22 THE ANATOMY OF SCIENCE 



guage has the properties of mathematics. As 

 arithmetic has its numbers and operations upon 

 numbers, so language has its words, and its 

 operations that we call grammar. It is out of 

 this great magma that have crystallized our 

 logic and our mathematics. 



After this long digression let us return to 

 the question of whether we shall say that arith- 

 metic is true or only that it is extremely inter- 

 esting and useful, or, as William James would 

 sa}^, that it is pragmatic. It is certainly useful, 

 and applicable to an enormous variety of prob- 

 lems, but is it applicable to all.? Consider the 

 simplest arithmetical proposition, 1 + 1=2; 

 does one liter of alcohol added to one liter of 

 water give two liters.? No, it gives hardly more 

 than one liter and nine tenths. If this failure of 

 arithmetic seems trivial, let us consider the case 

 of a large projectile which is moving away from 

 us at the rate of 100,000 miles per second, and 

 which in the same direction shoots out a smaller 

 projectile, which travels from it at the rate of 

 100,000 miles per second. Does the latter move 

 from us at the rate of 200,000 miles per second? 

 We shall see in a later chapter that such a veloc- 

 ity is not only not attained but is unattainable. 



Studvincr such failures of the arithmetical 



