198 SCIENTIFIC METHOD IN PHILOSOPHY 



demonstrable by the step-by-step method, and fail to be 

 true of infinite numbers. But so soon as we realise the 

 necessity of proving such properties by mathematical 

 induction, and the strictly limited scope of this method 

 of proof, the supposed contradictions are seen to contra- 

 dict, not logic, but only our prejudices and mental habits. 



The property of being increased by the addition of 

 1 i.e. the property of non-reflexiveness may serve to 

 illustrate the limitations of mathematical induction. It 

 is easy to prove that o is increased by the addition of 1, 

 and that, if a given number is increased by the addition of 

 1, so is the next number, i.e. the number obtained by the 

 addition of 1 . It follows that each of the natural numbers 

 is increased by the addition of 1. This follows generally 

 from the general argument, and follows for each particular 

 case by a sufficient number of applications of the argu- 

 ment. We first prove that o is not equal to 1 ; then, 

 since the property of being increased by 1 is hereditary, 

 it follows that 1 is not equal to 2 ; hence it follows that 

 2 is not equal to 3 ; if we wish to prove that 30,000 is 

 not equal to 30,001, we can do so by repeating this 

 reasoning 30,000 times. But we cannot prove in this 

 way that all numbers are increased by the addition of 1 ; 

 we can only prove that this holds of the numbers attain- 

 able by successive additions of 1 starting from o. The 

 reflexive numbers, which lie beyond all those attainable 

 in this way, are as a matter of fact not increased by the 

 addition of 1. 



The two properties of reflexiveness and non-inductive- 

 ness, which we have considered as characteristics of 

 infinite numbers, have not so far been proved to be 

 always found together. It is known that all reflexive 

 numbers are non-inductive, but it is not known that all 

 non-inductive numbers are reflexive. Fallacious proofs 



