The Principles of Mathematics (1903)

§ 495

In addition to his work on cardinal numbers, Frege has, already in the Begriffschrift, a very admirable theory of progressions, or rather of all series that can be generated by many-one relations. Frege does not confine himself to one-one relations: as long as we move in only one direction, a many-one relation also will generate a series. In some parts of his theory, he even deals with general relations. He begins by considering, for any relation f(x, y), functions F which are such that, if f(x, y) holds, then F(x) implies F(y). If this condition holds, Frege says that the property F is inherited in the f-series (Bs. pp. 55--58). From this he goes on to define, without the use of numbers, a relation which is the equivalent of some positive power of the given relation. This is defined as follows. The relation in question holds between x and y if every property F, which is inherited in the f-series and is such that f(x, z) implies F(z) for all values of z,belongs to y (Bs. p. 60). On this basis, a non-numerical theory of series is very successfully erected, and is applied in Gg. to the proof of propositions concerning the number of finite numbers and kindred topics. This is, so far as I know, the best method of treating such questions, and Frege's definition just quoted gives, apparently, the best form of mathematical induction. But as no controversy is involved, I shall not pursue this subject any further.(§ 495 ¶ 1)

Frege's works contain much admirable criticism of the psychological standpoint in logic, and also of the formalist theory of mathematics, which believes that the actual symbols are the subject-matter dealt with, and that their properties can be arbitrarily assigned by definition. In both these points, I find myself in complete agreement with him.(§ 495 ¶ 2)