22 Presidential Address 



Patriarch, are typical instances of nu- 

 merable, or rather innumerable, things. 

 The difficulty of enumerating them is 

 not that there is nothing to count, but 

 merely that the things to be counted 

 are very numerous. So are the atoms 

 in a drop of water, they outnumber 

 the drops in an Atlantic Ocean, and, 

 during the briefest time of stating their 

 number, fifty millions or so may have 

 evaporated; but they are as easy to 

 count as the grains of sand on a shore. 



The process of counting is evidently 

 a process applicable to discontinuities, 

 i. e., to things with natural units; you 

 can count apples and coins, and days and 

 years, and people and atoms. To apply 

 number to a continuum you must first 

 cut it up into artificial units; and you 

 are always left with incommensurable 

 fractions. Thus only is it that you can 

 deal numerically with such continuous 

 phenomena as the warmth of a room, the 

 speed of a bird, the pull of a rope, or the 

 strength of a current. 



But how, it may be asked, does discon- 



