1/6 SOMli PROBLEMS OF PHYSICAL CONTINUITY. 



equal to the square on the hypotenuse. When he came to 

 consider the case of a right-angled triangle with two equal sides, 

 he found himself face to face with the hard fact that the dia- 

 gonal and either of the sides were incommensurables. For a 

 system of philosophy which placed the essence of the universe 

 in a numerical relation, it was fatal to find two things which 

 refused to be expressed by any possible ratio of numbers. 



In a similar way, Mr. Russell takes it for granted that the 

 collection of possible points in any continuum must form a num- 

 ber. But obviously the number of these points cannot be 

 obtained by piling i upon itself, no matter how often the opera- 

 tion mav be repeated. Our ordinary numerals are quite incap- 

 able of expressing this number, because we can always imagine 

 something beyond the largest assigned numeral. This means 

 that no " finite " number can meet the case. 



" Irrational " numbers have long been used in arithmetic 

 in order to indicate the ratio of incommensurable lengths. 



Acceptinj4 the view that a length is composed of points, the existence 

 of incommensurahles proves that every finite length must contain an infi- 

 nite number of points^ The property of being unable to be counted is 

 characteristic of infinite collections, and is a source of many of their 

 paradoxical qualities. So paradoxical are their qualities that until our own 

 day they were thought to constitute logical contradictions.* 



But what reason is there to assume that the incommensurable 

 is a real number, and not rather a symbol of something that 

 cannot be expressed in numbers? A French inathematician 

 of note, C. A. Laisant,t waxes indignant that anyone should 

 question whether the ntmiber \' 2 exists. 



Autant se demander si 2 existe, je sais bien ce que c'est que 2 arbres. 

 2 anes ou 2 kilometres ; mais 2 tout seul, comme nombre abstrait, n'existe 

 qu'a I'etat de creation du cerveau et de signe representatif. De meme 

 \/2 a ime existence pareille, c'est un signe visible, qui represente une 

 notion nettement definie ; c'est la traduction precise d'une quan- 

 tite concrete, si je I'applique au metre pour unite, puisque je sais construire 

 la longueur -^2. La seule difference, c'est que je ne pourrai appliquer 

 le symbole d'un nombre incommensurable qu'a des grandeurs continues 

 par essence, aussi bien que je nc peux appliquer le symbole d'une fraction 

 qu'a des quantites divisibles. 



But it is stretching the language of arithmetic too far to 

 say that V^ is the exact translation of a concrete c|uantity. It 

 certainly may represent a definite length in continuous quantity, 

 but in the discrete mediuin of numbers it can never be fully 

 and accurately defined. The comparison with fractional terms 

 will not do. Fractions define the divisibility of objects in 

 numerical terins that leave nothing to the imagination ; but when 

 you have exhausted your physical endurance in defining \f2 

 in terms of numerals, you still have room for enquiry. 



Ought we not rather to say that the incommensurable num- 

 ber is a mathematical convention which symbolises a physical 

 experience with regard to the " continuous " — i.e., that you can 



* P. 164. 



t" La Mathematique," p. i^. 



