SOME PROBLEMS OF PHYSICAL CONTINUITY. IJJ 



go on dividing it for ever approximating to the actual value of 

 V2? The two processes are parallel, but the discrete medium 

 of arithmetic is not flexible enough to express all that can be 

 put into the continuous line. 



The mathematician is in fact deceived by the analogies of 

 the physicist and the metaphysician. The scientist deals with . 

 continuity as it strikes the senses in rcrmn natura ; the meta- 

 physician is free to consider all its fundamental aspects. Both 

 can appeal to the possibility of an infinite (i.e., indefinite ) num- 

 ber of points in the actual lines they perceive ; though they all 

 confess that no available instrument has helped them, or can 

 help them, to produce the infinite series of points, as distinct 

 parts of the continuum, in the way that the microscope breaks 

 up a uniform surface of some sea dust into organic animals. 



Mathematics can define the continuous only by means of 

 the numbered points wdiich it is able to postulate as the starting 

 places of its researches. The vital factor in the definition of 

 the mathematical continuum is that it contains i)oints arranged 

 in a certain order. The analogy of the other sciences would 

 indicate that these points are infinite in number. But this infinity 

 (in the sense of " indefiniteness ") cannot be turned into an 

 infinite number, with mathematical precision, until the possi- 

 bilit}' of " counting " such an infinity is shown. To point to 

 " irrational " numbers (numbers which cannot be enumerated) 

 is not so much a proof of the need of postulating the existence 

 of the infinite number as an indication that certain aspects of 

 physical continuity cannot be rationally represented in arithmetic. 



And this is perhaps only a particular instance of the warning 

 that the great French philosopher and mathematician, H. Poin- 

 care,* gives as to the limits of mathematical speculation. 



L'esprit a la faculte de creer des symboles, etc'est ainsi qu'il a con- 

 struit le continu mathematique, qui n'est' qu'un s\'steme* particulier de 

 synilioles. La puissance n'est limitee que par la necessite d'eviter toute 

 contradiction; mais l'esprit n'en use que si I'experience lui en fournit une 

 raison. 



The reason in this case would seem to be the need of ex- 

 pressing mathematically the fact that there is no arithmetical 

 equivalent for certain aspects of experienced continuity. For 

 whilst mathematics has done much to lessen the logical diffi- 

 culties of physical continuity, especially during the last fifty 

 years, it has its own limitations, which cause it to need help 

 from the other sciences and from practical experience, in order 

 to express the full meaning of the continuous. 



* ■' La Science et I'Hypothese," p. 40. 

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