THE VALUE OF SCIENCE 553 



change. Two of these internal changes a and /?' shall be regarded as 

 indistinguishable: (1) if they are so naturally, that is, if they are too 

 close to one another; (2) if a' is capable of correcting the same ex- 

 ternal change as a third internal change naturally indistinguishable from 

 /?'. In this second case, they will be, so to speak, indistinguishable by 

 convention, I mean by agreeing to disregard circumstances which might 

 distinguish them. 



Our continuum is now entirely defined, since we know its elements 

 and have fixed under what conditions they may be regarded as indis- 

 tinguishable. We thus have all that is necessary to apply our defini- 

 tion and determine how many dimensions this continuum has. We 

 shall recognize that it has six. The continuum of displacements is, 

 therefore, not equivalent to space, since the number of dimensions is 

 not the same; it is only related to space. Now how do we know that 

 this continuum of displacements has six dimensions? We know it by 

 experience. 



It would be easy to describe the experiments by which we could 

 arrive at this result. It would be seen that in this continuum cuts 

 can be made which divide it and which are continua ; that these cuts 

 themselves can be divided by other cuts of the second order which yet 

 are continua, and that this would stop only after cuts of the sixth 

 order which would no longer be continua. From our definitions that 

 would mean that the group of displacements has six dimensions. 



That would be easy, I have said, but that would be rather long; 

 and would it not be a little superficial ? This group of displacements, 

 we have seen, is related to space, and space could be deduced from it, 

 but it is not equivalent to space, since it has not the same number of 

 dimensions; and when we shall have shown how the notion of this 

 continuum can be formed and how that of space may be deduced from 

 it, it might always be asked why space of three dimensions is much 

 more familiar to us than this continuum of six dimensions, and con- 

 sequently doubted whether it was by this detour that the notion of space 

 was formed in the human mind. 



§ 2. Identity of Two Points 



What is a point? How do we know whether two points of space 

 are identical or different ? Or, in other words, when I say : The object 

 A occupied at the instant a the point which the object B occupies at 

 the instant /?, what does that mean? 



Such is the problem we set ourselves in the preceding chapter, § 4. 

 As I have explained it, it is not a question of comparing the positions 

 of the objects A and B in absolute space; the question then would 

 manifestly have no meaning. It is a question of comparing the posi- 

 tions of these two objects with regard to axes invariably bound to my 

 body, supposing always this body replaced in the same attitude. 



