Page:Mind (New Series) Volume 8.djvu/113

 CARVETH READ, Logic, Deductive and Inductive. 99 no pains to do what we can, but let us also recognise clearly what we cannot, and cease to waste time upon it." The spirit and temper of the book in this regard are indicated in a sentence on p. 194, in which the author is referring to those difficulties of ascertaining "exact equality" and "immediate sequence" of phenomena which are due to the limitation of human faculty. " It is right," he says, " to touch upon this well-known sceptical topic ; but to insist much upon it is not a sign of good sense." That this treatment has advantages from some points of view has just been allowed ; but it must be admitted that in some cases Mr. Bead's " short way with the sceptics " does not seem to lead to results satisfactory even from those points of view. Take, e.g., his account of the relation between Logic and Mathe- matics, or his treatment of the Laws of Thought, and of the connexion between Causation and Co-existence. On this view Mathematics seems to be almost (though perhaps not quite) a Science co-ordinate with Logic, and is described as dealing through- out with abstractions. Thus Logic is not all-em bracing, not the Science of Sciences, after all and Mathematics has the air of being separated by an impassable chasm from the world of con- crete fact. " Mathematics," it is said, " treats of the relations of all sorts of things considered as quantities," while Logic " treats of the relations of all sorts of things, but not as to their quantity " (p. 7) (this is indeed somewhat qualified by the recognition that " Logic may be said to be in some respects ' prior to ' or ' above ' Mathematics as usually treated," p. 8). Again, " As to Co-existences, the Geometrical do not belong to Logic" (p. 146), and " Geometrical Co-existence, when it is not a matter of definition, ... is de- duced from the Definitions and Axioms " (p. 233). The process of Geometrical proof " is purely Deductive ;. . . Diagrams are used not as facts for observation, but merely to fix our attention ;. . . no inference is required from the special case to all similar ones ; for they are all proved at once " (p. 166). And we even have the "falling of absolutely true dice" contrasted as "mere mathe- matical abstractions " with " concrete events " (p. 248). " The Mathematical Axioms again," we are told, " apply to Time, Space, Mental Phenomena, and Matter and Energy ; whereas the Law of Causation is only true of concrete events," etc. (p. 225). These contentions (though they certainly stop short of some obvious difficulties) are at least disputable, and to a large extent they appear to be even obviously paradoxical. Surely all the rules of Logic apply in mathematical reasonings if quantities as such are not subject to logical treatment, it would seem that all the world of phenomena in some of its aspects (and those the most definite and precise) is extra-logical. And when it is said that " the Law of Causation cannot be derived from the Mathe- matical Axioms, nor these from the Logical " (p. 226), it may be replied that it is not only Mathematical principles which cannot