CONCEPTIONS AND METHODS OF MATHEMATICS 459 



any purely philosophical speculations. But unfortunately we can- 

 not dismiss the matter in this way; for it has happened not infre- 

 quently that the most eminent men, including mathematicians, 

 have differed as to whether a given piece of reasoning was exact or 

 not; and, what is worse, modes of reasoning which seem absolutely 

 conclusive to one generation no longer satisfy the next, as is shown 

 by the way in which the greatest mathematicians of the eighteenth 

 century used geometric intuition as a means of drawing what they 

 regarded as necessary conclusions. 1 



I do not wish here to raise the question whether there is such a 

 thing as absolute logical rigor, or whether this whole conception of 

 logical rigor is a purely psychological one bound to change with 

 changes in the human mind. I content myself with expressing the 

 belief, which I will try to justify a little more fully in a moment, 

 that as we never have found an immutable standard of logical rigor 

 in the past, so we are not likely to find it in the future. However 

 this may be, so much we can say with tolerable confidence, as past 

 experience shows, that no reasoning which claims to be exact can 

 make any use of intuition, but that it must proceed from definitely 

 and completely stated premises according to certain principles of 

 formal logic. It is right here that modern mathematicians break 

 sharply with the tradition of a priori synthetic judgments (that is, 

 conclusions drawn from intuition) which, according to Kant, form an 

 essential part of mathematical reasoning. 



If then we agree that " necessary conclusions " must, in the present 

 state of human knowledge, mean conclusions drawn according to 

 certain logical principles from definitely and completely stated 

 premises, we must face the question as to what these principles 

 shall be. Here, fortunately, the mathematical logicians from Boole 

 down to C. S. Peirce, Schroder, and Peano have prepared the field 

 so well that of late years Peano and his followers 2 have been able 

 to make a rather short list of logical conceptions and principles upon 

 which it would seem that all exact reasoning depends. 3 We must 

 remember, however, when we are tempted to put implicit confidence 

 in certain fundamental logical principles, that, owing to their extreme 

 generality and abstractness, no very great weight can be attached 

 to the mere fact that these principles appeal to us as obviously 



1 All writers on elementary geometry from Euclid down almost to the close 

 of the nineteenth century use intuition freely, though usually unconsciously, in 

 obtaining results which they are unable to deduce from their axioms. The first 

 few demonstrations of Euclid are criticised from this point of view by Russell in 

 his Principles of Mathematics, vol. i, 404-407. Gauss's first proof (1799) that 

 every algebraic equation has a root gives a striking example of the use of intuition 

 in what was intended as an absolutely rigorous proof by one of the greatest and at 

 the same time most critical mathematical minds the world has ever seen. 



2 And, independently, Frege. 



3 It is not intended to assert that a single list has been fixed upon. Different 

 writers naturally use different lists. 



