186 Proceedings of the Royal Irish Academy. 



universal class of ideal figures. The symbolic figure is thus a generic or 

 typical image of a set of logical premisses and terms ; and when so under- 

 stood, the conclusions are direct inferences from the premisses ; they are 

 universal, and therefore useful. Intuition thus provides us with a kind of 

 symbolism which is absolutely indispensable for rapid thinking. Modern 

 logical geometry simply attempts to abstract the premisses from the 

 associated symbols. The use of diagrams in geometry is thus the same 

 as the use of diagrams in elementary logic (e.g., Euler's diagrams), viz. to 

 individualize the conceptions, as far as possible. 



This also applies to number. We may use the image of twelve points as 

 a generic image of the number twelve ; but to identify twelve with twelve 

 intuited points is absurd. Twelve is a property common to an endless set 

 of classes ; and when numbers are so considered, i.e., universally as connotative 

 or predicative of sets of classes, arithmetic becomes a science. I would ask 

 those who say that all numbers are intuited, whether they can intuite the 

 number ten million. It can easily be defined by powers and products. Or, 

 in geometry, how can you prove by intuition, or even grasp (when proved), 

 that a cubic surface has 27 right lines lying on it ? 



Kant's distinction between analytic and synthetic propositions is 

 connected with this discussion. Here I need only observe, without further 

 criticism, that if 5 + 7 = 12 is not involved in the definitions of 5, 7, and 12, 

 then we are at liberty to define these terms (5, +, 7, 12) without redundancy, 

 so that 5 + 7 = 12 will follow analytically. The Kantian view closely analysed 

 and taken literally leads to contradiction. The truth latent in Kant's 

 distinction is that knowledge proceeds from simple conceptions to more 

 complex ones, which logically presuppose the simpler ones, but cannot without 

 contradiction be identified with them,* 



V. 



In all Deductive Science there is constantly taking place a process of 

 generalization, which is often useful, as leading to wide application, and is at all 

 events inevitable . In geometry this generalization takes place as follows : — 

 First, we have empirical mensuration, the geometry of the ancient Egyptians 

 and of modern educationists. Secondly, we have Euclidean geometry, in which 

 the universality of the reasoning is recognized, but bound up with images, and 

 so obscured. Euclid's geometry is mixed, and his axioms and definitions are 

 logically inadequate.f Some of his propositions almost follow logically from 



* The writer has considered this more fully in Hermathena, No. xxxiii. (1 907) (' ' An Old Problem 

 in Logic "). 



t See Russell's Pritic'njles of Mathematics, Ch. xlvii. 



