468 MATHEMATICS 



made this relatively small number of observations, the remaining 

 results would be obtained deductively. Finally, we may suppose 

 the points given by their coordinates, in which case the complete 

 answer to our question may be obtained by the purely deductive 

 method of analytic geometry. 



According to the modified form of Kempe's definition which I 

 have just stated, mathematics is not necessarily a deductive science. 

 This view, while not in accord with the prevailing ideas of mathe- 

 maticians, undoubtedly has its advantages as well as its dangers. 

 The non-deductive processes, of which I shall have more to say 

 presently, play too important a part in the life of mathematics to 

 be ignored, and the definition just given has the merit of not exclud- 

 ing them. It would seem, however, that the definition in the form 

 just given is too broad. It would include, for instance, the deter- 

 mination by experimental methods of what pairs of chemical com- 

 pounds of the known elements react on one another when mixed 

 under given conditions. 



VI. Axioms and Postulates. Existence Theorems 



If, however, we restrict ourselves to exact or deductive mathe- 

 matics, it will be seen that Kempe's definition becomes coextensive 

 with Peirce's. Here, in order to have a starting-point for deductive 

 reasoning, we must assume a certain number of facts or primitive 

 propositions concerning any mathematical system we wish to study, 

 of which all other propositions will be necessary consequences. 1 

 We touch here on a subject whose origin goes back to Euclid and 

 which has of late years received great development, primarily at 

 the hands of Italian mathematicians. 2 



It is important for us to notice at this point that not merely these 

 primitive propositions but all the propositions of mathematics may 

 be divided into two great classes. On the one hand, we have pro- 

 positions which state that certain specified objects satisfy certain 

 specified relations. On the other hand are the existence theorems. 

 which state that there exist objects satisfying, along with certain 

 specified objects, certain specified relations. 3 These two classes of 

 propositions are well known to logicians and are designated by them 



1 These primitive propositions may be spoken of as axioms or postulates, ac- 

 cording to the point of view we wish to take concerning their source, the word 

 axiom, which has been much misused of late, indicating an intuitional or empirical 

 source. 



2 Peano, Fieri, Padoa, Burali-Forti. We may mention here also Hilbert, who, 

 apparently without knowing of the important work of his Italian predecessors, 

 has also done valuable work along these lines. 



* Or we might conceivably have existence theorems which state that there 

 exist relations which are satisfied by certain specified objects; or these two kinds 

 of existence theorems might be combined. If we take the point of view explained 

 in the second footnote on p. 467, all existence theorems will be of the type men- 

 tioned in the text. 



