Fundamental Principles of Scientific Inquiry, 371 



o£ infinite numbers. Poincare argued' 34 " that it is impossible 

 to assert anything about a class, and in particular anything 

 about the number of its members, until every member of 

 the class is defined in words ; and as only a finite number 

 of entities can ever be defined in words, it is impossible to 

 know anything about an infinite class, so that there can be 

 no knowledge of infinite numbers. The argument, as it 

 stands, is not valid against the theory of infinity, for in 

 order to make an assertion about a class it may not be 

 necessary to have definitions of all the members sepa- 

 rately ; often a general proposition about all members 

 can be asserted or postulated, and is enough for the 

 purpose. Poincare, indeed, seems to have overlooked 

 the fact that if his argument were sound it would also 

 destroy the whole theory of infinite series, on which most 

 higher pure mathematics is based ; for the convergence of 

 a series depends on the proposition that all the remainders 

 left after n, n + 1, n + 2, . . . terms are, for some value of n, 

 numerically less than a fixed quantity e. These remainders 

 are infinite in number, and hence it would be impossible, 

 if Poincare's assumption were granted, ever to prove that 

 a series is convergent. This result is, of course, quite 

 unacceptable. But Poincare's argument would go even 

 further than this. Nobody has had time in his life to 

 construct definitions of every member of a class of a 

 million members, and as a number is merely a property of 

 a, class it should be impossible to prove that, for instance, 



1 000 001 2 = 1 000 002 0C0 001. 



Thus the argument would also invalidate most of arithmetic. 

 If, therefore, we believe that the propositions of arithmetic 

 have some meaning and are true, we must abandon Poincare's 

 objection to the theory of infinite numbers. 



But while we cannot accept the argument of Poincare in 

 this case, it is clear that it is valid in cases where our only 

 source of information about the members of a class is 

 empirical; for the total number of observations any man 

 has made in his life is finite, and hence his experience 

 alone .can never tell him anything about all the members 

 of an infinite class of entities. An}' proposition about 

 such a class, or about all its members, is necessarily either 

 wholly a priori or else an inductive generalization, and 

 aieither directly known nor obtainable from experience by 



* 'Science et Methode/ pp. 192-214(1908). 

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