460 AMBROSE J. MCNICHOLL 



matical system without any reference whatsoever to reality, on 

 the basis of mental operations, and of notions and axioms 

 freely chosen in view of a determinate system. Hilbert would 

 push this formalising tendency even further, basing both logic 

 and mathematics on pre-logical and pre-mathematical symbols, 

 and treating mathematics as sheer calculus without any regard 

 to interpretation. Godel, however, claims to have shown that 

 the non-contradictory character of a purely formal mathematics 

 cannot be shown, so that no system would be possible even in 

 pure mathematics. 



Signs of a welcome swing away from this conventionalism 

 and towards realism are apparent in the views of the Intui- 

 tionists — Brouwer, Weyl, Heyting — who maintain that mathe- 

 matics may not be reduced to logic, and is not a purely formal 

 science, but based upon relations with experience. By intuition 

 is meant the ability of the mind to grasp the structure of com- 

 plex situations, and of the process of thought, anterior to all 

 determinate forms of thought, whether philosophical, logical 

 or mathematical. For mathematics, the basic intuition is of the 

 pure relationship of serial order, from which the primary 

 notions may be derived by a process of construction. 



In philosophy proper, the most decisive influence today is 

 that of Phenomenology , and it is noteworthy that Husserl was 

 led to philosophy from his study of the foundations of arith- 

 metic, in an attempt to combat psychologism. His notion of 

 science is far closer to the traditional one than that of most 

 of his contemporaries, although it remains to some extent 

 formalistic; but an essential difference is that, for him, science 

 must abstract completely from the real world, and be grounded 

 upon an immediate intuition of ideal essences within our own 

 subjective consciousness, and the method, in philosophy at 

 least, must be descriptive and analytic. Undue emphasis on 

 the constructive activity of the mind in determining the sig- 

 nification of the contents of consciousness led Husserl towards 

 idealism; but Scheler opened up new vistas for a realistic phe- 

 nomenology by upholding an affective intuition which reveals 

 the world of values. Hartmann, while developing the realism of 



