Note 153 — Chapter XXXII · PHILEBUS. (continued 2)

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.”