AND PLANTS 83 



axiomatic 1 . Having done all this, he makes you believe 

 his story about the sum of the three angles of a triangle by 

 showing you that unless you believe it you must either 

 deny one of his axioms, which your instinctive recognition 

 of its truth compels you to believe, or you must refuse to 

 accept one of his definitions. Now, if you can deny the 

 axioms to which Euclid appeals, the laws of your thought 

 are so different from those which govern the thought of 

 normal men and women, that you must make beliefs of 

 your own : neither Euclid nor any normal person can help 

 you. And if you refuse to accept the definitions, you 

 do not invalidate Euclid's statement, you only insist upon 

 talking about something else. 



I am not going to discuss the very difficult question 

 how it has come about that the truth of Euclid's axioms 

 is self-evident to all of us, and I shall not discuss the 

 way in which he was led to conceive his definitions of 

 straight lines and the rest. But, however these things 

 may have arisen, the definitions at least have now to be 

 accepted without any conscious appeal to experience, and 

 the process, by which we deduce the properties of a plane 

 triangle from them, involves no appeal to experiment, 

 either for its performance or its subsequent verification. 

 Euclid's statement is a description of a mental process, 

 and it would be just as true as it is, though it would 

 not be quite so interesting to many of us, if there were 

 nothing in our experience remotely like the plane tri- 

 angles he describes : every time we went through the 

 mental process he tells us to perform, we should still 

 arrive at exactly the same result, and his proposition 

 would still be exactly true. 



1 Speaking broadly. I do not, of course, wish to enter here on a 

 detailed discussion of the question whether they are all axiomatic. 



G 2 



