135 Compare, respecting this Platonic view, Republic, v. pp. 478-479, and pp. 583-585, where Plato contrasts the παναληθὴς or γνησία ἡδονή, which arises from the acquisition of knowledge (when the mind nourishes itself with real essence), with the νόθη (p. 587 B) or ἐσκιαγραφημένη ἡδονή, εἴδωλον τῆς ἀληθοῦς ἡδονῆς, arising from the pursuits of wealth, power, and other objects of desire.
The comic poet Alexis adverts to this Platonic doctrine of the absolutely pleasurable, here, there, and everywhere, — τὸ δ’ ἡδὺ πάντως ἡδύ, κἀκεῖ κἀνθάδε, Athenæ. viii. 354; Meineke, Com. Frag. p. 453.
In the Phædrus (258 E), we find this same class of pleasures, those which cannot be enjoyed unless preceded by some pain, asserted to be called for that reason slavish (ἀνδραποδώδεις), and depreciated as worthless. Nearly all the pleasures connected with the body are said to belong to this class; but those of rhetoric and dialectic are exempted from it, and declared to be of superior order.
The pleasure of gaining a victory in the stadium at Olympia was ranked by Greeks generally as the maximum of pleasure: and we find the Platonic Sokrates (Republ. v. 465 D) speaks in concurrence with this opinion. But this pleasure ought in Plato’s view to pass for a false pleasure; since it was invariably preceded by the most painful, long-continued training.
The reasoning of Sokrates in the Philêbus (see especially pp. 46-47) against the intense and extatic pleasures, as being never pure, but always adulterated by accompanying pain, misfortune, disappointment, &c., is much the same as that of Epikurus and his followers afterwards. The case is nowhere more forcibly put than in the fourth book of Lucretius (1074 seq.): where that poet deprecates passionate love, and points out that pure or unmixed pleasure belongs only to the man of sound and healthy reason.
Plato could not have defended this small list of Pleasures, upon his own admission, against his opponents — the Pleasure-haters, who disallowed pleasures altogether.
The catalogue of pleasures recognised by Plato being so narrow (and much of them attainable only by a few persons), the amount of difference is really very small between him and his pleasure-hating opponents, who disallowed pleasure altogether. But small as the catalogue is, he could not consistently have defended it against them, upon his own principles. His opponents could have shown him that a considerable portion of it must be discarded, if we are to disallow all pleasures which are preceded by or intermingled with pain — or which are sometimes stronger, sometimes feebler, according to the relations of contrast or similarity with other concomitant sensations. Mathematical study certainly, far from being all pleasure and no pain, demands an irksome preparatory training (which is numbered among the miseries of life in the Axiochus136), succeeded by long laborious application, together with a fair share of vexatious puzzle and disappointment. The love of knowledge grows up by association (like the thirst for money or power), and includes an uncomfortable consciousness of ignorance: nay, it is precisely this painful consciousness which the Sokratic method was expressly intended to plant forcibly in the student’s mind, as an indispensable antecedent condition. Requital doubtless comes in time; but the outlay is not the less real, and is quite sufficient to disentitle the study from being counted as a true pleasure, in the Platonic sense. Nor could Plato, upon his own principles, defend the pleasures of sight, sound, and smell. For though he might justly contend that there were some objects originally agreeable to these senses, yet all these objects will appear more or less agreeable, according to the accompanying contrasts under which they are presented, while, in particular states of the organ, they will not appear agreeable at all. Now such variability of estimate is among the grounds alleged by Plato for declaring pleasures to be false.137
136 See the pseudo-Platonic dialogue Axiochus, pp. 366-367. Compare Republic, vii. 526 C, vi. 504 C.
The Sokratic method, in creating consciousness of ignorance, is exhibited not less in the Xenophontic Memorabilia (iv. 2, 40) than in various Platonic dialogues, Alkibiades I., Theætêtus, &c. We read it formally proclaimed by Sokrates in the Platonic Apology.
Aristotle repeats the assertion contained in the Philêbus about the list of painless pleasures — ἄλυποι γάρ εἰσιν αἵ τε μαθηματικαὶ, &c. (Ethic. Nikom. x. 2, 1173, b. 16; 7, 1177, a. 25.) He himself says in another place (vii. 13, 1153, a. 20) that τὸ θεωρεῖν sometimes hurts the health, and if he had examined the lives of mathematicians, especially that of Kepler, he would hardly have imagined that mathematical investigations have no pains attached to them. He probably means that they are not preceded by painful appetites such as hunger and thirst. But they are preceded by acquired impulses or desires, which in reference to the present question are upon same footing as the natural appetites. A healthy and temperate man, leading a regular life and in easy circumstances, knows little of hunger and thirst as pains: he knows them only as appetites which give relish to his periodical meals. It is only when this periodical satisfaction is withheld that his appetite grows to a painful and distressing height. So too the φιλομαθής; his appetite for study, when regularly gratified to an extent consistent with health and other considerations, is not painful; but it will rise to the height of a most distressing privation if he be debarred from gratifying it, excluded from books and papers, disturbed by noises and intrusions. Kepler, if interdicted from pursuing his calculations, would have been miserable. Jason of Pheræ was heard to say that he felt hungry so long as he was not in possession of supreme power — πεινῇν, ὅτε μὴ τυραννοῖ, Aristot. Politic. iii. 4, 1277, a. 24; thus intimating that the acquired appetite of ambition had in his mind reached the same intensity as the natural appetite of hunger.
137 Plato, Philêbus, pp. 41-42. In the Phædon (p. 60 B) Sokrates makes a striking remark on the inseparable conjunction of pleasure with pain generally.
Sokrates in this dialogue differs little from these Pleasure-haters.
How little the Sokrates of this dialogue differs, at the bottom, from the fastidious pleasure-haters, may be seen by the passage in which he proclaims that the life of intelligence alone, without the smallest intermixture of pleasure or pain, is the really perfect life: that the Gods and the divine Kosmos have no enjoyment and no suffering.138 The emotional department of human nature is here regarded as a degenerate and obstructive appendage: so that it was an inauspicious act of the sons of the Demiurgus (in the Timæus139) when they attached the spherical head (the miniature parallel of the Kosmos, with the rotatory movements of the immortal soul in the brain within) at the summit of a bodily trunk and limbs, containing the thoracic and abdominal cavities: the thoracic cavity embodying a second and inferior soul with the energetic emotions and passions — the abdominal region serving as lodgment to a third yet baser soul with the appetites. From this conjunction sprang the corrupting influence of emotional impulse, depriving man of his close parallelism with the Kosmos, and poisoning the life of pure exclusive Intelligence — regular, unfeeling, undisturbed. The Pleasure-haters, together with Speusippus and others, declared that pleasure and pain were both alike enemies to be repelled, and that neutrality was the condition to be aimed at.140 And such appears to me to be the drift of Plato’s reasonings in the Philêbus: though he relaxes somewhat the severity of his requirements in favour of a few pleasures, towards which he feels the same indulgence as towards Homer in the Republic.141 When Ethics are discussed, not upon principles of their own (οἰκεῖαι ἀρχαὶ), but upon principles of Kosmology or Ontology, no emotion of any kind can find consistent place.
138 Plato, Philêbus, p. 33 B.
139 Plato, Timæus, pp. 43 A, 44 D, 69 D, 70-71. The same fundamental idea though embodied in a different illustration, appears also in the Phædon; where Sokrates depicts life as a period of imprisonment, to which the immortal rational soul is condemned, in a corrupt and defective body, with perpetual stream of disturbing sensations and emotions (Phædon, pp. 64-65).
Aristotle observes, De Animâ, i. p. 407, b. 2:— ἐπίπονον δὲ καὶ τὸ μεμίχθαι τῷ σώματι μὴ δυνάμενον ἀπολυθῆναι, καὶ προσέτι φευκτόν, εἴπερ βέλτιον τῷ νῷ μὴ μετὰ σώματος εἶναι, καθάπερ εἴωθέ τε λέγεσθαι καὶ πολλοῖς συνδοκεῖ.
We find in one of the Fragments of Cicero, quoted by Augustin from the lost work Hortensius (p. 485, ed. Orelli):— “An vero, inquit, voluptates corporis expetendæ, quæ veré et graviter dictæ sunt à Platone illecebræ et escæ malorum? Quis autem bonâ mente præditus, non mallet nullas omnino nobis à naturâ voluptates esse datas?” This is the same doctrine as what is ascribed to Speusippus.
140 Aristot. Ethic. Nikom. vii. 14, p. 1153, b. 5; x. 2, p. 1173, a. 8; Aulus Gellius, ix. 5. “Speusippus vetusque omnis Academia voluptatem et dolorem duo mala esse dicunt opposita inter se: bonum autem esse quod utriusque medium foret.”
Compare Plato, Philêbus, pp. 43 D-E, 33 B.
To whom does Plato here make allusion, under the general title of the Fastidious (οἱ δυσχερεῖς) Pleasure-haters? Schleiermacher (note to his translation, p. 487), Stallbaum, and most critics down to Dr. Badham inclusive, are of opinion, that he alludes to Antisthenes — among whose dicta we certainly read declarations expressing positive aversion to pleasure — μανείην μᾶλλον ἢ ἡσθείην Diog. L. vi. 3; compare ix. 101, and Winckelmann, Frag. Antisthen. xii. Mr. Poste, on the contrary, thinks it improbable that Antisthenes is alluded to (see p. 80 of his Philêbus). I confess that I think so too. Mr. Poste points out that these δυσχερεῖς are characterised by Plato (p. 44 B), as μάλα δεινοὺς λεγομένους περὶ φύσιν:— whereas we are informed that speculations on φύσις were neglected by Antisthenes, who confined his attention to τὰ ἠθικά. This is a strong reason for believing that Antisthenes cannot be here meant; and there are some other reasons also.
First, in describing the δυσχερεῖς, Plato notes it as one among their attributes, that they hold in thorough detestation the indecorous pleasures (τὰς τῶν ἀσχημόνων ἡδονάς, ἃς οὓς εἴπομεν δυσχερεῖς μισοῦσι παντελῶς, p. 46 A). Now this is surely not likely to have been affirmed about Antisthenes. It was the conspicuous characteristic of the Cynic sect, begun by Antisthenes, and carried still farther by his pupil Diogenes, that they reduced to its minimum the distinction between the decorous and the indecorous.
Next, we may observe that these δυσχερεῖς, whoever they were, are spoken of with much respect by Plato, even while he combats their doctrine (p. 44 C). I think it not likely that he would have spoken thus of Antisthenes. We are told that there prevailed between the two a great and reciprocal acrimony. And this sentiment is manifested in the Sophistês (p. 251 B), where the opponents whom Plato is refuting are described with the most contemptuous bitterness — and where Schleiermacher, and the critics generally, declare that he alludes to Antisthenes. The passage in the Sophistês represents, in my judgment, the probable sentiment of Plato towards Antisthenes: the passage in the Philêbus is at variance with it.
I imagine that the δυσχερεῖς to whom Plato makes allusion in the Philêbus, are the persons from whom his nephew and successor Speusippus derived the doctrine declared in the first portion of this note. The “vetus omnia Academia” of Aulus Gellius is an exaggerated phrase; but many of the old Academy, or companions of Plato, probably held the theory that pleasure was only one form of evil, — especially the pythagorising Platonici, adopting the tendencies of Plato himself in his old age. That Speusippus was among the borrowers from the Pythagoreans, we know from Aristotle (Eth. Nikom. i. 4, 1096, b. 8).
Now the Pythagorean canon of life, like the Orphic (both of them supposed by Herodotus to be derived in great part from Egypt — ii. 81), was distinguished by a multiplicity of abstinences, disgusts, antipathies, in respect to alimentation and other physical circumstances of life — which were held to be of the most imperative force and necessity; so that offences against them were of all others the most intolerable. A remarkable fragment of the Κρῆτες of Euripides (ed. Dind., vol. ii. p. 912) describes a variety of this purism analogous to the Orphic and Pythagorean:— Πάλλευκα δ’ ἔχων εἴματα, φεύγω γένεσίν τε βρότων, καὶ νεκροθήκης οὐ χριμπτόμενος· τὴν τ’ ἐμψύχων βρῶσιν ἐδεστῶν πεφύλαγμαι. Compare Eurip. Hippol. 957; Alexis Comicus, ap. Athenæ, iv. p. 161. See the work of M. Alfred Maury, Histoire des Religions de la Grèce Antique, vol. iii. pp. 368-384.
It appears to me that the δυσχερεῖς, to whom Plato alludes in the Philêbus, were most probably pythagorising friends of his own; who, adopting a ritual of extreme rigour, distinguished themselves by the violence of their antipathies towards τὰς ἡδονὰς τὰς τῶν ἀσχημόνων. Plato speaks of them with respect; partly because ethical theorists, who denounce pleasure, are usually characterised in reverential terms, as persons of exalted principle, even by those who think their reasonings inconclusive; partly because these men only pushed the consequences of Plato’s own reasonings, rather farther than Plato himself did. In fact they were more consistent than Plato was: for the principles laid down in the Philêbus, if carried out strictly, would go to the exclusion of all pleasures — not less of the few which he tolerates, than of the many which he banishes.
These pythagorising Platonici might well be termed δεινοὶ περὶ φύσιν. They paid much attention to the interpretation of nature, though they did so according to a numerical and geometrical symbolism.
141 Plato, Republic, x. p. 607.
Forced conjunction of Kosmology and Ethics — defect of the Philêbus.
In my judgment, this is one main defect pervading the Platonic Philêbus — the forced conjunction between Kosmology and Ethics — the violent pressure employed to force Pleasures and Pains into the same classifying framework as cognitive Beliefs — the true and the false. In respect to the various pleasures, the dialogue contains many excellent remarks, the value of which is diminished by the purpose to which they are turned.142 One of Plato’s main batteries is directed against the intense, extatic, momentary enjoyments, which he sets in contrast against the gentle, serene, often renewable.143 That the former are often purchasable only at the cost of a distempered condition of body and mind, which ought to render them objects shunned rather than desired by a reasonable man — this is a doctrine important to inculcate: but nothing is gained by applying the metaphorical predicate false, either to them, or to the other classes of mixed pleasures, &c., which Plato discountenances under the same epithet. By thus condemning pleasures in wholesale and in large groups, we not only set aside the innocuous as well as others, but we also leave unapplied, or only half applied, that principle of Measure or Calculation which Plato so often extols as the main item in Summum Bonum.
142 We read in Campbell’s Philosophy of Rhetoric (Book i. ch. 7, pp. 168-170) some very good remarks on the erroneous and equivocal assertions which identify Truth and Good — a thesis on which various Platonists have expended much eloquence. Dr. Campbell maintains the just distinction between the Emotions and Will on one side, and the Understanding on the other.
“Passion” (he says) “is the mover to action, Reason is the guide. Good is the object of the Will; Truth the object of the Understanding.”
143 Plato, Philêbus, p. 45 D. ἐν ὕβρει μείζους ἡδονάς, οὐ πλείους λέγω, &c.
So in the Republic, also, ἡδονὴ ὑπερβάλλουσα is declared to be inconsistent with σωφροσύνη (iii. 402 E).
Directive sovereignty of Measure — how explained and applied in the Protagoras.
In this dialogue as well as others, Measure is thus exalted, and exalted with emphasis, at the final conclusion: but it is far less clearly and systematically applied, as far as human beings are concerned, than in the Protagoras. The Sokrates of the Protagoras does not recognise any pleasures as false — nor any class of pleasures as absolutely unmixed with pain: he does not set pleasure in pointed opposition to the avoidance of pain, nor the intense momentary pleasures to the gentle and more durable. He considers that the whole course of life is a perpetual intermixture of pleasures and pains, in proportions variable and to a certain extent modifiable: that each item in both lists has its proper value, commensurable with the others; that the purpose of a well-ordered life consists, in rendering the total sum of pleasure as great, and the total sum of pain as small, as each man’s case admits: that avoidance of pain and attainment of pleasure are co-ordinate branches of this one comprehensive End. He farther declares that men are constantly liable to err by false remembrances, estimates, and comparisons, of pleasures and pains past — by false expectations of pleasures and pains to come: that the whole security of life lies in keeping clear of such error — in right comparison of these items and right choice between them: that therefore the full sovereign controul of each man’s life must be vested in the Measuring Science or Calculating Intelligence.144 Not only all comprehensive sovereignty, but also ever-active guidance, is postulated for this Measuring Science: while at the same time its special function, and the items to which it applies, are more clearly defined than in any other Platonic dialogue. If a man be so absorbed by the idea of an intense momentary pleasure or pain, as to forget or disregard accompaniments or consequences of an opposite nature, greatly overbalancing it — this is an error committed from default of the Measuring Science: but it is only one among many errors arising from the like deficiency. Nothing is required but the Measuring Science or Intelligence, to enable a man to make the best of those circumstances in which he may be placed: this is true of all men, under every variety of place and circumstances. Measure is not the Good, but the one condition which is constant as well as indispensable to any tolerable approach towards Good.
144 This argument is carried on by Sokrates from p. 351 until the close of the Protagoras, p. 357 A. ἐπειδὴ δὲ ἡδονῆς τε καὶ λύπης ἐν ὀρθῇ τῇ αἱρέσει ἐφάνη ἡμῖν ἡ σωτηρία τοῦ βίου οὖσα, τοῦ τε πλέονος καὶ ἐλάττονος καὶ μείζονος καὶ σμικροτεροῦ καὶ ποῤῥωτέρω καὶ ἐγγυτέρω, ἆρα πρῶτον μὲν οὐ μετρητικὴ φαίνεται, ὑπερβολῆς τε καὶ ἐνδείας οὖσα καὶ ἰσότητος πρὸς ἀλλήλας σκέψις; … Ἐπεὶ δὲ μετρητική, ἀνάγκῃ δήπου τέχνη καὶ ἐπιστήμη.
Yet Plato in the Philêbus, imputing to the Hedonistic theory that it sets aside all idea of measure, regulation, limit, advances as an argument in the case, that Pleasure and Pain in their own nature have no limit (Philêbus, pp. 25-26 B, 27 E. Compare Dr. Badham’s note, p. 30 of his edition).
The imputation is unfounded, and the argument without application, in regard to the same theory as expounded by Sokrates in the Protagoras.
At the end of the Philêbus (p. 67 B) Plato makes Sokrates exclaim, “We cannot put Pleasure first among the items of Good, even though all oxen, horses, and other beasts affirm it”. This rhetorical flourish is altogether misplaced in the Philêbus: for Plato had already specified it as one of the conditions of the Good, That it must be acceptable and must give satisfaction to all animals, and even to all plants (pp. 22 B, 60 C), as well as to men.
How explained in Philêbus — no statement to what items it is applied.
In the Philêbus, too, Measure — The Exact Quantum — The Exact Moment — are proclaimed as the chief item in the complex called — The Good.145 But to what Items does Sokrates intend the measure to be applied? Not certainly to pleasures: the comparison of quantity between one pleasure and another is discarded as useless or misleading, and the comparison of quality alone is admitted — i. e., true and false: the large majority of human pleasures being repudiated in the lump as false, and a small remnant only being tolerated, on the allegation that they are true. Nor, again, is the measure applied to pains: for though Plato affirms that a life altogether without pains (as without pleasures) would be the truly divine Ideal, yet he never tells us that the Measuring Intelligence is to be made available in the comparison and choice of pains, and in avoidance of the greater by submitting to the less. Lastly, when we look at the concession made in this dialogue to Gorgias and his art, we find that Plato no longer claims for his Good or Measure any directive function, or any paramount influence, as to utility, profit, reputation, or the greater ends which men usually pursue in life:146 he claims for it only the privilege of satisfying the aspiration for truth, in minds wherein such aspiration is preponderant over all others.
145 Plato, Philêbus, p. 66 A. μέτρον — τὸ μέτριον — τὸ καίριον.
146 Plato, Philêbus, p. 58 B-D.
Comparing the Philêbus with the Protagoras, therefore, we see that though, in both, Measuring Science or Intelligence is proclaimed as supreme, the province assigned to it in the Philêbus is comparatively narrow. Moreover the practical side or activities of life (which are prominent in the Protagoras) appear in the Philêbus thrust into a corner; where scanty room is found for them on ground nearly covered by the speculative, or theorising, truth-seeking, pursuits. Practical reason is forced into the same categories as theoretical.
The classification of true and false is (as I have already remarked) unsuitable for pleasures and pains. We have now to see how Plato applies it to cognitions, to which it really belongs.
Classification of true and false — how Plato applies it to Cognitions.
The highest of these Cognitions is set apart as Dialectic or Ontology: the Object of which is, Ens or Entia, eternal, ever the same and unchangeable, ever unmixed with each other: while the corresponding Subject is, Reason, Intelligence, Wisdom, by which it is apprehended and felt. In this Science alone reside perfect Truth and Purity. Where the Objects are shifting, variable, mixed or confounded together, there Reason cannot apply herself; no pure or exact truth can be attained.147 These unchangeable Entities are what in other dialogues Plato terms Ideas or Forms — a term scarcely used in the Philêbus.
147 Plato, Philêbus, p. 59 C. ὡς ἢ περὶ ἐκεῖνα ἔσθ’ ἡμῖν τό τε βέβαιον καὶ τὸ καθαρὸν καὶ τὸ ἀληθὲς καὶ ὃ δὴ λέγομεν εἰλικρινές, περὶ τὰ ἀεὶ κατὰ τὰ αὐτὰ ὡσαύτως ἀμικτότατα ἔχοντα — ἢ δευτέρως ἐκείνων ὅ τι μάλιστά ἐστι ξυγγενές· τὰ δ’ ἄλλα πάντα δεύτερά τε καὶ ὕστερα λεκτέον. 62 A: φρονῶν ἄνθρωπος αὑτῆς περὶ δικαιοσύνης, ὅ, τι ἔστι, καὶ λόγον ἔχων ἑπόμενον τῷ νοεῖν … κύκλου μὲν καὶ σφαίρας αὐτῆς τῆς θείας τὸν λόγον ἔχων.
Though pure truth belongs exclusively to Dialectic and to the Objects thereof, there are other Sciences which, having more or less of affinity to Dialectic, may thus be classified according to the degree of such affinity. Mathematics approach most nearly to Dialectic. Under Mathematics are included the Sciences or Arts of numbering, measuring, weighing — Arithmetic, Metrêtic, Static — which are applied to various subordinate arts, and impart to these latter all the scientific guidance and certainty which is found in them. Without Arithmetic, the subordinate arts would be little better than vague guesswork or knack. But Plato distinguishes two varieties of Arithmetic and Metrêtic: one purely theoretical, prosecuted by philosophers, and adapted to satisfy the love of abstract truth — the other applied to some department of practice, and employed by the artist as a guide to the execution of his work. Theoretical Arithmetic is characterised by this feature, that it assumes each unit to be equal, like, and interchangeable with every other unit: while practical Arithmetic adds together concrete realities, whether like and equal to each other or not.148
148 Plato, Philêbus, p. 56 E.
It is thus that the theoretical geometer and arithmetician, though not coming up to the full and pure truth of Dialectic, is nevertheless nearer to it than the carpenter or the ship-builder, who apply the measure to material objects. But the carpenter, ship-builder, architect, &c., do really apply measure, line, rule, &c.: they are therefore nearer to truth than other artists, who apply no measure at all. To this last category belong the musical composer, the physician, the husbandman, the pilot, the military commander, neither of whom can apply to their processes either numeration or measurement: all of them are forced to be contented with vague estimate, conjecture, a practised eye and ear.149
149 Plato, Philêbus, p. 56 A-B.
Valuable principles of this classification — difference with other dialogues.
The foregoing classification of Sciences and Arts is among the most interesting points in the Philêbus. It coincides to a great degree with that which we read in the sixth and seventh books of the Republic, though it is also partially different: it differs too in some respects from doctrines advanced in other dialogues. Thus we find here (in the Philêbus) that the science or art of the physician, the pilot, the general, &c., is treated as destitute of measure and as an aggregate of unscientific guesses: whereas in the Gorgias150 and elsewhere, these are extolled as genuine arts, and are employed to discredit Rhetoric by contrast. Again, all these arts are here placed lower in the scientific scale than the occupations of the carpenter or the ship-builder, who possess and use some material measures. But these latter, in the Republic,151 are dismissed with the disparaging epithet of snobbish (βάναυσοι) and deemed unworthy of consideration.
150 Plato, Gorgias, pp. 501 A, 518 A. Compare Republic, i. pp. 341-342.
151 Plato, Republic, vii. p. 522 B.
Dialectic appears here exalted to the same pre-eminence which is assigned to it in the Republic — as the energy of the pure Intellect, dealing with those permanent real Essences which are the objects of Intellect alone, intelligible only and not visible. The distinction here drawn by Plato between the theoretical and practical arithmetic and geometry, compared with numeration or mensuration of actual objects of sense — is also remarkable in two ways: first, as it marks his departure from the historical Sokrates, who recognised the difference between the two, but discountenanced the theoretical as worthless:152 next as it brings clearly to view, the fundamental assumption or hypothesis upon which abstract arithmetic proceeds — the concept of units all perfectly like and equal. That this is an assumption (always departing more or less from the facts of sense) — and that upon its being conceded depends the peculiar certainty and accuracy of arithmetical calculation — was an observation probably then made for the first time; and not unnecessary to be made even now, since it is apt to escape attention. It is enunciated clearly both here and in the Republic.153
152 Xenophon, Memorab. iv. 7, 2-8. The contrast drawn in this chapter of the Memorabilia appears to me to coincide pretty exactly with that which is taken in the Philêbus, though the preference is reversed. Dr. Badham (p. 78) and Mr. Poste (pp. 106-113) consider Plato as pointing to a contrast between pure and applied Mathematics: which I do not understand to be his meaning. The distinction taken by Aristotle in the passage cited by Mr. Poste is different, and does really designate Pure and Applied Mathematics. Mr. Poste would have found a better comparison in Ethic. Nikom. i. 7, 1098, a. 29.
153 Plato, Philêbus, p. 56 E. οἱ δ’ οὐκ ἄν ποτε αὐτοῖς συνακολουθήσειαν, εἰ μὴ μονάδα μονάδος ἑκάστης τῶν μυρίων μηδεμίαν ἄλλην ἄλλης διαφέρουσάν τις θήσει — where it is formally proclaimed as an assumption or postulate. See Republic, vii. pp. 525-526, vi. p. 510 C.
Mr. John Stuart Mill thus calls attention to the same remark in his instructive chapters on Demonstration and Necessary Truth (System of Logic, Book ii. ch. vi sect. 3).
“The inductions of Arithmetic are of two sorts: first, those that we have just expounded, such as One and One are Two, Two and One are Three, &c., which may be called the definitions of the various numbers, in the improper or geometrical sense of the word Definition; and, secondly, the two following Axioms. The sums of Equals are equal, the differences of Equals are equal.
“These axioms, and likewise the so-called Definitions, are (as already shown) results of induction: true of all objects whatsoever, and as it may seem, exactly true, without the hypothetical assumption of unqualified truth where an approximation to it is all that exists. On more accurate investigation, however, it will be found that even in this case, there is one hypothetical element in the ratiocination. In all propositions concerning numbers a condition is implied without which none of them would be true, and that condition is an assumption which may be false. The condition is that 1 = 1: that all the numbers are numbers of the same or of equal units. Let this be doubtful, and not one of the propositions in arithmetic will hold true. How can we know that one pound and one pound make two pounds, if one of the pounds may be troy and the other avoirdupois? They may not make two pounds of either or of any weight. How can we know that a forty-horse power is always equal to itself, unless we assume that all horses are of equal strength? One actual pound weight is not exactly equal to another, nor one mile’s length to another; a nicer balance or more exact measuring instruments would always detect some difference.”
The long preliminary discussion of the Philêbus thus brings us to the conclusion — That a descending scale of value, relatively to truth and falsehood, must be recognised in cognitions as well as in pleasures: many cognitions are not entirely true, but tainted in different degrees by error and falsehood: most pleasures also, instead of being true and pure, are alloyed by concomitant pains or delusions or both: moreover, all the intense pleasures are incompatible with Measure, or a fixed standard,154 and must therefore be excluded from the category of Good.
154 Plato, Philêbus, pp. 52 D — 57 B.
Close of the Philêbus — Graduated elements of Good.
In arranging the quintuple scale of elements or conditions of the Good, Plato adopts the following descending order: I report them as well as I can, for I confess that I understand them very imperfectly.
1. Measure; that which conforms to Measure and to proper season: with everything else analogous, which we can believe to be of eternal nature. — These seem to be unchangeable Forms or Ideas, which are here considered objectively, apart from any percipient Subject affected by them.155
2. The Symmetrical, Beautiful, Perfect, Sufficient, &c. — These words seem to denote the successive manifestations of the same afore-mentioned attributes; but considered both objectively and subjectively, as affecting and appreciated by some percipient.
3. Intelligent or Rational Mind — Here the Subject is brought in by itself.
4. Sciences, Cognitions, Arts, Right Opinions, &c. — Here we have the intellectual manifestations of the Subject, but of a character inferior to No. 3, descending in the scale of value relatively to truth.
5. Lastly come the small list of true and painless pleasures. — These, being not intellectual at all, but merely emotional (some as accompaniments of intellectual, others of sensible, processes), are farther removed from Good and Measure than even No. 4 — the opining or uncertain phases of the intellect.156
The four first elements belong to the Kosmos as well as to man: for the Kosmos has an intelligent soul. The fifth marks the emotional nature of man.