The Principles of Mathematics (1903)

§ 163

Finally, it is important to remember that, on the theory adopted in Chapter XIX, a given magnitude of a given kind is a simple concept, having to the kind a relation analogous to that of inclusion in a class. When the kind is a kind of existents, such as pleasure, what actually exists is never the kind, but various particular magnitudes of the kind. Pleasure, abstractly taken, does not exist, but various amounts of it exist. This degree of abstraction is essential to the theory of quantity: there must be entities which differ from each other in nothing except magnitude. The grounds for the theory adopted may perhaps appear more clearly from a further examination of the case.(§ 163 ¶ 1)

Let us start with Bentham's famous proposition: Quantity of pleasure being equal, pushpin is as good as poetry. Here the qualitative difference of the pleasures is the very point of the judgment; but in order to be able to say that the quantities of pleasure are equal, we must be able to abstract the qualitative differences, and leave a certain magnitude of pleasure. If this abstraction is legitimate, the qualitative difference must be not truly a difference of quality, but only a difference of relation to other terms, as, in the present case, a difference in the causal relation. For it is not the whole pleasurable states that are compared, but only—as the form of the judgment aptly illustrates—their quality of pleasure. If we suppose the magnitude of pleasure to be not a separate entity, a difficulty will arise. For the mere element of pleasure must be identical in the two cases, whereas we require a possible difference of magnitude. Hence we can neither hold that only the whole concrete state exists, and any part of it is an abstraction, nor that what exists is abstract pleasure, not magnitude of pleasure. Nor can we say: We abstract, from the whole states, the two elements magnitude and pleasure. For then we should not get a quantitative comparison of the pleasures. The two states would agree in being pleasures, and in being magnitudes. But this would not give us a magnitude of pleasure; and it would give a magnitude to the states as a whole, which is not admissible. Hence we cannot abstract magnitude in general from the states, since as wholes they have no magnitude. And we have seen that we must not abstract bare pleasure, if we are to have any possibility of different magnitudes. Thus what we have to abstract is a magnitude of pleasure as a whole. This must not be analyzed into magnitude and pleasure, but must be abstracted as a whole. And the magnitude of pleasure must exist as a part of the whole pleasurable states, for it is only where there is no difference save at most one of magnitude that quantitative comparison is possible. Thus the discussion of this particular case fully confirms the theory that every magnitude is unanalyzable, and has only the relation analogous to inclusion in a class to that abstract quality or relation of which it is a magnitude.(§ 163 ¶ 2)

Having seen that all magnitudes are indivisible, we have next to consider the extent to which numbers can be used to express magnitudes, and the nature and limits of measurement.(§ 163 ¶ 3)