60ME i'KUULEMS OF PHYSICAL C( )NTIN L^l 1 ^ . 175 



conversely, tinite numbers are " inductive," whilst infinite are 

 not. A reflexive number he detiues as one which is not increased 

 by the addition of I ; " inductive " numbers are all those that 

 can be reached by successive additions of i. Beyond these are 

 all the infinite numbers. 



The first of the infinite numbers has no immediate predecessor, be- 

 cause there is no greatest finite number ; thus no succession of steps from 

 one number to tlie next will ever reach from a finite to an infinite one. and 

 tile step-by-step metliod of proof fails.* 



When we have realised the existence of these properties of 

 number, we discover, if Mr. Russell judges correctly, that the 

 supposed contradictions of infinite series are really only shocks 

 to our ])rejudice, not to sound logic. 



But all this necessitates a new definition of " number," 

 which Mr. Russeil claims to be the work of a great mathematical 

 genius, Gottlob Trege of jena. There nnist be no counting in 

 Mr. Russell's " number," as is obvious from w^hat has been said. 

 In practical life two collections have the same number of terms 

 when there is a one-one relation between all the tenns of one 

 collection and those of another. There is a certain similarity 

 between number and colour. " The number of terms in a 

 given class is defined as meaning " the class of all classes that 

 aie similar to the given class." This definition is held to have 

 the supreme merit of showing that it is not physical objects, but 

 classes or the general terms by which they are defined, of which 

 numbers can be asserted. 



It must be admitted that Mr. Russell feels the full force 

 of the instinctive oreiudice that "ill '^reet this definition; people 

 will be at first inclined to resent its oddity as well as the peculiar 

 behaviour of inlinite numbers. But 



Numbers, in fact, must satisfy the formul.T of arithmetic; any indu- 

 bitable set of objects fulfilling this requirement may be called numbers. 

 So far. the simplest set known to fulfil this requirement is the set intro- 

 duced by the above definition. In comparison with this merit, the question 

 whether the objects to whicii tlie definition applies are like or unlike the 

 vague ideas of numbers entertained 1)y those who cannot give a definition 

 is of very little importance. 



From these last sentences hardl\ anyone will be found to 

 dissent, if the implied facts are accurate. It is undeniable that 

 a definition of number must satisfy the formul?e of arithmetic. 

 But where the formulae themselves are open to some doubt and 

 discussion, no convincing theory can be built merely upon one 

 interpretation of a debated theory. 



And here Mr. Russell seems to have had an experience like 

 to that of the Pythagoreans narrated by himself. These philo- 

 sophers held that " things are numbers," and they apparently 

 conceived the continuum as a series of measurable atoms with 

 empty space in between. But unfortunately for this philo- 

 sophical system, Pythagcjras discovered the proposition that the 

 sum of the s([uares on the sides of a right-angled triangle is 



* P 197 



