CONCEPTIONS AND METHODS OF MATHEMATICS 457 



are obtained by direct intuition of time and space without the aid of 

 empirical observation. This view seems to have been held by such 

 eminent mathematicians as Hamilton and DeMorgan; and it is a 

 very difficult position to refute, resting as it does on a purely meta- 

 physical foundation which regards it as certain that we can evolve 

 out of our inner consciousness the properties of time and space. 

 According to this view the idea of quantity is to be deduced from 

 these intuitions; but one of the facts most vividly brought home to 

 pure mathematicians during the last half-century is the fatal weak- 

 ness of intuition when taken as the logical source of our knowledge 

 of number and quantity. 1 



The objects of mathematical study, even when we confine our 

 attention to what is ordinarily regarded as pure mathematics are, 

 then, of the most varied description; so that, in order to reach a 

 -satisfactory conclusion as to what really characterizes mathematics, 

 one of two methods is open to us. On the one hand we may seek 

 some hidden resemblance in the various objects of mathematical 

 investigation, and having found an aspect common to them all we 

 may fix on this as the one true object of mathematical study. Or, 

 on the other hand, we may abandon the attempt to characterize 

 mathematics by means of its objects of study, and seek in its methods 

 its distinguishing characteristic. Finally, there is the possibility of 

 our combining these two points of view. The first of these methods is 

 that of Kempe, the second will lead us to the definition of Benjamin 

 Peirce, while the third has recently been elaborated at great length 

 by Russell. Other mathematicians have naturally followed out more 

 or less consistently the same ideas, but I shall nevertheless take the 

 liberty of using the names Kempe, Peirce, and Russell as convenient 

 designations for these three points of view. These different methods 

 of approaching the question lead finally to results which, without 

 being identical, still stand in the most intimate relation to one an- 

 other, as we shall now see. Let us begin with the second method. 



II. Peirce' s Definition 



More than a third of a century ago Benjamin Peirce wrote : 2 

 Mathematics is the science which draws necessary conclusions. Accord- 

 ing to this view there is a mathematical element involved in every 

 inquiry in which exact reasoning is used. Thus, for instance, 8 a 

 jury listening to the attempt of the counsel for the prisoner to prove 

 an alibi in a criminal case might reason as follows: "If the witnesses 



1 I refer here to such facts as that there exist continuous functions without 

 derivatives, whereas the direct untutored intuition of space would lead any one 

 to believe that every continuous curve has tangents. 



2 Linear Associative Algebra. Lithographed 1870. Reprinted in the American 

 Journal of Mathematics, vol. iv. 



3 This illustration was suggested by the remarks by J. Richard, Sur la philoso- 

 phie des mathmcatiques. Paris, Gauthier-Villars, 1903, p. 50. 



