[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.