APPENDIX 



MATHEMATICAL AND PHYSICAL NOTIONS' 



INFINITY 



What is really meant when the mathematician uses 

 the concept of infinity in his operations ? Suppose that 

 we take a line of finite length and divide it into halves, 

 and then divide each half into halves, and so on ad 

 infinitwn. We make cuts in the line, and these cuts 

 have no magnitude, so that the sum of the lengths into 

 which we divide the line is equal to the length of the 

 undivided line. We can divide the line into as many 

 parts as we choose, that is, into an " infinite " number 

 of parts. 



Suppose that we are making a thing which is to 

 match another thing, and suppose that we can make 

 the thing as great as we choose. If, then, no matter 

 how great we make the thing, it is still too small, the 

 thing that we are trying to match is infinitely great. 



Substitute " small " for " great," and this is also 

 a definition of the infinitely small. 



Clearly the idea of infinity does not reside in the 

 results of an operation, but in its tendency. It inheres 

 in our intuition of striving towards something, but not 

 in the results of our striving. 



1 It must be understood that some of the things dealt with in these appen- 

 dices are very hard to understand by the reader acquainted only with the 

 results of biological science. We urge, however, that they are all relevant if 

 biological results are to be employed speculatively. 



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