THE POSITIVE THEORY OF INFINITY 187 



of numbers to enable him to deduce his theorems. We, 

 since our object is philosophical, must grapple with the 

 philosopher's question. The answer to the question, 

 " What is a number ? ' which we shall reach in this 

 lecture, will be found to give also, by implication, the 

 answer to the difficulties of infinity which we considered 

 in the previous lecture. 



The question " What is a number ? " is one which, 

 until quite recent times, was never considered in the 

 kind of way that is capable of yielding a precise answer. 

 Philosophers were content with some vague dictum such 

 as, " Number is unity in plurality." A typical definition of 

 the kind that contented philosophers is the following from 

 Sigwart's Logic ( 66> section 3) : " Every number is not 

 merely a plurality y but a plurality thought as held together 

 and closed^ and to that extent as a unity T Now there is 

 in such definitions a very elementary blunder, of the 

 same kind that would be committed if we said " yellow 

 is a flower ' because some flowers are yellow. Take, for 

 example, the number 3. A single collection of three 

 things might conceivably be described as " a plurality 

 thought as held together and closed, and to that extent 

 as a unity" ; but a collection of three things is not the 

 number 3. The number 3 is something which all collec- 

 tions of three things have in common, but is not itself 

 a collection of three things. The definition, therefore, 

 apart from any other defects, has failed to reach the 

 necessary degree of abstraction : the number 3 is something 

 more abstract than any collection of three things. 



Such vague philosophic definitions, however, remained 

 inoperative because of their very vagueness. What 

 most men who thought about numbers really had in 

 mind was that numbers are the result of counting. " On 

 the consciousness of the law of counting," says Sigwart 



