Rogers — The Logical Basis of Mathematics. 185 



IV. 



The impotence of the Law of Contradiction, taken alone, was seen by 

 Kant, who, in mathematics, snbstitutes intuition for logic, sensibility for 

 nnclerstancling. 



The Kantian view is also liable to a misunderstanding, from which 

 Kant and his followers have only half escaped. For intuition is immediate 

 experience of particulars ; and thus mathematics has no universality if it is 

 all intuition. In geometry the proof that uses a particular figure applies to 

 that figure only, because that is the only figure intuited ; in arithmetic the 

 rules will have no universality, e.g., 5 + 7 = 12 will be true only for the set 

 of 12 points or marbles we are looking at ; and since addition, number, and 

 so forth have a specialised intuited meaning for the given figure and for the 

 given points or marbles, it appears that, after all, these propositions are 

 identically true ; they are only analyses of given perceptions. Hence the 

 intuitional view, which is most naturally interpreted thus, contradicts itself, 

 because it only means saying that a given intuition is that intuition and 

 nothing else. The inconsistency is this, that the intuitionist claims to 

 proceed by intuition, and really uses nothing but the principle of contradic- 

 tion. Moreover, this makes mathematics an absolutely useless science, 

 because it prevents it from leaving immediate experience. In reply it may 

 be said that, in a priori intuition, we see the universal in the particular. This, 

 however, is abandoning the genuine intuitionist standpoint (because intuition 

 is immediate perception), for it means that we universalize our given 

 perceptions ; we form the conception of a ' class ' of entities not perceived, 

 possessing formal properties similar to those of the intuited object. From 

 those universal premisses we deduce further conclusions by logic. Thus 

 logic is indispensable not only to mathematics, but to every form of science. 

 It is now commonly recognized by thinkers that the so-called 'inference 

 from particular to particulars ' is really syllogism based on hypothetic 

 generalization. Skilled mathematicians who have thought deeply enough to 

 see that mathematical knowledge is not merely immediate perception of 

 particulars, sometimes endeavour to escape by the Kantian theory of a 

 Schema of Space. This Schema is, however, both particular and universal, 

 and yet neither. Hence it lands us in the Lockian difficulty of ' abstract 

 ideas,' which, as Berkeley observes, is self-contradictory. 



In geometry one commonly uses mental images or figures often badly 

 constructed. A bad figure (or any figure), however, would be useless if the 

 proofs were merely intuitional or immediate — a fact noticed by the late 

 Provost Salmon. The figure imaged or drawn is really a symbol of a 



