THE UNITY OF SCIENCE 13 



ascertained through sense-experience, nor do they subse- 

 quently test or verify their conchisions by comparing them 

 with sense-experience. In short, their method is not, in the 

 ordinary sense, inductive ; it does not consist in the heaping up 

 of particular observed instances of a given phenomenon. 

 Whether mathematical knowledge is or is not ultimately based 

 upon experience is a philosophical question which need not be 

 discussed here. At all events, it manifestly is not derived 

 directly from experience as chemical or biological knowledge 

 is. The geometrician does not collect material triangles of 

 brass, iron, wood, chalk, and the like, and measure them be- 

 fore arriving at generalizations about all kinds of triangles, 

 before he feels confident of the universal truth of his the- 

 orems about those figures. On the contrary, he reasons with 

 entire confidence and assurance about relations that would 

 hold good in worlds which probably do not exist and of which, 

 certainly, nobody ever had or can have any sense-experience 

 whatever — ideal worlds, namely, of non-Euclidian space. 

 Mathematics and logic, then, deal with necessary relations 

 that hold between abstract concepts, or between propositions, 

 and for the attainment of knowledge — of, indeed, the clearest 

 and most certain kind of knowledge — they do not need to 

 wait upon the slow processes of sensory observation, or ex- 

 perimentation with material things, or the laborious accumula- 

 tion of individual instances. Now, if all the knowledge which 

 man has any occasion to require could be got by the math- 

 ematician's procedure — by the method of the abstract sciences 

 — we should manifestly be entitled to say that all the sciences 

 were one as to their method. But since it is not so, we have 

 already one important cleavage, one breach of continuity, 

 running across the fabric of science. About this a good deal 

 more might be said; but in the interest of brevity, it is neces- 

 sary to refrain from saying it. Instead, let us turn to the 

 sciences of phenomena, and from this point on consider ex- 

 clusively in what the unification of these alone would consist. 

 To narrow the question to the limits of our present course, 



