NUMBER AND MAGNITUDE 637 



Mathematical Magnitude and Experiment 



If we want to know what the mathematicians mean by a continuum, 

 it is useless to appeal to geometry. The geometer is always seeking, 

 more or less, to represent to himself the figures he is studying, but 

 his representations are only instruments to him ; he uses space in his 

 geometry just as he uses chalk; and further, too much importance 

 must not bo attached to accidents which are often nothing more than 

 the whiteness of the chalk. 



The pure analyst has not to dread this pitfall. He has disen- 

 gaged mathematics from all extraneous elements, and he is in a 

 position to answer our question : " Tell me exactly what this con- 

 tinuum is, about which mathematicians reason." Many analysts who 

 reflect on their art have already done so M. Tannery, for instance, 

 in his Introduction a la theorie des Fonctions d'une variable. 



Let us start with the integers. Between any two consecutive sets, 

 intercalate one or more intermediary sets, and then between these sets 

 others again, and so on indefinitely. We thus get an unlimited num- 

 ber of term?, and these will be the numbers which we call fractional, 

 rational, or commensurable. But this is not yet all; between these 

 terms, which, be it marked, are already infinite in number, other terms 

 are intercalated, and these are called irrational or incommensurable. 



Before going any further, let me make a preliminary remark. The 

 continuum thus conceived is no longer a collection of individuals ar- 

 ranged in a certain order, infinite in number, it is true, but external 

 the one to the other. This is not the ordinary conception in which it 

 is supposed that between the elements of the continuum exists an inti- 

 mate connection making of it one whole, in which the point has no 

 existence previous to the line, but the line does exist previous to the 

 point. Multiplicity alone subsists, unity has disappeared " the 

 continuum is unity in multiplicity," according to the celebrated for- 

 mula. The analysts have even less reason to define their continuum 

 as they do, since it is always on this that they reason when they are 

 particularly proud of their rigor. It is enough to warn the reader that 

 the real mathematical continuum is quite different from that of the 

 physicists and from that of the metaphysicians. 



It may also be said, perhaps, that mathematicians who are con- 

 tented with this definition are the dupes of words, that the nature of 

 each of these sets should be precisely indicated, that it should be ex- 

 plained how they are to be intercalated, and that it should be shown 

 how it is possible to do it. This, however, would be wrong; the only 

 property of the sets which comes into the reasoning is that of pre- 

 ceding or succeeding these or those other sets ; this alone should there- 

 fore intervene in the definition. So we need not concern ourselves 

 with the manner in which the sets are intercalated, and no one will 



