METAPHYSICS AND THE OTHER SCIENCES 239 



the Aristotelian doctrine that there is and can be no such thing as an 

 actually existing infinite collection, but which of us would care to 

 defend that time-honored position to-daj'? Similarly with continuity 

 all of us were probably once on a time instructed that whereas " quan- 

 tity" is continuous, number is essentially "discrete," and is indeed 

 the typical instance of what we mean by the non-continuous. To-day 

 we know that it is in the number-series that we have our one certain 

 and familiar instance of a perfect continuum. Still a third illustration 

 of the transforming light which is thrown upon old standing meta- 

 physical puzzles by the increasing formal development of mathe- 

 matics may be found in the difficulties attendant upon the conception 

 of the " infinitely little," once regarded as the logical foundation of 

 the so-called Differential Calculus. With the demonstration, which 

 may be found in Mr. Russell's important work, that "infinitesimal," 

 unlike "infinite," is a purely relative term, and that there are no 

 infinitesimal real numbers, the supposed logical significance of the 

 concept seems simply to disappear. Instances of this kind could easily 

 be multiplied almost indefinitely, but those already cited should be 

 sufficient to show how important are the metaphysical results which 

 may be anticipated from contemporary mathematical research, and 

 how grave a mistake it would be to regard existing metaphysical con- 

 struction, e. g., that of the Hegelian system, as adequate in principle 

 to the present state of our organized knowledge. In fact, all the mate- 

 rials for a new Kategorienlehre, which may be to the knowledge of our 

 day what Hegel's Logic was to that of eighty years ago, appear to lie 

 ready to hand when it may please Providence to send us the meta- 

 physician who knows how to avail himself of them. The proof, given 

 since this address was delivered, by E. Zermelo, that every assem- 

 blage can be well ordered, is an even more startling illustration of 

 the remarks in the text. 



It remains to say something of the relation of metaphysical specu- 

 lation to the various sciences which make use of empirical premises. 

 On this topic I maybe allowed to be all the more brief, as I have quite 

 recently expressed my views at fair length in an extended treatise 

 (Elements of Metaphysics, Bks. 3 and 4), and have nothing of conse- 

 quence to add to what has been there said. The empirical sciences, 

 as previously defined, appear to fall into two main classes, distin- 

 guished by a difference which corresponds to that often taken in the 

 past as the criterion by which science is to be separated from philo- 

 sophy. We may study the facts of temporal sequence either with a 

 view to the actual control of future sequences or with a view to 

 detecting under the sequence some coherent purpose. It is in the 

 former way that we deal with facts in mechanics, for instance, or in 

 chemistry, in the latter that we treat them when we study history for 

 the purpose of gaining insight into national aims and character. We 



