THE VALUE OF SCIENCE 85 



could have succeeded also, by giving other analogous fillips. Should we 

 not always have been able to justify these fillips by the same reasons? 

 One could at most have said to us : ' Your fillips are doubtless legiti- 

 mate, but you abuse them; why move the exterior objects so often? ' 



To sum up, experience does not prove to us that space has three 

 dimensions ; it only proves to us that it is convenient to attribute three 

 to it, because thus the number of fillips is reduced to a minimum. 



I will add that experience brings us into contact only with repre- 

 sentative space, which is a physical continuum, never with geometric 

 space, which is a mathematical continuum. At the very most it would 

 appear to tell us that it is convenient to give to geometric space three 

 dimensions, so that it may have as many as representative space. 



The empiric question may be put under another form. Is it im : 

 possible to conceive physical phenomena, the mechanical phenomena for 

 example, otherwise than in space of three dimensions ? We should thus 

 have an objective experimental proof, so to speak, independent of our 

 physiology, of our modes of representation. 



But it is not so ; I shall not here discuss the question completely, I 

 shall confine myself to recalling the striking example given us by the 

 mechanics of Hertz. You know that the great physicist did not believe 

 in the existence of forces, properly so called; he supposed that visible 

 material points are subjected to certain invisible bonds which join them 

 to other invisible points and that it is the effect of these invisible bonds 

 that we attribute to forces. 



But that is only a part of his ideas. Suppose a system formed of 

 n material points, visible or not; that will give in all 3n coordinates; 

 let us regard them as the coordinates of a single point in space of 3ra 

 dimensions. This single point would be constrained to remain upon a 

 surface (of any number of dimensions < 3n) in virtue of the bonds of 

 which we have just spoken; to go on this surface from one point to 

 another, it would always take the shortest way; this would be the 

 single principle which would sum up all mechanics. 



Whatever should be thought of this hypothesis, whether we be allured 

 by its simplicity, or repelled by its artificial character, the simple fact 

 that Hertz was able to conceive it, and to regard it as more convenient 

 than our habitual hypotheses, suffices to prove that our ordinary ideas, 

 and, in particular, the three dimensions of space, are in no wise imposed 

 upon mechanics with an invincible force. 



§ 6. Mind and Space 

 Experience, therefore, has played only a single role, it has served as 

 occasion. But this role was none the less very important; and I have 

 thought it necessary to give it prominence. This role would have been 

 useless if there existed an a priori form imposing itself upon our sen- 

 sitivity, and which was space of three dimensions. 



