144 SCIENCE AND METHOD. 



Cantor undertook to introduce into mathematics an 

 actual infinity — that is to say, a quantity which is not 

 only susceptible of passing all limits, but which is 

 regarded as having already done so. He set himself 

 such questions as these : Are there more points in 

 space than there are whole numbers .? Are there more 

 points in space than there are points in a plane ? etc. 



Then the number of whole numbers, that of points 

 in space, etc., constitutes what he terms a transfinite 

 cardinal number — that is to say, a cardinal number 

 greater than all the ordinary cardinal numbers. And 

 he amused himself by comparing these transfinite car- 

 dinal numbers, by arranging in suitable order the 

 elements of a whole which contains an infinite number 

 of elements ; and he also imagined what he terms 

 transfinite ordinal numbers, on which I will not dwell 

 further. 



Many mathematicians have followed in his tracks, 

 and have set themselves a series of questions of the 

 same kind. They have become so familiar with trans- 

 finite numbers that they have reached the point of 

 making the theory of finite numbers depend on that 

 of Cantor's cardinal numbers. In their opinion, if we 

 wish to teach arithmetic in a truly logical way, we 

 ought to begin by establishing the general properties 

 of the transfinite cardinal numbers, and then distin- 

 guish from among them quite a small class, that of the 

 ordinary whole numbers. Thanks to this roundabout 

 proceeding, we might succeed in proving all the propo- 

 sitions relating to this small class (that is to say, our 

 whole arithmetic and algebra) without making use of 

 a single principle foreign to logic. 



This method is evidently contrary to all healthy 



