642 SCIENCE AND HYPOTHESIS 



to its power is the necessity of avoiding all contradiction; but the 

 mind only makes use of it when experiment gives a reason for it. 



In the case with which we are concerned, the reason is given by 

 the idea of the physical continuum, drawn from the rough data of 

 the senses. But this idea leads to a series of contradictions from each 

 of which in turn we must be freed. In this way we are forced to 

 imagine a more and more complicated system of symbols. That 

 on which we shall dwell is not merely exempt from internal contra- 

 diction, it was so already at all the steps we have taken, but it is 

 no longer in contradiction with the various propositions which are 

 called intuitive, and which are derived from more or less elaborate 

 empirical notions. 



Measurable Magnitude. So far we have not spoken of the measure 

 of magnitudes; we can tell if any one of them is greater than any 

 other, but we cannot say that it is two or three times as large. 



So far, I have only considered the order in which the terms are 

 arranged; but that is not sufficient for most applications. We must 

 learn how to compare the interval which separates any two terms. On 

 this condition alone will the continuum become measurable, and the 

 operations of arithmetic be applicable. This can only be done by the 

 aid of a new and special convention; and this convention is, that in 

 such a case the interval between the terms A and B is equal to the 

 interval which separates C and D. For instance, we started with 

 the integers, and between two consecutive sets we intercalated n in- 

 termediary sets; by convention we now assume these new sets to be 

 equidistant. This is one of the ways of defining the addition of two 

 magnitudes ; for if the interval AB is by definition equal to the inter- 

 val CD, the interval AD will by definition be the sum of the intervals 

 AB and AC. This definition is very largely, but not altogether, arbi- 

 trary. It must satisfy certain conditions the commutative and as- 

 sociative laws of addition, for instance; but, provided the definition 

 we choose satisfies these laws, the choice is indifferent, and we need 

 not state it precisely. 



Remarks. We are now in a position to discuss several important 

 questions. 



(1) Is the creative power of the mind exhausted by the creation 

 of the mathematical continuum? The answer is in the negative, and 

 this is shown in a very striking manner by the work of Du Bois Eey- 

 mond. 



We know that mathematicians distinguish between infinitesimals of 

 different orders, and that infinitesimals of the second order are in- 

 finitely small, not only absolutely so, but also in relation to those of 

 the first order. It is not difficult to imagine infinitesimals of frac- 

 tional or even of irrational order, and here once more we find the 

 mathematical continuum which has been dealt with in the preceding 



