The Principles of Mathematics (1903)

§ 158

We may now sum up the above discussion in a brief statement of results. There are a certain pair of indefinable relations, called greater and less; these relations are asymmetrical and transitive, and are inconsistent the one with the other. Each is the converse of the other, in the sense that, whenever the one holds between A and B, the other holds between B and A. The terms which are capable of these relations are magnitudes. Every magnitude has a certain peculiar relation to some concept, expressed by saying that it is a magnitude of that concept. Two magnitudes which have this relation to the same concept are said to be of the same kind; to be of the same kind is the necessary and sufficient condition for the relations of greater and less. When a magnitude can be particularized by temporal, spatial, or spatio-temporal position, or when, being a relation, it can be particularized by taking into a consideration a pair of terms between which it holds, then the magnitude so particularized is called a quantity Two magnitudes of the same kind can never be particularized by exactly the same specifications. Two quantities which result from particularizing the same magnitude are said to be equal.(§ 158 ¶ 1)

Thus our indefinables are (1) greater and less, (2) every particular magnitude. Our indemonstrable positions are:(§ 158 ¶ 2)

  1. Every magnitude has to some term the relation which makes it of a certain kind.(§ 158 ¶ 3)

  2. Any two magnitudes of the same kind are one greater and the other less.(§ 158 ¶ 4)

  3. Two magnitudes of the same kind, if capable of occupying space or time, cannot both have the same spatio-temporal position; if relations, can never be both relations between the same pair of terms.(§ 158 ¶ 5)

  4. No magnitude is greater than itself.(§ 158 ¶ 6)

  5. If A is greater than B, B is less than A, and vice versâ.(§ 158 ¶ 7)

  6. If A is greater than B and B is greater than C, then A is greater than C[109].(§ 158 ¶ 8)

Further axioms characterize various species of magnitudes, but the above seem alone necessary to magnitude in general. None of them depend in any way upon number or measurement; hence we may be undismayed in the presence of magnitudes which cannot be divided or measured, of which, in the next chapter, we shall find an abundance of instances.(§ 158 ¶ 9)

§ 158 n. 1. It is not necessarily in (5) and (6) to add A, B, C being magnitudes, for the above relations of greater and less are what define magnitudes, and the addition would therefore be tautological.