εἰσάγεινλόγον, but I think he means that the special form of it which he quotes from Phanias was due to Polyxenus. However this may be, the really important point is that the argument ascribed to Polyxenus, like that put down more vaguely to "the sophists," does not turn on an indefinite "regress."You could not use either of these "sophisms" to show that there must be a "fourth" or "fifth" man, and Alexander shows himself to be quite aware of this. Hence I think that we must at least come to the conclusions that —

|65| (a) The argument from the "regress" is only one special form of a type of reasoning popularly known as the "third man."

(b) This type of reasoning was clearly quite well known in the time of Aristotle, since he would not otherwise have referred to it by a nickname. Even the special form which brings in the "regress" was no novelty when Plato wrote the Parmenides, since he makes Socrates allude to it in passing as something that requires no detailed explanation in a much earlier dialogue (Republic, 597 c).

(c) The version of the "third man" specially due to Polyxenus does not bring in the "regress," and therefore cannot be what Plato has in view in the Parmenides.

If I am asked from whom then did the argument about the "regress" come, I have to answer that I do not know. But one thing at least is significant. In the Parmenides this argument is used twice, once, as we have seen, against the notion of sensibles as "participating in" Forms, and a second time against the notion of sensibles as copies of Forms. That is, it is used against the Pythagorean as well as against the Platonic formula. This suggests that the argument is very possibly originally anti-Pythagorean, and that the employment of it against the μιμησις formula may go well back to the fifth century. In fact, it belongs to the same class of reasonings as those of Zeno against infinite divisibility and has all the appearance of coming from the same source. I see no anachronism therefore in supposing that it comes from Zeno himself, and is just the sort of objection that would probably have been made by him and Parmenides to the youthful Socrates if he expounded to them the doctrine which the Phaedo represents him as formulating in his early manhood. |66| Indeed I shall be surprised if Zeno had not already used the "sophism" against the Italian "friends of Forms."

In the face of these results, it is not unreasonable to raise the question whether, in spite of his modern interpreters, Aristotle is really thinking of the "regress" at all when he urges that the most "finished" discourses of the Academy lead to the difficulty about the "third man." He might be referring to one of the "third man" arguments which do not bring in the "regress."It is true that Alexander seems to have taken the same view as the modern interpreters, since his explanation of this remark identifies the objection meant by Aristotle with that which he raises at Met. A, 991 a 1, which is a simple reproduction of the Parmenides passage (see Alexander in Metaphysica, 991 a 1). But against this I would set the consideration that none of the other passages in which Aristotle speaks of the "third man" seems to have any connection with the "regress."

At Met. Z, 1039 a 2, there is a passing reference in connection with an argument to prove that no "universal" is an οὐσία, an individual substance, and that consequently all "universals" are attributes, not things (οὐδὲν σημαίνει τῶν κοινῖ κατηγορουμένων τόδε τι ἀλλὰ τοιόνδε). If you deny this, Aristotle says, "the third man and other difficulties will arise."There is nothing here to show that he is thinking of the "regress," and it is more natural to suppose that he is not. The sense seems to be simply this. Suppose that a "universal predicate" really is the name of a this or individual thing. Then when I say "Socrates is a man" what this is denoted by the word "man"? Not the Form, for the Platonists themselves, at whom the argument is aimed, say that Socrates is not the Form, but only "partakes of" it. And not a determinate individual, Socrates or another, since "Socrates is Plato" would obviously be |67| false, whereas "Socrates is a man" is true, and "Socrates is Socrates" manifestly, though true, does not mean the same thing as "Socrates is a man."Thus if the word "man" in the supposed proposition denotes an individual this at all, it must denote a third man, who is neither Socrates nor Man with a big M, and this, it is assumed, is absurd. It is quite clear, I think, that this is all that Aristotle means here.

A second passage occurs in the very doubtfully authentic book Met. K, 1059 b 8. Here again there is no question of a "regress," and the argument is exactly that which I have supposed to be intended by Polyxenus, that if there are Forms answering to all universals, there ought also to be men and horses and the like intermediate between the Forms and the sensible things, just as the "mathematicals" are intermediate between Forms and visible diagrams. The writer's words are: "Even if one postulates the Forms there is a difficulty about the question why it is not with other things of which there are Forms as it is with mathematicals. I mean that they place the mathematicals between the Forms and sensibles as a third class over and above the forms and the things in our world (οἷον τρίττα τινὰ παρὰ τὰ εἴδη τε καὶ τὰ δεῦρο, b 7), but there is no third man or third horse besides the Form and the particulars (τρίτος δ ͗ ἄνθρωποσ οὐκ ἔστιν οὐδ ἵππος πα ͗ αὐτόν τε καὶ τοὺς καθ ͗ἕκαστον)."

The one other reference to the "third man" in the Aristotelian corpus is in the work On Sophistical Refutations, which is in effect an unfriendly examination of the formal logical paradoxes of the Megarian school.

At 178 b 36, in an account of the fallacies of figure of speech — i.e. fallacies which arise from confusing one "figure of predication" or "category" with another — |68| Aristotle includes among them the argument that "there is a third man over and above the Form and particular men" (ὅτι ἔστι τις τρίτος άνθρωπος παρ ͗ αὐτὸν καὶ τοὺς καθ ͗ ἕκαστον). This argument, he says, is one of the fallacies of "figure of speech" because it turns on treating a general term such as "man" as if it stood for τόδε τι, a this, whereas it really stands for a τοιόνδε, a tale or such. That is, in Aristotelian language, it mistakes an attribute or predicate for a substance — a substance being by definition just that which can only be subject, never predicate, in a proposition. The reference again is manifestly not to fallacious appeals to the "regress" — it would be quite impossible to regard these as "in dictione" — but to the simplest form of "third man" argument. Alexander rightly says in explaining the passage that the argument meant is that according to which when we say "a man is walking" we are speaking neither of Man nor of a determinate and known man. I.e. the fallacy lies in treating the words "a man," when they really mean "one and only one of the members of the class man, but I do not know which member," as if they meant "this particular man whose name I could give if I chose." Aristotle's own remark on the logical error is that "it is not the isolating (or exponing, τὸ ἐκτίθεσθαι) of the universal which leads to the 'third man', but the assumption that the 'exponed' universal is a this" — i.e. a particular existent (S.E. 179 a 3). The paradox, that is, is not due simply to the legitimate insistence on the distinction between "some man or other" and "this particular man," but to the further illegitimate assumption that "some man or other" is an object of the same type as Zeno, Socrates, or Plato, though different from them.

We are justified then in saying that though, as Alexander tells us, Aristotle had made some use of the |69| argument from the alleged impossibility of the "regress" in his lost work περὶ ἰδεῶν, there is no passage in his extant works in which the τρίτος ἄνθρωπος need be understood as referring to the "regress. "It need not be understood so in his remark that certain Platonic arguments about the Forms lead up to the "third man"; it cannot be understood so in any of the other passages. I conclude then that Aristotle's allusions to the "third man" as a paradox implied by Plato's theories about Forms has nothing to do with the problem of the "regress."He only means that on the interpretation he always gives to Plato's language, viz. that the Form is a kind of particular existent, it would be a valid objection that the subject of such a proposition as "some man or other is walking" must also be another particular thing.

Before I proceed to deal more briefly with the second argument from the "regress," I will, to keep to the actual order of development in the Parmenides, examine the section which immediately follows that we have just dismissed, and which I called the Refutation of Idealism (Parmenides 132 b-c).

Socrates now makes the suggestion that all the difficulties about the unity or multiplicity of the Form may be avoided if we look on Forms merely as "subjective," as "ideas in our own heads," or, in his own words, as "thoughts" (νοήματα) which are not "in" things at all, but only "in souls" (ἐν ψυχαῖς), i.e. in the minds that think the thoughts. If a Form is just a "thought" and is not really "in" anything but the mind which has the thought, it seems obvious that my thought of "man" is the same thought whether I think that Socrates is a man or that Zeno is a man. So we seem here to have an account of Forms which allows of the "presence" of one Form to many particulars without leaving an opening |70| for an opponent to urge that the Form cannot be really one if the particulars are really many. For now all that will be meant by saying that the one Form is present to many things will be that we can think the same predicate of each of them — and this seems to be a fact of everyday experience. Such a doctrine clearly amounts to what in modern days is called "Idealism" in the strict and proper sense of a much-abused word — the view that the "unity" or "common nature" of a class, and similarly the relations which connect existents ("double of," "cause of," "husband of," and the like) are the "work of the mind" or are "put by the mind" into a "raw material" supplied by sense.

I shall therefore use the name "Idealism" for the view which is thrown out in this section of the Parmenides, merely adding that in some degree or other this view has deeply coloured most European philosophy from Locke's time to our own. By calling the section a Refutation of Idealism I mean, that is, a refutation — and to my mind the neatest and most unanswerable I know of — of the theory that unity and relational order are the "work of our minds" or "put by our minds" into experience. The Platonist point is that we no more "put" the universal into things than we create "things" by perceiving them or thinking about them. We discover a pre-existing order just because it is there to discover. (It is true that Plato also held that order is the "work of the mind" in the sense that it has been "put into" things by God, but he did not hold that God's knowledge that things are relationally ordered is the logical prius of their being so ordered.)

In view of the confidence with which it is often asserted on the strength of a glaring fallacy of ambiguity that Plato was an "Idealist" in some modern sense of the word, it should be noted that the present |71| passage is the only one in all his works where it is ever suggested that a Form is an "idea in the mind" or a "mental state," and that the suggestion is only made to meet with a refutation which is unanswerable and is accepted as such by Socrates (132 c, ἀλλ ͗ οὐδὲ τοῦτο, φάναι ἔχει λόγον). This, of itself, should show that the interpretation of Plato which goes back to Philo the Jew, and still has its defenders, according to which a Form is a "thought in the mind of God," is untenable. It is true that in his Refutation of Idealism Plato is thinking, primarily at least, of thoughts in the minds of men, but the principle of his argument would be valid against the attempt to identify the universals which pervade the world, and give it its structure with processes in any mind whatsoever. Plato would have agreed in principle with the observations of Bolzano (Wissenschaftslehre, i. 113, 115): "It follows no doubt, from the omniscience of God, that every truth, even if it is neither known nor thought of by any other being, is known to Him as the Omniscient, and perpetually present in His understanding. Hence there is not in fact a single truth which is known to no one. But this does not prevent us from speaking of truths-in-themselves as truths in the notion whereof it is in nowise presupposed that they must be thought by someone. For, though to be thought is not included in the notion of such truths, it may still follow from a different ground, i.e. from the omniscience of God, that they must at least be known by God, if by no one else …". A thing is not true because God knows it to be true; on the contrary God knows it to be true because it is so. Thus, e.g., God does not exist because God thinks that He exists; it is because there is a God that this God thinks of Himself as existing. Similarly God is not almighty, wise, holy, etc., because He conceives Himself as such; e converso |72| He thinks Himself almighty, etc., because He really is so."

With Plato, then, an εἶδος or ἰδέα or Form is always the object of a thought, that of which someone thinks, not the process of thinking nor any psychological characteristic of that process, not knowledge, but something which is known. Thus the number 2, as we learn from the Phaedo, is a Form, but my ψυχή is not a Form, and still less is that which takes place in my ψυχή when I think about the number 2 a Form; 2 and my thinking about 2 are as distinct as my (dead) grandfather and my present thinking about him. The view which is here suggested only to be dismissed differs in holding that 2 is the same thing as my thinking about 2, or at least is so connected with my thinking about 2 that a proposition about 2 is only true when I, or some other thinker, happen to be thinking about 2, and because someone is thinking about 2.

This "Idealist" view, which identifies a Form with the νόημα or thought of the Form, can perhaps be fairly expressed in modern phraseology as follows. (I do not know if any writer puts the point exactly in this way, but readers of modern works on the "theory of knowledge" will, I believe, admit that my statement of it is an impartial expression of a widely disseminated doctrine.)

The universe is throughout made up of a multitude of process-contents (the doctrine called Mentalism by Sidgwick). Each specific mental process has its own specific "content," or more precisely each cognitive process has its specific "content," that which is thought in it, and these contents are, of course, propositions. No two processes have precisely the same "content," or, at any rate, the "content" is never the same if the "processes" differ in any way beyond occurring at |73| different points of absolute time or in different minds. On the other side the specific content only exists — the special proposition is only true — as an "aspect" of the corresponding process, and this seems to be the reason why those who hold views of this type always call the propositions which we think "contents." They mean that, e.g., a true proposition about the number 2, such as that 2 x 2 = 4, is related to my thinking about the number 2 in the same way in which the pleasantness is related to the consciousness of endeavour in an unthwarted conation, and they also usually mean something further. The suggestion is really a double one: (а) that identity, difference, causal relation and all the other types of relation recognised by science only are, and the propositions which assert them only are true, while someone is actually thinking that they are; and (b) — and this is an even more important point — that by saying that they are, or that the propositions which assert them are true, we actually mean that someone is thinking that they are. Few really competent thinkers indeed go the whole length of maintaining the position explicitly and consistently, but it ought, I think, to be held by anyone who accepts the principles of Kant's critical philosophy or believes with Green that relations are the "work of the mind," and it is hard not to suspect that it is latent in Mr. Bradley's view of the relation of the "that" and the "what" in experience. I know, of course, that the distinction of the "that" and the "what" may be insisted on by a philosopher, as, for instance, by Aristotle, who regards it not as a distinction of "aspects," but as falling entirely within the object of cognition or experience. And it is therefore possible that Mr. Bradley does not really mean what his language seems to me to imply. But his insistence that there is nothing at all in the Universe except "finite |74| centres of experience" tells the other way, as there seems to be no reason for accepting this doctrine except the allegation that to be means "to fall within the experience of a finite centre," apart from the assertion that the objects of thought are "aspects" of the process of thinking. (And compare the use made at p. 15 of Appearance and Reality of the argument that "primary qualities" depend for their perception on an "organ" to show that they are not "real," and the unqualified assertions on p. 144 of that work that "to be real, or even barely to exist, must be to fall within sentience," and that "there is no being or fact outside of that which is commonly called psychical existence".)

As I have said already, I do not see that the general character of the theory is altered by the substitution of God's mind for our minds as the ψυχή in which the process is supposed to go on. For the view in question is not simply that what is is always present to God's thought, that God actually thinks all true propositions, but that when you say "this is so" — e.g. when you say that the greater angle in a triangle is subtended by the greater side, or sin x is a periodic function, or that prussic acid is a poison — you mean that God knows that these things are so. The word "true" then ceases to have any meaning as applied to God's thinking, since the proposition "what God thinks is true" is reduced to the empty tautology that "what God thinks is what God thinks." The peculiarity of the theory is thus that of universals and relations in the real world as if it were a psychological question about the details of mental processes. The refutation put into the mouth of Parmenides shows the impossibility of Idealism if we mean by Idealism the doctrine that the knowing mind makes its objects in the act of knowing them, |75| or that what I think is an "aspect" of the process of thinking.

The reasoning proceeds thus. The view that Forms are "thoughts" itself implies, of course, that each thought is a thought of something, or about something. No thought is a thought of, or about, nothing at all. We sometimes say, to be sure, that we are "thinking of nothing," but that is only another way of saying that we are not thinking at all. You can no more be thinking and yet thinking of nothing than Alice could really meet "Nobody." Of course, you can think about the number 0, but 0 is not nothing but something; it is, e.g., the cardinal number of all the combatants at the battle of Salamis who are now living in London. On the process-content theory itself, then, there can be no process to which there is not a corresponding content. And this content is something determinate, or as Parmenides says, a τὶ or somewhat, different from the other somewhats which are the contents."of other and different processes. What you think of at all you think of as having a determinate character of its own, not as a featureless blank. (This is the element of truth which is distorted into an absurdity in the Hamiltonian dictum that "to think is to condition".) Thus the "content" of your thought, being a somewhat, is something that is or has being, a νόημα is always a νόημα of an ὄν τι (132 c). This was, of course, as a matter of historical fact the main tenet of Parmenides himself, who declared what is to be one on the ground that you can only think of τὸ ἐόν; you cannot think of anything else, because anything other than τὸ ἐόν must be μὴ ἐόν (what has no being), and μὴ ἐόν is merely an empty name to which no real thought corresponds. "It is possible for It to be, but it is not possible for nothing to be." As Plato was to show in the Sophistes the only way to meet the |76| paradoxes of Eleatic Monism is to deny the premise that "what is not is just nothing at all," and to insist that "what is not" in one sense "is" in another sense. The proposition that what is thought of is and its contra-positive that what is just nothing at all cannot even be thought of are unassailable. Fully expressed, the proposition that every thought is a thought of something that is means that, whatever you think of, you think of as being already so-and-so, already occupying a definite place and standing in definite relations to other constituents of a world which your thinking of it does not create. You never think of anything as having no other further reality, no other determination, beyond the mere fact that you are now thinking about it. There is no such thing as an ens rationis or as the mere "being for thought" of which some philosophers talk.

(To indicate more exactly what this means and what it does not mean, let me show how it bears on the familiar question of the "subjectivity of secondary qualities." It does not necessarily follow, from the principle that whatever is perceived or thought has a being which is not merely a "being for thought" or "for perception," that things have colours when no eye is looking at them. The sort of realism implied by what Parmenides has just said would be quite consistent with the view that colours depend for their existence on our eyes, and that the colours of the things in this room no longer exist when it is left empty. What the doctrine denies is that the existence of the colours is dependent on our minds. It may or may not be that our eyes help to create the colours; it is false that our minds make them by attending to them. The mere fact that we may attend to details in a scene which we had at first overlooked proves that whether or not, e.g., colours depend for their existence on a physical relation to a retina, |77| they do not depend on a psychical relation to a mind. Whether they exist where there is no eye to see them or not, when seen they are qualities of the objects we see, not qualities of our minds. However we answer the question what becomes of them when there is no eye to see them, it is at least certain that colours are not "subjective," they do not exist "in" the mind, but, in the only sense such a phrase can have, "without the mind".)

It follows then that the "content" of the process in which you think of a Form is always one something. It is "some one specific somewhat which that thought thinks as present in all the instances" (ὃ ἐπὶ πᾶσιν ἐκεῖνο τὸνόημα ἐπὸν νοεῖ , μίαν τινὰ οὖσαν ἰδέαν, where must of course be taken with ὅ and not with νόημα). Parmenides means, to put the point in more modern language, that even on the "Subjectivist" or "psychologising" or "Idealistic" view, there are determinate universal meanings, though on this view these meanings are held to be the "other aspect" of the occurrence of specific mental processes. He next adds that, since each of these meanings is a universal, each of them must be what Socrates calls a Form, a point of identity in the particulars of existence, a "common nature."Next we combine with the result thus deduced the Socratic premise that a particular derives its existence entirely from its "participation" in a Form or Forms, in other words that it is just a bundle of universal predicates and relations, and what follows? I.e. what follows if we assert (1) that a thing is just a complex of universals and (2) that universals are "the work of the mind"? Well, it follows that if things are made of universals and relations (which is what Socrates is maintaining) and if further these universals and relations only are as "aspects" or characters of mental states, then |78| either everything is made of mental states and all things think (πάντα νοεῖν, or else that there are "unthinking thoughts" (ἢ νοήματα ὄντα ἀνόητα εἶναι. Some good scholars have rendered the phrase "unthought thoughts," but I submit that this is impossible Greek at least for Plato. The only place in good Attic Greek where ἀνόητος means anything like "not thought" is Phaedo 80 b, where the soul is said to be νόητον, "apprehended by thought," but the body ἀνόητον — i.e. apprehended not by thought but by sense-perception, and there, as Professor Burnet remarks in his edition of the dialogue, Plato is making a pun; ἀνόητον gets an otherwise impossible meaning from the anti-thesis with νόητον. The regular meaning of the word in ordinary classical Attic is "silly," and this is enough to show that its literal sense was felt to be "unthinking.")

The alternatives then are these: either all things whatever — including steam-engines as Mr. Bradley once observed apropos of Mill's version of Idealism — are mental processes, or there are thoughts which are not mental processes. The first alternative is transparently absurd; the second contradicts the very doctrine from which it has been deduced, which was that for every "content" there is a process which is inseparable from it. An umbrella, for example, is not a complex of mental processes, though Mr. Spencer does somewhere talk of performing the feat of making the set of visual states which he calls his umbrella move past the set of visual states he calls the sea and sky. On the other hand "unthinking thoughts," thoughts which are all "content" without any process, are impossible according to Subjectivism itself. The plain conclusion is that the whole attempt to treat the objects of thinking as "aspects" of the process of thinking leads to impossible results (οὐδὲ τοῦτο ἔχει λόγον). |79|

It may be worthwhile at this point to leave the text of the Parmenides and ask whether after all we cannot escape this admission by a way of which Plato has not thought. Certainly the existence of "unthinking thoughts" seems quite impossible even on the premises of the Mentalist himself. But what of the other alternative that "all things think"? Common sense regards it as absurd, and so do Parmenides and Socrates in our dialogue. Yet many things which common sense is prone to call absurd seem to be true, e.g. in mathematics, and a fair-minded controversialist would probably allow that it is no disproof of a doctrine in theology to say that it looks absurd to untutored common sense. No one who knew his business would go to the "man in the street" to learn whether there are in God three personae in one substantia or whether the rational soul is derived by generation from one's parents. So there seems to be no intrinsic reason why a metaphysical proposition which sounds paradoxical to the "man in the street" should not be true. And, to say nothing of our professed Pampyschists, Dr. McTaggart has vigorously maintained that the Universe consists exclusively of souls. So it may be as well to ask whether, in spite of Parmenides, either "mental states" or "souls" may be the only things there are. I do not myself think we can make either assertion. To begin with, on any theory, it could only be of the particular existents in the Universe that we could say that they were all states of mind, or all souls, and the Universe contains much besides its particular existents. Suppose that all particular existents are souls. Then the Universe includes not only these souls but their various attitudes to one another, and no one will say that if A and B are souls, A's love for B is a third soul, and B's recognition of A's love a fourth. We get rid of this particular difficulty if we say |80| not that all particular existents are souls, but that they are mental states. But this view, too, has to face equal difficulties. It involves, of course, the denial that there are such things as minds or selves which have or own the states. This denial, however, though I myself think it philosophically bad, is made by men of eminence, and I will not dispute it here. But what about, e.g., "the hopelessness of A's love of B" or "the absurdity of C's opinions about D's philosophy." These, at any rate, can hardly be mental states, but they are as much constituents of the Universe as A's love of B or C's opinions about D themselves.

Even so, we have only touched the fringes of the real difficulty. Assuming problematically the more moderate position that souls (or, if you prefer it, mental states) are the only particular existents in the Universe, we have to ask, in this society of souls (or mental states), what do the souls (or states) think of and know? Do they only know, can they only think of, the propositions of Psychology? In our own case, we certainly suppose ourselves to know propositions about many particular existents which are not propositions of Psychology, and unless all these propositions without exception are false, there must be particular existents which are not souls nor yet mental states. Thus I may believe that there is at this moment a round pebble lying on my garden path, or that the pen with which I write these words was made by Messrs. Macniven & Cameron, and these propositions, which assert the existence of the stone and the pen, certainly do not convey psychological information about souls or mental states. Even if stones and pens have souls or mental states, it is pretty clear that they are not souls or mental states, and that a statement about the weight of the stone or the hardness of the pen is not an assertion about a mental state. Again, we |81| believe the gravitation-formula to be a statement which is true, or nearly so, of a certain relation between certain particular existents, but the relation which it expresses is not a relation between minds or states of mind.

Thus if Pampsychism only means that everybody has a soul or a mental life, it implies, rather than denies, that there are non-mental particular existents. But if it means that all existents are minds or mental states, and all the relations between them relations falling within the purview of Psychology, it seems to be proved false by the existence of the other sciences.

If we finally try to maintain the other alternative offered to us by Parmenides in a modified sense, by holding that things may be thoughts without being my thoughts in particular, because it is always possible that what I am not actually thinking of is always being actually thought of by other men, or by God, we are really no better off. That things which I have no ground for supposing to be actually thought of by any being but God may yet be real existents seems to be clear from the simple fact that an unknown body may cause perturbations in the behaviour of a known one. Neptune existed, not merely before Adams or Leverrier discovered Neptune, but before anyone had observed the perturbations in the periodic motion of Uranus which led to the discovery. It would be gratuitous to assert that because the perturbations existed before we discovered them there must have been non-human astronomers who did know about them. And though it may be reasonable on other grounds to believe in an omniscient God who, being omniscient, did know about the perturbations and their cause before we suspected either, it seems nonsense to say that God's knowledge of the existence of Neptune is what we mean by the existence of Neptune. For we should then have to say |82| that what Adams and Leverrier discovered was not Neptune but the fact that God knew about Neptune. So, as Bolzano says, "There is a God" does not mean that "God thinks that there is a God." We might make this point even clearer by asking what an atheist means when he says "There is no God." He cannot mean (a) "I, A. B., think there is no God," for if he meant that he could prove his proposition by merely proving his sincerity in making it. But no sane man thinks you can prove a proposition to be true by merely proving that you honestly believe it. Nor can he mean (b) "Men in general think there is no God."It is just because he knows they think there is a God that he gives himself the trouble of trying to reason them out of their mistake. And he assuredly does not mean {c) that "God thinks there is no God," for if he means that, what has become of his atheism? Again, even if every proposition which is true is thought by someone, it is certainly not true that whatever is thought by anyone is true, and this of itself shows that to be true is not the same thing as to be thought true by someone. And though both the pro- positions "whatever God thinks is true," and "whatever is true is thought by God" may be true, yet "to be true" cannot mean "to be thought true by God," for this would lead at once to a vicious regress. "God is," e.g., would have to mean "God thinks He is," and this again would not merely imply but mean "God thinks that He thinks that He is," and so on. Hence the real meaning of the statement "God is" would be unknown and unknowable, at least to a human intelligence.

Thus it seems clear that neither to be nor to exist can mean the same thing as to be thought of, and, as we have no empirical reason for believing that whatever is or exists is also thought of, we cannot deny that there may be any number of existents the existence of which |83| is not known to any mind, unless we can, on independent grounds, assert the existence of at least one omniscient mind. In that case it would be true that whatever is or exists is actually known, not because it is any part of the meaning of being or existence to be known, but as Bolzano says, because there is an would not in the least follow from the existence of an omniscient mind that all the other existents known by that mind are themselves mental. There is no more reason to think that a mind can only know minds than to suppose that an eye can only see eyes or a nose only smell noses.

It is an interesting question from what quarter the suggestion that Forms may be νόηματα, "thoughts," originally came. It is certainly very unlikely that Plato should have invented this gratuitous false interpretation merely for the sake of refuting it, but it is not at all easy to say with whom the idea originated. Proclus, if he knew, keeps his information to himself, and most modern expositors seem to think they have done their duty when they have made a reference to Berkeley. Grote, however, with his usual scholarship and conscientiousness, really tries to solve the problem. He observes (Plato and the other Companions of Socrates, vol. iii. p. 64, n. 2, ed. 1885) that Aristotle expressly alludes to the same view at Topics, 113 a 25, where he says that εἰ τὰς ἰδέας ἐν ἡμῖν ἔφησεν εἶναι, if your opponent in a dialectical encounter has maintained that the Forms are "in us," i.e. are states of our minds, you might meet him by arguing that his thesis leads to the simultaneous affirmation of contradictories {e.g. as a believer in Forms he must admit ex hypothesi that Forms are changeless, but if they are "in us" they change their position as we move about). A few pages |84| further on (op. cit. p. 74, n. 2) Grote connects the thesis that Forms are "thoughts" in "souls" with the doctrine that qualities (the word is, of course, a piece of Aristotelian Categorienlehre) are ψιλαὶ ἔννοιαι, "mere notions. "Simplicius says in a scholium on Aristotle's Categories, 8 b 25, that this subjectivist view was specially held by the Eretrian school of Menedemus, of whom we really know nothing except that they, like the Megarians, were famous for formal dialectic and that they must have been influenced by Eleaticism, since it is recorded of Menedemus (Diogenes Laertius, ii. 135) that he refused to recognise negative propositions. On the scanty evidence we possess, Grote's conjecture that Plato's Refutation of Idealism is meant to refer to this view seems to me the best that can be made.3 Antisthenes, as usual, has been suggested as the object of the criticism on the strength of the saying ascribed to him, "I can see a horse, but I never saw horse-ity."This is less likely. Antisthenes was perhaps dead when the Parmenides was written, even if the mot in question is authentic, not to add that the point of the alleged saying is not that "horse-ity" is a thought, but that is an empty name.

Socrates now offers another suggestion which leads to a second appeal to the impossibility of the "regress."He suggests that the difficulty about the Unity or Plurality of the Form may be escaped by thinking of Forms as παραδείγματα, fixed "models" or "types" of which sensible particular existents are "imitations" or "representations" (ὁμοιώματα). The precise meaning of the statement that the particular existent "partakes" in the Form will then be that it is a "likeness" or "copy" |85| of the "type," and it is easy to argue that there is no reason why any number of "likenesses" may not be "copies" of one "type," just as any number of impressions may be struck from one die or any number of engravings reproduced after the same original. It must be carefully borne in mind that in this new formulation of the theory the relation between the particular existent and the Form is not merely similarity or resemblance, but the relation of copy to original. The particular does not merely resemble the Form, but further is derivative from and dependent on it. It is this further relation of derivation which gives Parmenides an opening for a fresh application of the objection to an infinite regress.

There are many interesting questions about the relation between the new formulation of Socrates' theory and that which had been given earlier in the dialogue, which I am obliged to pass over as irrelevant to the purpose of this paper. I will merely note that the "imitation" version of the relation of particular to Form was, as we have learned from Aristotle, the Pythagorean one, and apparently older in date than the "participation" formula. Parmenides does not admit that the change in phraseology leads to any improvement in sense. He sets himself to argue (1) that the new version of Socrates' theory is still open to the objection that it leads to the "regress," and (2) that it has the still graver fault of leading by rigid logical consequence to a pure agnosticism. It is only with the first of these criticisms that I am to deal here. The argument of Parmenides is briefly as follows. If a particular existent is a "likeness" of a Form, then not only must it be like that Form, but the Form must be like it, since "being like" is, as we should now say, a symmetrical relation, a relation which is its own converse. |86| But, according to the theory itself, whenever two things are like one another, they are so because they "partake of" one and the same Form. Hence, since we have just admitted that particular and Form are like one another (e.g." that the Form of "man" is like Zeno or Socrates), our own theory requires us to hold that the particular and its Form both "partake of" a second Form. That is, employing the explanation just given of what is now supposed to be meant by "participation," the particular and the Form of which it is a copy, must both be copies of a second Form. And in the same way we shall argue that the second Form, the first Form, and the particular existent, are all like one another, and are therefore, on our own premises, copies of a third Form, and so on without end. The only way to avoid this "regress" is to deny the proposition "if A is like B, B is always also like A," and so to make it possible to hold that a particular existent is like a Form and yet the Form not like the particular. As this seems hopelessly paradoxical, it appears that we must say "it cannot be a virtue of likeness that things participate in Forms" (133 a).

Now, as to this argument, the alleged "regress" is plainly a vicious one, since the point of the reasoning is that we cannot even state what we really mean when we say, e.g., "Socrates is like the Form of Man," without going through in succession all the terms of an endless series. Also, on his own premises, the reasoning of Parmenides seems wholly sound, and we are thus driven, as he says, to admit that the puzzle can only be solved if it is possible to hold that a particular existent and a Form are not, on the theory under examination, like one another in the same sense in which two particular existents which are members of the same class are like one another. More precisely, what we need to be able to say is that the relation between Form and |87| particular existent symbolised by calling the second a "likeness" of the first is asymmetrical. Fortunately this position, which Parmenides calls paradoxical, is quite easily defensible. Proclus says truly that the solution of the difficulty is this. The relation of likeness which holds between two copies of the same original in virtue of the fact that they are copies of the same original does not hold between copy and original. Thus, though the resemblance between two engravings may justify the belief that they are copies of the same painting, it does not follow that this painting and the engravings are alike in any sense which would justify us in believing that all three are copies of a still older painting. As Proclus puts it, the copy is a copy of its original, but the original is not a copy of the copy. The relation really meant by Socrates when he spoke of particulars as "likenesses" of Forms was not mere likeness in some point or other, a symmetrical relation, but the kind of likeness which there is between an original and a copy, likeness plus derivation, and this relation is asymmetrical. Parmenides only proves his point because Socrates is so "young" and unpractised in formal logic that he allows the proposition "sensibles are likenesses (ὁμοιώματα) of Forms" to be reworded in the shape "sensibles are like Forms."The fallacy becomes manifest in a simple case. My carte-de-visite photograph and my living face may be like one another, but the likeness is not such that it could be argued "This photograph is a likeness of you, ergo, by conversion, you are a likeness of it."You can argue that since my reflection in a looking-glass is like me, therefore I am like it, but you cannot argue that since it is the reflection of me, I am the reflection of it. This is how Socrates permits Parmenides to argue when he allows him to substitute for the statement that a sensible thing is a likeness of a |88| Form the very different and much less specific statement that a sensible thing is like a Form.

When it is argued that since two sensibles which are like one another are, ex hypothesi, both "copies" of the same Form, therefore a Form and its "copies," being like one another, must all be "copies" of another Form, everything turns on the question whether "like" bears the same meaning throughout the premises. In point of fact it does not, and this is where the fallacy comes in. No particular existent is like a universal in the same way in which two instances of the same universal are like each other. Thus two green leaves are like one another in the sense that they both have the same colour, but a green leaf and the colour green are not like one another in this sense, since green has no colour but is a colour, the leaf is not a colour but has the colour green. Two men are alike in exhibiting the same type of bodily or mental structure, but John Smith and the human organism, or John Smith and "the human mind," are not alike in this sense, since the bodily or mental organisation characteristic of men is not itself a body or a mind. To take a case which touches the doctrine of Forms as expounded by the Platonic Socrates even more closely, two pairs of things, say a pair of gloves and a Parliamentary "pair," are alike in having the same cardinal number; there are two gloves, there are two members of Parliament. But a pair of gloves and the number 2 are not thus alike, for 2 is not a pair. There are two gloves, but not "two units" in 2, since 2 is not two numbers but one number, though Aristotle could not see this and is very wroth with Plato for having said that numbers are not generated by addition.

Let it be carefully noted what these examples show. They do not show that aForm, or universal, and a set of particular existents are not in some way "like" one |89| another. They do not, for instance, show that the Form of man and Socrates may not both be "copies of" or "partake in" some Form. But they do prove that the Form of man and Socrates cannot both be "copies" of the Form of man, and it is this absurdity which Parmenides was trying to extract from the statements of Socrates. He wanted to show that what Socrates calls the Form of man is really not one Form at all, but an endless hierarchy of Forms of man of ascending orders, and in fact, a "well-ordered series of type ω."Unless he can show this he has not proved that there is a vicious "regress" implied in saying that two men are alike because they both "imitate" or embody the same Form. If it is true that the particular man and the Form of man both "imitate" a further Form, which is not the Form of man, that is a harmless truth. The regress to which it gives rise is only an endless chain of implications. But if it were true that there is not one Form of man but an endless series of them, you would never be able to say what it is of which two particular men are "copies" or embodiments, and this is the pretended objection to the theory of Forms. Just so it creates no difficulty in arithmetic that if there is a finite integer, say 2, there must be another integer which comes next after 2, and another which comes next after that, and so on without end. But all arithmetic would come to an end if instead of one number 2 there was an infinity of 2's, so that 2 came an infinite number of times after itself.

I hope then that I have made it clear that the vicious regress which follows logically enough from the premises used by Parmenides does not follow from the assertions of Socrates of which the premises of Parmenides are an ingenious perversion. So far, the principle of the theory of Forms, that the making of |90| intelligible propositions, and consequently all science, depends on the pervasion of the Universe by universal types of structure and schemes of relation which are neither particular existents nor inventions of the knowing mind remains unshaken by the criticisms we have passed in review. But it is clear from the way in which Socrates receives these criticisms without attempting to answer them, as well as from the express declaration of Parmenides at 135 c that the failure of Socrates to repel his assaults is due to his lack of practice in dialectic, that Plato means us to understand that though the theory is at bottom sound and rests on a right perception of the character of scientific knowledge, its originators were not possessed of the logical equipment required to formulate it in a way which would ensure it against grave objections. For this purpose the theory required to be reshaped by a master of logic and pure mathematics, and the reshaping was the task of Plato's maturest thought. The form in which the theory finally emerges from his hands was never embodied in his dialogues. In them he remained true to the words he twice wrote to Dionysius II that there would never be a σύγγραμμα Πλάτωνος, an exposition of the philosophy of Plato. But its general outlines can be still recovered by careful study of the unsympathetic and often not very intelligent polemic of Aristotle as well as from the indications preserved in the remaining fragments of later Platonists.


1^ See my article on the subject in Classical Review (xi. 81 ff.).

2^ This doctrine must be carefully distinguished from the statement given in all works on symbolic logic that "true propositions are implied by all propositions." The reference here is to "material implication"; what the philosophers referred to in the text mean is apparently "formal implication," — a very different thing.

3^ Only if so, the doctrine must have been older than Menedemus, who belongs to the early third century.

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III. Forms and Numbers: A Study in Platonic Metaphysics

This paper was originally presented in 1926. [With the whole of what follows I must ask my reader to compare Milhaud, Les Philosophes-Géomètres de la Grèce, bk. i. c. 2, bk. ii. cc. 4, 5; Burnet, Greek Philosophy, Part I. 320-324; Stenzel, Zahl und Gestalt bei Platon und Aristoteles. I have to apologise for repeating much which is common property, but it is necessary to do so if I am to make it quite clear exactly where these writers seem to me to have stopped short of what seems the precise truth. (I reprint these pages as they were originally written. The modifications to the second edition of Dr. Stenzel's book (1933), which confessedly bring his views rather nearer to those I am urging, would require a separate essay for their discussion.)]

We all know the famous chapter in Aristotle's Metaphysics (A, 6) where Aristotle sums up the Platonic doctrine about the ἀρχαί and calls attention to its points of disagreement with Pythagoreanism. As all my readers will, doubtless, recollect, Aristotle holds that the differences between the otherwise very similar doctrines are two: (1) the Pythagoreans say that the constituents of number are the unlimited (ἀπειρον) and limit (πέρας), Plato that they are "the one" and the "great-and-small," or, as it is alternatively called, the "indeterminate duality" (ἀόριοστος δυας). (There are really two points of difference here; the One, which, as we know from other passages, was regarded by the Pythagoreans as derivative, being the simplest "blend" of their ἀρχαί, πέρας and the ἄπειρον, appears in the Platonic version as itself one of the underived "constituents" of number, the "formal" constituent, as Aristotle calls it; the other, |92| or "material" constituent, is a duality of some kind.) Aristotle's language (Metaphysics A, 987 b 25) shows that this is the peculiarity which strikes him as specially remarkable. (2) Though both parties agree that things are somehow made of numbers, the Pythagoreans simply identify these numbers with the things we perceive by our senses: Plato distinguishes the two, and further interposes an intermediate class of "mathematical objects" (μαθηματικά) between them.

It is obvious, as most recent scholars have seen, that Aristotle is not talking here about anything of the nature of a "senile aberration." He identifies this doctrine unreservedly with the teaching of Plato, and this must mean that no other "Platonic theory" was known in the Academy all through the twenty years between Aristotle's entrance there and Plato's death. If he has not explained what the formula means more fully, the reason must be that he believed himself and the contemporaries for whom he was discoursing to understand its sense without any explanation on his part.

Naturally enough, the meaning is not so obvious to us, who have to rely for our knowledge of Plato's teaching in the Academy on chance observations of Aristotle himself eked out by a very few statements of contemporary Academics preserved by his later commentators. It does not follow that the true interpretation cannot be recovered with pains and industry. In fact, my object in this paper is to show that three recent scholars, M. Milhaud in his Les Philosophes-Géomètres de la Grèce, (1900), Prof. Burnet in Early Greek Philosophy, Part I. (1914), and Dr. Julius Stenzel in his important recent work, Zahl und Gestalt bei Platon und Aristoteles (1924), have been, as the children say, very "warm" in their search for the key to the puzzle. But I believe none of them has ever quite tracked down |93| the quarry, probably because none of them has adequately interpreted the one passage in Plato which is more nearly than any other the explanation, Epinomis, 990 c 5-991 b 4. M. Milhaud ignores this important page altogether, except for the passing citation of a single phrase, presumably on the ground that it comes from a "spurious" dialogue. Prof. Burnet makes some use of the first half of it (op. cit. p. 322); Dr. Stenzel, with a sound perception of its importance, quotes and comments on nearly the whole of it (op. cit. 91 ff.) and would, I believe, have given the full interpretation, but for the want of mathematical knowledge which he candidly confesses in his Preface. Yet the mathematics required for complete understanding of the whole are really very elementary; if they were not, I should not venture to attempt the solution which I now propose to students of Plato for their judgment and censure.

I must make a beginning with the familiar passage of Aristotle. The thing which strikes him as singular is not simply that Plato, like the Pythagoreans, should have attempted a derivation of numbers from two components, but that he should have made one of these components, the one which Aristotle calls in his own terminology the "matter" of numbers, a duality, and that this duality should have been a "great and small." And quite clearly this is the point from which investigation should start. Why was Plato dissatisfied with the simpler statement that the "matter" of number is an ἀπειρον? And again why, if it is to be a couple of some kind, is it a couple of the great and small? If we could only identify the particular problems which have suggested the general formula, we might be able to answer these questions. In the light of the passage from the Epinomis it is possible, I believe, to identify the problem or problems almost beyond a doubt, and thus to |94| penetrate to Plato's meaning by reconstructing a piece of mathematical history.

Before I proceed further, however, I must mention two explanations which I feel bound to dismiss as insufficient, though on the right lines, so far as they go. It is quite insufficient to say, in the well-known words of Plato's disciple, Hermodorus (Simplicius in Physica, 247, Diels; Greek Philosophy, Part I. 330) that "those things which are spoken of as having the relation of great to small all have the 'more and less', so that they can go on to infinity in the direction of the 'still greater' and the 'still less'." This may explain why an ἄπειρον may be called a duality; it does not explain why it must be called so. The point is made in so many words in the Philebus (24 a ff.) where Plato is careful to make Socrates work with the old Pythagorean antithesis of the ἄπειρον and πέρας, if it were all that is meant, there seems no reason why Plato should have made any modification in the formula, or why, if he had done so, Aristotle should not dismiss the change, as he does the substitution of the word μέθεξις for μίμεσις, as a mere change of language. Also, the remark throws no light on a point on which Milhaud, Burnet, and Stenzel are all, rightly as we shall see, agreed, that the Platonic formula is somehow connected with the doctrine of "irrational" numbers. If there were no numbers but the rationals, it would still be true, as Hermodorus says, that there can be an infinity of, e.g., lengths greater and again smaller than a given length, of notes "sharper" or "flatter" than a given tone. Hermodorus may, no doubt, have known that his words do not give the full explanation of the Platonic doctrine. It is quite possible that he went on to explain further, but if he did, Simplicius must have cut his excerpt short before reaching the principal point, and that is hardly likely in a man |95| of his intelligence. Or Hermodorus may not have thought fit to say all he could have said. He has certainly not told us all we want to know before we can see why Plato should have been dissatisfied with the Pythagorean formula.

Again, when Milhaud, Burnet, Stenzel, all look for the explanation of the formula in the conception of the value of an "irrational" bysuccessive approximations to a "limit," they are plainly on the right track, as the passage of the Epinomis we have to deal with shortly demonstrates. Their explanation comes much nearer being the whole truth than the remarks quoted from Hermodorus by Simplicius. Yet it is not the whole of the truth. The thought of "convergence to a limit," important as it is, does not really explain why it should be necessary to replace the ἄπειρον by a "duality" and why the "duality" should be a "great and small." An example or two will make this clear. Consider the series 1, 1/2, 1/4, 1/8 … 1/2" …. We see at once that the series "converges to the limit" 0, since, by taking n large enough we can make the difference 1/2" – 0 less than any assigned rational fraction σ, however small. But though the endless sequence of the terms is a good example of an ἄπειρον, it is not clear how it can be an example of a "great-and-small." Since each term is one-half the preceding term, the series proceeds in a single direction, that of "the small" or "defect." Now consider the series formed by taking the sums of 1, 2, 3 … n … terms of our first series, 1, 1 + 1/2, 1 + 1/2 + 1/4, … 1 + 1/2 + 1/4 + … 1/2" …. This again "converges to the limit" 2, as we all know. But again, the series proceeds in a single sense. Each term, 3/2, 7/4 … is greater than the preceding. Thus our series is emphatically not a "great and small. "The inevitable inference is that when Plato replaced the Pythagorean ἄπειρον by |96| the "duality" of the "great and small," he was thinking of a specific way of constructing infinite convergent series which his interpreters seem not to have identified. I propose to show what the method was, by indicating the precise problem from which Plato was starting.

What the problem was we are all but told in so many words in the Epinomis, to which we must now turn. (It is irrelevant for the immediate purpose to enter into a discussion of the authenticity of the dialogue, though I may confess here my own conviction that it is genuine. Those who have adopted the ascription to Philippus of Opus usually recognise that the author is an immediate scholar of Plato, specially competent in mathematical matters, and that the work was issued from the first along with the Laws. Even so much is sufficient ground for holding that we may accept the matter of a mathematical passage from the dialogue as genuinely Platonic with reasonable confidence. If our exegesis should make it appear that the passage actually gives the clue to Plato's language about the "great and small," then, I submit, reasonable confidence passes into complete assurance. Incidentally also, such a result would, I take it, put the authenticity of the dialogue beyond question. That a "stylometrist" already determined to bring out a different result, examining the few pages of the dialogue under the microscope, should succeed in detecting some small peculiarities of the diction, as compared with that of Laws i.-xii., would prove nothing on the other side. If any slight departure from a stylometric average is proof of spuriousness, what single page of any author is safe from the first critic who has his reasons for wishing to get rid of it?)

I come then to the critical passage of the Epinomis. We must begin by recalling the context in which it is set. The ostensible purpose of the whole dialogue is to |97| answer the question what scientific studies are indispensable in a member of the "nocturnal council," the standing Committee of Public Safety, as we might call it, which watches over the general well-being of the community of the Laws. We are first told that the maintenance of a high standard of public piety will be the first concern of this council. That piety may be wholesome and rational, the Olympians are to be replaced as the primary objects of the public cultus by the heavenly bodies, the "great works" which, by the strict conformity of their apparently mazy dance to mathematical law, most specially declare the wisdom of the Creator. The fundamental business of the authorities who enforce this cult of the host of heaven is to impress it on men's minds that the heavenly bodies are not capricious creatures, like the fabled Olympians, but move in accord with law. And to satisfy men of this, it is necessary to ascertain the rhythmic periods of the movements of each "planet" and express them in terms of the period of any other. Consequently, the members of the council must not only be astronomers, they must also be thoroughly versed in all the preliminary knowledge which the astronomer will need for the execution of the task just mentioned. (Thus, exactly as in the Republic, before we come to astronomy itself we are conducted through the stages of a preliminary mathematical training. But there is this difference between the Republic and the Epinomis — it is just the difference between the mathematical science of the age of Socrates and that which Plato, with the work of Eudoxus and Theaetetus before him, was hoping to inaugurate — that, whereas the Republic specifies three preliminary sciences, arithmetic, plane geometry, solid geometry, the Epinomis introduces a new and extended conception of number which has the effect of bringing the |98| whole of the prolegomena to kinematics under the single head of arithmetic. Arithmetic, as now conceived, is the whole of what is strictly science in the "pure" mathematics.)

The speaker now proceeds to develop his views in a page which defies all formal grammar, probably not so much because it is badly "corrupted," though there are one or two points at which we are tempted to emend — as because the syntax is that of thought, and the words have never been subjected to revision with a view to circulation. If they are notes of Plato, "transcribed from the wax" with scrupulous piety after his death, this is intelligible; it is harder to understand, if they were deliberately set down by any one who meant them to be read as they stand. To avoid unnecessary prolixity, I will therefore merely give such a resume of the general sense as remains unaffected, whatever view we take of the text and grammar of the various clauses. We shall need, he says, various μαθήματα, first one which deals with numbers simply as numbers (αὐτοὶ ἀριθμοί), not as embodied in anything, and studies the "generation of the odd and even" and the character (δύναμις) they impart to nature (990 c 5-8). Next we must study what has been very ludicrously called "mensuration" (γεωμετρία), but is really an art which assimilates to one another numbers which are not similar in their own nature, by reference to surfaces (or areas). This art is a more than human miracle in the eyes of those who can appreciate it (990 d 1-6). Then comes another art which deals with numbers "raised to the third power and similar to volumes" (τοὺς τρὶς ηὐξημένους καὶ τῇ στερεᾷ φύσειὁμ οίους), and once more makes similar a second class of numbers not naturally similar. This is what the inventors who first hit on it called "gauging" (στερεομετρία). This again is a miracle in the eyes of those who understand how |99| "all nature" (ὅλη ιἡ φύσις) is moulded in form and kind (fεἶδος γένος ἀποτυποῦται) as the function (δύναμις) and its converse (ἡ ἐξ ἐναντῖας ταύτῃ) move about "the double" in each progression (i.e. geometrical, arithmetical, harmonic, 990 d 6-991 a 1). The simplest form of "the double" is the ratio 2/1, from this we get, in geometrical progression, the second power 4/1, and the third 8/1. With this third term 8/1, "the double" has advanced to "volume and the tangible." If we treat 1 and 2 as the end terms of an arithmetical or harmonic progression and insert the arithmetical mean 3/2 and the harmonic mean 4/3 (or, if, to get whole numbers for all the terms, we consider the A.P. 6, 9, 12 and the H.P. 6, 8, 12) we have the secret of music (991 a 1-b 4).

The general drift of the argument may be considered before we come to detail. The connection in thought is this. To compare the various astronomical periods with one another, we need arithmetic. We begin the study with the arithmetic of the integers or natural numbers, and we must remember that it is explicitly announced that (1) the integers are to be studied as "pure," not as "embodied," and (2) that the study is to involve an account of their γένεσις or derivation. The point of the first statement is to guard against the confusion, into which it is so easy to fall, between an integer and a collection of which it is the cardinal number. What is meant is that, though every pair consists of two things, every triplet of three, and so on, the number 2 is not a pair of numbers, nor the number 3 a triplet. This is important for two reasons. It makes it clear that though there may be many pairs of things, there are not many 2's but only one 2, the number characteristic of each and every pair, and again, that though a pair, e.g., of gloves is made up of two gloves, 2 is not made up of two 1's. 2 is not two 1's, but one 2, and it does not |100| "contain" any 1's. We see therefore that the standing Aristotelian criticisms, which regularly assume that there are as many 2's as there are pairs, as many 3's as triplets, and again that integers are generated by the summation of 1's, are irrelevant as criticisms of Plato; they are no more than dogmatic affirmations of a non-Platonic, and manifestly false, theory of the nature of the integers. We see also that in the account of the "generation" of integers of which the Epinomis speaks, they are meant to be generated in some other way than by summation of 1's, that the integer (n + 1) will not be defined as the sum of the integer n and the integer 1. The point could not be stated by Plato in just this way, because the numerical notation at his disposal did not enable him to use a general symbol, like our (n + 1) or Peano's improved symbol (n +), to stand for "the integer immediately after a given integer." His way of stating it is to say, as we know from Aristotle he did say, that "numbers" cannot be "added" (are not συμβλητοί).

The transition from the remarks about arithmetic to the comments on "geometry" and "stereometry" has an obvious motive which is not expressed in words. Arithmetic, we are told, must be cultivated because it will be required for the determination of the periods of the heavenly bodies. But it is not the fact, and the Epinomis anticipates the knowledge that it is not the fact, that all the ratios we shall have to consider in our astronomy, the ratio, e.g., of the lunar month to the year, or day, or of one planet's period to that of another, can be accurately stated as "ratios of one integer to another." The number of days in the lunar month, or of lunar months in the year is not a whole number, and we must be prepared to face the possibility that it is not a rational fraction. Hence the astronomer will need in his calculations to manipulate "surds." He will require |101| to estimate such ratios as that of the side to the diagonal of a square, that of the diameter of a circle to its circumference, of the diameter of a sphere to the edge of each of the regular solids inscribed in it, and of each of these edges to the rest. We may note in this connection that the two last-mentioned problems are actually discussed, in connection with the constructions for the inscriptions of the regular solids, "the figures of Plato," in the 13th Book of Euclid's Elements, and we may be quite sure that the author of the Timaeus was deeply interested in them. We may fairly assume that when Theaetetus completed the Pythagorean geometry by discovering the constructions for the icosahedron and octahedron, he did not neglect to make this determination of the magnitude of edges part of his investigation. It is clear, then, that our astronomer will need to be able to determine the values of irrational quadratic and cubic roots, and to determine them with as close an approximation as his problems demand. Since such values were actually found by geometrical constructions, the common view was that the determination of them belongs not to arithmetic but to geometry, and so long as arithmetic is conceived of simply as the study of the integers, geometry must, of course, be regarded as a wholly distinct science, since it is full of "incommensurable magnitudes," but there are no "incommensurable integers," the position pertinaciously defended by Aristotle. The Epinomis insists, on the other hand, that the real scientific problem has nothing in itself to do with the "measuring of land" or the "gauging of solids," but is numerical. In other words, when we have learned how to evaluate the square and cube roots of the integers, we have, in principle, solved the problem of determining the length of the side of a regular polygon of given area, or the edge of a regular solid of given volume. The rest |102| is no more than a special application of our arithmetical discovery. (The insight shown by this view may well be illustrated by a very similar remark which occurs somewhere in Couturat's work De l'infini mathématique. Most of us commonly think of π as "the ratio of the circumference of a circle to its diameter," and again of e as the basis of the system of natural logarithms, i.e. of a series devised for the practical purpose of facilitating calculations. But the numbers π and e have a much more general significance than this. As Couturat says, even if we never had to survey a circular area or to make elaborate calculations, we should come upon e in the prosecution of analysis by the discovery that ex is the function of x which is its own derivative, and we should discover π from such purely numerical considerations as that π/4 is the limiting value to which the sum 1 – 1/3 + 1/5 – 1/7 … converges as the number of the terms summed increases indefinitely.) Thus we may say that the passage of the Epinomis under our consideration is historically important as the literary record of the first discovery of the "real numbers," if we are careful to bear in mind that the writer confines his attention only to those real numbers which are necessitated by the geometrical problems familiar to him, the quadratic and cubic irrationals. He does not envisage a series which would contain the whole of the algebraic numbers, still less has he any conception of the "transcendental numbers." (The problem of the quadrature of the circle had, as we know, already been raised in the fifth century, but naturally enough, no one in Plato's time was in a position to say that π might not turn out to be a quadratic surd. That it is not an algebraic number of some kind was only finally proved by Lindemann in our own days.)

So far, then, we have reached the result that astronomical |103| problems force on us the extension of arithmetic by the discovery of a method of evaluating quadratic and cubic "surds," and a corresponding enlargement of our conception of number which will enable us to include these "surds" among numbers. What the required method of evaluation is to be is at least hinted in the words which follow. We are now told that if we take the simplest of all numerical ratios, that of the "double" 2/1, and its reciprocal, the "half," 1/2, and study them in the light of the doctrine of progressions, they will disclose the whole secret of science. The latter part of this explanation, which deals with music, calls for no comment. It points out simply that 1, 3/2, 2 is an arithmetical progression, 1, 4/3, 2 an harmonic, and that the ratios 3/2 : 1, 4/3 : 1 correspond to the fundamental melodic intervals of the scale, the fifth and the fourth, so that we can get as many octaves as we wish by merely repeating them thus, 1, 3/2, 2, 3, 4 …, 1, 4/3, 2, 8/3, 4 …. Here we are not going beyond familiar Pythagorean ground. The immediate meaning of what precedes is, of course, that 1, 2, 4, 8 …. is a geometrical series composed of the powers of 2, and is the simplest example of the proposition that the areas of similar polygons are in the ratios of the second powers of their sides, those of the volumes of similar solids in those of the third powers of their edges. But there is clearly more intended than this, and we must discover what that more is before we shall see the connection between what had been said about the generalisation of number and these remarks about geometrical progression. The "double" had been in the fifth century the subject of one disturbing problem, that of the "common measure of the side and diagonal," where the problem is to know what must be the length of the side of a square if its area is to be double that of a given square. In Plato's |104| own time it gave rise to another problem, with which Plato is traditionally said to have been concerned, the "Delian problem" of finding the length of the edge of a cube whose volume is double that of a given cube. In other words, the fifth century had been concerned with the question what is the "square root" of 2, the fourth was trying to find the "cube root" of 2 {3√2}, and I suggest that the language of our passage is meant as a definite allusion to this. The underlying thought would thus be that the theory of arithmetic will only be complete when we have learned how to give a numerical expression for √2 and 3√2 — and thus, by the way, solved the "Delian problem" — and have then proceeded to generalise a method for the evaluation of the rest of the quadratic and cubic "irrationals."

If these were the special origins of Plato's conception, it ought not to be difficult to determine what kind of method he has in view, and then to answer our former question about the reason for the name "great-and-small." There was already in existence in the latter part of the fifth century a rule for making approximations to the value of √2, the rule to which Plato apparently alludes in Republic 546 c, where he makes Socrates speak of 7 as the "rational diameter of 5." The meaning is that since 72 = 49 and 52 = 50/2, 7/5 is an approximate value of √2. The "diagonal" (διάμετρος) of 5, that is the length of the diagonal of a square whose side is 5, is, by the Pythagorean theorem, 5√2, and this is an "irrational," but 7/5 approximates fairly closely to it, since 72 = 2 x 52 – 1. Of course it would be possible to suppose that such approximations were originally discovered empirically. One might, for example, write out a list of the "squares" of the integers from 1 to 100, and then pick out, by inspection, every pair of values which would satisfy the equation y2 – 2x2 ± 1. This would |105| yield us the pairs of integral values x = 1, y = 1; x = 2, y = 3; x = 5, y = 7; x = 12, y = 17; x = 29, y = 41; x = 70, y = 99. But there is a general rule, given by Theon of Smyrna (Hiller, p. 43 f.) for finding all the integral solutions of the equation, or, as the Greek expression was, for finding an unending succession of "rational diameters," that is, of increasingly accurate rational approximations to √2, the "ratio of the diagonal to the side." The rule, as given by Theon, is this. We form two columns of integers called respectively "sides" and "diagonals." In either column we start with 1 as the first term; to get the rest of the "sides," we add together the nth "side" and the nth "diagonal" to form the n + 1th "side"; in the column of "diagonals," the (n + 1)th "diagonal" is made by adding the nth "diagonal" to twice the nth side. Fortunately also Proclus (In Remp. Kroll ii. 24-25, 27-29; ibid., Excursus, ii. p. 393 ff.) has preserved the recognised demonstration of this rule; it is a simple piece of geometry depending only on the identity (a + b)2 + b2 = 2(a/2)2 + 2(a/2 + b)2, which forms Euclid's proposition II. 10. Since the students of the history of Greek geometry seem agreed that the contents of Euclid II. are all early Pythagorean, there is no reason why the rule given by Theon should not have been familiar not only to Plato, but to Socrates and his friends in the fifth century. The probability is that they were acquainted with it, and thus knew how to form an endless series of increasingly close approximations to one "irrational," √2.1

|106| We note at once that the ratios obtained by forming the "sides" and "diagonals" are identical with what we call in modern language the successive "convergents" to √2, formed by expressing the "irrational" as an unending continued fraction. I hope the reader acquainted with a little elementary algebra will pardon me if I explain this point briefly for the benefit of students of Plato whose school mathematics have been neglected through no fault of their own.

To express a quadratic surd as an unending fraction, we start from the identity (√a + b)(√a – b) = a – b2. Thus in the case where √a is to be √2, we can put b = 1, and the identity becomes
(√2 + 1) + (√2 – 1) = 2 – 1, i.e. √2 – 1 = 1 / √2 + 1.

We then proceed as follows:
√2 = 1 + (√2 – 1) = 1 + 1 / √2 + 1 = 1 + 1 / 2 + (√2 – 1)
= 1 + 1 / 2 + 1 / (√2 + 1) = 1 + 1 / 2 + 1 / 2 + (√2 – 1)
= 1 + 1 / 2 = 1 / 2 + 1 / (√2 + 1).

Where we see that by substituting 1 / (√2 + 1) for (√2 – 1), whenever the second expression recurs, we shall ultimately get the unending fraction
√2 = 1 + 1 / 2 + 1 / 2 + 1 / 2 + 1 / 2 + 1 /…

The successive "convergents" to the value of this fraction are formed by "stopping it off" at the first, second, |107| … nth step in the formation. Thus they are the rational numbers 1, 1 + 1/2, 1 + 1 / 2 + 1/2

and so forth. The reader will readily see that they correspond precisely to the series of "diagonals" and "sides" given by Theon, the numerator and denominator of each being one of the pairs of integral solutions of the equation y2 = 2x2 ± 1.

On examining the way in which the unending "continued fraction" and its "convergents" are formed, we at once note the following points:

(a) Each "convergent" is a nearer approximation to the required value than the one before it. Thus 17/12 is a nearer approximation to √2 than 7/5, since if we take 2 x 52 as = 72, we are wrong by 1 in 50, but, if we take 2 x 122 as = 172, we are only wrong by 1 in 288.

(b) The convergents are alternately rather less and rather greater than the value to which they are approximations. Thus 7/5 is less than √2, since 72 = 2 x 52 – 1; 17/12 is > √2, since 172 = 2 x 122 + 1. 7 and 5 are, in fact, solutions in y and x of the equation y2 = 2x2 – 1; 17 and 12 are solutions of y2 = 2x2 + 1.

(c) The interval, or absolute distance, between two successive "convergents" steadily decreases, and by taking n sufficiently large, we can make the interval between the nth and n + 1)th convergent less than any assigned rational fraction σ, however small, and can therefore make the interval between the nth convergent and the required "irrational" smaller still than σ.

(d) The method is manifestly applicable to any "quadratic" surd, since it rests on the general formula
(√a – b) = a – b2 / (√a + b). |108]

It is most readily applicable when a, whose "square root" is required, is an integer of the form m2 + 1, since in that case, by taking b = m, we reduce our fundamental formula to the simple form (√a – 1) = 1 / (√a + 1).

Thus we get at once such results as that
√5 = 2 + 1 / 4 + 1 / 4 + …
√17 = 4 + 1 / 8 + 1 / 8 + …

But we can also use it with a little more trouble to yield, e.g.,
√3 = 1 + 1 / 1 + 1 / 2 + 1 / 1 = 1 / 2 + … and similar results.

The general character of the procedure is thus that in the expression of √a as an "unending continued fraction," by forming the series of "convergents" we pin down √a between two values, one of which is a little too small and the other a little too large, but the difference between the too small and the too large is decreasing at every step and can be made less than any fraction we like to assign, though we never quite get rid of it, because we cannot actually arrive at a last convergent. To put it another way, in approximating to √2 by this method, we are not merely approximating to a "limit," we are approximating to it from both sides at once; √2 is at once the upper limit to which the series of the values which are too small, 1, 7/5, … are tending, and the lower limit to which the values which are too large, 3/2, 17/12, … are tending. This, as it seems to me, is manifestly the original reason |109| why Plato requires us to substitute for the ἄπειρον as one thing, a "duality" of the great and small. √2 is an ἄπειρον, because you may go on endlessly making closer and closer approximations to it without ever reaching it; it never quite turns into a rational number, though it seems to be on the way to do so. But also, it is a "great-and-small" because it is the limit to which one series of values, all too large, tends to decrease, and also the limit to which another series, all too small, tends to increase.

The meaning of what is said in our passage of the Epinomis about plane geometry will thus be that the real problem of the study is to evaluate all quadratic surds (√3, √5, etc.), by the same method which has proved successful in the case of the "double"; they are all, in modern phraseology, to be expressed as unending "continued fractions," and our conception of number is to be enlarged to include these "irrationals," which by the proposed method can be made rational to within whatever "standard" we like to adopt. It is the indispensability of providing a means of checking the interval within which the "error" of an approximation falls which is the real reason for replacing the single ἄπειρον by a "duality."

Thus, for example, when we know that √2 lies somewhere between 1 and 3/2, our work is not really done. We are not to say that it simply is one of the "infinitely numerous" values between 1 and 3/2. By taking the next pair of "convergents," 7/5 and 17/12 we can exclude it from that part of the interval (3/2 – 1) which lies between 1 and 7/5 and again from that part of it which lies between 18/12 and 17/12. The alternation of the too small which is steadily increasing and the too great which is steadily decreasing is demanded if we are to estimate the amount of error incurred by taking |110| a given approximation as the true value of our "irrational."

It may be worth while to note here that this absolute necessity for the revision of the Pythagorean formula would not have existed if the Greek arithmeticians had possessed our method of developing irrational "square roots" as unending decimal fractions. When we employ this method to evaluate √2 and get the result that √2 = 1.41421, … we are approaching our "irrational" only from one side, that of the "too small." Any approximation got by taking ni> significant figures to the right of the decimal point will be too small, because there are always still more significant figures that can be added. Yet we are able to assign a limit to the amount of the error. Thus I can say at once that the value 1.4 is too small by an amount which lies somewhere between .01 and .02. And yet, even so, we have not quite got away from the "duality." If I merely said that 1.4 is "too small," this would leave much too wide a margin of error, since the possible error might be anything between .00000 … 1, when I may suppose as many 0's as I please, provided only that the number is finite, and .009999 … where again, the 9's may be as numerous as you please, so long as their number is finite. If we are to make any accurate estimate of the error, still more, if we are to be able to diminish it ad libitum, we must be able to confine our approximation between a μέγα and a μικρόν. We do this habitually in our elementary calculation, when we follow the rule that if we wish to get an approximation right to n "significant places," we must first work it out to (n + 1) places, and then, if the (n + 1)th figure is greater than 5, increase the nth by 1; for example, if we wish to give the value of π "to four places," we must not write 3.1415, though 5 is really the fourth figure in the "decimal"; |111| we must write 3.1416, since the full calculation would give 3.14159 ….

The task which the Epinomis would impose on the "geometer" would, in the absence of a numerical notation resting on the principle of position, be a formidable one. We can conceive two possible ways of executing it. One would be the purely empirical one of forming a table of the successive "squares" of integers by actual multiplication and then picking out on inspection suitable pairs. Thus, to find √3, we might try to pick out from such a table the pairs which satisfy one of the equations y2 = 3x2 – 2,
y2 = 3x2 + 1." The solution of the first equation would give the first, third, fifth, … those of the second the second, fourth, … "convergents" to the continued fraction 1 + 1 / 1 + 1 / 2 + …. But this procedure would not only be exceedingly tedious, in view of the great number of multiplications involved; it would have the further difficulty that the requisite equations are hard to detect except in the most favourable cases. We may, therefore, feel fairly sure that a fourth-century student of the problem would attempt to establish the equation by finding a geometrical construction on the basis of Euclid, Elements II., as we know from Proclus was actually done for √2. But these constructions themselves would often be difficult to discover, and I have not been able as yet to learn whether any such constructions can be shown to have been actually known in the fourth century. Is there any proof that there was a known construction of this kind for √3? Perhaps some special student of the history of mathematics may be able to answer the question. If there was not, we must understand the Epinomis as simply indicating a |112| programme for Academic mathematicians of the future."2 In any case, it should be noted that the principle of the method, the pinning down of an irrational between a "too large" and a "too small" which are made to approach one another indefinitely, is the same which was employed for the finding of the areas of curvilinear figures and the volumes of solids with curved surfaces, as when, e.g., the area of the circle was treated as intermediate between that of a circumscribed and a similar inscribed polygon and these two areas then made to tend to equality by supposing the number of the sides of the polygon increased.

The "stereometer's" problem is next said to be in principle the same. His business is to express surd "cube roots" as limits of series of approximation which are alternately too large and too small. Here, again, we have the materials for a question which I should like to propose to special students of the history of mathematics. Had the mathematicians of the fourth or third century a method of extracting such "roots," and if they had, what was it? The restriction of the treatment of irrationals in Euclid X. to quadratic surds may possibly be evidence that no such method was in the possession of Euclid or his Academic precursors, since one cannot believe that they would not have utilised it, |113| if they possessed it, for the solution of the "Delian" and similar problems. (The solutions known to us are all geometrical, not arithmetical.) In any case, if the Academy anticipated, as the language of the Epinomis would naturally suggest, that the problem could be solved by a method analogous to the construction of endless continued fractions, their anticipations were premature. The problem resists this treatment for the simple reason that the product (3 √a – 3√b)(3√a + 3√b) is irrational. Yet we can exhibit cube roots in a form which displays the regular alternation of the "great" and the "small," though by a method unknown to the ancients. For we can in general write 3√x + y = 3√(x ⅹ 3√1 + y/x) and then proceed to expand (1 = y/x) 1/3 by the Binomial Theorem, since it is easy to show that (1 + y/x) ⅓ is convergent if y/x is <1. Hence, when once we have found 3√2, we can find in succession 3√3, 3√4 and the rest. For 3√2 itself, things stand rather differently, as we cannot throw (1 + 1)1/3 into the form demanded. We can, however, show that the series arising from the expansion of (1 + 1)1/3 is convergent by considering that the terms, apart from the first, form a series in which each term is numerically less than the preceding, and that they are alternately positive and negative. This proves that the sum of them converges to a limiting value, and consequently the sum of them with the addition of the first term, 1, is also convergent. In practice the method is not employed, for the reason that a considerable number of terms have to be calculated in order to secure any accuracy of approximation. We are not entitled to assume that the Academy actually possessed any method by which 3√2 could be calculated, and the Delian problem solved arithmetically, rapidly, and accurately. We should rather take the Epinomis to express the natural hope that the method which had |114| disposed of the fifth-century problem of "side and diagonal" would prove directly applicable to all the "irrationals" as yet recognised, quadratic and cubic alike.

I submit, then, that the character of the series of what we call the "convergents" to an endless continued fraction supplies the reason for the denomination of the irrational as μέγα καὶ μικρόν; in the power the method affords of restricting the value of the irrational within limits which can be made to approach one another as nearly as we wish, we have the motive for the correction of the earlier formula; in the anxiety to clear up the mystery of the "side and diagonal" we can see the starting-point of the conception. If this is so, we understand the origin of the formula guaranteed by Aristotle, that the part played by the "formal" element in a "number," the one, is to equalise (ἰσάζειν) the "great" and the "small."3 Since the series of convergents, alternately too small and too large, never actually comes to an end, there is always an "inequality" or tension between the "great" and the "small," and thus always a still unrationalised "matter" in the "number." But since we can make the interval between two consecutive convergents less than any assigned rational interval, the tension is steadily growing fainter as you pass along the series. It would come to rest in a complete "equality" if, per impossibile, two successive convergents could have an identical value. They would then not be two values, but one; the "interval" would be reduced to zero. This never actually happens, but it "all but happens"; you cannot come literally "as near as nothing" to a rational fraction which, when multiplied by itself, gives the product 2 or 3 or 5, but you can come nearer |115| than anything which is not literally 0 to such a fraction. Thus we might say of the "irrational" in the phraseology of the Phaedo, that though it never quite succeeds in being a rational, "it tries its very hardest to be one," and misses by immeasurably less than the traditional hair's breadth. This means that if we are to equate Forms with "numbers," as Aristotle assumes that Plato did, and also to say, with the Epinomis, that "irrational roots" are numbers, and therefore Forms, the relation the Phaedo assumes for "sensible things to Forms, must also exist among the Forms themselves." The "irrational" is a Form, but it is always trying, and never quite succeeding in the attempt, to exhibit the Form of a "rational." (The unending "decimal" tries its hardest to "recur.") This is, in principle, why the στοιχεῖα of the Forms are the στοιχεῖα of all things.

We may illustrate this point in a little more detail from the Timaeus. It is true that the speaker there is a fifth-century Pythagorean and that his repeated insistence on the provisional and tentative character of all his mathematical physics shows us that Plato does not wish to take the responsibility for precise details. But the existence of the dialogue itself is sufficient proof that the general type of view which pervades the dialogue is meant to be regarded as sound. There all the sensible characters of bodies animate and inanimate are made to be functionally dependent on the geometrical structure of their corpuscles. The structure of the corpuscles again is determined by that of their faces, and that of the faces by the structure of the two types of triangle from which they are built up. Now the triangles are determined in everything but their "absolute magnitude," which is asserted by Timaeus (57 d) to be variable within limits, by the triplet of "numbers" which gives the ratios of their sides to one another, and |116| this triplet in each case introduces irrationals. For the isosceles right-angled triangle, it is the triplet (1, 1, √2) and for the "right-angled scalene" adopted as the foundation of three of the five regular solids, the triplet (1, √3, 2). If you know how to "approximate" within as near an interval as you please to √2 and √3, you know how to form these triangles, and thus you are possessed of the secret on which all the physical characters of the realm of becoming depend. The στοιχεῖα of the "numbers," √2, √3, are, in the end, the στοιχεῖα of γιγνόμενα. It is worth noting that the two triangles might equally have been specified by giving the ratios between their angles. This would give the triplets (1, 1, 2), (1, 2, 3), where no irrational appears. In Speusippus, Fr. 4, where these triangles and their significance for solid geometry are described, these ratios are mentioned, but nothing is said about the others. This is significant, since the passage is given in the Theologumena Arithmetica as an extract from a work by Speusippus, the Pythagorean Numbers, based on the teaching of Philolaus. We may infer that the originators of the doctrine of the "elementary triangles" reached it by a consideration of the angles of the two figures, and if they knew, as they pretty certainly must have known, that the ratios of the sides, in both cases, introduced the "scandal" of the "incommensurable," they kept a decent silence on the point. Plato, on the other hand, calls express attention to the ratios of the sides, and is silent about those of the angles, but for the single remark that there is a certain "beauty" about these triangles, which presumably means that the ratios involved between their angles are the simplest λόγοι, of integer to integer, 1 : 1, 1 : 2, 1 : 3. Thus he makes Timaeus bold enough to insist on the very point which is embarrassing to the Pythagorean conception of number, |117| but, in the interests of historical verisimilitude, will not let him say that his geometrical ἀρχαί have still more ultimate arithmetical ἀρχαί, "surd" numbers. He allows the speaker to escape with the ambiguous remark that "God knows" what ἀρχαί there can be more ultimate than the triangles (Timaeus, 53 d).

In the Theaetetus (147 d) we are told that the Pythagorean mathematician Theodorus has just been explaining to the lad Theaetetus and his friends that 3, 5 … 17 have no rational square roots. This plainly does not mean that Theodorus merely explained that these numbers have no integral square roots; this is obvious and could have been said in a sentence. Nor can it mean that Theodorus actually possessed and explained the method of approximating to √3 and the rest. If a Pythagorean in the year 400 could have given such a method, there would have been no novelty about the Platonic view of number. We must suppose that Theodorus is supposed to be able to demonstrate the irrationality of the various numbers, √3, √5 …, without knowing how to construct them, and to show that the construction involves an endless series. The simplest way in which this might be done would be to employ the kind of reasoning by which it is proved in a theorem appearing in some MSS. at the end of Book X. of Euclid's Elements4 that √2 is irrational. The method is to prove that if there is a rational root, the absurd consequence follows that a certain fraction both is and is not stated "in its lowest terms." Thus, if we suppose that √3 = (1 + m/n) where m and n are integers and have no common factor, it is easy to prove that our equation demands that both m and n shall be even numbers, and therefore have the common factor 2, contrary to the hypothesis. The method could be used for any of the numbers stated to |118| shave been investigated by Theodorus. Also, in some cases, as Zeuthen has pointed out,5 it would be possible to apply the method given in Euclid X. 2 for the finding of a G.C.M., and to show that a given rational magnitude and a second magnitude under consideration have no G.C.M., and therefore the second must be incommensurable with the first. (Thus it is easy to see from the construction of Euclid II. 11 that the two segments into which a straight line is divided in the proposition have the ratio √5 – 1 : 2, and then, by using Euclid X. 2, that the two segments have no G.C.M.) Zeuthen reasonably infers that the irrationality of the expression √5 – 1, and consequently of √5. must have been known to Theaetetus in this way. (It does not follow, though it is possible, that Theodorus had actually demonstrated the fact. We might suppose that Plato is taking certain propositions, due to his own associate Theaetetus, and dramatically feigning him to have learned them from his old teacher, though in view of the admittedly Pythagorean character of the "geometrical algebra" of Euclid II, I think it probable that we may go a step beyond Zeuthen and ascribe the propositions in question to Theodorus.)

In the Meno again (83 a-e) the immediate object of the cross-questioning of Meno's page is to establish the point that √2 lies somewhere between 1 and 1.5. The Republic (546 c) shows that Socrates is familiar with the closer approximation 1.4 (7/5), and we may suppose that if it had been necessary for his purpose, he could have continued his elenchus until this had been made plain, i.e. that he knew the construction given by Proclus for approximation to √2.

To sum up, then, the special points I want to make are these:

|119| (a) The fundamental novelty about the Platonic theory is that it represents the first discovery, in an incomplete form, of the real numbers, as the ultimate determinants of geometrical structure, and so mediately of the physical characters of things.

(b) That the real numbers are conceived of as the common limit to which two "infinite" series "converge." The terms of the one series (that of the odd "convergents" of the complete double series) are always "too small," but the defect is steadily diminishing; the terms of the other (the series of the even "convergents") are all "too great," but the excess steadily diminishes. This is why the "material" constituent of number must be called the μέγα καὶ μικρόν.

(c) The origin of the whole theory is to be found in the discovery of the "side-and-diagonal" numbers which form an endless series of increasingly close approximations to √2. Plato holds that the business of "geometry" is to discover similar series for all the quadratic surds.

(d) The rise of solid geometry in the Academy, leading as it did to problems involving "cube roots," creates a demand for the formulation of a similar method of approaching the cubic surds alternately from the side of the too great and from that of the too small. Here again, the simplest case of the general problem is that of finding a series for 3√2 (the "Delian problem"). Theoretically the construction of such series of approximations to cube roots from the two sides alternately is readily performed, though it involves algebraical methods not possessed by the Greek mathematicians. Practically the method has little value, since the number of terms which must be taken into account to secure a moderately accurate approximation is inconveniently large.

|120| There is one further observation, more important than any of the foregoing, which can only be made here with the utmost brevity. The Platonic theory is inspired by the same demand for pure rationality which has led in modem times to the "arithmetisation of mathematics." The object aimed at, in both cases, is to get rid of the dualism between so-called "continuous" and "discrete" magnitude. The apparent mystery which hangs about the "irrationals" is to be dispelled by showing how they can be derived, by a logical process which is transparently rational at every step, from the integers and the "rational fractions," or λόγοι of integers to integers. It is precisely the same process, carried further, which we see in modern times in the arithmetical theory of the continuum, or in Cantor's further elaboration of an arithmetic of the "transfinite." In all these cases, the motive for the construction is to get rid of an apparent mystery by the discovery in the seemingly unintelligible of the principle of order of which the integer-series is the perfect and ideal embodiment. "Forms are numbers" because "order is Heaven's first law," and number is the type and pattern of order. The task which still awaits us is to consider the nature of the further and final step by which the conception of number as the determination of a "material" (the great-and-small), by "form" or "order" was extended to cover the case of the integers themselves. So far as we have gone, the integers and their order have figured as given data, but the harder problem remains to detect the elements of matter and form within the integer-series itself. We must not be surprised if we find that the Platonic theory, so far as we can discern its character, was less successful in dealing with this than with the easier problem of the rationalising of the irrational.

II. The present essay will not attempt to deal with the whole intricate problem of the Platonic theory of the integers and Aristotle's criticisms upon it. It would be absurd to dispose in a few pages of a subject which has cost M. Robin several hundred large pages for the mere full display of the materials for a conclusion. I can, at best, offer a vindemiatio prima effected by the intellectus sibi permissus. It will be enough to discover, if we can, what the theory must have been in its broad outlines to give rise to the kind of strictures it provokes from Aristotle. We shall gain something if we can see that these strictures are not mere explosions of willful petulance, and yet that the doctrines which provoked them represent a real advance in mathematical thought, in spite of features which rightly rouse grave misgiving. I believe it will appear from our discussion that the number-theory of the Academy is in much the same position as the Calculus before the purification of it from bad logic in the days of our fathers. The main ideas are sound and fertile, but the formulation given to them involves illogicalities. The remedy for such a situation is neither, with Aristotle, to dismiss the theory on the ground of these illogicalities, nor, like rash admirers of Plato, to pretend that the illogicalities are not there. It is to reconstruct the theory with a more exact and subtle logic, and so to show that its apparent paradoxes are incidental excrescences; the only way to get bad logic out of a philosophical theory is to have more logic, not less.

I start from the assumption, justified by the whole tenour of the Aristotelian criticism, that the doctrine of the One and the "great-and-small," apparently derived from the study of irrationals, was at once extended to |122| the integers themselves. It is always in this form, as a theory about integers, that Aristotle deals with it. We can readily see that the thought of the "real numbers" as limits of series of rationals, once introduced, would not be confined to irrational real numbers. For the limit to which an infinite series of rationals converges need not be an irrational. Thus, for example, the sum of n terms of the series 1 + 1/2 + 1/4 + 1/8 … converges to the value 2, that of n terms of the series 1 + 1/3 + 1/9 + 1/27 … to the value 3/2, and in general, obviously the "sum to infinity" of a decreasing geometrical progression with rational terms is rational. This would be apparent at once from consideration of the process of unending bisection to which Zeno had directed attention.

      | | | |
      C D E F

We see at once from a diagram that as the process continues, the successive points of bisection, C, D, E, F …, approach nearer and nearer to the terminal B, and that, since bisection never becomes impossible, we can make the interval between a point of bisection and B less than any assigned interval σ, however small σ may be, by merely continuing the bisection long enough. In other words the sum 1 + 1/2 + 1/4 + 1/8 … converges to the value 0 as the number of terms summed increases "indefinitely." It is on this consideration that Eudoxus based the principle of his method of "exhaustions," that if from the greater of two unequal magnitudes there be taken more than its half, from the remainder more than its half, and the same process be continued, a finite number of subtractions will leave us with a remainder less than the smaller of the two given magnitudes (Euclides X. i). The rule implies the knowledge that such sums as 1 – 1/3 – 1/9 – 1/27 …, 1 – 1/4 – 1/16 – 1/64 … converge to |123| limiting values which are rational and not = 0 (in the first case to 1/2 the second to 2/3.

In these particular cases, indeed, we do not see the μέγα καὶ μικρόν, since the sum of r + 1 terms of such a series is always less than the sum of r terms. But it is easy to give an example which is a μέγα καὶ μικρόν. Thus, take the sum 1 – 1/2 + 1/4 – 1/8 …. We see at once that it can be written as (1 + 1/4 + 1/16 …) – 1/2 (1 + 1/4 + 1/16 …), and that it is therefore 1/2 (1 + 1/4 + 1/16 …), and so converges to the value 4/3 x 1/2 = 2/3." These considerations prove, as I have said, that if a μέγα καὶ μικρόν is a "real number," the "real numbers" cannot all be "irrationals," and that the duty of a sound arithmetical theory is to provide for the derivation of the whole series, including its rational members, on a single principle.

This involves a consequence of the first importance, which has only been definitely realised in very modern times. The "real number" 2 cannot be the same entity as the integer 2, for the reason that the integers are presupposed as a series from which the "real numbers," including the "real number" 2, have to be derived. Hence no "real number" is an integer and no integer a "real number," though the distinction is concealed by our convenient and economical habit of making our numerical symbols do double duty. The reality of the distinction is seen as soon as we attempt to use the method we have been describing to furnish the necessary definitions of "real numbers." If I am going to introduce the sign √2, for example, I must define it, and if I define it by saying that it stands by definition for the value to which the "side-and-diagonal" series 1, 3/2, 7/5, 17/12 … converges, the very statement presupposes the integers, 1, 2, 3, 5, 7, 12, 17, as already given and known. Since the process of forming "convergents" thus presupposes the integers as the "matter" out of |124| which the convergents are formed by a proper selection, it manifestly cannot be employed to "derive" the series of the integers themselves. Yet, as the history of arithmetic proves, it was inevitable that there should be an initial confusion between the integers and the corresponding "real numbers," and that the necessity of making a rigid distinction should only be discovered by later reflection on the consequences of the confusion. This makes it intelligible why the theory of the derivation of number from the determination of the "great-and-small" should have been extended to the integers themselves. At the same time, it prepares us for the discovery that the simple transference of the notions of the "One" and the "great-and-small," in the form in which they are serviceable in the theory of the irrationals, to the logical derivation of the integers themselves will prove logically unsatisfactory. We are thus in a position to do what many students have found impossible, to be just at once to the Platonic theory as containing the germs of a true theory of whole number, and to the objections of Aristotle. We are neither called on to regard the Platonic theory as a "senile aberration" (the real meaning of the German euphemisms about Zahlenmystik), nor, in our admiration for Plato, to treat Aristotle as a mere carper or a pure fool, as I fear even C. Ritter, who has done more than any living German writer for the understanding of Plato, tends to do. It should really be equally incredible that Plato meant anything but a piece of rational mathematics by his theory, and that Aristotle was a mere envier of his master's reputation or a common ass. We shall not look in Aristotle for a mathematician's appreciation of mathematical new conceptions, but we shall look for, and we shall find, intelligible protests against something which, on the face of it, appears to conflict with |125| the logic of common sense. We shall be prepared to admit that the criticisms may so far hit the mark as to show that the Platonic new idea can only be "saved" by a recasting which removes some very fundamental Aristotelian objections. Aristotle, no doubt, was one of those numerous persons who cannot follow a piece of mathematical reasoning until it is turned into the language of "common sense," but we shall do well to remember the saying of W. K. Clifford that "algebra which cannot be translated into good English and sound common sense is bad algebra."6

The particular application I would make of this observation is this. It is quite clear that if the proposed conception of a number as a determination of the ἄπειρον of the "great-and-small" originated, as we have tried to argue, from consideration of the quadratic and cubic "irrationals," it was extended to cover the case of the integers themselves. It is on the application to the integers that Aristotle's criticisms directly bear. If the "numbers" in question had all been "irrationals," which Aristotle himself followed the older tradition in regarding as μεγέθη, or ποσὰ συνεχῆ, in contradistinction to ἀριθμοί or ποσὰ διωρισμένα, he would have been obliged to make the criticism, which he never does make, that the so-called ἀριθμοί of which the Academic analysis holds good are not ἀριθμοί at all, but μεγέθη, and that the definition thus confuses the two εἴδη of τὸποσόν, which are in fact ἀντιδιῃρημένα. Again, his apparent standing confusion of the ἀόριστος δυάς of the great and the small with the αὐτὸ ὅ ἐστι δυάς, the integer 2, would be so gross as to be wholly incomprehensible unless the integer 2 actually played the part of a factor in the derivation of "number," as it can only do if the integers are thought of themselves as having a δυάς as one of their factors. If |126| there is a logical confusion of two wholly distinct concepts in Aristotle's strictures, the only reasonable explanation is, though I fear it is one to the neglect of which in the past I must myself plead guilty, that the confusion had been made before him by the Academic champions of the theory themselves, and that his acquiescence in it therefore does not affect the value of his criticism as an argumentum ad homines. Thus when he tells us (Metaphysics M, 1082 a 12) that "the indeterminate dyad, as they say, lays hold of the determinate dyad and so makes two dyads," this ascription to the ἀόριστος δυάς of "doubling" — whether by multiplication or division — what it "lays hold of," where the confusion of it with the integer 2 is manifest, must be an actual piece of the Academic theory, as the words ὥς φασί imply. (And, in the absence of any evidence, we are not entitled to maintain that the confusion is not Plato's own but merely that of inferior disciples.)

So the type of criticism which is exemplified by the argument of Metaphysics M, 1084 a 23, that if 4 is the form of horse and 2 that of man, "man will be a part of horse," bad as it is, at least implies that according to the Platonic theory the integers 2 and 4 are among the "numbers" of which the "One" and the μέγα καὶ μικρόν are the factors, just as the complaint of the next lines (1084 a 25-27) implies that the Form-numbers par excellence are the integers from 1 to 10. It is still more important to realise the implication of the passage Metaphysics M, 1081 b 12-22, where Aristotle's point is that in a sound theory the integers must be logically derived in their "natural" order, 1, 2, 3, … but "if this is so numbers cannot be derived, as they derive them, from the dyad and the one. For 2 (the dyad) is a part of 3 and 3 of 4, and so on throughout the series. But what was derived from the 'first dyad' (the integer 2) and the indeterminate dyad was |127| the integer 4." The whole point of the criticism is that in the Academic theory, 4 can be defined without defining 3, i.e. the Academic deduction of the integers does not give them in their "natural" order. Aristotle's own theory that each number of the series is made from the preceding by adding 1 to it is a piece of very bad logic, but this does not affect the perfect soundness of his contention that a correct theory ought to define the integer (n + 1) in terms of n, and so yield the "natural" order, a point which is fully recognised by Russell when he calls the finite integers those which can be obtained by mathematical induction, and by Frege when he defines the whole series as "the successors of 0." The criticism would be meaningless if the Academy had not defined the numbers up to 10 in an order other than the "natural." This consideration of itself would be fatal to the diagram of the derivation given on p. 31 of Dr. Stenzel's penetrating study of the Platonic doctrine,7 which does succeed in getting the successive integers from 1 to 10 in the familiar order, and is thus shown to be un-Academic. Dr. Stenzel's method is, in fact, to start with 1, and to derive from it the two next integers 2 and 3. He then derives two more integers from each of the two thus obtained and regards the process as one to be continued indefinitely. We thus get the diagram {shown} and so on.
The obvious criticisms on this procedure are (a) that it disregards the plain indications of Aristotle that the numbers from 1 to 10, the numbers of the "decad," held |128| a privileged position in the Academic theory, and (b) that it is not clear by what sort of derivation the author supposes himself to be getting his new integers from an old one at each step. He apparently thinks that, e.g., the application of the "indeterminate dyad" to 1 produces 2 and 3, its application to 2, 4 and 5, and so on. But this is barely consistent with the express statement (Metaphysics M, 1081 a 14) that the function of the ἀόριστος δυάς is to be δυοποιός of whatever it "gets hold of." You cannot make 3 from 1 or 5 from 2 by any process of doubling or halving.8 Presumably Dr. Stenzel's thought is that, since his method somehow always derives from any integer n the two integers 2n, 2n + 1, the transition from n to 2n is always the work of the dyad which is δυοποιός. Exactly how the transition from n to 2n + 1 is to be described is not clear. I suppose it illustrates the function of the other constituent of number, the "One." But if the Platonic method of generating the odd numbers had been Dr. Stenzel's, it would have been a criticism Aristotle could hardly have missed that half the integers, on the theory itself, are got by the "addition" of a "unit." I find this irreconcilable with Aristotle's insistence on the "addition of 1 to 1" as a point which, it is implied, was not recognised in the Platonic theory, and also with his statement that the function of "the One" was supposed to be that of "equalising" something, a statement for which we have already seen the evidence. I am thus driven to conclude that, though all Dr. Stenzel says in his book about the importance of the definition of an ἄτομον εἴδος by the process of logical division is of the highest value, he has gone astray in supposing that we can directly discover |129| the Platonic theory of the logical structure of the integer-series from the teaching of the Sophistes and Philebus about διαίρεσις κατ ͗εἴδη. (Perhaps he has forgotten also that the Philebus is clear on the point that διαίρεσις does not proceed necessarily by dichotomy. We divide a "kind" into the fewest sub-kinds necessary at each step, but not necessarily, as the Sophistes had done, always into two and no more, Philebus 16 d.)

What is meant by "equalising" I think we can see best from the use of the same word in connection with the doctrine of "directive justice" in the fifth book of the Nicomachean Ethics. It will be remembered that the main thought there is that the commission of a wrong creates an inequality between two parties who were before, and ought to be, equal. A has now an advantage and B a disadvantage, and the business of a dicastery, in assessing its award, is to transfer to B's side of the account with A just so much as will cancel the inequality and leave the parties in the state of "equality" which existed before A's unlawful act. This is done by striking a kind of "arithmetical mean."9 So much is clear, whatever view we take on the special points about which interpreters have diverged from one another. There is another equality besides absolute or arithmetical equality, that which Aristotle calls "equality of proportion," and Plato in the Gorgias, "geometrical equality," but when "equality" is spoken of without qualification it means absolute equality, and when "equalising" is spoken of in the same unqualified way, what is meant is the taking of an arithmetical mean. The arithmetical mean between b and a, (a + b)/ 2 (where I assume b > a), is said to "equalise" this because the difference b – (a + b)/2 is identical with the difference (a + b)/2 – a. |130|

If we apply this to the case of numbers, we see at once that it implies that an odd number 2n + 1 is to be looked on not as the result of adding 1 to 2n, but as the result of taking the arithmetical mean between the two even numbers 2n and 2n + 2. These two numbers, being both even, are themselves the product of the activity of the "indeterminate dyad" which "doubles" something. The "equalisation" of a great and small is effected by the arithmetical mean, 2n + 1, because the mean "falls short" of the greater, 2n + 2 by the same interval by which it "exceeds" the less, 2n. It follows, I submit, that we are to think of the integers from 1 to 10 as "generated" in the following manner:

(1) The ἀόριστος δυάς, by doubling whatever it "lays hold of," originates the "power-series" 1, 2, 4, 8, as is clearly taught by what is said of the "double" in Epinomis, 990 e and Metaphysics M, 1081 a 2 x, b 20.10

(2) Next we "equalise" the "great and small" by taking the arithmetical mean between the two adjacent even numbers 2 and 4, viz. 3.

(3) The double of 3 is next produced by the action of the "dyad," and we get 6.

(4) We now get the "mean" between 4 and 6 and 6 and 8; this gives us 5, 7.

(5) The "dyad" lays hold of 5, and this gives us 10.

(6) Finally we obtain the "mean" between 8 and 10, 9.

The resulting order is thus 1, 2, 4, 8, 3, 6, 5, 7, 10, 9, and we note that the odd integers, apart from 1, are all got by what is really division by 2, 2n + 1 being |131| regarded as 1/2(4n + 2); the even integers are got, apart from 2 itself, which will be considered later, by multiplication by 2. Thus the series once more illustrates the phrase of Epinomis 990 e ἀεὶ περὶ τὸ διπλάαιον στρεφομένης τῆς δυνάμεως καὶ τῆς ἐξ ἐναντίας ταύτῃ. We must add to what we have already said about this phrase that the whole integer-series is produced by doublings and their inverses, halvings.

There is just a little uncertainty about the precise arrangement of the five last terms of the series. We might get 5 before 6, if we supposed that step (2) covers the insertion of a "mean" between 2 and 8, as well as a "mean" between 2 and 4. And 6 itself might again be got, not as 3 x 2 but as the "mean" between 4 and 8. Or again, we might try to get the odd numbers 3, 5, 9 immediately after the series 1, 2, 4, 8 by "limiting" the advance of the "doubling dyad." The third step would then be the doubling of 3 and 5, and the fourth and last the "limiting" of the "advance" of the dyad from 6. Thus we should get an order given by Robin (Théorie Platonicienne, p. 282), 1, 2, 4, 8, 3, 5, 9, 6, 10, 7. But the evidence seems to me to indicate that only odd numbers are derived by the process called τὸ ἰσάζειν, and against the last suggestion we have the double consideration that it destroys the symmetry of a regular alternation of multiplication and "equalisation," and also that it obtains the odd numbers by a method which Aristotle would have described as the addition of 1 to 1. If half the integer-series had actually been derived by the Platonists in this way, it seems to me that Aristotle could not have failed to make the telling objection that it is illogical to scruple at getting all the integers by the method you yourself employ for getting half of them. Yet he never makes any such point, but assumes rather that it is characteristic of the Academy to make no use |133| whatever of "addition of 1 to 1." And, in fact, on the view adopted here, this is the case. Unless we make an exception for 2, no term of the series has been simply derived from that which immediately precedes it in the "natural" order. Hence I believe that the order we have given is the one really intended. It is the same as one given by Robin (op. cit. 447-9), and I think the method of obtaining it described there is identical with that given above, though M. Robin does not notice the use of ἰσάζειν in the Nicomachean Ethics, and does not explicitly explain that one of the two operations involved is the taking of the "mean." Hence I did not myself see the superiority of this arrangement over that proposed op. cit. 282, until I happened to reflect on the use of ἰσάζειν in the Ethics.

If the reconstruction thus attempted is sound in principle, it should show us that there is real point in Aristotle's criticism. Several of his contentions must in fairness be allowed to be valid. (1) Thus though he is manifestly in error in supposing that the higher integers contain the lower as "parts" of themselves, and more generally in thinking that an integer n which is > 1 is a collection of n "units," he is wholly right in objecting to any number-theory which generates the integers in any but their "natural" order. To say nothing of the practical inconvenience of having to prove separately the endless "inequalities" which we should require to establish if we started with such a series as Plato's method would give us, it is a logical fault in the theory that it requires two distinct principles of method for the generation of the integers, the operation x 2 for the even, and the operation /2 for the odd. Both, no doubt, may be said to be operations with the "double," but they are different operations, one the inverse of the other, and operations, like entia, are not to be multiplied praeter necessitatem.

|133] And there really is a single formative principle which governs the logical structure of the whole series of finite integers; they have a definite type of order, different from any other, which it is important to recognise. It is only when they are arranged in the "natural" order that we see at once what this type is, that of a "well-ordered" infinite series, i.e. one in which there is a first term and in which every term has one and only one next succeeding term. That you could not discover this in an integer-series constructed by what seems to have been Plato's method is obvious. The order of the first five integers would, indeed, be unequivocally fixed as 1, 2, 4, 8, 3. But we might doubt whether the immediate successor of 3 should be got by "equalising" 2 and 8, so as to yield 5, or by applying the "dyad" to 3 so as to yield 6. It is not clear what the "integer next after 3" on this scheme is. Mathematicians recognise the soundness of Aristotle's point when they define the finite integers as those which can be reached by "mathematical induction," or "induction from n to n + 1." Aristotle's error in supposing that because a collection of n + 1 things contains n things and 1 thing over, the integer n + 1 contains the integer n and a 1 over does not affect the real force of his criticism. He is in precisely the same position as Berkeley and others who condemned the Calculus for employing the notion of "vanishing" magnitudes. They were quite right in saying that the theory of the Calculus, as formulated by its exponents, introduced vanishing magnitudes which are treated as somethings which are turning into nothings, and that to talk of such nothing-somethings is to talk nonsense. But the criticism really hit not the Calculus itself but only the inaccurate analysis its exponents had given of their own method. So Aristotle's criticism affects not the thought the Academics were trying to |134| express, but the faulty expression they gave to it. His appeal to the process of counting as indicating the lines on which a correct logical theory of whole numbers should proceed (Metaphysics M, 1080 a 30) is perfectly sound; unfortunately, it destroys his own doctrine of the generation of the integer-series by successive additions of 1, since we do not count, "one, one, one, one …," but "one, two, three …," as he has to admit in so many words.

(2) Again, it seems to me probable that Aristotle is not himself to be made responsible for the curious confusion of the number 2 with the ἀόριστος δυάς which runs through so much of his argumentation. When we compare the statements that 4 is produced "as they say," from the "first dyad and the indeterminate dyad" (Metaphysics M, 1081 b 21), that the indeterminate dyad, "as they say," laying hold of the determinate dyad makes two dyads, for its character by definition is to double what it lays hold of (τοῦ γὰρ ληφθέντος ᾖν δυοποιός, Metaphysics M, 1082 a 12-14), that ἡ ἀόριστος δυὰς δυοποιός ᾖν (Metaphysics M, 1083 b 35), it seems impossible to doubt that the Platonists, and presumably Plato himself, definitely regarded multiplication by 2 as the work of the ἀόριστος δυάς. And here there clearly is a bad mistake in logic. There is nothing "indeterminate" either about the process or about the result of "doubling." It is true, indeed, that multiplication in arithmetic is strictly, like addition, an operation on collections, not on the cardinal numbers of the collections. But the operation is one which has a perfectly determinate rule, and when both the collections involved are finite, as in the case of the "multiplication of 2 by 2," there is no ἀπειρία anywhere in data, process, or results. The data are collections of classes of two members each, the process is that of forming the "logical product" by combining each member of each of the two classes with |135| each of the other, and the logical product is a class with the determinate number 4. The combinations of couples which can be constructed out of the members of the couples (a, b) and (c, d) are (ac), (ad), (bc), (bd), and no others. Thus it should be the "first" or "determinate" dyad which is both multiplier and multiplied, and Aristotle is justified in making difficulties about the appearance of an "indeterminate dyad" in the character of multiplier. 2 is not an "indeterminate coefficient."

(3) And this leads to a further problem. What about the "first dyad," the integer 2, itself? If 4 results from the operation of the "indeterminate dyad" on 2, does 2 result from its operation on 1? Or must we regard 2, the "prime dyad," as an original datum for the theory, by the side of 1 and the ἀόριστος δυὰς? From the very passages of Metaphysics M which have just been cited above, when we remember that there are no similar allusions to a "laying hold of" 1 by the "dyad," it seems to me clear that the "doubling" of 2 was supposed to be the first example in the integer series of the working of the ἀόριστος δυὰς. And the same thing, as I still think, is the only possible explanation of the perplexing remark of A, 987 b 33, that Plato made one of the constituents of his numbers a "dyad" because numbers are so easily generated from it, as from a matrix — ἔξω τῶν πρώτων. The πρῶτοι, which form the exception to the statement, are, I think, certainly 1, the "unit," and 2, the "first dyad." We have seen already that all the others are very easily "generated" by combining the two operations of multiplying by 2 and finding an arithmetical mean. If the first operation of "doubling" you will admit is the multiplication of 2 by 2, of course 2 itself, like 1, will not be one of those so generated. And since, as Aristotle implies, the Platonic theory forbids you to generate 2 by "adding 1 to 1," obviously you cannot "generate it" by a |136| multiplication of 1 by 2, thus presupposing 2 as one of the factors which generate itself. You must assume that 2 is there as something "given," no less than 1. And this leaves you with three underived data instead of two, 1, the "indeterminate dyad," 2. These difficulties are really inseparable from the attempt to make "doubling" the function of the "indeterminate dyad," and are therefore justly chargeable not on Aristotle, to whose memory I would hereby make an amende honorable for former utterances, but on the πρῶτος εἰπών, Plato himself.

The real source of the trouble is a curious one. It is that 1 is a term of the series of "powers" of every number whatever, that, as we express it, x0 = 1, whatever number x may be, or, differently expressed, that logx 1 = 0, for all values of x. The mere statement of the point makes it clear that the first step to a theory which would avoid both the illogicality of confusing 2 with a couple of 1's, and the necessity of assuming 2 as well as 1 as a datum for the construction of the integer series, could not be taken until the notion of an arithmetical zero was clearly formed, that is, until it was understood that there is an "integer 0," and that 0 is the first of the whole numbers. But, though the Academic conception of the line as the "fluxion" of a point, like the Platonic name for the "point," ἀρχὴ γραμμῆς, shows that Plato and his followers had definitely conceived of a zero magnitude which is the geometrical analogue of the "number 0," the very fact that "the One" was treated as an underived constituent of number, this "One" being, as Aristotle's criticisms plainly show, regarded as identical with the number 1, proves that they had not thought of replacing 1 as the first term of the integer-series by 0. Hence the integer 2 creates a great difficulty for their number-theory. Once you have got 2, as we have seen, you can easily go on to "derive" all the other even |137| integers. But how are you to get your original 2? There seems nothing for it but to presuppose that it is given as a datum needing no derivation. And yet, even if one winks at the ascription of "doubling" to the wrong "dyad," the "indeterminate," one's logical sense must be shocked by having to hold that one is not to regard 1 x 2 = 2 on the same footing with 2 x 2 = 4. If one does assimilate the two propositions, one would be led straight to the identification of the 2 by which 1 is multiplied in the first equation with the "indeterminate" dyad which ἦν δυοποιός. The ambiguity about this "dyad," which seems at once to be and not to be the integer 2, is thus not due to a blunder on Aristotle's part; it is inherent in the Platonic account of number itself.

Yet the thought at the bottom of the theory is far too valuable to be surrendered. We cannot fall back, like Aristotle, on the view that an integer n is a "sum" of n 1's, since that is manifestly nonsense. No integer contains the integer 1 or any other; it is strictly true that integers are not συμβλητά, and that our language about adding them to one another is a convenient and inaccurate abbreviation. When we say that 7 + 5 = 12, what we really mean is that if we have a collection a with 7 members and another collection b with 5 members, and proceed to form the "additive class," "things which are either a or b," the cardinal number of this new class is 12. If there are 4 greater prophets and 12 minor prophets, there will be 16 persons in the collection of persons who are prophets, major or minor. But there is no number 12 or number 4 in the number 16. It is the two sets of prophets, not their cardinal numbers, which are "addible."

The importance of the Platonic theory, in spite of the easily recognised logical difficulty about the two |138| "dyads," is perhaps best brought out by stating it in a way which throws its apparent paradoxicality into the strongest relief. It refuses to employ addition as a means of generating numbers, but employs multiplication and its inverse operation, division. This shocks common sense, which regularly looks on multiplication as an equivalent to repeated addition, regarding x x y as an abbreviation for x + x + x to y terms. Yet the view implied in the Platonic theory is a sound one and that of common sense unsound, though we should never discover the fact in our practical operations with numbers. So long as we are concerned only with finite integers, it is true that multiplication may be treated as no more than a compendious substitute for repeated additions. But when we look into the logical character of the two operations, we shall see that it is not the same; logically multiplication is an independent operation with a principle of its own. In addition, as has been already explained, what we really do is to form a new collection which has for its members those and those only which were members of one or the other of two original collections. The fundamental logical process is that of forming the notion of the collection "a or b" from the notions of the collections a and b. In multiplication, on the other hand, when we multiply the collection a by the collection b, the underlying notion is that of the "multiplicative class," and this is generated on a totally different principle. We combine each member of a with each member of b to form a couple; the number of couples thus obtained is the number of the "product" of a by b, each couple being one of the members of the "multiplicative class."11 Thus 5 x 3 = 15. means that if |139| we have five things, a, b, c, d, e, and three things a, β, γ, 15 is the number of distinct couples in which one member is taken from the group a, b, c, d, e, and the other from the group a, β, γ. The logical process underlying addition is disjunctive, that which underlies multiplication is combinative. The extraordinary method by which the Platonists appear to have constructed the δεκάς arises from an over-hasty application of a truth which is, in itself, of the first importance.

To return to the construction of the integer-series itself. It is manifest that the curious process by which the successive integers are reached in an order which is not the "natural" one cannot stand against the criticism of common sense. The fatal objection to any such construction is that the "natural" series of the integers is a particularly important type of order and that no other arrangement of them exhibits the type in the same transparent way. Thus, suppose we have already completed the δεκάς and wish to continue the series. We shall first have to apply the "dyad" to 6, the lowest integer to which it has not already been applied, and this will give us 12. Then by "equalising" 10 and 12 we get 11. But the procedure obviously implies that we already know that there is one and just one empty space to be filled between 10 and 12, that there is a "number next after 10" and that it is also the number "next before 12." And this, in turn, implies that the integers in their "natural" order are already known before the process of derivation by means of the "dyad" begins. We cannot condemn "common sense," speaking through Aristotle, for finding such a construction incoherent. Indeed, if the contention of our former essay was |140| correct, and the method really originated from an attempt to determine the value of "irrational real numbers" in terms of the integers, the transference of it to the derivation of the integer-series itself was morally bound to lead to incoherence. If a "one" and a "dyad" are to be discovered as the constituents of the integers themselves, they cannot be identical with the "one" and the "dyad" which figure in the derivation of the algebraic "surds"; at best, we can only expect the analysis of the formation of integers to reveal constituents which function in an analogous manner. The confusion of the analogous with the identical has already met us as characteristic of Greek number-theories in the form of the identification of the real numbers 2, 3, … with the corresponding integers. It is only what we should expect that the same logical confusion should encounter us again in the ascription of the function of "doubling" to the "indeterminate dyad" instead of the "first dyad," which has led to so much of the Aristotelian criticism. On the other hand, we cannot regard Aristotle's justified objections to the theory as any justification of his own view that integers are generated by "addition of 1 to 1." If we neglect the confusion between integers and the collections of which they are the numbers already spoken of, there is still a simple difficulty which is enough to annihilate this "common-sense" explanation. If I say "how many are 1 and 1 and 1 and 1?" I am asking a question which I cannot answer unless I am already acquainted with the series 1, 2, 3, 4. To count, I have to know that when, e.g. in this illustration, a stop is made in the repetition of the words "and 1," the fourth "1" has been reached, and no other. Once more, the integer-series is tacitly assumed as already known in the very theory offered to describe its generation. One might have seen this from the simple reflection that we |141| do not learn to count as children by saying "one, one, one" …, but by saying "one, two, three …." I can only learn by counting on my fingers that "1 and 1 and 1 are 3," if I have already learned to count the fingers as "one, two, three …." Hence the first task of an arithmetical kind for a young child is to get the names "one, two, three …" in their right order stamped on its memory, and it is very amusing to listen to children as they acquire certainty about this order by repetition, and to observe how hard they find it not to miss out some of the terms or to invert their order.

If we analyse the actual method by which we can successively define each of the integers in terms of the preceding, we shall see that, in fact, it may be said to be dependent throughout on the notions of a "one" and an "indeterminate" dyad, or great-and-small. What functions as the "one" is the notion expressed in language by the indefinite article "a" or "an," and what functions as the "dyad" is the notion expressed by "any." These are not, of course, the same things as integers of any kind; in particular "a" does not mean "one"; it is a notion already possessed by the youngest child who can point to a thing and say "ball" or "dog," long before it has even begun to learn to count. Given these elementary notions we can proceed to derive the integers in order in the way regularly followed by modern philosophical writers. Roughly the procedure is as follows.

We begin by defining the first member of the series, 0. To do this we need, besides the notion of "any," the general logical notions of true and false and of a propositional function and its "arguments."12 By a |142| propositional function, for our purposes, we mean whatever can be significantly asserted about anything; we denote it by such symbols as φ, ψ, χ, written before a pair of brackets. By an argument to a function we mean anything of which the assertion in question can be made so as to yield a significant statement. Such arguments may be denoted by the symbols, a, b, c …, and when we wish to consider the function alone, leaving it an open question whether any argument can be supplied which converts it into a true statement, we may indicate this by using the symbols, x, y, w …. Thus, e.g., φ(x) may mean "is bald," ψ(y), "is infallible," etc. Then if a denotes Julius Caesar and b the Pope, φ(a) will be the proposition "Julius Caesar is bald," ψ(b) the proposition "the Pope is infallible." So much premised, we might define 0 by saying that if every proposition φ(a), φ(b) … is false, the number of the "values of x which satisfy φ(x)" is 0, by definition.

We may then go on immediately to define 1, in terms of the conceptions used in our definition of 0, thus: 1 is the number of "values" which satisfy φ(x) if (i) some proposition of the form φ(a) is true, and (2) "φ(b) is true" implies that b is identical with a. If this is the definition of 1, we can proceed at once to define 2 thus: 2 is the number of values of x which satisfy φ(x), provided that (1) some proposition of the form φ(a) is true, (2) some proposition of the form φ(b), where b is not identical with a, is true, (3) if is true, either c is identical with a or c is identical with b.

More generally, when once the integer n has been defined, we proceed to define its immediate successor n + 1 by the conditions that n + 1 is the number of values which satisfy φ(x) if (1) there are n distinct arguments a, bn, such that φ(a) is true, φ(b) is true… φ(n) is true; (2) is true and p is not identical with a, |143| not identical with b, … not identical with n; (3) if φqis true, either q is identical with a or is identical with b, ... or is identical with n, or is identical with p. Thus, what I mean by saying that there were three triumvirs is that the two propositions, "Octavian was a triumvir," "Antony was a triumvir" are true (Octavian not being the same person as Antony); that the proposition "Lepidus was a triumvir" is true, and that Lepidus was not Octavian nor yet Antony; that if there is any person of whom it is true that he was a triumvir, that person is identical with Octavian, or identical with Antony, or identical with Lepidus. E.g. it is true that Augustus was a triumvir, but that is because Augustus was the same person as Octavian; it is true that M. Aemilius was a triumvir, because M. Aemilius was Lepidus; or, to put the matter in a different way, n + 1 is the number of members of a collection if it includes a collection with n members, if it has also a member which is not a member of this collection of n members, and if whatever is a member of it is identical either with some member of the collection of n or with the member, not included in the collection of n, mentioned in the last clause.

All this is wholly trite and familiar and I only repeat it here for the sake of making the following points clear. (1) the notion "a" or "an" is not the same as the notion "1," as is shown by the fact that we have to use it as part of the definition of 1. To put it quite popularly, "There is one reigning King of England" means "there is a reigning King of England — George V — and there is no other." (2) The notion of "any" is not the same as the notion of "all." It means "any you may please to take," "any which turns up," not "each and all." Hence we can employ it in cases where it might be doubtful whether we could speak of "all," a definite |144| totality. In speaking, e.g., of "any integer" we do not commit ourselves to the view that there are propositions which can be significantly made about "all integers" or "every integer." We can speak significantly of "any region of space" without by implication denying Prof. Alexander's view that space is not a "whole." This is why there is no logical fallacy involved in the use of the notion "an" as a datum presupposed in the definition of the integer 1; it is also why the notion of "an" by itself, without that of undefined plurality, would not be sufficient for the definition of 1, and why, for example, the belief that "there is a God" is not the same as the belief that God is one.

If the Platonic line of thought was really that which has been indicated, we can see that though, in a sense, it is true that the "material" of the integers is a "great-and-small" or "indefinite duality," in fact, the concept of indefinite multitude, the actual working-out of the thought has committed the double confusion of this duality or dyad with the integer 2, and of the antithetic notion of "an" with the integer 1. Were there no numbers but the integers to take into account, the older Pythagorean analysis which made the constituents of number limit and the unlimited and treated the "unit" or integer 1 as itself the simplest combination of these constituents might fairly be held to be more satisfactory than the amended version of it which presupposes 1, and apparently 2, as initial data calling for no derivation, and so enables Aristotle's criticism to get home. This seems to me good ground for supposing that the necessity of the modification was due to the fact that Plato and his friends started from the consideration of something other than the integers, and that their theory of the derivation of the integers was a further development motived by the desire to transfer to number as a |145| whole a theory originally adapted to the special case of the irrationals. So long as the "real and rational" numbers are not distinguished from the integers, it is inevitable that, e.g., the integers 2, 3 and the irrational real numbers √2, √3 should be regarded as terms of one and the same series subject to one and the same law of derivation. The confusion which seems to play havoc with the theory is one which could not have arisen if it were the case that all the values to which infinite series converge were irrational. The integers and the "real numbers" would then have stood out as two distinct series and there would have been no temptation to bring both under a single formula. And here, again, we may do a tardy justice to the stalwarts of the older tradition who, like Aristotle, insisted that though there may be irrational geometrical magnitudes (μεγέθη), there are no irrational ἀριθμοί. If their distinction on this point between geometry and arithmetic had been obstinately sustained by the mathematicians, the development of modern mathematics would have been impossible. The one-to-one correspondence of points in a plane with couples of real numbers which is the very foundation of Descartes' analytical geometry would have been ruled out. Yet, at bottom, there is a real justification for their protest against the "new" arithmetic of the Academy, though they do not express the justification in the right way. What they are trying to say is something which is quite true, that the integers and the real numbers are two distinct series and that no term of the one is a term of the other. The important principle embodied in the insistence on this radical distinction between ἀριθμοί and μεγέθη is that the real numbers cannot be obtained from a set of logical data by the method which yields us the series of integers, the "inference from n to n + 1," "mathematical induction." |146| This is both important and obvious. "Inference from n to n + 1" presupposes that n + 1 has a determinate meaning, i.e. that in a series derived by this method every member has one determinate "immediately next" member, as is the case with the integers. The integer-series is the standing type of a "well-ordered" infinite series, satisfying the conditions that there is a first term of the whole series and that every term has one and only one term which is next after it. It might appear, at first sight, that the series of rational fractions does not conform to the type, since there is, e.g., no rational fraction which, in what we call the order of magnitude, comes next to a given fraction, But, as is well known, we may rearrange the rational fractions in such a way that the condition is satisfied. Thus if I begin the series thus, 0/1, 1/1, 0/2, 1/2, 2/1, 2/2, 0/3, 1/3, 2/3, 3/3, 3/3. 3/1, … it is manifest that I can continue the series endlessly and that by the law of its formation, every term has a definite next following term, and that no rational fraction will appear more than once, and none will be omitted. (The same thing would be true, if for brevity's sake one suppressed all the fractions with 0 for their numerator after 0/1 and reduced all others to their "lowest terms".) Thus the series of rational fractions can be made to correspond one-to-one to that of the integers, and so has the same "ordinal number." But the case is altered when we come to the real numbers. It is true that the "algebraical" real numbers, those which are the roots of equations with integral indices and rational coefficients, can be arranged as a well-ordered series, and that the real numbers known to the Academy, the quadratic and cubic surds, are only an infinitesimally small selection out of this multitude (Couturat, De l' Infini mathématique, 622 ff.). But when we take into account the whole "real number" series, including the "transcendent" |147| as well as the "algebraical" real numbers, the correspondence with the integer-series breaks down. The continuum of real numbers appears definitely not to be a well-ordered series. Attempts have been made, including one by the late Philip Jourdain in Mind, to show that it must be possible to arrange it as one, but to the layman, like myself, these proofs have a suspicious appearance of assuming the conclusion, under a thin disguise, as a premise for its own demonstration, and they do not seem to have convinced the mathematicians. And it is suggestive that their authors never seem to have ventured to put their view to the test by producing a sample segment of the continuum of real numbers rearranged by their own methods. We may probably take it as certain that the continuum of all the real numbers is a series of a type different from that of the integers or any series whose terms can be made to correspond one-to-one with the integers. It is, at any rate, certain that we cannot logically derive the series from that of the integers by the methods by which we can develop the integer-series itself or derive from it the series of rationals or that of algebraical real numbers, which have the same type as the integer-series itself. Neither Aristotle nor the Platonists he criticises could have known this, but it is the real justification of his stubbornness in opposition to them, though it is amusing to recollect that, as we saw in our former study, the only "irrationals" with which Aristotle was acquainted, the quadratic and cubic surds, can be derived from the integers by the methods to which he objects and that the whole series of algebraical real numbers has the same type of order as the integers. The strength of his position on this point only appears when we take into consideration the "transcendent" numbers of which he could have had no conception. (Of course I |148| am not forgetting the simple fact that a transcendent number can be expressed as the limit to which a series of rationals converges, as when we write π/4 = 1 – 1/3 + 1/5 – 1/7 ... ad infin. We could not calculate them, if this were not so. What I mean is that, on the one hand, such a series itself is not yielded by the method Plato appears to have contemplated, but presupposes acquaintance with the exponential and logarithmic functions, and, on the other, that the difference of order-type between the integers and the continuum of the real numbers is the vital difference embodied, though Aristotle could not know the fact, in his own contrast between ἀριθμοί and μεγέθη.)

To sum up, then, the main points I offer for consideration are these:

(1) The peculiar Platonic theory of the constitution of number probably originated in the conviction that the quadratic and cubic surds which figure in geometry, especially in the solid geometry created by Theaetetus, compel us to recognise "irrational numbers." As an historical point, we might conjecture that the prominence given to the topic of irrational μήκη in the Theaetetus may very possibly indicate that Theaetetus himself, in the Academy, influenced the development of Plato's convictions in this direction. (2) Probably the character of the theory was determined by the fact that it started from knowledge of a rule for finding the successive "convergents" to √2, regarded as a "continued fraction." The anticipation was that all quadratic and cubic "surds" could be treated by this method, and it is from the alternation of convergents which are alternately too small and too large that the notion of the "indeterminate duality" has arisen. |149|

(3) Owing to the fact that the values to which such infinite series converge need not be irrational, it was inevitable that the "rational" real numbers should be confused with the integers, and that the attempt should be made to extend the theory to the whole field of known numbers, the integers and the quadratic and cubic surds, regarded as a single series.

(4) But the extension to the integers involves a deformation of the theory which leads to confusion of "the One" and the "indeterminate dyad" with the integers 1 and 2, and the unfortunate assigning to this "dyad" of the function of "doubling," which should really have been connected with the "auto-dyad" or integer 2. It also involves deriving the integers in an arrangement other than their "natural" order, for two reasons. The powers of 2 have to precede the other integers of the decad, and further no odd number 2n + 1 can appear until 2n and 2n + 2 have already been obtained. A number 2n + 1 will always follow after 2n + 2.

(5) Hence Aristotle's criticisms, so far as they concern these features of the doctrine, are justified. There is also a certain justification, unknowable to Aristotle himself, for his insistence on the distinction between ἀριθμοί and μεγέθη.

(6) At the same time, the Platonic formulae, as applied to the integers themselves, correctly anticipate two important points of a true number theory, the necessity of a strict derivation of the integer-series from more fundamental logical notions and the logical independence of multiplication on addition.

(7) The identification of the forms (εἴδη) with numbers means that the "manifold" of nature is only accessible to scientific knowledge in so far as we can correlate its variety with definite numerical functions of "arguments."The "arguments" have then themselves |150| to be correlated with numerical functions of "arguments" of a higher degree. If this process could be carried through without remainder, the sensible world would be finally resolved into combinations of numbers, and so into the transparently intelligible. This would be the complete "rationalisation" of nature. The process cannot in." fact be completed, because nature is always a "becoming," always unfinished; in other words, because there is real contingency. But our business in science is always to carry the process one step further. We can never completely arithmetise nature, but it is our duty to continue steadily arithmetising her. "And still beyond the sea there is more sea"; but the mariner is never to arrest his vessel. The "surd" never quite "comes out," but we can carry the evaluation a "place" further, and we must. If we will not, we become "ageometretes."


1^ It may be noted in passing that the interest in finding the value of √2 is not prompted by purely "geometrical" considerations. Since one approximate value is 17/12, for 72 = 289 = 2 x 144 + 1 = 2 x 122 + 1, 17 is very nearly = 12 √2. If it were strictly true that √2 = 17/12, it would follow that 172 = 288, and therefore = 16 x 18. Thus, 16, 17, 18, would be a geometrical progression and we should have 18/16 or 9/8 = (17/16)2. This would enable us to divide the musical interval of the tone into two equal semi-tones. The irrationality of the "diagonal" is thus connected with a corresponding irrationality in music.

2^ See on the whole subject Zeuthen, Histoire des mathématiques dans l'antiquité et dans le moyen âge, 43-52, for the difficulties to be faced. The one such approximation to an irrational square root preserved in the literature before the time of Hero of Alexandria seems to be the value of √3 given by Archimedes in Dimensio Circuli, iii. He assumes that the value is intermediate between 265/153 and 1351/780 (op. cit. 49). Both fractions occur among the convergents to 1 + 1 / 1 + 1 / 2 + 1 / 1 + 1 / 2 + …, the expression of √3 as a "continued fraction," the former being the ninth and the latter the twelfth term of the series. It is not clear to me why the more accurate 989/571 was not taken as the value which errs by defect.

3^ Metaphysics M, 1081 a 24, where ὁ πρῶτος εἰπών definitely ascribes the expression to Plato.

4^ Heiberg, Euclides iii. App. pr. 27.

5^ Op. cit. p. 45.

6^ Common Sense of the Exact Sciences, p. 21.

7^ Zahl und Gestalt bei Platon und Aristoteles, (ed. i.), p. 31.

8^ One might add that Aristotle's language seems to make it plain that 2 was not supposed to be derived from 1 by "doubling," but was, like 1 itself, assumed as a datum. He clearly means to say that the first and simplest example of the "doubling" power of the "dyad" was supposed to be given by 2 x 2 = 4. See infra.

9^ Nicomachean Ethics 1132 b 10 ff.

10^ 1 is supposed to be given already as a datum needing no derivation. I have spoken, for brevity's sake, as though 2 is to be taken as the result of an application of the ἀόριστος δυάς to 1. This is the view which ought in consistency to be taken of 2, if the underived constituents of the integers are the "One" and the "indeterminate dyad." I shall give reasons later on for thinking that in fact the "dyad" was only used to obtain even numbers greater than 2, and that 2 itself was treated, though it should not have been, as a primitive datum of the theory.

11^Here and elsewhere in this essay, I am, of course, wholly unoriginal, merely repeating, in a rough and approximate way, what the reader will find put with full scientific precision in works like Whitehead's and Russell's Principia Mathematica, Russell's Principles of Mathematics, Frege's Grundgesetze der Arithmetik, and the like. To readers who know these works I apologise for dragging in what is so much more competently given there, on the ground that many persons who are interested in Plato seem too much frightened by the titles of such works as I have named to consult them for themselves.

12^>Here again I am, of course, simply borrowing as much as is necessary for my purpose of the terminology of Frege, Whitehead, and Russell, and am aiming at no more precision than is strictly necessary for my immediate purposes. I may give a general reference to the first chapter of the Introduction to Principia Mathematica, vol. i., and to Frege's essay Funktion und Begriff.

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This paper originally appeared in the journal Proceedings of the Aristotelian Society, XVIII., 1918.

We have seen in recent years a remarkable awakening of intelligent interest in the Neo-Platonist philosophy which our grandfathers and their fathers were content to deride without understanding. We have learned that the Neo-Platonists were neither magicians nor emotionalist schöne Seelen, but systematic philosophers addressing themselves to the philosopher's task of understanding the world in which he lives as seriously as Aristotle or Descartes or Kant. No one writing today on the history of Greek thought about God, man, and nature would be likely to mistake so great a metaphysician as Plotinus for an apologist for polytheism or a New England littérateur born out of due time. Still the rehabilitation of the Neo-Platonists has hardly so far led to an adequate appreciation of Proclus, by far the most important member of the school after its great founder, though an honourable step has been taken in this direction by Mr. Whittaker.

This neglect of Proclus is unfortunate in more ways than one. For the historian of thought his importance is hardly second to that of Plotinus himself. It is in the main from him that Christianity received the Neo-Platonic impress still distinguishable in orthodox theology under all the disguise of a formal Aristotelianism. It is true that before the date of Proclus Christianity had been deeply influenced by Neo-Platonic ideas |152| derived from Plotinus through such intermediaries as St. Augustine. But the main sources of the unmistakable Neo-Platonism in the great scholastic philosophers are two, the writings of the so-called Dionysius, themselves the work of some Christianised follower of Proclus, and the work De causis, commonly supposed in the Middle Ages, though St. Thomas was better informed, to be Aristotelian, though it is really a Latin rendering of an Arabic work based on the very treatise of Proclus to which I shall directly invite your attention. It was from these sources that the schoolmen of the golden thirteenth century derived the peculiar theory of causality upon which their conception of the Universe rests, and it is most instructive, as an illustration of the impossibility of drawing any real dividing line between ancient and modern thought, to find Descartes, in the very act of professing to construct a new way in philosophy, assuming as his fundamental principle and treating as evident "by the natural light" of the understanding just this same theory. Proclus again is the author of the only work that has come down to us in which the attempt is made to exhibit the main principles of the Neo-Platonic doctrine, in the strict order of their logical connection as a well-articulated whole. In many ways a student of Neo-Platonism would be well advised to begin his reading with the brief but pregnant Στοιχείωσις θεολογική or Elements of Neo-Platonism of which I am to speak this evening. He will find there, brought together in a compact form and expressed with a special view to logical precision, just the leading positions he requires to understand in order to find his way in the multitude of occasional essays we call the Enneads of Plotinus. He will also find that the style of the author presents far fewer difficulties. Proclus has, indeed, none of the splendid bursts of spiritual eloquence which at |153| times carry the reader of Plotinus off his feet. His is emphatically the sermo pedester. But, by way of compensation, he makes no such demands on his reader as Plotinus in his uninspired moments. He neither defies the rules of grammar nor perplexes his sentences by Plotinus' trick of incessantly interweaving with his own words imperfectly quoted phrases from Plato. If you approach him with a decent working knowledge of Greek, you will find his style on the whole less difficult and less encumbered with technicalities than Aristotle's, and not more arid than, say, that of John Stuart Mill in English. If he is dry to the taste of many readers, it is only for the same reason that Mill's Logic is dry to readers of the same class; the nature of his subject requires him to pursue a train of close argumentation to its logical issue, and affords no scope for the eloquence which appeals to the imagination.

The one real difficulty which besets the student of the is the wretched condition of the text.1 As regards this, in some ways his most important book, Proclus has been most unfortunate in his editors. To the schoolmen only part of the work was known in the Latin version of William of Morbeke. Since the invention of printing, so far as I am aware, only two texts have been published. The Greek was first printed in 1618 by Aemilius Portus at the end of his folio edition of the much longer but infinitely less valuable Six Books on the Theology of Plato. Of Portus it is hard to speak worse than he deserved. His Greek text was made from bad manuscripts and swarms with errors equally fatal to grammar and to sense. The Latin version which accompanies it proves that he was quite incapable of translating even an uncorrupt and simple Greek original |154| with any fidelity. The worst stumbling-block is created, however, by his wholly senseless method of punctuation. On the whole, Portus seems to have a preference for placing a full stop or even beginning a new paragraph in the middle of a sentence. The first translation into English was made from this execrable text in 1816 by that curious eccentric, Thomas Taylor, "the Platonist," and, so far as I know, there has never been a second until the praiseworthy but insufficient enterprise of Mr. Ionides. Taylor deserves great respect for his real devotion to Platonic studies, but as a translator he was badly handicapped by the hopeless badness of his text, and in an only less degree by his own want of Greek. His book is now very rare, and probably most of you may never have seen it.

In 1832 the Elements obtained a second editor, the notorious Frederic Creuzer, better known as responsible for the unsatisfactory but typographically beautiful Plotinus of the Oxford University Press. Creuzer was one of the mystagogues of Schelling's coterie, who professed to find the key to all philosophies in Orphic, Eleusinian, and Samothracian orgia of which they knew nothing and no one else much. He also stood in relations with Hegel, whose adulatory letters to him suggest some unpleasant doubt of that philosopher's competence to act as an expositor of things Hellenic. The whole brood of quacks to which Creuzer belonged, as I may remind you, was finally blown into the air by the scathing exposure of Lobeck's immortal Aglaophamus. Creuzer, had he been less incompetent, might perhaps have produced a decent text of the Elements, as he received from Schweighäuser the readings of a manuscript then at Strassburg which corrects many of the worst faults and fills up many of the lacunae of the inferior MSS. on which Portus had relied. Unfortunately, |155| Creuzer knew very little Greek, and thought he knew a good deal. Hence, though he inevitably made a great advance on Portus by adopting many of the readings furnished by Schweighäuser, he rejected to his notes scores of others which should have been adopted. A conscientious editor would, of course, have made or caused to be made a complete collation of Schweighäuser's MS., but it is not clear from Creuzer's prefatory account of his own proceedings whether he did more than record such readings as he thought fit. What is worse, he retained the sense-destroying para- graphing and punctuation of his predecessor, and, though he professed to have revised the Latin version of the Greek, very little examination shows that, while removing some bad errors, he introduced a goodly number on his own account, so that his rendering is often a mere trap for the confiding and unwary.2 The text has not been edited since Creuzer, and the Strassburg MS. was lost in the siege of 1870. At the present moment, any reader who is to understand his author requires to construct a working text for himself by the aid of such light as Creuzer's records of the readings of the Strassburg MS. afford.3 Fortunately, the usus et norma loquendi of the author enable this to be done with a reasonable approach to certainty in all matters of moment.

I propose in the following pages to give some general account of the method and contents of Proclus' work, leaving the task of criticism to others. It should, of course, be remembered that the doctrine expounded is not, except on one or two points, peculiar to the author. |156| His object is to give a compendious summary of the principles of the whole Neo-Platonic school, and in the main the doctrines expounded are those which were held in common by all the thinkers who looked back to Plotinus as the restorer of what they took to be the philosophy of Plato. There are perhaps only two points on which Proclus diverges from Plotinus, both duly recorded by Mr. Whittaker. Like most of the later members of the school, Proclus rejects the possibility, admitted by Plotinus, that the soul may lead a double life, lapsing, as the phrase was, from eternity into time and mutability only in her least worthy elements, while her higher and nobler activities remain in the purely spiritual world "unfallen." Proclus, like nearly all the Neo-Platonists from the time of Iamblichus, maintains in the last proposition of his book that the soul, when she "falls," falls wholly and in every part. This modification of the doctrine of Plotinus, as Mr. Whittaker has said, seems to have been due, at any rate in part, to reaction against what was felt as an over-strained idealism. Even to his devotees Plotinus seemed to be preaching an other-worldliness which was not quite wholesome. It is the same feeling which comes out in the commentary on the Republic, where Proclus makes a vigorous defence of the Homeric stories and the tragic drama against the censures of Plato's Socrates. But there was also a sound logical reason for the revision of the older view; the theory of Plotinus was felt to be inconsistent with the unity of human personality. It involved something like what our Anglo-Hegelians call a "faculty psychology," and it is on this ground that Proclus rejects it in the Elements. You cannot reconcile the unity of our mental life with the distinction between a "fallen" and an "unfallen" part in the soul.

More important, and much more difficult to interpret, 157| is a doctrine which appears in the very middle of the book, and affects the whole subsequent exposition, the doctrine of the divine Henads or Unities which Proclus identifies with the "gods many" of Hellenic religion. On the probable meaning of this doctrine — which appears to be peculiar to Proclus4 — and the reasons for insisting upon it, I hope to offer a suggestion in the right place. For the moment I must be content to introduce my digest of Proclus' metaphysics by some general remarks on the method and arrangement of his manual.

Perhaps I need hardly say that Proclus will wholly disappoint a reader who comes to him eager to hear about ecstasies and other abnormal psychological wonders. These things belong to the personal religious life, not to philosophy. Even in Plotinus, for all his personal saintliness, the passages where the mystical "rapt" is dwelt on are few and far between, and there is no suggestion that it is attended with any of the abnormal psychological excitements on which the adepts and illuminati lay stress, and in a logical exposition of the metaphysical doctrine of the school there is no occasion to mention ecstasy at all. In philosophy, as Mr. Whittaker has rightly said, the Neo-Platonists are from first to last rationalists. Like Descartes, they believe themselves to have found a strictly logical and coherent theory of God, Man, and Nature, and they are as ready as any other philosophers who have ever lived to expound their reasons for their convictions. The manner and method of Proclus are, in fact, much those of the great rationalists of the seventeenth century from Descartes to Leibniz and Locke. In method, in |158| particular, he recalls at once two famous names in modern philosophy, Spinoza and Hegel. Of Spinoza, he reminds us by the care with which his method is based on that of Euclid and the geometers, of Hegel by his insistence upon the grouping of notions in triads. These resemblances, however, must not blind us to equally important differences. As far as regards the use of the "geometric method" goes, its employment is, of course, not peculiar to Spinoza; nor is there really anything specifically geometrical about it. It is merely the method known to the ancient mathematicians as synthesis, the systematic exhibition of a body of truths in the order of increasing logical complexity, the simplest being placed first, and the more complex exhibited as a series of successive deductions from them. It is thus the natural method of any thinker who has to expound a system of true propositions, and is concerned not with the historical problem of showing how they were originally discovered, but with the purely logical problem of indicating the implications which hold between them. It used at one time to be thought that there was some special connection between the matter of Spinoza's Ethics, and the method adopted for exhibiting it. This is, of course, a mere misapprehension. Any body of demonstrable propositions can be thrown into the form of the "geometrical method." Spinoza had been preceded in its use by Descartes, who gave, in the Answers to the objections against his Meditations, a formal "geometrical" proof of the real distinction between body and mind, and Spinoza's own first use of the method was to employ it on an exposition of Cartesian doctrine with which, as he candidly avowed, he did not himself wholly agree. Proclus uses it precisely because it is the method now called the "hypothetical-deductive," originated by Zeno and explained at length by |159| Socrates in the Phaedo of Plato. It consists simply in putting forward a theory or hypothesis, or set of postulates, as the explanation of a group of "appearances."

The consequences of the "hypothesis" are deduced at length, for the purpose of seeing whether they accord with the "appearances."If they do, the appearances are said to be "saved"; if they do not, some other hypothesis must be discovered which will save them. In the case of Proclus the appearances to be "saved" are just the whole body of all that we know, or think we know, about the things in the Universe, and the justification of his philosophical postulates is that these known truths about minds, souls, and bodies are "saved" in their entirety by the postulates of Neo-Platonism. (There is, of course, no question of dismissing the "appearances" as illusions or transmuting and transforming them into something other than what they are. We are throughout kept faithful to the principle enunciated by Butler in what is perhaps the most weighty single sentence ever uttered by any philosopher: "Everything is what it is and not another thing.") Proclus believes that his postulates do in fact "save" all the appearances and are therefore true, but there is no miraculous virtue about the mere use of the method. If you start with false premises, it will not prevent you from drawing conclusions which are also false. Spinoza, too, understood this quite well, as is shown by his use of the method in an exposition of Descartes where it brings out results which are false in Spinoza's opinion precisely because it has relied on false premises. As a point of history, it was the geometers of the Alexandrian age who took over the synthetic method with much else from the philosophy of Plato, not the philosophers who borrowed it from the mathematicians.

As to the parallel with Hegel, again, it is instructive |160| to observe precisely how far it holds. Proclus, like Hegel, believes that the triadic arrangement reproduces in thought the order of the links by which the richest of realities, the ens realissimum, is connected with the poorest and meanest. Only, in spite of appearances, he really begins at the end of the ladder where Hegel left off. Hegel, you will remember, opens his Logic with the notion of Being, on the ground that it is the most empty, abstract, and insignificant of all concepts, and works up gradually through increasingly fully determined concepts to the Absolute Spirit, the most significant of all. Proclus, on the other hand, begins with an even simpler concept than Being, Unity, or the One — precisely because he believes it to be, like God in the philosophy of the Christian schools, the richest and fullest of all concepts — and works downwards from it through the successive series of Minds, Souls, and Bodies to what he regards as the poorest. Again the method by which successive triads are found is widely different from Hegel's, — to my own thinking, not for the worse. Hegel's method, as we know, was first to take a concept, next to discover a contradictory opposite for it, and then to look for a third which could be plausibly represented as contradicting the contradiction. The procedure of Proclus is less heroic, but more readily intelligible. He usually arrives at a triad by first calling attention to two members A and B, which are doubly disjunct. A, that is, has the characters x and y, B those of not-x and not-y. He then argues that if A and B are both found as terms of a serial order of connected concepts, they cannot be in immediate juxtaposition. There must be an intermediary which resembles A, let us say, in having the character y and B in having the character not-x. The full reason for insisting on this necessity of an intermediate link between |161| two doubly disjunct terms will only appear as we come to speak of the logically most important thing in the system, the Neo-Platonic doctrine of Causality. The importance of this theory can hardly be exaggerated, though it is one of the pleasant ironies of history that Proclus' exposition of it should have provided the Christian religion, which he so heartily disliked, with just the instrument it needed for the elaboration of its doctrine of God, and should a second time have given Descartes the basis of the argument for the existence of God without which he could not take the first step beyond the mere affirmation of his own existence.

It has been truly said by Mr. Whittaker that the general theory of the world which Proclus elicits from his initial postulates forms something like a via media between Leibniz and Spinoza. We have a Monism with an Absolute as the logical and causal prius of everything but itself, but just because Proclus goes farther back than Spinoza in his quest for a simple first principle, the Absolute is a theistic Absolute, a transcendent Deity who is the source of existents, their characters and the relations between them.

We meet with causa sui and substantia (if we may take the latter as a rendering of the Greek οὐσία), but they are not the Absolute; their place is a secondary one. There are also monads of various orders, but, since the Neo-Platonist theory of causality makes all causality transitive — even in the case of the causa sui — the monads are not "windowless" and we escape all the paradoxes connected with the Pre-Established Harmony and its ambiguous relation to God's "choice of the best." For the same reasons we are left free to accept at their full value all the familiar facts which tell so powerfully in favour of an Interactionist theory of the relation of Mind and Body. Moreover, the very |162| insistence on the transcendent character of the Deity and the transitiveness of Causality make it possible, against Spinoza, to assert the permanent reality of individual souls, and against Leibniz the genuine reality of brute inanimate matter. The Neo-Platonist philosophy thus aims at uniting coherently the strongest points in what are commonly thought the incompatible doctrines of Monism and Monadism. As the choice seems to lie between Monism and Monadism for all of us who can find no refuge either in Kantian Agnosticism or in some pure Materialism, the type of view represented by the epitome of Proclus may perhaps fairly claim to have more than a merely antiquarian interest.

I shall probably succeed best in the attempt to give an account of the Neo-Platonist metaphysics at once concise and reasonably intelligible to students of philosophy who have no previous acquaintance with Neo- Platonist literature by allowing myself to sit rather loose to the actual terminology and order of the propositions of Proclus, and to deal only with the central conceptions of the system. It will be found that the notions of chief importance in the development of the system are those of the transcendent character of the One, the ultimate source of the Universe, and its identity with the Good which, as Plato had taught, is at once the motive power throughout the life-history of the Universe and the goal or aim of all processes, the causal relation which connects the One with the various stages of its evolution, and every stage with every other, the principal stages of this evolution, or, as the Neo-Platonists call it, "progression," and the process of "inversion" or "reflection" which is always found associated with progression and serves to make the system formed by the One and its manifestations into a |163| complete and harmonious whole. If we take our main topics in this order, the One, Causality, Progression, Inversion, we shall not indeed be following the order of Proclus quite exactly, but we shall not depart very far from the main structural outlines of his work.

He begins then, as was natural to a Platonist who had the Philebus well-nigh by heart and had commented at enormous length on the Parmenides, with the earliest and most stubbornly persistent of all philosophical antitheses, that of the One and the Many, which had, in fact, dominated all Greek thought from the time that it was first insisted upon by Parmenides and Zeno. The two conceptions of unity and plurality are not, strictly speaking, coordinate. Logically, and therefore ontologically also, the One is antecedent to the Many because it is involved in the very conception of a Many or Aggregate or Assemblage (πλῆθος), that it is, to use the old Pythagorean and Euclidean definition πλῆθος μονάδων, an Assemblage of units. It is the same thought which leads Leibniz to begin his Monadology with the proposition that the complex presupposes the simple. A modern mathematical logician would hardly be satisfied with the form of Proclus' proof, which, as is common with him, is a reductio ad absurdum based on the alleged impossibility of an infinite regress. The real point is, however, independent of this assumption, and amounts to the contention that a well-ordered series must at least have a first term, though it need have no other, or again that all complexes, even if their degree of complexity be infinite, must, as Leibniz said, be complexes of individuals. To take other illustrations of the same principle, if logical classes are to "exist," there must be at least one thing which is not a class but an individual, if "classes of classes" are to "exist" there must be at least one class |164| which is a class not of classes but of individuals; if truths of what Russell calls the "first order" are to be possible, there must be at least one thing which is not a true proposition but an individual about which a proposition can be made; if there are to be truths of the second order, there must be at least one truth of the first order, and so on in indefinitum.

The propositions which follow furnish the basis for a Philosophy of Transcendence as opposed to all "immanence doctrines" of the ἓν καὶ πᾶν type. Whatever "partakes of the one," i.e. whatever can have oneness predicated about it, is in a sense "one thing," but in a sense also not-one or many. As our Anglo-Hegelians say, it is one in virtue of being a whole of parts, not-one or many in virtue of being a whole of parts. It is a unity but it is not Unity. The oneness we ascribe to such wholes must be something other than any of them. This is why Plato and the Platonists say that they are not "one" but "partake of" the One. What follows prepares the way for the enunciation of the theory of Causality. Whatever produces anything other than itself (i.e. is the source of its existence) is superior in kind to that which is produced. This is the principle implied in the characteristic Neo-Platonic conception of evolution as "progression" or "emanation" (a word, by the way, which is not with Plotinus and Proclus a technicality but an illustrative metaphor). It is also the principle denied by every philosophy which treats epigenesis as the final word in evolutionary theory. The proof of this proposition is interesting, and depends on the implied assumption that causality is a transitive relation and that its terms are substantival entities, not events. Either the entity produced by a causal agent is itself capable of producing something further, or it is not. If it is not, this very fact establishes |165| its inferiority to its own cause. If it is, its effects are either superior to, equal to, or inferior to itself. The second possibility may be excluded as it leads to the conclusion that there is no hierarchy of better and worse, no difference in levels of value, among things, and this is assumed to be plainly at variance with the "appearances."The third possibility is that evolution is a steady process of epigenesis by which the inferior gives rise to the superior. But this, too, is unthinkable. For if an agent could produce certain perfections in that on which he acts, he could equally have produced at least as much perfection in himself, since, ex hypothesis he had sufficient power, and his failure cannot be due to lack of will, since by a universal law all things tend to attain the Good as far as their powers reach. (This, it will be recollected, is the reason given by Descartes for holding that he is not himself the perfect being.) Universally, then, the cause from which anything derives at once its being and its specific character is higher in the scale of goodness than its own effects.

Further, from the Platonic principle that all Beings seek for the Good, and their whole life is determined by the pursuit of it along the lines possible to them in virtue of their various specific constitutions, it follows that the Supreme Good, the first term in an ordered hierarchy of goods, cannot itself be one of these Beings or the totality of them. Just because it is what all beings strive to obtain, it must be beyond them all. It must be, as Plato had put it in the famous passage of the Republic, ἐπέκεινα οὐσίας, "on the other side of Being." It cannot be a "good something" but must be just "the Good," that whose whole character is goodness and nothing else. Good is not a predicate of it; Good is it. It needs only the further step of identification of the Good, thus conceived, with the One, that is the identification of the |166| Universal End with the Universal Source, to convert the logic of Proclus into a theistic theology agreeing with that of the Christian Church, in looking on God as a transcendent Being distinct from the Universe or whole of creatures and internally simple, not like an Herbartian "real" or the "bare monad" from the poverty of His nature, but just because all the perfections which are found in diffusion among His creatures are wholly concentrated and interpenetrant in their source. This is, in fact, what the schoolmen mean when they tell us that Deus est suum esse, and again that each "attribute" of God is God Himself. We are specially warned against confusing the Good with the "self-sufficing" A self-sufficing being can, indeed, meet all its needs out of the plenitude of its internal resources; it can live, so to say, by the consumption of its own fat. But the very statement implies that such a being has needs, though it can always meet them. The Good, being "good" simpliciter, has no needs to meet. We must not mistake it for a magnified Stoic Cato. We may not even say of it, that it is "filled with good."It is Good, and therefore must be called, as Proclus more than once calls it, "more than full," ὑπερπλῆρες." The epithet seems meant to indicate the Neo-Platonist answer to the obvious question why there should be a Universe at all. Why should the Good not be alone to all eternity in a state of single and perfect blessedness? How comes there to be a world of creatures who aspire to it? The Neo-Platonist explanation is that which Plato had long before put into the mouth of Timaeus. Goodness is, of its very nature, a self-imparting or self-communicating activity. It cannot keep itself to itself, but must overflow, much as Christians have said the same thing of love. Unlike Christian theologians, Plotinus and Proclus do not represent the creative |167| activity in which Goodness finds its outlet as one of "free choice." To them this would have implied that Goodness might conceivably not have imparted itself to anything, and therefore might not have been wholly good. Finally, they agree with Spinoza that God acts ex legibus suae naturae, though, unlike him, they are stout assertors of Providence and Final or Intentional Causality, and are careful to treat Free Will (τὸ αὐτεξούςιον) as a reality. The difference between them and Spinoza is really much greater than their divergence from the thought of Christian scholastics. Indeed, this latter divergence is much reduced when we recollect that, according to the schools, neither free choice nor anything else can be universally asserted of God and of any creature. The difference from Spinoza goes deeper. For Proclus would understand by the "laws of God's nature," the law of Goodness, whereas in Spinoza it is no part of the nature of Deus-substantia to be good, and even the distinction between good and bad in human character and conduct comes perilously near being dismissed as an illusion in the famous appendix to the first Part of the Ethics.

The formal identification of the One with the Good — derived, of course, from Plato himself — which turns Proclus' "First Cause" into God is effected by the help of the famous definition of Eudoxus, "the good is that at which all things aim."Such a good or end of appetition is manifestly a principle of unification and cooperation. Health, for instance, is the body's good, and health is just the harmonious organisation of all the constituents and members of the body. Salus populi is the good of a society of men, and it is realised in virtue of the conatus or nisus, conscious or otherwise, of each member of the body politic after it. Wherever you find good you find it as a common object of |168| appetition to the members of a πλῆθος, and it is this nisus after one and the same end which makes the πλῆθος a unity-in-multitude. So, if the creatures really form a Universe — and it is the presumption involved alike in thought and action that they do — it is because all of them are striving up to "the measure of the light vouchsafed" towards a common principle of Good which is beyond and above them all. It sounds a paradox, but it is thus the fact, that the One is the unifying principle in the Universe just because it is itself not "in" the Universe but "beyond" it. The general line of thought is thus very similar to that which is followed by Professor Varisco in the last chapter of his Massimi Problemi, where he sets himself to argue that the question whether the Universe as a whole, has value (is good) or not depends upon the prior question whether, as he phrases it, "Being has other determinations than the concretes, in which case the traditional conception of Being is transformed into the Christian conception of God." Proclus answers this question affirmatively; it is a matter of terminology that what Professor Varisco speaks of as "determinations of Being other than the concretes," are called by Proclus ὑπερπλῆρες, the things "above" being.

At this point, it will be convenient for a moment to desert the actual order of our author's exposition, which is designed with a view to preparing for the distinctions to be drawn between Minds, Souls, and Bodies, and anticipate a little by explaining the doctrine of Causality upon which his further account of the Universe depends. Causality, as I said before, is always, to the Neo-Platonists, a transitive relation. It implies two related terms, the producer (τὸ παράγον) and the produced (τὸ παράγόμενονν), and these are never events.

The cause or producer is always an agent or the |169| activity of an agent; the effect produced may be the existence of an individual or a quality of an individual, or both. As the relation is not one between events, it is not necessary that it should involve temporal sequence, and the Neo-Platonists were thus free to maintain with Aristotle that the historical sequence of events has no beginning. That the Good is the Great First Cause, means with them simply that everything depends, both for its existence and its special character, on the Good; but for the Good there would be nothing. It does not mean that there was a time when the Good was not "overflowing," and there was no world of creatures. Further, the way in which the agent or cause works is by imparting its own characteristics to that of which it is the cause. This is, of course, because operari sequitur esse, and it is in virtue of being what it is, that a cause causes just such effects as it does and no others. The effect is thus "like" its cause, or an "image" of it, but since, as we have already seen, it is a cardinal point in the system that what is produced is always an inferior and imperfect image of what produces it, the causal relation is asymmetrical, and Proclus thus agrees with Russell on the fundamental importance of asymmetrical relations. As Proclus and Plotinus are fond of putting it, the cause is imperfectly "mirrored" in its effects. It irradiates them, but they are at best broken lights of it. The Neo-Platonists would have been only in imperfect sympathy with the numerous modern philosophers who have maintained that the relation of cause and effect is really identical with the logical relation of antecedent and consequent. They would have agreed that the cause is always the "reason why," since, in their view, the causal relation is always a case of "participation"; the effect is what it is because the cause is what it is. But they would never have admitted either |170| that temporal sequence is an illusion or that to complete insight it would be possible to reason from effect to cause with the same certainty as to effect from cause, precisely because they hold that the effect is not the cause but only "participates" in it, and therefore only mirrors it partially. On their view there is always more in the cause than is ever reflected in the effect. It is notable that Proclus is careful to warn us that the transitivity of the relation is not done away with, even in the case of things which may be said to be "self-caused" (αὐθυπόστατα), because they contain in themselves the source of their own motions. As we know from Plato, this is the case with all souls, and it is the defining proprium of a soul, in contradistinction from all other existents, that it has "its principle of movement within itself," or is "that which can move itself." Even here, the Neo-Platonists, following the lead of Aristotle, say that though the terms of the causal relation are identical, it is still a dyadic relation and transitive. For this reason causa sui cannot, as with Spinoza, be identified with the "great first cause."The One, because its Oneness is itself, is not causa sui. It is simply uncaused.

Strictly speaking, the phrases self-moving or self-caused (αὐτοκίνητος) must not even be used of Intelligence or Mind, for Intelligence or Mind is (as Aristotle had held) something which remains itself unmoved or unchanged, but gives rise to the internally initiated changes in the soul. Hence, by putting the causa sui at the head of his hierarchy Spinoza is, from the point of view of Proclus, opening his account of things in the middle. He can only take into his reckoning Souls and the Bodies which are moved by those Souls. He has left out of consideration all that is really of highest moment in the universal order.

A last point of fundamental importance in this doctrine |171| of Causality is that, as Proclus is careful to state, the higher up in the hierarchy a cause is, the lower down the scale are its effects felt. The reason is that what comes nearer to the Absolute One in the scale is, being a truer reflection of the One, a unifying principle of higher order than what is more remote. Hence the unifying power of the One or Good extends to the whole Universe. Everything in the Universe, down to the mere unformed matter which is the ideal lower limit of dispersion and lack of organisation, derives its being from the One or Good. As Socrates said, everything is, and is what it is, because it is best that it should be so. The activity of Mind does not reach so far down, precisely because Mind is not itself the supreme or divine principle, but merely its most immediate reflection. We can, indeed, satisfy ourselves of this by the simple consideration that Mind does not make matter. It is true that order and structure are everywhere put into matter, even into inanimate matter, by Mind. For Divine Providence extends to the inorganic as well as to the organic world, and again human intelligence, which within its own limits mirrors Providence, shows itself constantly at work shaping inorganic matter by the introduction of form. But Matter is not existentially dependent on Mind; it is something which from the point of view of Mind is vorgefunden as an instrument of expression, not created by Mind itself. There is no ultimate dualism in the system, since Minds and all the things which are existents are alike existentially dependent on the transcendent One or Good: but if you forget the One, and start with Mind as your ens realissimum, you will be led to such a dualism, just because Mind is found everywhere correlated with an object, not Mind, to which it is related alike as knower and as organising principle. It is just by not accepting |172| Idealism in the modern sense, by not equating the ens realissimum with Mind, that the Neo-Platonists avoid dualism.

Again, the Soul is a less adequate mirroring of the One than its Mind or Intelligence. In fact, the Soul directly mirrors Mind or Intelligence, and reflects the One only at second-hand. And again we see that the causal activity of the Soul ceases to show itself before that of Mind. The activity of the Soul consists, in fact, precisely in communicating to another its own proprium, life. It is just the principle of life, and what it does to things is to bestow life on them, to endow them with the special kind of unity and organisation characteristic of organisms. Now not all bodies are capable of receiving this kind of unity and structure, but only some. There is organic matter, but there is also inorganic matter, and on dead or "inorganic" matter the Soul can exercise no influence. It can mould to its own ends the protoplasm of which our bodies are formed; it cannot dwell in or "inform" stocks and stones. But Mind, as we have seen, can give form to inorganic matter. A cabinet-maker or a statuary can not merely beget sons and daughters, but he can also, because he is not only an animal but an intelligent one, fashion cabinets or statues out of boards and stones. So universally, the higher the rank of a causal agent the more far-reaching are its effects, and, in particular, we may say of the Good which stands outside the whole series of existents and is above "being" that there is a sense in which its effects extend beyond the realm of existents and affect what is "below" being. For, as we have seen, inorganic bodies fill the lowest place in the system of existents. The "bare matter" which we are constrained to think of as that which is common to them all is never found actually existing.

|173| It is like the limit of an infinite absolutely convergent series, to which each successive term makes a closer approximation, though it never appears itself as a term of the series, or, to be more precise, it is like the limit of an infinite series whose terms, though all positive, tend to zero. Thus, as Aristotle had held, such mere matter may be called μὴ ὄν, "the non-existent," and can only be conceived by way of negation. Just as God is implicitly thought of by Neo-Platonists and Christian schoolmen as a simple being, who is at the same time the subject of all positive predicates, "bare matter" is a simple being which is the subject of no positive predicates. Yet the One stands in causal relation even with this mere negation. It is because of the presence of the One that what exists is not this bare potentiality of being something, in other words, that there really is something and not nothing. The idealist of the modern type is naturally tempted to call this shadowy universal "substrate" or "first matter," which is nothing in particular, a "creation" or "fiction" of our minds, but the Aristotelian and Neo-Platonic thought seems to me the truer. Of course we only arrive at the notion by a process of comparison and abstraction, but comparison, if we consider it rightly, only discovers, it never creates. That there is something common to the most elementary existents, which is never found itself actually existing, is a discovery. If the "something common" were not really there, no process of comparison would ever conduct us to it; comparison would be, as the Anglo-Hegelians say abstraction is, always falsification.

It is a corollary of this conception of causality that a predicate may be said to be contained in its subject in any one of three ways. Since a cause is mirrored in its effect, i.e. its activity consists in imparting its own character, so far as that is possible, to the effect, whatever is |174| characteristic of the effect may already be said to be contained in the cause. It is not there exactly as it is in the effect, since the effect is an imperfect image of the cause. In the cause the character of the effect is present "in a more perfect manner," in intimate conjunction with other characters which do not appear in that particular effect, but only in other effects of the same cause. To borrow Leibniz' metaphor the effects of a cause are perspectives, each reproducing the cause from one special point of view. In the effect itself the character in question is said to exist καθ ͗ὕπαρξιν — i.e. it is strictly only of the effect that we can predicate the character in question. (The expression καθ ͗ὕπαρξιν is obviously coined by analogy with the use of the verb ὑππάρχει τῷ in Aristotelian logic, where A ὑππάρχει τῷ B is the standing way of saying B is an A, or A is predicable of B.) In the effects of the effect, the same character again will be found imperfectly mirrored or represented, or, as Proclus says, "by participation."In scholastic Latin these distinctions are carefully kept up. The characters of an effect are said to be "formally" in itself, but "eminently" in its cause; Descartes' familiar assertion that what is thus "formally" in the object of an idea is "objectively" in the idea itself is a simple special case of the presence of a character "by participation" in the effects of the cause which has the character "formally," since the idea is held by Descartes to be caused by its object. The special Cartesian proof of the existence of God from my possession of the idea of God thus is proved by its very terminology to be a simple reproduction of Neo-Platonism as put into technical form by Proclus. Presumably all this Neo-Platonism reached Descartes through the medium of scholastic philosophy in his early days at Rennes, and this is why he supposed what he had been taught as a schoolboy |175| to be so evident by the natural light of the understanding.

We can now formulate very briefly the general Neo-Platonic conception of the world of existents and its relation to its single internally simple, transcendent cause, or source, the Good. It is of the nature of the Good to overflow its own banks, to bestow itself on something else, and this is the real answer to the questions, why there are existents at all, and why they form an ordered and connected universe. The overflow is by way of representation; the Good gives rise to a system of existents which imperfectly mirror or image its own goodness, and they in turn to "appearances" which imperfectly mirror them. It is to be noted that the imperfection of the mirroring, as Proclus tells us, is due to the inevitable defects of the mirror. If the actual world is not perfect, this is not due to any withholding of perfection from it by the Good. The Good is present to all things in its super-plenitude, but they cannot receive all that it has to give. They receive "according to their own constitution."It is with perfection as the Scottish divine admired by Johnson and Boswell said that it is with happiness: the quart pot and the pint pot are both full, but they do not contain the same measure; each is as full as it can hold. This is, of course, an inevitable inference from the general conception of causation as a process of imaging, or, what is the same thing, the principle that there is always greater excellence in the cause than its effect. (Fully thought out the principle would have led Christian theologians who accepted Neo-Platonism as their philosophic basis to an Arian doctrine of the Trinity: The Son, being the "image of the Father," would have been "inferior to the Father" not only "as touching His manhood" but also "as touching His godhead".) The doctrine of Plotinus had |176| been that the immediate "image" of the Good is Mind (νοῦς), and the immediate image of Mind is Soul (ψυχή), and that Mind is thus the highest member of the chain of actual existents. I.e. Mind is the highest kind of individuality which we find as an actual existent. Mere Soul, as we know it, e.g. in ourselves when we are at the mercy of irrational passion or impulse, or again in the immature who have not yet "found themselves," or still more as we discern it in the lower animals, is still less of a real complete and stable unity. The triad thus formed by the One, Mind and Soul is the only example of a triad in the Enneads. One must note carefully also that Mind and the objects of its thought (the world as apprehended by science) together make up the whole of what can be properly called real existents (ὄντα), and that Mind and its objects are inseparable. "The objects of Mind (τὰ νοητά) do not subsist outside Mind" was the doctrine thought by Porphyry to be peculiarly characteristic of Plotinus, and it was precisely his stubborn doubts about this tenet which delayed his entrance into the school of Plotinus until he had written a criticism which was in turn refuted to his own satisfaction by an earlier disciple. By this doctrine it is not meant that the objects known by Mind are themselves mental in the sense that they are made of "mental states" or "processes."What is meant is that the distinction between the epistemological subject and the epistemological object is not regarded as characteristic of the interior life — if we may call it so — of the Absolute One. It emerges first in the first image of that life, which is the life of Mind. The life of Mind is always a knower's attitude towards a known — the concept and the thinker of the concept are inseparable, not — to borrow a distinction familiar to readers of Professor Ward — in a psychological, but in an epistemological sense. |177| Actual existence then consists of Mind and what Mind knows. When we come to the life of the mere soul, not as yet rationalised, we are at a lower level. It is, compared with the waking vision of science, a sort of confused dream. Like the dreamer, the Soul, as mere soul, is itself perplexed and confused, and there is the same confusion in the object of its cognition and striving. It is a thing itself not realised moving about in a world unrealised. It belongs and its world belongs to "becoming" — the region where everything is perpetually baffling us by proving not to be what it seemed to be — not to "being."

Proclus refines something on the original statement of the doctrine. Within the primary triad itself we have to distinguish a subordinate triad. On inspection, Being, which Plotinus had treated as equivalent to Mind, breaks up into the triad of Being, Life, and Mind. For many things are, which are lifeless, and again many living things are not minds. But of this, as of the other triadic constructions which figure in rather confusing multitude in the elaborate Six Books on the Theology of Plato, it is not necessary to say much in a mere brief sketch like the present. Roughly, the successive more and more imperfect reflections or images of the Good may be said to be, in order of increasing imperfection, Mind, and Soul (the former being eternal both in its nature and its activity, the latter eternal in nature but temporal in its activity, and both together making up "what is"), and finally Body — temporal at once in nature and in its activities, which is what "seems," though we must remember that what "seems" really does "seem."Body as such has its place in the system: it has not to be "transmuted" or "absorbed" into something else as a condition of recognition.

Further, we must add that as Proclus conceives the |178| world, each member of this series gives rise to something other than itself in two different ways, or along two different lines. The source of this conception is manifest. There are many individual minds, souls and bodies in the Universe, and it does not occur to the Neo-Platonists to explain away this plurality of individuals of different types as an illusion. It is a fact which must take its place as a fact in an adequate philosophy. Hence Proclus conceives of Mind and Soul, not merely as units each of which can be "imaged" by a unit of a lower type, Mind by Soul and Soul by Body, but as first terms of series. His doctrine is that in each such series the first member generates a series of beings of the same type as itself, though each, according to its distance in serial order from the first term, is a less adequate representative of the type. There are thus, at the level of Mind, a whole series of more or less exalted Minds, and similarly, at the lower levels, a whole series of Souls and a whole series of Bodies of greater or less worth and dignity. The first member of each such series is called ἀμέθεκτος, mparticipable,5 that is unpredicable, because it is in the strictest sense only capable of appearing in a proposition as subject and can never be predicate. (Even Descartes at the stage of reflection reached by examination of the cogito can only say I am a mind, not I am Mind.) The rest of the series are the "participated" minds or "souls."Thus, in the case of minds, the first member of the series is Mind with the capital M, the other are the minds of the various beings who are said each to "have" a mind.

This theory is obviously applicable to the Good or One, no less than to Mind or Soul. If Mind gives rise not only to Soul, but to a plurality of Minds, the |179| Supreme One or Absolute Good must be thought of likewise as giving rise to a series of Ones which Proclus calls the "divine" Henads or Unities and also simply "the Gods."As the Good is God, so in his system the Henads are "Gods" in the plural, related to God as the minds of you and me are related to the entity we call Mind. This doctrine is, as I have said, the peculiar property of Proclus and its interpretation has caused some trouble. It has sometimes been spoken of as a mere device for saving the face of dying Hellenic polytheism. This, however, is not to my thinking its real raison d'être, though Proclus has filled many weary pages of his Theology of Plato and commentary on the Parmenides with ingenious attempts to identify and classify the Henads and to show that with some forcing they may be read into the traditional theology. I think the origin of the theory more likely to be what I have indicated already. Some explanation had to be found for the existence of individual Minds and Souls, some reason why this plurality should be real and why there is not just one Mind, one Soul and one Body. The doctrine of the series of Minds and Souls is already suggested by Plotinus, who always treats individual human Souls as existing, so to say, with the same right and on the same level as the Anima Mundi. It originates in the justified refusal of the Neo-Platonists to treat personal individuality as a kind of illusion and reduce human persons to the status of "modes" of a single Deus-substantia. When the theory had been thus thought out for the case of Mind and Soul, it was a mere exigence of logic to extend it to the first member of the supreme triad. It is thus, as it seems to me, the logical completion of a line of thought inherited by the whole school from Plotinus. It is a rather more difficult thing to feel sure of the interpretation to be placed on |180| the doctrine. Mr. Whittaker suggests the highly ingenious comparison with the modern conception of the stars as centres of planetary systems, but avoids committing himself to an opinion about Proclus' own intention. I think one may venture at any rate on a tentative suggestion. Just as the Imparticipable One is identified with the Good, Proclus tells us that the various divine Henads or Gods are ἀγαθότητες — "goodnesses," and that each of these "goodnesses," which are all comprised eminenter — or, as his own phrase is, ἀγαθότητες — in the One, forms one of the Henads or Gods. He also connects the doctrine with the well-known passage in the Phaedrus where Socrates speaks of different classes of men, statesmen, warriors, poets, as under the protection of a particular deity. The real meaning of this, according to Proclus, is that each different type of individual mind is linked to the One in a two-fold way; it is a member of the series of Minds, and the first term of this series, Mind, is derived from the One; also this special mind is a mirroring or image of a special Henad in the series of "divine numbers," and this Henad belongs to the series headed by the One.

On the strength of such passages one might suggest that what Proclus has in his mind is a doctrine of the attributes of God like that of Philo, or, again, of the great scholastics. The scholastics speak of a plurality of these attributes, goodness, wisdom, power, and say of each that God's wisdom is God, God's power is God, and the like, as may be read at length in the first book of St. Thomas' Summa contra Gentiles. This may be how we ought to understand what Proclus says about the gods or Henads and their relation to the One God."6 |181| They are, I take it, the "perfections" or "excellences" which in God exist, according to scholastic philosophy, in a way compatible with God's absolute simplicity, but in His works are found displayed to a great extent separately, some of the works revealing more particularly the wisdom, others the power, and yet others the goodness of their author. The notion must, of course, be carefully distinguished from Spinoza's theory in which just what is characteristic of each attribute is that you cannot say "God's extension is God" or "God's thought is God," and cannot conceive God as really simple at all.

Thus finally, including the Henads which are "above being" and bodies which are properly speaking "below" it, we may say that the One or Good appears as the source of four orders, Gods, minds, souls, bodies, and that as the four orders form a hierarchy of "images" or "reflections" of the ens realissimum, so each order itself is a hierarchy of "reflections" of its own initial members. The whole system is, in modern language, a well-ordered series of well-ordered series.

There remains, however, yet another fundamental doctrine on which I have not yet touched — the theory of ἐπιστροφή or reflection backwards. "Reflection" has commonly been used in English to translate the word, but with us the expression is ambiguous and I have already been obliged to employ it to illustrate what Proclus means by the progression of Henads, Minds, and the rest from the One. "Inversion," especially for some of its mathematical associations, would really be a better word. We have already seen that the One is thought of as being at once the source of all existence and the end or good which all existents tend towards by the law of their being.

Once more, we may remind ourselves that the |182| thought is derived directly from the definition given by Eudoxus, the astronomer, and adopted by Aristotle, that the good is that οὗ πάντα ἐφίεται, that which all things "go for." And the "all things" do not mean simply the "sentient creation."The thought is that in everything, sentient or insentient, animate or inanimate, there is a real nisus towards systematic organisation or unity. You see this nisus, a Neo-Platonist would have said, at different levels in the cohesion of the particles of a homogeneous body, in chemical affinities, in the attraction of the plant for the insect, in the sexual life of the animal kingdom, the family and social aspirations of man, the life-long struggle of the thinker after an organised and coherent body of knowledge, or of the saint after the disciplined life of holiness. In all these instances, what we discern is marked by two characteristics. The nisus is not, as Spinoza, being a mere naturalist, supposed, towards self-preservation, but, to use Dr. Ward's expression, towards betterment of some kind, attainment of a good which is the specific good of the creature exhibiting the nisus Dr. Ward's remark that a creature which, as we commonly say, eats to fill its skin gets as a consequence a better skin to fill, exactly hits off this aspect of the Neo-Platonist view of the fundamental conatus or "will to be" in things. Again — and Proclus would say that this is an immediate consequence of the identity of the transcendent One with the Good — the nisus because it is always a turning back or inversion of the process by which a thing is derived from its cause, is always a movement towards simplification. It is always a tendency towards the assertion of unity and individuality. A vertebrate is more truly one creature than a sponge, the inner state of cognition in a man of science or feeling and will in a saint is one of simplicity, as compared with the state of |183| a man whose mind is a medley of confused and unsystematised beliefs, or the battle-ground for perplexed and divided counsels or passions. This is the ultimate source of the mediaeval conception of growth in grace as a steady "reduction of the soul to its ground." Unless we have the theory of ἐπιστροφή or inversion well in mind we are bound to go astray when we try to understand what a writer like Dante or a Kempis really means by extolling sancta simplicitas as the highest and best state for a Christian. I suppose we can all feel the beauty of such a line as that famous one about the anima semplicetta che sa nulla, or the tenderness of a nameless English mystic's adjuration never to try to melt the "cloud of unknowing that is between thee and thy God," or of Ruysbroek's description of the "noughting" of the soul, but, apart from Neo-Platonism, we cannot really understand what all this meant.

Two points are specially important in connection with this conception of ἐπιστροφή: (1) The fundamental conatus of everything is the nisus to reverse the process of its production — to return to its immediate source. Macaulay quotes, as typical of the nonsense of "Satan" Montgomery, a badly expressed line, to the effect that "the soul aspiring seeks its source to mount." To Macaulay this seemed unmeaning, but it is only what Donne or Vaughan would have said better about the soul and about everything else. But there is a vital distinction between two classes of things, those which in being reflected back upon their source are also reflected upon themselves, and those which are incapable of reflection or inversion into themselves. This distinction is as fundamental for Neo-Platonic philosophy as the distinction (on which the difference between infinite and finite assemblages depends) between series which can be "mirrored" within themselves and series which cannot |184| is for modern mathematics. The point is that some things contain the principle from which they proceed within themselves in a way in which others do not. "Soul," for example, is thus related to its principle, Mind or Intelligence. As Plato had said in the Sophistes and Timaeus, νοῦς always exists in ψυχή. Of course it might be said that, on the other hand, there are many ψυχαί, those of animals, for example, which do not think. I suppose the rejoinder would be that even animal instinct, as Professor Stout has argued at great length, not only produces results which are justified at the bar of reason, but is found everywhere working under intelligent guidance, as we see from the regular adaptation of instinctive trains of action to the special requirements of the individual situation and the modifiability of instinct by experience. There is intelligence involved in a cat's pursuit of a mouse, though we might hesitate to say that the intelligence belongs to the cat in the same way in which my intelligence belongs to me. But when you come to deal with bodies the case is altered. Bodies, it is held, are the "images" of souls, and, as Plato had argued, all bodily movement is produced directly or indirectly by the prior "motions" of a soul. And bodies share in the universal tendency to reflection back into their proximate cause. Body is only seen at its fullest and completest when it is an animate body. From the standpoint of ancient physics it would seem natural and right to look upon inanimate bodies as having the function of feeding and sustaining plant life, and plants that of providing sustenance for animals, who in their turn minister to the needs of man. And the Platonist view was that the prime elements of all bodies, animate or inanimate, are the same. A living body does not differ from a lifeless one by being made of ultimately different stuff, but simply by being organic to a soul — |185| by being the body of that soul. So that the general facts about the so-called "three kingdoms" of the popular natural histories would be just an example of the process of ἐπιστροφή on a grand scale.

But bodies, in being reflected back into their cause, are not reflected back into themselves. Self-reflection, inversion into self, is characteristic only of what is bodiless. Proclus offers a curious formal proof of this. The argument is that reflection or inversion into self is a relation of a whole to itself as a whole. Whatever is thus inverted, he says, must be in contact as a whole with itself as a whole, that is, it must be directly existentially present as a whole to itself. But this relation cannot hold between wholes which are aggregates of distinct parts. In them, each part is present to itself in a direct way in which no two non-identical parts are present to each other. It is this relation, only possible to wholes which are simple units, in the sense that they are not made of separable parts, which constitutes knowledge. Hence in being "inverted" into Mind, a soul not only knows Mind, but knows itself. And Mind similarly, when in contemplation it is turned back upon the One or Good, is in the very same act reflected into itself and knows itself. Bodies, because they cannot be thus directly present to themselves, know neither themselves nor their causes. The analogy between this doctrine and Leibniz' distinction between "bare" monads and souls strongly suggests that the one has been modeled on the other. From the same source, I venture to think, comes Locke's well-known language about "ideas of reflection." According to Locke these are ideas of the mind's own activities got by the mind in taking note of our own "operations about our ideas."That the mind has this power of inversion, by which its own activities become objects for its contemplation, Locke assumes as something |186| which no one will deny. I suggest that both the assumption and the name "ideas of reflection" are due to the same cause which produced the polemic of the First Book of the Essay against innate principles, the general and widespread influence of Neo-Platonism on the English philosophical writers contemporary with Cudworth and Henry More. The eighteenth century saw the gradual decrease of this influence; in its latter third, writers like Gibbon had lost all sense of the meaning of Neo-Platonic language, and we thus find the Decline and Fall of the Roman Empire treating as gibberish doctrines which are referred to, for example, by Bacon as perfectly familiar and intelligible.

One should note that, easy as it would have been to treat the doctrine of ἐπιστροφή as the basis of an anti-rational mysticism — and this, I would suggest, is very much what M. Bergson does in his doctrine of the élan vital — it does not occur to the Neo-Platonist to do so. In spite of the familiarity of the school with the psychological facts about "rapts" and "ecstasies," no Neo-Platonist ever regarded these states as revealing philosophical truths. Plotinus expressly compares the state of the ecstatic with the position of a priest who has passed the veil that screens the holy of holies and left all the images of the temple behind him, and dwells on the point that it leaves behind it no memory of what it was. And when Proclus speaks of Mind as reverting in self-contemplation to its principle the One, he is not referring to "rapts" at all. There is an agnostic side to his doctrine which reappears in the orthodox schoolmen. Nothing, he says, is ἄπειρον, indeterminate, in itself, but everything is ἄπειρον, not fully determinable or fathomable, by anything that stands lower in the universal hierarchy than itself. The higher, though more rational in its own constitution, is something of a |187| mystery to the lower, much, we might say, as a man is Unknowable, but this only means that since it is something more than Mind, Mind can only know it by the reflection of itself it has stamped on Mind. The One is, he says, in itself ἐνιαίως "after the fashion of unity," but in Mind only νοερῶς, "after the fashion of Mind."That is, I take it, Mind is not the highest and most perfect type of individuality.

The utter individuality of God, the source alike of Mind and everything else, is proper to God; but Mind, since we have minds, is the most truly individual thing we can understand. We can see that God is something even more individual — a more perfectly articulated and yet absolutely individual being — than Mind itself, but what it is like to be something more than Mind, we, not having the experience of it in ourselves, cannot say. In general, the higher is only known to the lower by its effects on the lower itself, because it is in self-knowledge that we have to come to the knowledge of what is higher than self. The "negative theology" or "way of negation" — so salutary a protection against the extravagances of ignorant imagination to those who understand its real meaning, so utter a puzzle to moderns like William James and the Pragmatists, who have criticised it without knowledge of its history — is all contained in this doctrine of the necessary limitations of our knowledge. It is the real defence of sober thought against that "wild license of affirmation about God" with which Matthew Arnold, in reckless defiance of facts, charged mediaeval theologians, and we may more reasonably charge a good many of our popular scientific writers who would refuse to call their ultimate reality by so old-fashioned a name as "God."

(2) The other point on which there may be room for a |188| word or two is also an example of Neo-Platonic sobriety. The doctrine of ἐπιστροφή must not be interpreted in the light of modern theories of the Deus-substantia type about the unreality of finite selves and finite things in general. The existence of a plurality of finite individuals of different types is to the Neo-Platonists, as to Plato himself, an ultimate premise. Each individual has his good or end and "reverts" to it, but the process is conditioned throughout by the specific nature of the individual. He "reverts," or unites himself with his Good, in the way his nature permits. The bonds of individuality are not burst in the process. Bodies, in the process of inversion, do not cease to be bodies or souls to be souls. So with Mind; Mind in attaining full knowledge of itself also discerns its immediate source, God, the One, — but it does not become God or a god. "Every thing is what it is and not another thing," and in the process of ἐπιστροφή it does not cease to be what it is, though no doubt it may discover that it is much more than it had at first supposed. The reversion or inversion of Mind into the One does not mean that Mind becomes God, but that in self-contemplation it learns to know God, so far as God is comprehensible to any of His creatures. There is no question of an Absolute in which finite individuality of any kind is transmuted and transformed into the irrecognisable. Proclus, in fact — though limits of time will not permit me to follow him — professes to be able to prove the everlastingness, both a parte ante and a parte post, of every individual capable of self-inversion, that is, of every individual which is not a body. The demonstration follows the usual lines of the old rational psychology, attacked by Kant, and need not delay us. What interests me more personally is a reflection suggested by the Neo-Platonic insistence on transitive causality.

|189| It is frequently said nowadays that the fault of the old orthodox theology lay in its devotion to a "transcendent" deity. To be in earnest with transcendence, we are commonly told, means to exclude all possibility of any real relation between God — or whatever else a man likes to call the Supreme Being — and other beings. Religion, as a personal matter, because it means intimate personal relation with the Supreme, requires a doctrine of "immanence."Against this fashionable view I wish to suggest that it is, in point of fact, just the "immanence" philosophies which have always found it impossible to have any theory of the relation of their ἓν καὶ πᾶν with the individuals we know. Either the Deus-substantia has to become an empty name for a mere aggregate, without any individuality of its own, or the individuals have, by elaborate logical sophisms, to be made into mere illusions.

It is palpable, as it seems to me, that this inability to recognise the reality of individuals other than the Ens summum is certain to be fatal to the philosophies of Mr. Bradley and Professor Bosanquet as they stand. I do not, of course, mean that these philosophies ought to be rejected, or will be rejected, because we do not like their reduction of our own individuality to an illusion. I mean rather that on careful scrutiny the arguments of these distinguished philosophers reveal themselves as variations of one single contention which turns out, on close examination, to be a petitio principii. My own growing feeling is — and I believe it is by no means peculiar to myself — that if Mr. Bradley and Professor Bosanquet discover their own individuality to be unreal, the reason is that they set out from the start with a parti pris. Naturally they do not find what they are unconsciously determined not to see. To myself it seems obvious that if there is a real supreme principium |190| individuationis, it must be, as the Neo-Platonists held, an end as well as a source, and must, therefore, of course, stand "outside the Universe," and that it is just because it is "outside" that direct and intimate personal relations with it are possible to all of us, if indeed they are possible. This means, of course, that I feel bound to hold as a point of general theory that transitive causality and transitive asymmetrical relations are ultimate in logic. I can see no vestige of ground in logic for the assumption, tacitly or expressly made in so much of the thinking of the generation before my own, that there are no relations of one-sided dependence. Herbart's protest against Kant's assumptions about the ubiquity in the Universe of "reciprocal action" seems to me as unanswerable as it has remained unanswered. To be more precise, the particular doctrine about which I feel the greatest difficulty in Professor Bosanquet's system of thought is his theory of Causality.

What gravels me is not so much his assertion that the relation of Cause and Effect is at bottom identical with that of Antecedent and Consequent. The ancients, who called both Cause and Reason Why αἴτιον, in a sense accepted this, and I could make shift myself, perhaps, to regard Causality as a special case of the more general relation. My difficulty is with the further assertion that in a really true hypothetical proposition Antecedent and Consequent are simply convertible. This, of course, means that there are no ultimate and unanalysable relations of one-sided dependence. But why should there not be?

So, again, the assertion that time is not real is, I suppose, a consequence of the same view, since, if time is real (unless it can be shown that events recur in cycles), the relation before-after is transitive and asymmetrical, as all relations which generate series appear |191| to be, This is why I feel myself that if we are not to declare ourselves frank Irrationalists, we must hold that a philosophy of the general type of Neo-Platonism is at least nearer the truth than Spinozism or those versions of Hegelianism which have had the widest currency in our Universities for the last generation.


1^ Now remedied by the edition and translation of Professor E. R. Dodds. Proclus, The Elements of Theology, Clarendon Press, 1933.

2^ Creuzer re-edited the text, many years later, and no more satisfactorily, for the firm of Didot. We have had to wait until 1933 for Professor Dodds to produce a text founded on real knowledge of the MSS.

3^ Happily, since the appearance of Professor Dodds's text, this statement is no longer true.

4^ Prof. Dodds has, however, made out a strong case for the view that the actual author of this doctrine was Syrianus, the immediate teacher of Proclus. See Elements of Theology, p. xxiv. 257 ff.

5^ (This doctrine of ἀμέθεκτα, however, is not original with Proclus, but comes to him from Iamblichus.)

6^ But for some deserved strictures on the undue assimilation of the doctrine to that of the scholastics made here, cf. E. R. Dodds, Proclus, the Elements of Theology, 270-71.

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V. The Analysis of Επιστήμη [Episteme] in Plato's Seventh Epistle

This paper was originally presented in 1912.

The question as to the genuineness of the seventh of the thirteen letters contained in our Plato manuscripts is one of considerable importance for the History of Philosophy. If the letter is really the work of Plato, it throws a flood of light on the philosopher's early life, particularly on the nature of his relations with Socrates and the causes by which Plato was led to abandon the career of public life as an Athenian statesman, which would have been normal in a man of his birth and endowments, for the vocation of a φιλόσοφος. It is further, on the hypothesis of genuineness, of great value as giving us an authoritative version of the reasons by which Plato was led in later life to attempt an active intervention in the affairs of Sicily, and his grounds for holding that the death of Dion was no fatal defeat for his "cause." It is not, however, my object at present to examine the whole literary problem created by the Epistles in general, or by our letter in particular. In view of the present state of the controversy, I feel justified in assuming that the evidence, both from style and from historical allusions, is overwhelmingly in favour of the view that the seventh Epistle is, as it purports to be, a document belonging to the year immediately after the assassination of Dion by Callippus, and presumably, therefore, a genuine letter of Plato. There remains, |193| however, a difficulty which is neither historical nor literary but philosophical, and it is this difficulty which I propose to discuss, and, if possible, to remove. The most remarkable passage in the letter (342a-344d), a digression which professes to give the justification for the well-known Platonic view that "philosophy" cannot profitably be communicated by books, but only by the direct and long-continued personal intercourse of mind with mind in the common pursuit of truth, has given great offence to students who have professed themselves unable to perceive its drift, or to feel sure that what meaning it has is genuinely Platonic. I therefore propose, to the best of my ability, to argue that the whole section has a definite purpose, that its leading contentions are in principle sound, and that good Platonic authority can be adduced for them. An examination of the passage is the more indispensable that Constantin Ritter, who strongly defends the genuineness of the letter as a whole, regards this passage as a later interpolation,1 and that Mr. H. Richards, in his recent Platonica, not only speaks of its presence as possibly an "insuperable difficulty" for those who ascribe the letter to Plato, but loads the digression itself with terms of disparagement, such as "rigmarole," "skimble-skamble stuff," and the like.

I will divide my treatment of the subject into two main parts. First I shall urge briefly, against the theory of later interpolation, that the digression is strictly relevant, and indeed necessary, in the connection in which it stands, and next, in answer to the charge of unintelligibility, I will translate and interpret the passage in question as carefully as I can, adducing such parallels from the Platonic dialogues as suggest themselves to me. |194]

I. Relevancy. — That a statement showing why it is futile to convey the deepest truths by means of written handbooks (τέχναι) is relevant to the matter in hand is clear from what has been said immediately before. The object is to prove that the reprehensible proceedings of Dionysius are no legitimate fruit of the Academic philosophy. For Dionysius had by no means heard the whole Platonic doctrine from its author in the days of their friendly intercourse. Indeed, he had not even asked for a full exposition, for "he professed that he already knew for himself and was sufficiently master of many things in it, and those the most important."2 Since then Plato has heard that Dionysius has actually written a handbook (τέχνη) on the subjects of their conversation, in which he has gravely misrepresented its purport (γεγραφέναι αὐτὸν περὶ ὧν τότε ἤκουσε, συνθέντα ὡς αὑτοῦ τέχνην, οὐδέν τῶν αὐτῶν ὧν ἀκούοι. That is, he wrote the handbook on his own initiative, took the credit for its contents to himself, and totally misrepresented Plato's views on the matters which had been discussed). The writer of the letter does not know whether this report is true, though he knows of others who have done the same thing, οἵτινες δὲ οὐδ ͗ αὐτοὶ αὑτούς — apparently a colloquial way of saying that these writers were "nobodies." He will only remark that all past and future writers of books "who profess knowledge of the matters to which I devote myself (περὶ ὧν σπουδάζω), cannot possibly, in my opinion, understand one whit of the business. There is not, and God forbid there should ever be, any work of my own on the subject,3 for it cannot be expounded (ῥητὸν γὰρ οὐδαμῶς ἐστιν, |195| — a play on the mathematical meaning of ῥητὸν, 'exponible') like other branches of learning (μαθήματα). It is only from long-continued personal association in the business and in a common life4 (ἐκ πολλῆς συνουσίας γιγνομένης περὶ τὸ πρᾶγμα αὐτὸ καὶ τὸ συζῆν, that suddenly, as it were, the fire bursts forth from one soul and enkindles (sc. in the other) a spiritual flame which thenceforth feeds itself" (341c). Consequently, the letter goes on, if it were really possible to put the "essence of Platonism" into a written book, the proper person to write the book would be Plato himself, the very man who would feel most distress if the work were ill done. Hence "if I had thought these matters could be adequately written about for the general public, and were exponible5 to it, what nobler task could I have had in life than to write things of so much service to mankind and bring Nature to light for all to look upon?" (341 c). But the author's opinion is that there are very few men who can be benefited by being urged to study such subjects, and they are just those who can be trusted to discover truth for themselves with a little preliminary guidance (διὰ σμικρᾶς ἐνδείξεως). In most men the reading of philosophical works either leads to an undeserved contempt for what they find above their intelligence or to idle vanity about their attainments. Thus far the introductory page on which the much-decried bit of analysis follows. The writer then goes on to make a special application of his general thesis to the case of Dionysius. He remarks that the most appropriate test of capacity for philosophy in a "prince" is to give him some initiation |196| into the subject and then observe the effect, and discover into which of the three classes just enumerated he falls. Such a πεῖρα he says he applied to Dionysius. He had one philosophic conversation with him, and found that the young king asked for no more, but claimed to be henceforth qualified to "philosophise" on his own account, and is even reported to have composed a written work on φύσις.6 This gives the opportunity for a long restatement of the familiar Platonic view that written works are an inadequate vehicle of philosophical education. The central part of this restatement is taken up by an attempt to show that the doctrine in question necessarily follows from the nature of the problems of ἐπιστήμη on the one side, and the character of written and spoken language on the other. It is this passage (342a–344d) to which Mr. Richards takes special exception (though all the time regarding it apparently as an integral part of the letter). The writer then returns to the subject of Dionysius and his book (344d), and tells us, in effect, the final verdict to which the application of his πεῖρα to Dionysius has led him. This is, as we should expect, highly unfavourable (344c–345c). Now it seems to me that the whole passage from the end of 341a, where we first hear of the application of the πεῖρα to Dionysius, down to 345c is so closely coherent that it would be impossible to excise any minor part of it and yet retain the rest. We cannot, for instance, hold that the letter as a whole is genuine, and only the particular digression about ἐπιστήμη and |197| language, to which Mr. Richards so strongly objects, spurious. For where, in that case, are we to look for the close of the interpolation? It cannot be placed before the return to the case of Dionysius and his book at 344d, because the allusions in 344 b to the ὀνόματα, λόγοι, and ὄψεις which serve as intermediaries of communication of knowledge are hardly intelligible apart from the full account of what is meant which was given on pages 342, 343, and further because the metaphor of the kindling of the flame of φρόνησις and νοῦς by the personal intercourse of questioner and answerer in that section is a conscious reversion to the exactly similar metaphor of 341c. Nor can we reasonably regard the interpolation as ending at 344d 2, where the letter reverts to the subject of Dionysius, because, when the long digression has been removed, the words with which the next paragraph opens, τούτῳ δὴ τῷ μύθῳ τε καὶ πλάνῳ ὁ συνεπισπόμενος εὖ εἴσεται κτλ. have nothing left to which they can refer. It is the digression itself (342c–344d 2) which is the μῦθος and πλάνος in question. Consequently Mr. Richards' view that the digression on ἐπιστήμη is senseless cannot be held by anyone who regards the seventh letter in the main as genuine, unless he is prepared to excise not only the piece of analysis contained in 342–343 but the whole of the pages which deal with the interview in which Plato applied the πεῖρα to Dionysius and the results which he obtained.

Thus I think we may fairly say that if there is an interpolation at all it must extend to the whole of the pages which Ritter wishes to excise; every allusion to the supposed "book" of Dionysius must go out. Let us see then what the result of so considerable an excision would be. The genuine Platonic letter will run as follows, if we epitomise its argument from the beginning of 340, where Plato is explaining that in coming to |198| Syracuse for the third time he was sacrificing his own judgement to that of Dion and Archytas, who thought well of the "philosophical" qualifications of Dionysius (339c): "I undertook the journey with grave secret misgivings. When I reached Syracuse I thought it my first duty to find out whether Dionysius had really been 'set on fire' by philosophy, as was reported, or not.7 The most becoming way of ascertaining this in the case of a prince (ὄντως τυράννοις πρέπων, 340b) is to make it quite clear to him that philosophy is a big thing and only to be acquired by great labour of the mind (ὅτι ἔστι πᾶν τὸ πρᾶγμα οἷόν τε [340ξ] καὶ δι᾽ ὅσων πραγμάτων καὶ ὅσον πόνον ἔχει, ibid.). Such discourse, in fact, acts as a spur to the noble mind and incites it to learn how to walk alone without a guide. The true votary is led on to mould his life in accord with philosophy in all the affairs of business, and to order his daily conduct in a way which will make him quick to learn, slow to forget, able to reason soberly for himself (340d). But those who are not true 'lovers of wisdom' at heart, but cherish a mere skin-deep interest in 'points of view' (δόξαι, 340d), will soon grow weary of the prolonged course of study and the daily discipline of the 'affections and lusts', and give up the quest. Some of them will try to persuade themselves that they are already masters and need pursue no further (ὡς ἱκανῶς ἀκηκοότες εἰσὶν τὸ ὅλον, καὶ οὐδὲν ἔτι δέονταί τινων πραγμάτων, 341a). The proposed test, applied to those who have not the diligence to follow the path to its goal, brings out the difference between the man who blames his teacher for his failure to make progress and him who rightly, lays the blame on his own inability to live as a philosopher must (πάντα τὰ πρόσφορα ἐπιτηδεύειν τῷ |199| πράγματι, ibid.). It was in this spirit that I spoke as I did then to Dionysius." Hereupon, according to Ritter, follows immediately (345c ff.) the account of the growing unfriendliness of Dionysius to Dion, on which Plato remarks that it enabled him "to see accurately how much Dionysius cared about philosophy, and gave him the right to feel indignant," and the narrative then runs straight on with an account of Plato's desire to be allowed to go home and the reluctance of Dionysius to part with him.

Now is it not clear that, as thus restored (?), the continuity of the letter is violated? The elaborate account of the kind of "test" (πεῖρα) appropriate to the case of a prince has led nowhither. We are neither told whether it was actually applied, nor, if it was, how it resulted. We have been led to expect information as to whether Dionysius confessed his inability to live up to the standard demanded of the philosopher, or tried to lay the blame for his failure on Plato, or finally posed as having already learned all that Plato had to teach, and being therefore free from the obligation to live any longer sub disciplina. This information we get in the letter as it stands, but not in the mutilated form which Ritter regards as original. Indeed, in Ritter's version, Plato is made to ascribe his discovery of the uselessness of further attempts to make a "philosopher" out of Dionysius not to the application of the carefully described πεῖρα, but solely to the prince's unworthy treatment of Dion.

This incoherence is so glaring that I venture to say that if the letter had come down to us without the whole passage which Ritter excises, critics would have declared with some reason that there must be a considerable lacuna in the text at the very point where Ritter begins his excision. Some account of the application of |200| the πεῖρα to Dionysius would have been rightly felt to be indispensable. But when once so much as this is granted, we have to allow that the coherency of the whole "interpolation" with itself, and particularly the intimate connection of its beginning and end with what immediately precedes and follows, forbid our resorting to any theory of a minor "interpolation" covering only the two pages 342–343. There is no reasonable way of dealing with the text by excision short of cutting out the whole of what Ritter regards as "interpolated," and the whole, as we have just seen, cannot be removed without making the letter hopelessly incoherent. Either then the whole letter is spurious, a view which Ritter himself in my opinion properly rejects, or the whole is genuine, and Plato must take the blame for any unintelligibility there may be in the pages which deal with ἐπιστήμη and the reasons which make written books unsatisfactory as instruments for imparting it. This brings me to my second point, the careful consideration of the incriminated passage, and the exposition of its real meaning. I proceed then to offer a translation of the whole passage 341b-345c, with such interspersed comments, marked in the text by square brackets, as seem requisite. I will only premise that the reader remembers that the studies which, according to Plato, are particularly effective in producing the intellectual and moral elevation requisite for philosophy are mathematics and dialectic, the former, as commonly taught, being a propaedeutic for the latter, and that it was precisely by a course of geometry that, according to ancient tradition, the training of Dionysius for the work of kingship was begun. Consequently we shall not be surprised to find that the account of the difficulties attending the attainment and communication of ἐπιστήμη is specially concerned with the philosophy of the mathematical |201| sciences, though allusion is also made to ethical inquiries. If we bear this in mind, we shall discover that many of the alleged obscurities of the passage vanish of themselves.

II. Interpretation. — "To be sure, I did not give a complete exposition, nor did Dionysius ask for one. He professed, in fact, to know and be adequately possessed of many, and those the most important, matters for himself by reason of what he had imperfectly heard from others. I hear that since then he has even written a work about what he heard at the time, in the form of a Handbook to Philosophy (τέχνη) by himself, which differs utterly from what he then heard. About this I have no certain knowledge, but I know for certain that others have composed writings on the same topics, though who they are is more than they know themselves.8 Yet I can state this much of all writers in the past or future who profess knowledge of the matters to which I devote myself, whether on the ground of having learned them from me or others, or as a discovery of their own. It is, in my opinion, impossible that they should understand one whit of the business. At least, there is not, and God forbid there should ever be, any written work of my own about it. [Plato in the main kept his word. We learn a little about the topics described here as περὶ ὧν σπουδάζω from the Timaeus, Philebus, and Laws, but for the most part we are dependent on the reports of Aristotle.] For the matter is not 'exponible' in speech, like other branches of study. It is only after long fellowship in the business itself [sc. the pursuit of the philosophic life; the reiterated τὸ πρᾶγμα seems to mean something like 'the grand |202| concern'] and in life together that, so to say, a light is kindled in one soul by the fire bursting forth from the other,9 and, once kindled, thereafter sustains itself. [The meaning is that there are μαθήματα — the rules of composition as taught by Isocrates would be a case in point, and so would 'geometry and the kindred τέχναι ͗, as taught in our school books with avoidance of all reference to the philosophical problems they suggest — which can be learned from a manual or τέχνη. Philosophy, being a 'way of life', or 'habit of mind', cannot be compressed into any such spoken or written summary of rules and results. It is only by living daily the life the Master lives, and by lying long open to the influence of his personality, that, sooner or later, the soul of the pupil 'takes fire'. After that moment the sacred fire, which required at first to be fed by the example and precept of another, 'feeds' itself. The philosopher is thus not a 'crammer' but a trainer of men. Cf. the well-known views on the relation between teacher and taught, and the necessity for intimate first-hand intercourse between the riper and less ripe in mind in Republic, 518b; Phaedrus, 274-276; Theaetetus, 149-151.]

"And yet I am sure of one thing, that if these matters were to be expressed in spoken or written discourse, it had been best done by myself, and further that I should feel more distress than another if the written exposition were a bad one. Had I thought them 'exponible' and adequately communicable in writing for the public, to what nobler work could I have given my life than that of writing what would be so serviceable to mankind, and of throwing the light of day on Reality for all men? [This sounds arrogant, but the arrogance is only in the sound. It is Plato's apology for giving the public |203| hitherto only 'discourses of Socrates', not a set exposition of his own philosophy. If the Platonic philosophy could have been imparted to the world at large in a book, Plato was obviously neglecting his duty in not writing that book.] But I do not regard the so-called dialectical exposition of these matters10 as good for any but the few who are capable of discovering them for themselves with the help of a little guidance. As for the rest of mankind, it would fill some of them out of all measure with a mistaken contempt and others with a vain and empty conceit of the sublimity of what they had learned." [I.e. some would not understand a written work on φύσις and would despise it as jargon; others, who mistakenly thought they understood it all, would plume themselves on their fancied intellectual penetration and forget that as yet they were only beginners and not proficients. Readers of Herbert Spencer among ourselves illustrate the point. Some of them are led to despise all philosophy as high-sounding jargon; others give themselves airs of superiority on the strength of the supposed intellectual power shown in professing to understand what "the general" are baffled by.]

[The upshot so far, then, is that the result of the πεῖρα was unsatisfactory. Dionysius, by his boasts of his knowledge, had shown himself to belong to the class of men whose philosophy is only skin-deep, inasmuch as he insisted that he had already reached a point at which he had nothing more to learn from Plato, and had even, if reports could be trusted, attempted to expound a philosophy of his own in a book.]

"It has occurred to me to say something more at length about these matters αὐτῶν I take to be neuter as in |204| the preceding τὴν ἐπιχείρησιν περὶ αὐτῶν λεγομένην and the following λεγομένην αὐτῶν, though it might conceivably be masculine, referring to the persons just spoken of, for whom the study of philosophy is not desirable]. There is a truth adverse to the pretensions of those who would commit any such matter to writing, a truth which I have often uttered before, but must, as it seems, repeat again now."

[Now follows Mr. Richards' 'rigmarole', about which I would make one preliminary remark. In the author of the Laws and Philebus, who, as he says himself in the Theaetetus, 172 d, regarded it as a mark of the 'liberal' character of philosophical discourses that the speaker is not controlled by a brief, or hampered by a time-limit, but can digress at any moment and to any length he regards as desirable, the passage must not be condemned simply because it is a digression, and to our taste a lengthy one, but only if its contents can be proved unworthy of their alleged author.]

"For everything that is [ἑκάστῳ τῶν ὄντων, i.e. for every concept which is an object of scientific contemplation] there are three intermediaries by which the knowledge of it must be imparted [the notion of 'imparting' is suggested by the παρα in παραγίγνεσθβι], and we may reckon the knowledge itself as a fourth thing, the object of it, which is the Knowable or Real Being, being counted as fifth; thus, (1) a name; (2) a discourse [λόγος, the example below will show that he is thinking more particularly of the λόγος τῆς οὐσίας or definition, the 'discourse' which tells you what the thing called by the name is]; (3) an image (εἴδωλον); (4) a knowledge. If you wish to understand what I am saying, you may take one example and conceive all the rest analogously. There is something which we call a circle. Its name is the very word we have just uttered. Next comes its |205| 'discourse' [or definition, λόγος], a complex of names and verbal forms [ῥήματα].11 E.g. the 'discourse' of the thing which has the names 'round', 'circle' is 'that which has its boundary in every direction equidistant from its middle point'. Third is what we draw and rub out, fashion on the lathe and destroy again. These affections do not belong to the 'circle itself' with reference to which all these operations are performed, clearly because it is something different from what is thus constructed.12 Fourth, there are knowledge and understanding (νοῦς) and true judgement (δόξα ἀληθής) about all this. These must again all be reckoned as one condition which inheres neither in sounds, nor in the shapes or colours of bodies, but in minds [ψυχαῖς ἐνόν. I.e. knowledge and belief inhere neither in the visible diagrams and models, nor in the spoken words of discourse; they are states of mind]. Hence it is manifest that this condition is something different both from the real circle [αὐτοῦ τοῦ κύκλου τῆς φύσεως, where I may be pardoned for remarking that κύκλου depends on φύσεως, not vice versa, αὐτὸς ὁ κύκλος, ὁ ἐν τῇ φύσει κύκλος, ἡ τοῦ κύκλου φύσις are all synonymous in Platonic language], and from the afore-mentioned three things. [That is, the 'circle itself', which is the object of the geometer's knowledge, is neither a name, nor corporeal thing, nor a psychical thing or 'state of mind'; it is strictly what the scholastics call a universale ante rem, like all the Platonic εἴδη.]

"But of all the rest understanding (vow) approximates most closely to the fifth [real being, the object of |206| scientific thought], in affinity and likeness; the others stand at a further remove. We must hold the same view of straight and curved figure, of colour, of the good, beautiful, and right, of every body artificially made or naturally generated, of fire and water and their likes [sc. not only of bodies but of the popularly recognised στοιχεῖα of bodies, which are, of course, not στοιχεῖα at all for Plato], of all organisms and all tempers of the soul, as well as with reference to all action and passion; in all these cases, unless one in some sort acquires the first four, he will never partake fully of the knowledge of the fifth."

[The application of all this to the problem of the communication of knowledge is obvious. To communicate knowledge about an object of thought, you have (1) to use a name for it, (2) to explain what the name means by stating its equivalence to a certain 'discourse' or 'definition', (3) to illustrate your 'discourse' by actual appeal to diagrams or models, — or, we may add, to the memory-images of them, which like the diagrams and models themselves fall under the general head of εἴδωλα. It is only by these intermediaries that you succeed at last in producing in another a genuine knowledge about the purely conceptual objects of genuine scientific knowledge, and we are to see directly that every stage of the process is attended by grave sources of error.]

"Furthermore, all these processes, thanks to the imperfection of language (διὰ τὸ τῶν λόγων ἀσθενές), are used no less in showing what the object is like than in showing what it is (τὸ ποῖόν τι περὶ ἔκαστξν – ἑκάστν τὸ ὂν ἑκάστου). Hence no man of understanding will ever venture to put his concepts into language (εἰς αὐτό = εἰς τὸ τῶν λόγων ἀσθενές), least of all into language which cannot be altered, as is the case with writings."

|207| [So far the sense is quite free from difficulty, and wholly Platonic. The point is that it is a hard and barely possible thing to communicate profound mathematical and 'dialectical' truth by a written textbook illustrated with diagrams. That Plato distrusted these last we know independently from Phaedo 92 d; Cratylus 436 d; Republic 510. And for this very reason he requires in the passage of the Republic that 'dialectic' shall be devoid of the help of diagrams (511 b, αἰσθητῷ παντάπασιν οὐδενὶ προσχρώμενος), much as the philosophical mathematicians of our own day demand a "geometry without diagrams."diagrams."13 This throws light upon the one apparent |208| obscurity of the present passage. What, it may be said, has the alleged ultimate source of all difficulties, the inadequacy of language, τὸ τῶν λόγων ἀσθενές, to do with the danger of reliance on diagrams? That there is such a danger we know very well. A diagram is always imperfect, and often suggestive of error, and moreover a diagram can never exhibit a problem or construction in its generality. Thus, to take the simplest examples, Euclid never proves a point which is vital in his very first proposition, viz. that the two circles described from the end-points of his given straight line have a point of intersection not lying on the given line. He is apparently contented to take this for granted on the strength of the diagram, though it can be, and ought to be, proved. In i. 2 his figure only illustrates one of several possible "cases" of the general problem, and so again in i. 47 the proposition would be equally true of squares constructed on the inner sides of the straight lines which compose the triangle, but the diagram does not represent this case. But what connection have these considerations with the "inadequacy of discourses"? Simply, as I suppose this, it is the inadequacy of ψιλοὶ λόγοι to render mathematical reasoning generally intelligible which drives the mathematician to eke out his "discourse" by appeals to always imperfect and often misleading diagrams and models. The ordinary man will not take in what is meant by the statement, nor be able to fix his attention on the steps of the reasoning, without some such helps for his imagination. So, again, the propositions of Euclid V. hold good of all ratios between magnitudes of any kind, those of Euclid X. for any surd magnitudes of the types discussed, but as aids to the imagination the writer regularly employs two special kinds of magnitudes, lengths and areas. And in X., we see that this habit has reacted on his |209| thought and language; he constantly speaks of surd "lines" and "rectangles" as though the propositions he is enunciating held exclusively of lines and areas.14 The same considerations account for the allusion to the distinction between the Sv and the irotiv ti and the stress which our passage goes on to lay on the confusion between them as a source of fallacy. The way in which the distinction appears in geometry is well illustrated by some remarks made by Proclus in connection with the classification of Euclid's propositions as "theorems" and "problems."The distinction between the two, he tells us, had not originally been universally admitted in the Academy. Speusippus had regarded all geometry as composed exclusively of "theorems," Menaechmus had held that all propositions of geometry are "problems" (Comment. in Euclid. 77-78).

Thus the dispute, since Speusippus had taken sides in it, may not unreasonably be held to go back to the lifetime of Plato himself, as Speusippus only survived his uncle by a few years, and was already an elderly man when he became head of the Academy.

Proclus goes on to trace the effect of it upon the views of later thinkers, and tells us that "Posidonius and his school defined the one as a proposition in respect of |210| which it is asked whether something exists or not, the other as a proposition in which it is asked 'what is it?' or 'of what sort is it?'" (op. cit. p. 80), a view which exactly corresponds to the modern doctrine that "problems" are existence-theorems.15 It is in accord with this view that when Proclus comes to comment on Euclid's first "theorem," Prop. I. 4, he justifies the arrangement by which the proposition is preceded by the familiar "problems," I. 1-3, in the following way. "How was Euclid to instruct us about the συμβεβηκότα καθ ͗ αὑτό of the triangles, and the equality of their angles and sides, without first constructing the triangles and providing for their genesis? Or how could he have assumed sides equal to sides and straight lines equal to other straight lines if he had not already considered this by way of problems and achieved the finding of the equal straight lines? For suppose him to say, before constructing them (i.e. before showing how a triangle or a straight line equal to a given straight line can be constructed), 'If two triangles have the following property (σύμπτωμα), they will also in all cases have such-and-such a second property'; would it not be easy for anyone to retort on him, 'But do we know whether a triangle can be constructed at all?' Or suppose him to go on, 'If the two triangles have two sides of the one equal to two sides of the other' — might not one raise the question, 'But is it possible that there should be two equal straight lines?' … It is to anticipate such objections that the author of the Elements has furnished the construction |211| of triangles (sc. in I. i) — for his procedure is equally applicable to all three kinds of triangle — and the genesis of equal straight lines. This last he has achieved in a double form. He both constructs such a line in general (I. 2), and constructs it by cutting off a segment from an unequal line (I. 3). Thus he reasonably makes the theorem follow on these constructions" (op. cit. 234- 235). In the light of Proclus' account of the antiquity of the question about the distinction between theorems and problems, it seems natural to me to suppose that it is to this that the language of our passage alludes. The sense then will be simply that the effects of the inadequacy of language are felt in problems and theorems alike.]

"I must, however, repeat the lesson I am now giving. Every one of the circles which are drawn or fashioned on the lathe by actual manual operation (ἐν ταῖς πράξεσι) is full of the opposite of the Fifth, I mean, it is everywhere in touch with the Straight. But 'the circle itself', so we say, has neither more nor less in it of the nature of its opposite. [I.e. no actually described physical disc is ever absolutely circular. At any point you please you can make it coincide throughout a finite distance with a physical 'straight' line. So, one might also say, in the diagrams of our Euclids, the so-called circle and tangent can always be seen by anyone to coincide throughout a perceptible distance. But 'the circle itself', the 'mathematical' circle of which Euclid is speaking, only coincides with the tangent at a 'mathematical' point.] So we say that none of them [viz. the 'physical' figures referred to] has a fixed name. There is no reason why those we actually call round should not have been called straight; yes, and the straight ones round. The names would be just as fixed if we interchanged them. [E.g., to revert to the example I have just given, we may |212| imagine a diagram in which the so-called tangent coincides throughout a visible interval with the so-called circle. Then, the so-called tangent is not really straight, nor the so-called circular arc really circular. So if you were to cut out this bit of the diagram and consider it by itself, it would not matter which of the visible lines you called the arc, and which the tangent.]16

"Further the same must be said of the 'discourse', since it is made up of names and verbal forms (ῥήματα). There is no sufficiently fixed fixity in it. [μηδὲν ἱκανῶς βεβαίως εἶναι βέβαιον. This is an obvious consequence. If a mathematical term gives rise to ambiguity owing to the possibility of its being taken as standing for something we can see and draw, a fortiori the same must be said of a definition or other proposition containing many terms.] And there is no end to be said (μυρίος δὲ λόγος) in the same way (αὖ) of the ambiguity (ἀσαφές) of each of the four [sc. ὀνόματα, λόγοι, εἴδωλα, δόξαι]. But the principal point is that we mentioned just now. The ὄν and the ποῖόν τι are two different things. But when the soul is trying to know the What (τί), not the ποῖόν τι, each of the four presents it, in word or in fact [λόγῳ τε καὶ κατ ͗ ἔργα, where κατ ͗ ἔργα refers to the construction of the sensible diagram or model], with that after which it is not seeking,17 and thus renders what is being stated |213| and demonstrated open to refutation by the senses, and so fills everyone, speaking roughly, with confusion and perplexity. Accordingly in matters as to which, from our evil upbringing, we have not so much as learned the habit of seeking for truth, but are satisfied by any image of it which presents itself, we do not make one another ridiculous in the process of question and answer, by our ability to tear in pieces and refute the four; but in cases in which we are obliged to answer about the fifth and point it out, anyone who has the will and the ability is equal to overthrowing ≪ his antagonist ≫ and can make the exponent ≪ of a truth ≫ in speech or writing or answers to questions [i.e. dialectical inquiry] appear to most of his hearers to know nothing about the matter on which he is undertaking to write or speak, since the audience often forget that it is not the mind (ψυχή) of the writer or speaker which is being refuted, but only one or other of the four, thanks to its ill constitution.18 It is the process through |214| them all, the transition forwards and backwards in the case of each, that last hardly gives birth to knowledge of the well-constituted in a well-constituted . But if ≪the mind's≫ constitution be ill (and this is most men's case both with respect to acquiring knowledge and with respect to acquiring what is called character), and in part also corrupted, Lynceus himself could not make such men see."

[We may, I think, illustrate the whole passage in the following way: Let us suppose a mathematician to be dealing with a branch of his study which owes its fundamental principles to the researches of the Academy, the Geometry of the Conic Sections. Let us further suppose that he is aiming at proving the existence-theorem that there are three such curves and no more; a task which, of course, involves the establishment of a correct definition of each of the three by reference to some exclusive property — an answer to the question τίἐστι; It is at once obvious that any ambiguity of language may introduce great difficulties into the communication of his investigations on such points to others. But the same thing is equally true of the diagrams by which he attempts to remedy the imperfections of language, and make his meaning definite for his hearers. Thus, for example, if we formed our notions of the ellipse from the diagrams of the textbooks, we should probably be struck by the inequality of the two axes of the curve, which the diagrams usually make prominent, and we might be led to think this inequality a universal characteristic of ellipses. This would lead to entire misconception of the relation of the ellipse to the |215| circle, since we should fail to see that the circle is only one special case of an ellipse, in which the eccentricity of the curve is = 0, a mistake which was actually committed by Herbert Spencer when he contrasted circular orbits as "homogeneous" with elliptic orbits as "heterogeneous." Or, again, the teacher might wish to prefix to his treatment of the separate 'conics' a general account of the properties of the 'general conic'. For the benefit of his readers or hearers, he would probably illustrate his propositions by diagrams. But in any one diagram the figure will not be a 'general conic', but definitely either parabolic, elliptical, or hyperbolic, and this may lead the beginner to confuse the properties of conics as such with those of the special conic figured in the diagram. Or, yet again, and this illustrates the remarks about the ease with which a smart ἀντιλογικός can make the mathematician look ridiculous, the lines which in a figure represent the asymptotes to an hyperbola can be seen to be such as would soon meet the line which stands for the curve, if both were produced, and the side and diagonal of the "square," measured by our rough appliances, will seem to have a common measure. Hence an ἀντιλογικός who insists on regarding the figure as the actual object of which a proposition is enunciated can readily make it seem that the geometer is uttering paradoxes which an appeal to the senses will explode. It is apropos to remember that the subtle arguments of Zeno were for centuries supposed to be idle though ingenious sophisms which might be set aside by such an appeal, though in fact they go down to the roots of mathematical philosophy. Thus we can understand what the writer means when he says that the whole series of intermediaries by which knowledge is imparted, name, definition, diagram or model, belief as to the teacher's meaning, is attended at every stage by |216| possibility of misapprehension, and that it is only by a repeated dwelling on each of the four and comparison of it with the rest (e.g. repeated transition from written text to diagram, and back again from diagram to text) that the truth which the teacher is struggling to express at last dawns on his pupil's mind. For this reason alone, it would follow that, as has been already maintained, long personal association in the pursuit of truth is necessary if one philosopher-mathematician is to train up another. The same difficulty would meet us, as the writer says, in the attempt to impart knowledge of any kind but for the fact that in our daily life we are commonly content with something far short of the ideal. We think it enough that those to whom we communicate our ideas should form a mere rough-and-ready approximation to our meaning. This is, e.g., all that is commonly arrived at when a man makes a speech on a social or political topic. He does not expect his audience to take in all that he means, or to frame very precise notions of the "liberty," "order," "progress," and so forth of which he speaks. He is satisfied if they catch his meaning "there or thereabouts."But it is the great merit of the μαθήματα as a mental discipline that a "there or thereabouts" standard of comprehension is not tolerated in "geometry and the kindred arts."

So much for the sense of the passage. A word on the reference to Lynceus the sharp-sighted with which it ends. Mr. Richards finds this nonsensical, and asks whether Lynceus was supposed to be able to infect others with his gift of vision. I think this a serious misunderstanding of a phrase which in itself is simple enough. Everyone knows that if, e.g., you take a country walk with a keen-sighted friend, he is likely to call your attention to all sorts of minute or distant objects which you can see well enough for yourself after your attention |217| has been directed to them, but would otherwise have passed by unnoticed. It is in this sense that Lynceus is spoken of as able to make other men see things. And it is in this same ability to call a pupil's attention to what he would otherwise have overlooked that a good teacher may properly enough be compared with a keen-sighted companion whose range of vision is longer than one's own.]

"In a word, if there is no affinity19 between a man's mind and our study (τὸ πρᾶγμα) mere quick receptivity or good memory will never create such an affinity [with οὐκ ἂν ποιήσειεν we have clearly to understand συγγενῆ, so that the construction is τὸν μὴ συγγενῆ οὔτε εὐμάθεια οὔτε μνήμη ποιήσειεν ἂν συγγενῆ]. For it absolutely refuses to make its appearance except in a kindred soul. Hence neither those who have no natural attachment and |218| affinity to righteousness and whatsoever else is fair, though perhaps quick to learn and steadfast to retain other knowledge of various kinds, nor yet those who have this natural affinity but are slow to learn and forgetful, — none of these, I say, will ever fully master the truth about virtue and vice."20 (≪ I say 'virtue and vice', ≫ because both must be learned together, and similarly truth and falsehood about Real Being as a whole have to be learned together, and this, as I said at first, demands much time and practice (τριβή)). It is only in consequence of a reciprocal friction of them all, names, discourses, visual and other perceptions, with one another and the testing of them by kindly examination, and question and answer practised in no spirit of vainglory (ἄνευ φθόνων), that the light of sound judgement (φρόνησις) and understanding (νοῦς) flashes out on the various problems with all the intensity permitted to human nature.21

"Wherefore every worthy man (πᾶς ἀνὴρ σπουδαῖος) will beware with all caution of bringing worthy matters into |219| the range of human rivalry and perplexity by writing of them. In one word, one must learn from what has been said that when one sees written compositions by an author, whether laws written by a legislator, or writings of any other kind soever, these were not the matters the writer deemed worthiest, if indeed he is himself a man of worth; such things [sc. the things the writer deemed σπουδαιότατα] are laid up in the fairest place the man possesses.22 If he indeed committed them to writing as things of greatest worth and moment, 'why then, thereafter', not gods but men 'bereft him of his wits'.

"Well then, he who has followed this discursive tale will know full well that whether Dionysius, or any other man, greater or less, wrote down any of the highest and primary truths of Nature, he must have had no sound instruction nor understanding of anything of which he wrote, at least not according to my conviction. Else he would have had the same awe of these themes as I have, and not have exposed them to the eyes of discord and uncomeliness.23 He could not have been impelled to write them as memoranda for his private use (there is no danger that one whose mind has once compassed the truth should forget it, for it is contained in the briefest words). It must have been done, if it were done at all, from an unseemly vanity, whether the purpose was to claim the knowledge as his own, or to prove his participation in an education of which he was unworthy |220| if he was greedy of the reputation of having participated.

"Now if Dionysius was affected thus by our only interview (perhaps it might be so, though how it happened 'Goodness only knows', to use the Theban phrase, for I went over the subject in the way I have described, and that once only, and on no second occasion),24 he who would learn how the effect fell out as it did, must ask himself why we did not go over matters a second and third time, and yet oftener. Does Dionysius after a single hearing fancy that he has knowledge, and has he really sufficient knowledge, whether from discoveries of |221| his own or from previous instruction by others? Or does he think my exposition worthless (φαῦλα τὰ λεχθέξτα)? Or, for this is the third possibility, does he think the subject beyond him and admit that he has not the ability to live in the practice of wisdom and virtue? If he thinks the exposition worthless, he will find himself in disagreement with numerous witnesses whose judgement in such matters is of much greater weight than that of a Dionysius. If he holds that he has discovered the truth or been instructed in it, and that it is at any rate worthy of a liberally educated mind, how could he ever have been so ready to affront one who had been his master and guide in these studies, unless he is a very singular fellow? The nature of the affront I can explain from personal experience."

The concluding paragraphs of our passage from 344 a 1 on do not seem to me to require any particular explanation or elucidation. They merely restate for us the view which had long before been expressed in the Phaedrus of the one reasonable ground on which written works on φιλόσοφια may be defended, viz. that they serve to refresh a man's fading memories of the actual converse in which two minds have followed up the trail of truth, and urge that that ground is not pleadable on behalf of the work ascribed to Dionysius. Either his character or his judgement is proved to be bad by his seeking no second interview with Plato, and his boasts of his own proficiency, — and Plato inclines strongly to the view that the defect is not merely one of judgement. I have therefore to confess that if my interpretation of the two pages against which Mr. Richards' strictures are specially directed is substantially accurate, I see nothing either in the matter or in the manner of the whole digression which interferes with the ascription of the letter as we have it to Plato. |222| And I think I have already shown that (a) the whole passage as I have rendered it is too much of a unity for one-half of it to be excised without the excision of the remainder, and (6) that if the whole is excised the result is immediately a sensible gap in the continuity of the letter. Whence I conclude that for those of us who lack the face to condemn the whole letter on the flimsy grounds which have been urged against it, apart from the objection to this particular section, the only alternative is to recognise the whole as from the hand of Plato. We may feel that Plato's hand has lost something of its cunning; there are unnecessary repetitions in the narrative, and the grammatical construction is sometimes loose, though both charges may be made with equal truth against, e.g., Laws vii.-ix. But these are no remarkable faults in a man well over seventy, and most of us, even in our prime, do not carry out in correspondence, and should not employ even in a published pamphlet to which we had purposely given the form of a personal letter, the same strict rules of composition as in a formal treatise or essay.

I will add only one further remark, which seems to me of some weight. The seventh epistle, whether by Plato or not, bears every mark of being a genuine document of the date at which it purports to have been written. The Epinomis, indeed, has been quoted as a like example of imitation, but the argument is worthless until some better evidence than that of a φασιν in Diogenes Laertius has been produced against the Platonic authorship of the Epinomis. Moreover, even those who think the Epinomis an imitation at least account for its exact correspondence with the style of Plato by assigning it to an immediate intimate and personal companion of the master. If we are to ascribe the seventh letter to another than Plato, I do not see how |223| we can avoid a similar view of its origin. It will have to be regarded as a manifesto in the name of the Academy, produced at the latest within a very short time after Plato's death. Indeed, I think, we may go further. Such a manifesto could hardly have been written after the direction of Sicilian affairs had been put into the hands of Timoleon, and it is scarcely more likely to have been later than the horrible catastrophe which ended the tyranny of Dionysius in Locri (346). Thus we are driven to date the supposedly spurious letter before Plato's death, and to assume that, even if he did not actually write it, it was composed in his name and with his approval. It follows that if the philosophical digression which we have been examining is "rubbish," it was rubbish which imposed on Plato and was taken by him for an expression of his own theories. I, for one, cannot believe in such a theory. The only rational view of the matter, to my mind, is that a letter which purports to have been written by Plato shortly after the death of Dion in 353, cannot have been written later than 346, and is indistinguishable in style from Plato's latest work, really was written by Plato, and that its philosophical part is therefore very unlikely to be "rubbish" and very likely to be excellent sense.


1^ See his Neue Untersuchungen über Platon, Essay 7; Platon, i. p. 8.

2^ 341 b, πολλὰ γὰρ αὐτὸς καὶ τὰ μέγιστα εἰδέναι τε καὶ ἱκανῶς ἔχειν προσεποιεῖτο διὰ τὰς ὑπὸ τῶν ἄλλων παρακοάς, sc. from the imperfect accounts he had received before Plato's own arrival.

3^ Cf. Ep. ii. 314 c, οὐδ᾽ ἔστινσύγγραμμα Πλάτωνος οὐδὲν οὐδ᾽ ἔσται. If these are not, as I believe them, to be, the actual words of Plato, they must be regarded as copied from the present passage, which, in that case, must have been regarded as genuine by the writer of Ep. ii.

4^ That is, from the effort to lead the βίος φιλόσοφος is common.

5^ For the repeated play on the word ῥητὸν cf. Republic 546 b; here the word has primarily its literal sense, but there is a pretty clear allusion to the specially mathematical sense to which magnitudes commensurable with a "proposed" standard are said to be ῥητἁ (Euclid, Elements, x. def. 3).

6^ It would probably be wrong to take the reiterated allusions to φύσις as the topic of the conversation with Dionysius and of the alleged τέχνη as concerned specially with in the Aristotelian sense, φύσις throughout the passage apparently represents the subject matter of φιλόσοφια or ἐπιστήμη in general; that is, it means τὸ ὄν or τὰ ὄντα. So when we read in Theaetetus 173 c, of the mind as πᾶσαν πάντῃ φύσιν ἐρευνωμέν η τῶν ὄντων ἑκάστου ὅλου, the context shows that this includes ethical reflexion as well as the astronomy and geometry which have just been mentioned.

7^ I.e. whether what he had learned from Plato on his former visit had kindled a genuine desire to know more. Cf. 338 d ff. φιλόσοφια throughout the letter means virtually the theory and practice of the Academy.

8^ We cannot expect to know who the persons thus referred to are, except that they seem to be pupils with whose performances Plato was not wholly satisfied. Readers with a sense of humour would naturally like to think of Aristotle, but to identify the culprits at all is to be wise above what is written.

9^ 341 c ἐξαίφνης, οἷον ἀπὸ πυρὸς πηδήσαντος ἐξαφθὲν φῶς, ἐν τῇ ψυχῇ γενόμενον αὐτὸ ἑαυτὸ ἤδη τρέφει. The context shows that the ψυχή is that of the disciple which catches fire from the soul of the Master.

10^ For the meaning here given to τὴν ἐπιχείρησιν περὶ αὐτῶν λεγομένην compare Sophistes, 239 c; Laws, 631 a, 722 d; Aristotle, Topics, 111 b 16, 139 b 10, and the regular technical sense of ἐπιχείρημα in the Topics. The full expression is ἐπιχείρησις τοῦ λόγου in Ep. viii. 352 e.

11^ I have rendered ῥήματα here 'verbal forms' on the authority of Sophistes 262 b, where βαδίζει τρέχει καθεύδει, and τἆλλα ὅσα πράξεις σημαίνει are said to be ῥήματα, but it is, of course, possible that the sense is more general, and that we should translate 'predicative phrases', so as to include adjectives used predicatively.

12^ Cf. Republic 510 d-e, for the point of the distinction between a mathematical diagram or model, and the concept which it is meant to illustrate.

13^ It may be thought that I am here forgetting that it is not 'geometry and the kindred τέχναι ͗, but dialectic which the Republic forbids to employ diagrams. But the fact is that the "philosophy of mathematics," and the treatment of mathematical problems in the light of such a philosophy, form a most important part of what Plato calls 'dialectic'. It is not in its subject-matter, but in its method and its success in reaching certain first principles, as contrasted with positions merely assumed for the purpose of argument, that dialectic is discriminated from the other μαθήματα, and it is the business of dialectic to justify by deduction from ἀνυπόθετοι ἀρχαί the unproved assumptions which the ordinary mathematician uses simply as — to employ a technical term which the evidence of Proclus proves to be as old as the first generation of the Academy — αἰτήματα, assumptions which are not self-evident, but merely demanded for the sake of argument. Cf. Proclus, Comment. in Euclid. 188. 8, {Greek omitted}. The reference is to Analytica Posteriora, i. 76 b 23-34.
  To avoid misconception about the character of 'dialectic' it is worthwhile to note that the biological researches of Speusippus known as the "Ομοια or HomologiesDe Speusippi Academici scriptis, 1911), to have been intended in the first instance as a contribution to 'dialectic' (because they subserve the process of classification), and again that the theory of surd magnitudes in Euclid X., in the same way, aims at making a classification of types of surds (see op. cit. Prop, 111, Corollary, which was very probably, as Heiberg suggests, the original conclusion of the book). The work is therefore primarily a piece of dialectic as described in the Philebus, as is only natural when we remember that it completes and embodies the researches of the Academicians Theaetetus, Socrates, and Eudoxus (see Plato, Theaetetus 147 d, [Aristotle], De lineis insecabilibus, 968 b 16 ff., Proclus, op. cit. 66-68 on these mathematical pioneers. The passage of the Theaetetus itself represents the study of surds as from the first a problem in classification.

14^ This may be illustrated by the distinction kept up throughout the book between the surd expressions called respectively μέση and μέσον. The μέση (the full expression is μέσον εὐθεῖα) is defined at X. 21 as the straight line whose square is equal to the rectangle under two straight lines which are only δυνάμει σύμμετροι. Thus, e.g., √a√b would be a μέση since (if a and b are integers, and b not a square number) a and √b are δυνάμει μέσον σύμμετροι. The μέσον (full expression μέσον χωρίον) is the rectangle under two μέσαι which are σύμμετροι whether μήκει or only δυνάμει (X. 24). Hence √ab√cd satisfies the definition of a if you regard it as the square root of abcd; it satisfies that of a μέσον if you regard it as the product of √a√c and √b√d (premising, as before, that c and d are not square numbers). The apparently hard and fast distinction between the type μέση and the type μέσον really depends only on the arbitrary selection of a straight line or a rectangle to symbolise a surd magnitude. The irrelevancy of the εἴδωλον is here the source of a confusion of thought.

15^ Friedlein's text makes Proclus say that it was the problem which Posidonius regarded as investigating "what the thing is" (op. cit." p. 80, l. 22, τὸ δέ πρόβλημα πρότασιν ἐν ᾗ ζητεῖται τι ἐστιν ἢ ποῖόν τι). As this contradicts not only Proclus' previous statement that the doctrine of Posidonius was derived from "Zenodotus who belonged to the succession of Oenopides" and taught that "a theorem asks what is the σύμπτωμα predicated of the matter under consideration," but also his further account of the views of Posidonius himself, I can only suppose the word πρόβλημα to be a mistaken gloss on to τὸ δέ.

16^ Or is it simply meant that we might — since names are a matter of "convention" — have used the word "straight" to mean what we actually call "curved"? This is true, but I do not see how it can be regarded as giving rise to any difficulty.

17^ These are perhaps the most difficult words in the whole of the passage. What are we to understand by this substitution of the irrelevant question for the relevant? I think we may explain the matter as follows. The mind is said to be inquiring into the "what" of something, and the reference to a diagram or model as the source of the confusion shows that the something is a mathematical line or figure of some kind. Its τί or "what" will therefore be its defining characteristic, and to "inquire into" this τί will presumably mean to "construct" the curve or figure or what not, to establish its "existence."You will be committing the inconsequence spoken of if you attempt to demonstrate the συμπτώματα of the line, curve, figure, before showing how to construct it, i.e. before proving that it is one of the objects contained in the geometer's universe of discourse. Thus Proclus' already-quoted observations about the reason for placing Euclid I. 4 after I. 1-3 amount to urging that Euclid would have committed the fault in question if he had investigated certain συμπτώματα of triangles in which two sides and the included angle of the one are equal to two sides and the included angle of the other, without having proved by implication (as Proclus assumes he has done) that a pair of such triangles can exist.
  A more modern illustration would be attempting to establish some theorem about the tangent to a given curve at any point before you have ascertained that the curve has a tangent at any point.

18^ I.e. the man who is attempting to express a true proposition may employ ill- selected terms, or a badly expressed or otherwise imperfect definition, or may attempt to illustrate his meaning by a diagram which does not represent the relations with which he is dealing adequately. You may then make it appear that he is uttering a paradox or that what he says is at variance with what can be seen in the diagram. But you have not in this way proved that what he meant to assert is false. Thus, for example, Aristotle's "refutations" of the Platonic views about the εἰδητικπὶ ἀριθμοί turn in effect on denying that a surd expression such as is a number. ("All number is composed of ones," etc.) This, however, is no criticism of Plato's meaning, but only of his terminology. So, again, it is no disproof of the existence of "transfinite" numbers to say that the arithmetic of "transfinites" contradicts the axiom that "the whole is greater than the part." The contradiction merely shows that the axiom only holds good of collections with a finite number of terms, and is, therefore, as it stands, badly expressed. Or, finally, you do not refute a proof that a given curve has no tangent at a given point (say the "origin") by drawing a diagram in which it looks as if there were a tangent at that point. You only show that the diagram does not represent what it stands for correctly.

19^ For the thought that Philosophy demands kinship between the Reality known and the mind that knows it, see Republic 490 b, οὐδ᾽ ἀπολήγοι τοῦ ἔρωτος, πρὶν αὐτοῦ ὃ ἔστιν ἑκάστου τῆς φύσεως ἅψασθαι ᾧ προσήκει ψυχῆς ἐφάπτεσθαι τοῦ τοιούτου—προσήκει δὲ συγγενεῖ κτλ. As the Republic passage shows that the thought is closely coloured by the imagery of the ἰερὸς γάμος, the allusion in συγγενής may not impossibly be to the Attic Law of the heiress, by which the hand of the orphan ἐπίκληρος fell of right to the next-of-kin. In any case, the meaning is that mere quickness in learning and good memory will never make a philosopher. A special elevation of soul and a peculiar gift of insight is indispensable. Historically we can trace back the thought that the highest intelligence is akin in a special way to the worthiest objects of knowledge to the Orphic belief that while the body to which a soul is temporarily assigned consists of materials drawn from its physical surroundings, the soul itself, being a divinity, comes from "heaven." Cf. Xenophon, Cyropaedia, viii. 7. 20 (in the course of an argument for immortality manifestly drawn for the most part from the Phaedo), διαλυομένου δὲ ἀνθρώπου δῆλά ἐστιν ἕκαστα ἀπιόντα πρὸς τὸ ὁμόφυλον πλὴν τῆς ψυχῆς (earth to earth, etc.) αὕτη δὲ μόνη οὔτε παροῦσα οὔτε ἀπιοῦσα ὁρᾶται, where the suggestion plainly is that the soul also departs at death, πρὸς τὸ ὁμόφυλον, to its cognate gods {Translation: "When the spirit is set free, pure and untrammelled by matter, then it is likely to be most intelligent. And when man is resolved into his primal elements, it is clear that every part returns to kindred matter, except the soul; that alone cannot be seen, either when present or when departing."}; Aristophanes, Clouds, 229, εἰ μὴ κρεμάσας τὸ νόημα καὶ τὴν φροντίδα λεπτὴν καταμείξας ἐς τὸν ὅμοιον ἀέρα. (where the allusion is to the theory of Diogenes of Apollonia — also mentioned in the Phaedo as one which had interested Socrates, — that air is, as we should now say, the vehicle of intelligence); cf. also the lines of the Orphic plate from Petelia, εἰπεῖ: "Γῆς παῖς εἰμι καὶ Οὐρανοῦ ἀστερόεντος, αὐτὰρ ἐμοὶ γενος οὐράνιον" κτλ., and those found at Thurii, καὶ γὰρ ἐγὼν ὑμῶν γένος ὄλβιον εὔχομαι εῖναι κτλ. The humorous parallel of Plato's Timaeus between the shape of the human head and the shape of the οὐράνός is a fanciful expression of the same central idea.

20^ εὐμάθία and μνήμη are similarly demanded as qualifications of the philosopher at Republic 486 c-d, but there, as here, they form only a very small part of Plato's requirements. He also demands a passion for 'truth' or 'reality', indifference to the satisfactions of appetite, "high-mindedness" and freedom from all pettiness of soul (ἀνελευθερία) and from all unworthy fear, and various other qualities which are here aptly summed up in the requirement of a "natural affinity" to the object of philosophical study.

21^ The metaphor of the sudden breaking out of the flame forms, of course, a link with 341 c-d, but it is ingeniously prepared for by an almost untranslatable play on words. It has just been said that much τριβή 'practice' will be required for the acquisition of philosophy. The writer then echoes the word τριβή in the τριβόμενα of the next sentence, where he speaks of the process of repeated alternations of attention between the words and λόγοι of a demonstration and the diagrams which illustrate it, as one of "friction." This leads naturally to the image of the insight which results from the process as a light kindled by "friction." This gradual preparation through an apparently scarcely conscious figurative use of a common word for a full-blown metaphor has always seemed to me characteristically Platonic. The point of the remark that the philosopher has to learn at once τὸ ἀληθές and τὸ ψευδές is, of course, that in the use of dialectic it is precisely by seeing where unsatisfactory hypotheses are false that we are led on to a truer one.

22^ κεῖται δέ που ἐν χώρᾳ τῇ καλλίστῃ τῶν τούτου, that is, ἐν τῇ ψυχῇ, or more precisely ἐν τῷ νῷ; the truths which are of most moment in the opinion of a good and wise man are written on the "fleshly tables of the heart," not on tablets of wax or sheets of parchment. He bears them about with him and does not need to store them in a library.

23^ εἰς ἀναρμοστίαν καὶ ἀπρέπειαν ἐκβάλλειν. If Dionysius had really any understanding of philosophic truth, he would have avoided casting it before swine. He would not have exposed it, by circulating his σύγγραμμα, to the disordered and lewd minds of his court circle.

24^ In this difficult sentence I would punctuate thus: εἰ μὲν οὖν ἐκ τῆς μιᾶς συνουσίας Διονυσίῳ τοῦτο γέγονεν, — τάχ᾽ ἂν εἴη, γέγονεν δ᾽ οὖν ὅπως, 'ἴττω Ζεύς,' φησὶν ὁ Θηβαῖος: διεξῆλθον μὲν γὰρ ὡς εἶπόν τε ἐγὼ καὶ ἅπαξ μόνον, ὕστερον δὲ οὐπώποτε ἔτι — ἐννοεῖν δὴ δεῖ … τίνι πότ᾽ αἰτίᾳ τὸ δεύτερον καὶ τὸ τρίτον, πλεονάκις τε οὐ διεξῇμεν: That is I take, εἰ μὲν οὖν … γέγονεν as a protasis to which ἐννοεῖν δὴ δεῖ κτλ. forms the apodosis. The general meaning is "I only spoke with him once; hence the question arises why we did not go over the ground again and again."The three possible explanations of Dionysius's conduct in not asking for a second interview, viz. that he thought himself after the first conversation competent to carry on his studies by himself, that he thought what he had heard disappointing, that he felt unequal to the demands of philosophy on his intellect and character, correspond to the three possible results of the πεῖρα suitable to a prince enumerated on p. 340.
  As for points of detail, I assume with some hesitation that γέγονεν δ᾽ οὖν ὅπως, ἴττω Ζεύς stands for ἴττω Ζεύς ὅπως γέγονεν, "we may leave it to Zeus to know how it happened," i.g, "only God can say how it came about." Hence I have removed Burnet's comma after ὅπως. In the apodosis I would also remove the comma after τὸ τρίτον, since Plato is insisting that there was only one interview, and the οὐ must therefore negative the whole clause τὸ δεύτερον καὶ τὸ τρίτον, πλεονάκις τε. The only point now left obscure is the meaning of the τοῦτο of 345 a 1. What is it that might be supposed to have happened, God knows best how, to Dionysius after his interview with Plato? Apparently the τοῦτο means the "compassing of the truth" referred to in the previous sentence (ἐὰν ἅπαξ τῇ ψυχῇ περιλάβῃ), so that the whole sense is, "if Dionysius really understood that part of the subject which I put before him at our only interview, — perhaps he did, though God knows how he could, — why did he not seek further instruction?" Two explanations, viz. that what, on the supposition, he already knew was enough to qualify him to philosophise for himself in future, or that he was disappointed in what Plato had to say, are then examined and dismissed, and the implied, though unexpressed conclusion is that the only remaining alternative, viz. that he felt his unfitness for the vocation of a philosopher, must be the true one. (I should say that I owe this explanation of the reference of τοῦτο to a suggestion of Prof. Burnet.)

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Alfred Edward Taylor
Eminent British Idealist philosopher
Fellow, Merton College, Oxford, Professor of Logic and Metaphysics, McGill University, Professor of Moral Philosophy, University of St. Andrews and University of Edinburgh.



Taylor, A. E. Philosophical Studies. London: MacMillan & Co., Ltd, 1934. This work is in the Public Domain.