This general doctrine is illustrated by two particular cases — Speech and Music. The voice (or Vocal Utterance) is One — the voice is also Infinite: to know only thus much is to know very little. Even when you know, in addition to this, the general distinction of sounds into acute and grave, you are still far short of the knowledge of music. You must learn farthermore to distinguish all the intermediate gradations, and specific varieties of sound, into which the infinity of separate sounds admits of being distributed: what and how many these gradations are? what are the numerical ratios upon which they depend — the rhythmical and harmonic systems? When you have learnt to know the One Genus, the infinite diversity of individual sounds, and the number of subordinate specific varieties by which these two extremes are connected with each other — then you know the science of music. So too, in speech: when you can distinguish the infinite diversity of articulate utterance into vowels, semi-vowels, and consonants, each in definite number and with known properties — you are master of grammatical science. You must neither descend at once from the One to the Infinite Multitude, nor ascend at once from the Infinite Multitude to the One: you must pass through the intermediate stages of subordinate Forms, in determinate number. All three together make up scientific knowledge. You cannot know one portion separately, without knowing the remainder: all of them being connected into one by the common bond of the highest Genus.[17]
[17] Plato, Philêbus, p. 18 C-D. καθορῶν δὲ ὡς οὐδεὶς ἡμῶν οὐδ’ ἂν ἓν αὐτὸ καθ’ αὑτὸ ἄνευ πάντων αὐτῶν μάθοι, τοῦτον τὸν δεσμὸν αὖ λογισάμενος ὡς ὄντα ἕνα καὶ πάντα ταῦτα ἓν πως ποιοῦντα, μίαν ἐπ’ αὐτοῖς ὡς οὖσαν γραμματικὴν τέχνην ἐπεφθέγξατο προσειπών.
Plato’s explanation does not touch the difficulties which he had himself recognised as existing.
Such is the explanation which Plato gives as to the identity of One and Many. Considered as a reply to his own previous doubts and difficulties, it is altogether insufficient. It leaves all those doubts unsolved. The first point of enquiry which he had started, was, Whether any Universal or Generic Monads really existed: the second point was, assuming that they did exist, how each of them, being essentially eternal and unchangeable, could so multiply itself or divide itself as to be at the same time in an infinite variety of particulars.[18] Both points are left untouched by the explanation. No proof is furnished that Universal Monads exist — still less that they multiply or divide their one and unchangeable essence among infinite particulars — least of all is it shown, how such multiplication or division can take place, consistently with the fundamental and eternal sameness of the Universal Monad. The explanation assumes these difficulties to be eliminated, but does not suggest the means of eliminating them. The Philêbus, like the Parmenidês, recognises the difficulties as existing, but leaves them unsolved, though the dogmas to which they attach are the cardinal and peculiar tenets of Platonic speculation. Plato shows that he is aware of the embarrassments: yet he is content to theorize as if they did not exist. In a remarkable passage of this very dialogue, he intimates pretty clearly that he considered the difficulty of these questions to be insuperable, and never likely to be set at rest. This identification of the One with the Many, in verbal propositions (he says) has begun with the beginning of dialectic debate, and will continue to the end of it, as a stimulating puzzle which especially captivates the imagination of youth.[19]
[18] Plato, Philêbus, p. 15 B-C.
[19] Plato, Philêbus, p. 15 D. φαμέν που ταὐτὸν ἓν καὶ πολλὰ ὑπὸ λόγων γιγνόμενα περιτρέχειν πάντῃ καθ’ ἕκαστον τῶν λεγομένων ἀεὶ καὶ πάλαι καὶ νῦν. καὶ τοῦτο οὔτε μὴ παύσηταί ποτε οὔτε ἤρξατο νῦν, ἀλλ’ ἔστι τὸ τοιοῦτον, ὡς ἐμοὶ φαίνεται, τῶν λόγων αὐτῶν ἀθάνατόν τι καὶ ἀγήρων πάθος ἐν ἡμῖν.
The sequel (too long to transcribe) of this passage (setting forth the manner in which this apparent paradox worked upon the imagination of youthful students) is very interesting to read, and shows (in my opinion) that Stallbaum’s interpretation of it in his note is not the right one. Plato is here talking (in my judgment) about the puzzle and paradox itself: Stallbaum represents Plato as talking about his pretended solution of it, which has not as yet been at all alluded to.
Plato seems to give his own explanation without full certainty or confidence: see p. 16 B. And when we turn to pp. 18-19, we shall see that he forgets the original difficulty which had been proposed (compare p. 15 B), introducing in place of it another totally distinct difficulty, as if that had been in contemplation.
It is nevertheless instructive, in regard to logical division and classification.
But though the difficulties started by Plato remain unexplained, still his manner of stating them is in itself valuable and instructive. It proclaims — 1. The necessity of a systematic classification, or subordinate scale of species and sub-species, between the highest Genus and the group of individuals beneath. 2. That each of these subordinate grades in the scale must be founded upon some characteristic mark. 3. That the number of sub-divisions is definite and assignable, there being a limit beyond which it cannot be carried. 4. That full knowledge is not attainable until we know all three — The highest Genus — The intermediate species and sub-species; both what they are, how many there are, and how each is characterised — The infinite group of individuals. These three elements must all be known in conjunction: we are not to pass either from the first to the third, or from the third to the first, except through the second.