638 SCIENCE AND HYPOTHESIS 



doubt the possibility of the operation if he only remembers that " pos- 

 sible " in the language of geometers simply means exempt from con- 

 tradiction. But our definition is not yet complete, and we come back 

 to it after this rather long digression. 



Definition of Incommensurables. The mathematicians of the Ber- 

 lin school, and Kronecker in particular, have devoted themselves to 

 constructing this continuous scale of irrational and fractional num- 

 bers without using any other materials than the integer. The math- 

 ematical continuum from this point of view would be a pure creation 

 of the mind in which experiment would have no part. 



The idea of rational number not seeming to present to them any 

 difficulty, they have confined their attention mainly to defining incom- 

 mensurable numbers. But before reproducing their definition here, I 

 must make an observation that will allay the astonishment which this 

 will not fail to provoke in readers who are but little familiar with the 

 habits of geometers. 



Mathematicians do not study objects, but the relations between 

 objects; to them it is a matter of indifference if these objects are 

 replaced by others, provided that the relations do not change. Matter 

 does not engage their attention, they are interested by form alone. 



If we did not remember it, we could hardly understand that Kro- 

 necker gives the name of incommensurable number to a simple symbol 

 that is to say, something very different from the idea we think we 

 ought to have of a quantity which should be measurable and almost 

 tangible. 



Let us see now what is Kronecker's definition. Commensurable 

 numbers may be divided into classes in an infinite number of ways, 

 subject to the condition that any number whatever of the first class is 

 greater than any number of the second. It may happen that among 

 the numbers of the first class there is one which is smaller than all 

 the rest; if, for instance, we arrange in the first class all the numbers 

 greater than 2, and 2 itself, and in the second class all the numbers 

 smaller than 2, it is clear that 2 will be the smallest of all the num- 

 bers of the first class. The number 2 may therefore be chosen as the 

 symbol of this division. 



It may happen, on the contrary, that in the second class there is one 

 which is greater than all the rest. This is what takes place, for ex- 

 ample, if the first class comprises all the numbers greater than 2, 

 and if, in the second, are all the numbers less than 2, and 2 itself. 

 Here again the number 2 might be chosen as the symbol of this 

 division. 



But it may equally well happen that we can find neither in the 

 first class a number smaller than all the rest, nor in the second class a 

 number greater than all the rest. Suppose, for instance, we place in 

 the first class all the numbers whose squares are greater than 2, and in 



