Page:Mind (New Series) Volume 9.djvu/369

LOGICAL, THEORY OF THE IMAGINARY. 355 distinction which exists between the processes of formal and material thought and their relations to concrete sciences, such as mathematics. The view we shall present is an expansion of the following remark of Grassmann's in the EMcitung to tlir Aiindchiiutujdehre of 1844 : " Die formalen Wissenschaften betrachten entweder die allgemeinen Gesetze des Denkens, oder sie betrachten das Besondere durch das Denken gesetzte, ersteres die Dialektik (die Logik) letzteres die reine Mathematik ". This passage contrasts logical inference based on universal laws of thought with mathematical resting on the particular. In principle, the calculus of Boole's Laws of Thought is identical with the ordinary logic. No inference can be drawn in the former which cannot also be drawn in the latter. It has all the weakness of formal logic in dealing with material consequence. It may be regarded as the limiting case of material inference. If we compare such a calculus of logic with the caculus of the Ausdehnuiigslehre, we shall find that the leading characteristic of the former is that it treats the terms with which it deals as self-identical units, which may coincide or not but between which no other relation can exist. The equations by which it is distinguished are : x 2 x. x (1-x) = 0. On the other hand the Ausdehnunyslehre presents equations the opposite of this : a* = 0. a b = b a. It is evident that the literal symbols in the first set of equations have their value residing in themselves ; those of the second set have their value in relation to each other, and in the character of that relation. The one calculus views the units with which it deals, as identical, self-related, coin- ciding or not coinciding but otherwise unrelated to each other. The other calculus regards its objects as existing only in relation, as constituted by relation to something different and out of that relation becoming zero. The connexion between these two algebras is not external or contingent. They are united by reason of the necessary synthesis of thought with objects of experience. Ordinary mathematics employs both processes of inference. In imagin- ary expressions the absolute disconnexion which the abstract use of the negative in ordinary logic involves is overcome by means of the opposite principle of relativity and necessary synthesis.