Page:Mind (New Series) Volume 9.djvu/364

 350 G. ,T. STOKES : make use of it only in a tentative manner ; second, those who have considered it as undefinable and uninterpretable, and build upon this supposed fact a special theory of reasoning ; third, those who, viewing it as capable of definition, have sought for the definition in the ideas of geometry ". As an example of the first class Prof. Macfarlane instances the astronomer Airy, and as an example of the second the view put forward by Boole in his Laws of Thought (p. 68), who bases on the non-interpretability of the symbol N / 1 in mathematics, a claim to dispense with the interpretability of the intermediate results in other processes of reasoning. . Prof. Macfarlane does not adduce expressly any instances of the third class. It may I suppose be considered as representing the common opinion on the subject. The following attempt at a logical interpretation of the mathematical symbol was suggested to me by the considera- tion of the function evolved by Boole in pursuance of the view referred to above. Here, however, the process is re- versed and an attempt made to explain an uninterpretable symbol by an intelligible logical relation. With the doubtful exception of Carnot, whose discussion of the subject in his Geometric de Position touches very closely the view here advocated, the greatest names in mathematics are identified exclusively with attempts at finding a geometrical interpretation for the imaginary roots of unity. De Morgan who assigns a geometrical meaning to "double" and "triple" algebra says expressly of "single algebra" or what I have above called pure algebra that the symbol J 1 is in it unmeaning. The same view seems to have been held by Clifford. It is expressly stated in the Common Sense of the Exact Sciences. More recently this view has been reasserted by Mr. Russell in his Foundations of Geometry, who however adds that he is "unacquainted with any satisfactory philo sophy of imaginaries in pure algebra". Essentially the same standpoint is adopted by Mr. Whitehead in his recently published Universal Algebra. This position seems to me to be essentially paradoxical, and the difficulties inherent in it very great. Whoever adopts this view is obliged to hold that in pure, or, to use Dr. Morgan's term, "single" algebra impossible or imagin- ary quantities are an anomaly, and that they receive what- ever meaning they have as something tacked on from the outside by this application to a particular subject-matter. This would simply be an unaccountable process in any logical theory of the movement of thought. Moreover it evades precisely the point which has to be explained, viz., how an