¹ , which is being used in this context in the restricted sense of coming to have propositional knowledge. The acquisition of inarticulate skills, though well within the scope of the word in ordinary usage, is tacitly excluded.

² My interpretation has much in common with those offered by the following:

Cornford, F. M., Principium Sapientiae (Cambridge, 1952), Ch. IV, “Anamnesis.”Google Scholar

Guthrie, W. K. C., Plato: Protagoras and Meno (London, 1956), 107–114.Google Scholar

Bluck, R. S., Plato's Meno (Cambridge, 1961), 8–17.Google Scholar

Gulley, N., Plato's Theory of Knowledge (London, 1962), Ch. I, “The Theory of Recollection.”Google Scholar

Crombie, I. M., An Examination of Plato's Doctrines, Vol. II (London, 1963), 50–52, 136–41.Google Scholar

To each of these works I shall refer hereafter merely by the author's name.

³ Though Greek mathematicians were occasionally misled by their diagrams to assume some proposition not listed in their axiom-set (e.g. a continuity postulate, needed for the proof of I, 1, etc. in Euclid: cf. T. L. Heath, The Thirteen Books of Euclid's Elements, I² [Oxford, 1925], 235 and 243), they would not dream of citing the sensible properties of a diagram as a reason in a proof. One cannot imagine a sentence like, “This must be true because that is the way it looks (or, measures) in th e diagram,” in a Greek mathematical text.

⁴ The only required changes would be the substitution of “figuring” and “arithmetic” for “doing geometry” and “geometry” at 85E.

⁵ Apart from being so much drier than Plato's example, the main loss resulting from the substitution would be the boy's mistakes; but we could easily make room for these, e.g. by having him make a wrong guess to begin with and then find out by the same method that (and why) his guess was wrong. A graver defect in my example is that it would not show nearly as well as Plato's the gap that may exist between discovery and proof; finding out that 13 + 7 = 20 by the above method would bring one much closer to seeing why this must be so than the slave could have come to seeing the why of the theorem at the end of the interrogation.

⁶ That a + (b + c) = (a + b) + c was used, without being mentioned, in the example.

⁷ A. E. Taylor, Plato, the Man and his Work (London, 1937) [hereafter “Taylor”], 137. Taylor then refers to “the suggestions provided by Socrates’ diagrams and questions” (my italics), apparently failing to realize that the logical status of suggestions provided by questions is entirely different from that provided by sense-experience. In the Meno Plato speaks of recollected opinions as suggested (“awakened”) by questions (86A 6; cf. Phaedo 73A 7), not by sense-experence; the latter point is first made in the Phaedo (73C 6ff.).

⁸ The equality of the sides was mentioned at the start (82C), that of the angles was not, but would have been admitted right off by the boy: the concept, equality of angle, would have been familiar, and Socrates would have had no difficulty in getting the boy to say that all 4 angles of a square must be equal.

⁹ It is not unreasonable to assume with Guthrie (110) that the figure would be only “roughly” drawn, so that the two triangles would be visibly unequal. But nothing is made of this in the interrogation. Socrates has other ways of getting across the idea that the properties of the squares, triangles, etc., he is talking about are those that a figure would have if it instantiated the concept, square. See next note.

¹⁰ Subtler suggestions to the same effect are conveyed by the form in which the questions are put almost from the very start: “Now could not such a figure be either larger or smaller?” (83C 2–3) puts the inquiry in the domain of possibility, where it is kept by the next question, “Now if this side were 2 ft. long and that [side] the same, how many feet would the whole be?,” which puts the specification of size in the hypothetical mode and asks what would happen if this were the case. The same modalities are signalled by the syntactical form of the sequel: optative with ἄν (“indefinite supposition”) in the apodosis at 82C 5–6, and imperfect indicative with ἄν (suppositio irrealis) at C 8 (Cf. E. S. Thompson, The Meno of Plato [London, 1901], ad loc). In the next question (D 1–2) the γíγνεται expresses a logical consequence (this is what would result “because [on the hypothesis] it is 2 ft. long that way too.”) The interrogation continues on the hypothetical plane until the second break at 84D: the question remains what would have to be the case to satisfy the conditions laid down at 82E 1–2 and what would follow if we were to suppose with the slave that the required line is 3 or 4 feet long. After the break the syntax is again well stocked with optatives with ἄν to re-establish a framework of inference (exploration of logical consequences) rather than factual observation. —The English reader should remember that the modalities and logical connectives do not always come through even in excellent translations. Thus 82B 10 - C 3 becomes in Guthrie, “It has all these 4 sides equal? . . . And these lines which go through the middle of it are also equal?” Here the δν (“in inferences, then, therefore,” Liddell and Scott, Lexicon, s. v. III) has dropped out in the first question; in the second one would miss the fact that a participial form (ἔχδν) so links it with the preceding question as to keep it wthin the field of force of the δν. A more literal translation would be: “Is inot a square, then, a figure having all these four sides equal? . . . And having these lines that go through the middle equal also?”

¹¹ So it would be obviously wrong to say that the lad “began by not knowing something and ended by knowing it” (Taylor, 138, my italics), rather than that he would have ended by knowing it.

¹² And cf. 85E 1–2: the slave-boy “will do the same thing [as he has done in the preceding interrogation] in the case of the whole of geometry and of all other sciences” (“the same thing,” τατὰ τατα, here has the same reference as the same expression at G 10-11, where the reference is clearly to his performance in answering Socrates' questions).

¹³ Cf. n. 3 above.

¹⁴ The received translations, down to Guthrie's (“And the spontaneous recovery of knowledge that is in him is recollection, isn't it?”), put Plato in the position of saying that the subject already has the knowledge he recollects, thus flatly contradicting his earlier assurances that the boy did not know, and still does not know, the theorem he has discovered, but has only a true belief of it (85C 2–10). Surely all we can get from the wording in 85D 6 is that the “recollected” knowledge is being “recovered” from inside a person's own mind—not that it is already there as knowledge. The commentators frequently represent Plato as holding that what we come to know by “recollecting” is already present in us in the form of latent knowledge. But Plato never uses this expression (or variants of it, like “potential”) in the context of the theory of recollection. He does not picture our souls as being always in a state of “virtual” omniscience, but as having once “learned” everything (86A 8, where does not mean “has been for ever in a state of knowledge” [Guthrie], but “has been for ever in the condition of having [once] acquired knowledge”: cf. μεμαθυκίας τῆς ψυχῆς, 81D 1), and then lost this knowledge (95C 6-7; 86B 2-3; and cf. especially Phdo 76B 5-C 3), while retaining the ability to recover it. By “the truth of things being always in the soul” (Meno 86B 1-2) and “knowledge and right reason being in” us (Phdo 73A 9-10, ) Plato can only mean that all men have (i) some (not, all) knowledge, (ii) the ability to make correct judgments ( = to perceive logical relations) and, therefore, (iii) the ability to extend their knowledge (by persevering in inquiry) without any preassigned limit (81D 2-4).

¹⁵ : “until you tether them by working out the reason” (Guthrie). “Cause” for aitia here (Jowett, Meridier, Bluck) is misleading, since modern philosophical usage reserves the term for relations which instantiate laws of nature, never for purely logical conditions. Thus to speak of the premises of a syllogism as the of the conclusion (Aristotle, Post. An. 71B 22) would be the crudest sort of category-mistake if Aristotle's term did mean what we understand by “cause.” In some contexts, as in Aristotle's “four causes,” the canonical mistranslation will no doubt have to be perpetuated. But readers of Plato, at least, can be spared some confusion if the mistranslation is avoided when avoidable, as it is certainly in the Meno. To tolerate “chain of causal reasoning” for , and illustrate by a mathematical diorismos (Gulley, 14-15) which involves no causal reasoning whatever, is disconcerting.

¹⁶ Cf. Bluck ad loc: “No mention has been made earlier, at least in so many words, of an , but this reference is clearly to 85C 9-D 1.”

¹⁷ And cf. 82E 12-13.

¹⁸ In the reference to , 82D 4.

¹⁹ Examples in F. Ast, Lexicon Plalonicum, s. v. . When the Theory of Ideas is introduced (along with ) stand for the mode of their apprehension in sharp opposition to sense-perception: Phdo 65C 2-3, 79A 3, Phdr. 249B, Parm. 130A, Soph. 248A 11.

²⁰ H. Diels and W. Kranz, Die Fragmente der Vorsokratiker ⁵ (Berlin, 1934-37), Frag. B 8, 30-31. (All subsequent references to Presocratic fragments will be to this work.) The same metaphor in lines 14 and 37 of the fragment, with Dike and Moira taking the place of Ananke, symbolizing the rational appropriateness of the bond , while Ananke stands for its inexorable necessity.

²¹ Frag. B1 (twice), B3.

²² Frag. B7 (twice).

²³ Examples in Ast, op. cit.

²⁴ i.e. of an axiomatized science. Though great progress in axiomatization was made in Plato's own life-time, (cf. the references to Leo and Theudius in Proclus, Comment, in Eucl. [G. Friedlein], 66, 19-22 and 67, 12-16), there is no reason to think there had been no earlier work along the same lines. The distinction between primitive and derivative propositions in geometry would certainly have been well established by the end of the fifth century.

²⁵ I am extrapolating from the line of argument followed by Socrates in the Phaedo (74B 4 ff.: from certain judgments we have been making since our childhood it is inferred that , 74 E 9)

²⁶ ‘Real,’ not ‘nominal,’ definitions, which are the prime object of Socratic inquiry in many dialogues, including the Meno, where Socrates starts by diverting Meno from ‘Is virtue teachable?,’ to ‘What is virtue?,’ as the logically prior one and insists repeatedly that we cannot know any of virtue's properties or ) until we have come to know its essence (): 71B 3-8; 86D 2-E 1; 100B 4-6. Cf. R. Robinson, Plato's Earlier Dialectic ² (Oxford, 1953), 50-51, where the same point is made strongly and backed with a plethora of additional references.

²⁷ When someone proposes a false definition there are two ways of disproving it in the Socratic dialogues:

(1) Find cases which, as he admits, instantiate the definiens, but not the defmiendum, or the latter, but not the former.

(2) Find propositions known to him which contradict the definition. Socrates could not hope to demonstrate the true definition by the same, or analogous, methods:

(1) He could not go through all the cases of the definiens to show they all exemplify the definiendum.

(2) A statement of what X is could not be proved by entailment from other statements about X which are known to be true, since Socrates holds (cf. the preceding note) that if the essence of X is not known nothing else can be known about X (though, of course, there could be many true beliefs about it). Hence, though Plato does not say so, it would follow that, while argument can disprove incorrect answers to the ‘What is X?’ question, it cannot prove the correct one.

²⁸ If one rereads the interrogation in our text in the light of these two paragraphs, one will see how deductive inference and analytic insight into concepts are called into play just as far as they can within the practical limitations of the occasion (dealing with a boy utterly ignorant of the vocabulary and method of geometry, and getting results with a speed consistent with the dramatic tempo of the dialogue). Thus the correction of the two mistakes 83A-E is for all practical purposes a proof that the two erroneous propositions (that the side is 4 feet or that it is 3 feet) are inconsistent with the theorem that the area of a square with side x feet long must be x ² square feet. Given more time Socrates could surely have got the boy to grasp a formal proof of this theorem of a sufficiently rigorous sort to pass contemporary mathematical standards. So far from giving this proof, Socrates does not even give a general statement of the theorem, and for the simple reason that even to get the boy to understand such a statement would take longer than the dramatic time-budget allows. For the same reason he does not take time to dot the i's and cross the t's of items which are matters of conceptual insight. Thus the only feature of a square mentioned at the start is the equality of its sides, this being enough to get the boy's mind moving in the right direction toward the major objective, i.e. to come in view of the concept of superficial (in contradistinction to linear) magnitude, since everything in the sequel will depend on the boy's ability to see the difference between the size of an area, with its two parameters of length and breadth, and that of one-dimensional magnitudes. The boy cannot even understand Socrates' question, let alone get into position to attempt its solution, until he gets some inkling of this difference. When the question is first put to him at 82 C 5-6, “Now look at it this way: If this line were 2 feet long and that line also 2 feet, how big would be the whole [square, i.e. its area]?,” he is stumped.

²⁹ Untypical not only for common usage (as is obvious), but also for Plato's own: so far from thinking “teaching” (rightly understood, as dialectic) incompatible with “learning,” he distinguishes (Gorg. 454E) rhetoric from “teaching” (διδασκαλιῆς, 455A 1) as producing respectively “belief” (πίστις) and “knowledge” (έπιστήμη) and, conversely (Tm 51 E) nous from true belief as produced respectively by “teaching” (διδαχῆς) and “persuasion.”

³⁰ Cf. A. E. Taylor's comment on Tm 51D 3 in Commentary to Plato's Timaeus (Oxford, 1928), 338-39.

³¹ Cf. Leibniz's use of expressions like “prendre de chez soi,” “tirer de son propre fonds,” for our coming to know necessary truths, and of the mind (or the understanding) as the “source” of these truths: Nouveaux essais sur l'entendement humain, Book I, Chapter I, Section 5.

³² Cf. the empiricist theory of the origin of knowledge which is mentioned as a part of the teaching of the natural philosophers in the Phaedo (96B).

³³ This is precisely what Leibniz takes to be the point of the expressions in n. 31 above, alluding specifically to Platonic anamnesis: “. . . on doit dire que toute l'arithmétique et toute la géométrie sont innées et sont en nous d'une manière virtuelle, en sorte qu'on les y peut trouver en considérant attentivement et rangeant ce qu'on a déjà dans l'esprit, sans se servir d'aucune vérité apprise par l'expérience ou par la tradition d'autrui, comme Platon l'a montré [in the interrogation of the slave-boy in the Meno],” loc. cit., my italics.

³⁴ This would follow, regardless of their other doctrines, from their denial of plurality.

³⁵ E.g. Paul Tannery, Pour l'histoire de la science grecque (Paris, 1877), La Géométric grecque (Paris, 1877), 124; H.-G. Zeuthen, “Sur les livres arithmétiques des Éléments d'Euclide,” Oversigt det Kongelike Danske Videnskabemes Selskabs, Forhandlinger, 1910, 395 ff. at 432-34; F. M. Cornford, Plato and Parmenides (London, 1939), 58-61; H. Hasse and H. Scholz, “Die Grundlagenkrisis der griechischen Mathematik,” Quellenhandbücher der (Berlin, 1928). Contra: B. L. van der Waerden, “Zenon und die Grundlagenkrisis der griechischen Mathematik,” Math. Annallen 117 (1940-41), 141 ff.; G.E.L. Owen, “Zeno and the Mathematicians,” Proc. Arist. Soc., 1958, 199 ff.

³⁶ Pr. Anal. 67A 21-22; cf. also Post. Anal. 71A 1-B 8: 99B 25-34. Cf. H. Cherniss, Aristotle's Criticism of Plato and the Academy (Baltimore, 1944), 69 ff and notes.

³⁷ According to A. E. Taylor it had been “the mathematician-saint Pythagoras” himself who had converted the theological doctrine of the transmigration of the soul “into a theory of the a priori character of mathematics,” Plato, 186, n. 2. For a sane discussion of the historical question see L. Robin, “Sur la doctrine de la reminiscence,” Rev. des Études Grecques 32 (1919), 451-61; but Robin is confused on the point to which I called attention in n. 11 above: he says that Plato “suppose que nous naissons avec des connaissances toutes faites . . . les seules qui soient dignes de ce nom,” 460.

³⁸ Cf. Gulley, 18.

³⁹ Nouveaux essais sur l'entendement humain, Book I, Chapter I, Paragraph 5.

⁴⁰ An earlier draft of this paper was included in the John Locke Lectures on “Mysticism and Logic in Greek Philosophy” which I delivered in Oxford in 1960 and will be eventually published. I wish to thank all those who have critized that draft, most particularly Mr. Yukio Kachi.