Continuity 23 



tinuity apply to number? The natural 

 numbers, i, 2, 3, etc., are discontinu- 

 ous enough, but there are fractions to 

 fill up the interstices; how do we know 

 that they are not really connected by 

 these fractions, and so made continuous 

 again? 



(By number I always mean commensur- 

 able number; incommensurables are not 

 numbers: they are just what cannot be 

 expressed in numbers. The square root 

 of 2 is not a number, though it can be 

 readily indicated by a length. Incom- 

 mensurables are usual in physics and are 

 frequent in geometry; the conceptions of 

 geometry are essentially continuous. It 

 is clear, as Poincare says, that "if the 

 points whose co-ordinates are commen- 

 surable were alone regarded as real, the 

 in-circle of a square and the diagonal 

 of the square would not intersect, since 

 the co-ordinates of the points of inter- 

 section are incommensurable.") 



I want to explain how commensurable 

 fractions do not connect up numbers, nor 

 remove their discontinuity in the least. 



