THE CHALLENGE TO THE TRADITIONAL IDEAL OF SCIENCE 459 



guage, and to be, not a body of truths, but an activity, the 

 analysis of linguistic forms, in order to uncover confusions that 

 have their source in our manner of speaking, and to reveal that 

 the traditional problems of philosophy are in fact pseudo- 

 problems. Some kind of Behaviorism is frequently assumed in 

 connection with such theories, and the prevailing atmosphere 

 is nominalistic, although this is not so anti-metaphysical as 

 it is with the Logical Positivists who work out a semiotic theory 

 consistent with an assumed physicalism. 



For Physicalists and Analysts, the subject-matter of science 

 seems to be restricted either to facts, whether physical or physi- 

 ological, and to the language in which such facts are stated. 

 The influence of multi-valued logical systems contributes to 

 undermine the conviction of an absolute truth, and logic itself 

 has been, to a great extent, not only symbolized, but also con- 

 ventionalised. The difficulties presented by a seemingly a priori 

 and absolute mathematics were conveniently disposed of by 

 Wittgenstein, who showed how mathematical propositions could 

 be treated as tautologies. 



Recent studies on the foundations of mathematics point 

 mainly in the direction of Formalism, and the axiomatic ap- 

 proach, again influenced by the new logic, tends towards Con- 

 ventionalism. The mathematical sciences are generally regarded 

 as hypothetico-deductive systems, purely formal in themselves, 

 without any direct reference to reality, whose elementary no- 

 tions are left undefined, and whose axioms and theorems are 

 established by convention. Klein may be taken as representa- 

 tive of the new approach to geometry, which is said to treat, 

 not of real space, but of relations of position in any ordered 

 multiplicity, in so far as these can be expressed in a coherent 

 system, where the only principles allowed are those that de- 

 termine such relations, and the fundamental concept is that of 

 " group," applying to a series of operations, rather than quan- 

 tity or number. Most theorists however regard arithmetic as 

 more fundamental than geometry, and stress its affinity with 

 logic. The Logicists — Cantor, Frege, Dedekind, Russell — 

 reduce mathematics to logic, and try to construct a mathe- 



