The Principles of Mathematics (1903)

Note to Chapter XIX.

The work of Herr Meinong on Weber's Law, already alluded to, is one from which I have learnt so much, and with which I largely agree, that it seems desirable to justify myself on the points in which I depart from it. This work begins (§ 1) by a characterization of magnitude as that which is limited towards zero. Zero is understood as the negation of magnitude, and after a discussion, the following statement is adopted (p. 8):(Ch. 19 n. 1 ¶ 1)

That is or has magnitude, which allows the interpolation of terms between itself and its contradictory opposite.(Ch. 19 n. 1 ¶ 2)

Whether this constitutes a definition, or a mere criterion, is left doubtful (ib.), but in either case, it appears to me to be undesirable as a fundamental characterization of magnitude. It derives support, as Herr Meinong points out (p. 6 n.), from its similarity to Kant's Anticipation of Perception[110] But it is, if I am not mistaken, liable to several grave objections. In the first place, the whole theory of zero is most difficult, and seems subsequent, rather than prior, to the theory of other magnitudes. And to regard zero as the contradictory opposite of other magnitudes seems erroneous. The phrase should denote the class obtained by negation of the class magnitudes of such and such a kind; but this obviously would not yield the zero of that kind of magnitude. Whatever interpretation we give to the phrase, it would seem to imply that we must regard zero as not a magnitude of the kind whose zero it is. But in that case it is not less than the magnitudes of the kind in question, and there seems no particular meaning in saying that a lesser magnitude is between zero and a greater magnitude. And in any case, the notion of between, as we shall see in Part IV, demands asymmetrical relations among the terms concerned. These relations, it would seem, are, in the case of magnitude, none other than greater and less, which are therefore prior to the betweenness of magnitudes, and more suitable to definition. I shall endeavour at a later stage to give what I conceive to be the true theory of zero; and it will then appear how difficult this subject is. It can hardly be wise, therefore, to introduce zero in the first account of magnitude. Other objections might be urged, as, for instance, that it is doubtful whether all kinds of magnitude have a zero; that in discrete kinds of magnitude, zero is unimportant; and that among distances, where the zero is simply identity, there is hardly the same relation of zero to negation or non-existence as in the case of qualities such as pleasure. But the main reason must be the logical inversion involved in the introduction of between before any asymmetrical relations have been specified from which it could arise. This subject will be resumed in Chapter XXII.(Ch. 19 n. 1 ¶ 3)

Ch. 19 n. 1 n. 1. Reine Vernunft, ed. Hartenstein (1867), p. 158.