NUMBER AND MAGNITUDE 639 



the second all the numbers whose squares are smaller than 2. We 

 know that in neither of them is a number whose square is equal to 2. 

 Evidently there will be in the first class no number which is smaller 

 than all the rest, for however near the square of a number may be to 2, 

 we can always find a commensurable whose square is still nearer to 2. 

 From Kronecker's point of view, the incommensurable number of 2 

 is nothing but the symbol of this particular method of division of 

 commensurable numbers; and to each mode of repartition corresponds 

 in this way a number, commensurable or not, which serves as a sym- 

 bol. But to be satisfied with this would be to forget the origin of these 

 symbols; it remains to explain how we have been led to attribute to 

 them a kind of concrete existence, and on the other hand, does not 

 the difficulty begin with fractions? Should we have the notion of 

 these numbers if we did not previously know a matter which we con- 

 ceive as infinitely divisible i.e., as a continuum ? 



The Physical Continuum. We are next led to ask if the idea of 

 the mathematical continuum is not simply drawn from experiment. 

 If that be so, the rough data of experiment, which are our sensations, 

 could be measured. We might, indeed, be tempted to believe that this 

 is so, for in recent times there has been an attempt to measure them, 

 and a law has even been formulated, known as Fechner's law, accord- 

 ing to which sensation is proportional to the logarithm of the stimulus. 

 But if we examine the experiments by which the endeavor has been 

 made to establish this law, we shall be led to a diametrically opposite 

 conclusion. It hap, for instance, been observed that a weight A of 10 

 grammes and a weight B of 11 grammes produced identical sensa- 

 tions, that the weight B could no longer be distinguished from a 

 weight C of 12 grammes, but that the weight A was readily distin- 

 guished from the weight C. Thus the rough results of the experiments 

 may be expressed by the following relations: A = B, B = C, A< C, 

 which may be regarded as the formula of the physical continuum. 

 But here is an intolerable disagreement with the law of contradiction, 

 and the necessity of banishing this disagreement has compelled us to 

 invent the mathematical continuum. We are therefore forced to con- 

 clude that this notion has been created entirely by the mind, but it is 

 experiment that has provided the opportunity. We cannot believe that 

 two quantities which are equal to a third are not equal to one another, 

 and we are thus led to suppose that A is different from B and B from 

 C, and that if we have not been aware of this, it is due to the imper- 

 fections of our senses. 



The Creation of the Mathematical Continuum : First Stage. So 

 far it would suffice, in order to account for facts, to intercalate between 

 A and B a small number of terms which would remain discrete. What 

 happens now if we have recourse to some instrument to make up for 

 the weakness of our senses? If, for example, we use a microscope? 



