METAPHYSICS AND THE OTHER SCIENCES 231 



pure and the empirical sciences may be roughly indicated by saying 

 that the latter class comprises all those sciences which yield infor- 

 mation about the particular details of the temporal order of events 

 physical and psychical, whereas the pure sciences deal solely with the 

 general characteristics either of all truths, or of all truths of some 

 well-defined class. More exactly we may say that the marks by 

 which an empirical is distinguished from a pure science are two. 

 (1) The empirical sciences one and all imply the presence among 

 their premises of empirical propositions, that is, propositions which 

 assert the actual occurrence of some temporal fact, and depend upon 

 the witness of immediate apprehension, either in the form of sense- 

 perception or in that of what is commonly called self-consciousness. 

 In the vague language made current by Kant, they involve an appeal 

 to some form of unanalyzed "intuition." The pure sciences, on the 

 other hand, contain no empirical propositions either among their pre- 

 mises or their conclusions. The principles which form their premises 

 are self-evidently true propositions, containing no reference to the 

 actual occurrence of any event in the temporal order, and thus in- 

 volving no appeal to any form of "intuition." And the conclusions 

 established in a pure science are all rigidly logical deductions from 

 such self-evident premises. That the universality of this distinction 

 is still often overlooked even by professed writers on scientific method 

 seems explicable by two simple considerations. On the one hand, it 

 is easy to overlook the important distinction between a principle 

 which is self-evident, that is, which cannot be denied without explicit 

 falsehood, and a proposition affirmed on the warrant of the senses, 

 because, though its denial cannot be seen to be obviously false, 

 the senses appear on each fresh appeal to substantiate the asser- 

 tion. Thus the Euclidean postulate about parallels was long falsely 

 supposed to possess exactly the same kind of self-evidence as 

 the dictum de omni and the principle of identity which are part 

 of the foundations of all logic. And further Kant, writing under 

 the influence of this very confusion, has given wide popularity to 

 the view that the best known of the pure sciences, that of mathe- 

 matics, depends upon the admission of empirical premises in the 

 form of an appeal to intuition of the kind just described. Fortunately 

 the recent developments of arithmetic at the hands of such men 

 as Weierstrass, Cantor, and Dedekind seem to have definitely refuted 

 the Kantian view as far as general arithmetic, the pure science of 

 number, is concerned, by proving that one and all of its propositions 

 are analytic in the strict sense of the word, that is, that they are 

 capable of rigid deduction from self-evident premises, so that, in 

 what regards arithmetic, we may say with Schroder that the famous 

 Kantian question " how are synthetic judgments a priori possible? " 

 is now known to be meaningless. As regards geometry, the case ap- 



