ABSTRACT
This chapter is concerned with a question that is strictly philosophical rather than logical. It considers to what extent the ideal of logical rigour has been attained. As the abstraction becomes more complete, thereby achieving greater disconnexion from any given set of particular occasions, the method of science passes from classification to causal investigation, and from causal investigation to measurement. It follows that no reference to the actual world is involved in any mathematical proposition. The sciences in which classification plays an important part are at the other extreme; they have achieved the least complete abstractions. Counting, therefore, presupposes the notion of similarity, and is less logically simple, since counting requires an order but similarity does not. Hence, the definition of number by means of similarity leads to greater generality. In the attempt to carry out a strictly rigorous deduction of the general properties of classes and relations from the fundamental logical premisses certain contradictions became apparent.
