232 METAPHYSICS 



pears to a non-mathematician like myself more doubtful. Those 

 who hold with Schroder that geometry essentially involves, as Kant 

 thought it did, an appeal to principles not self-evident and depend- 

 ent upon an appeal to sensuous "intuition," are logically bound 

 to conclude with him that geometry is an "empirical," or as W. K. 

 Clifford called it, a "physical" science, different in no way from 

 mechanics except in the relative paucity of the empirical premises 

 presupposed, and to class it with the applied sciences. On the other 

 hand, if Mr. Bertrand Russell should be successful in his promised 

 demonstration that all the principles of geometry are deducible from 

 a few premises which include nothing of the nature of an appeal to 

 sensuous diagrams, geometry too would take its place among the 

 pure sciences, but only on condition of our recognizing that its 

 truths, like those of arithmetic, are one and all, as Leibniz held, 

 strictly analytical. Thus we obtain as a first distinction between the 

 pure and the empirical sciences the principle that the propositions 

 of the former class are all analytical, those of the latter all synthetic. 

 It is not the least of the services which France is now rendering to 

 the study of philosophy that we are at last being placed by the 

 labors of M. Couturat in a position to appreciate at their full worth 

 the views of the first and greatest of German philosophers on this 

 distinction, and to understand how marvelously they have been 

 confirmed by the subsequent history of mathematics and of logic. 



(2) A consequence of this distinction is that only the pure or 

 formal sciences can be matter of rigid logical demonstration. Since 

 the empirical or applied sciences one and all contain empirical pre- 

 mises, i. e., premises which we admit as true only because they have 

 always appeared to be confirmed by the appeal to " intuition," 

 and not because the denial of them can be shown to lead to false- 

 hood, the conclusions to which they conduct us must one and all 

 depend, in part at least, upon induction from actual observation of 

 particular temporal sequences. This is as much as to say that all 

 propositions in the applied sciences involve somewhere in the course 

 of the reasoning by which they are established the appeal to the 

 calculus of Probabilities, which is our one method of eliciting general 

 results from the statistics supplied by observation or experiment. 

 That this is the case with the more concrete among such applied 

 sciences has long been universally acknowledged. That it is no less 

 true of sciences of such wide range as mechanics may be said, I 

 think, to have been definitely established in our own day by the 

 work of such eminent physicists as Kirchhoff and Mach. In fact, 

 the recent developments of the science of pure number, to which 

 reference has been made in a preceding paragraph, combined with 

 the creation of the "descriptive " theory of mechanics, may fairly 

 be said to have finally vindicated the distinction drawn by Leibniz 



