The positive portions of the book display sound analytic judgment, while the critical parts, besides being keen and indicative of the author's wide range of reading, carefully restrict themselves to the essential points of the theories attacked. His consideration of the arguments advanced by mathematicians must acquit him of the charge of "one-sidedness" frequently urged against logicians who discuss the philosophy of mathematics. Everywhere Dr. Husserl is clear, in thought as well as in expression, a characteristic which, when we remember the abstruseness of the subject and the traditional bent of the German mind for involved sentences, should be doubly appreciated. His intentional disregard of a terminology, which often repels those not skilled in the craft, renders the pages accessible to mathematicians as well as to philosophers.
The first part of the work deals with psychological questions connected with the concepts, plurality (Vielheit), unity (Einheit), and sum (Anzahl), while the second treats of the symbolical ideas of plurality and sum, and shows how the fact that we are restricted to symbolical number-concepts in arithmetic, determines its character. The author rejects the logical method, which is so strongly advocated by many writers. For him number is the result of psychical processes (p. 130). Notions like unity and plurality cannot be logically defined, but rest upon ultimate psychical data. In this sense they may be designated as form-concepts or categories (p. 91).
Dr. Husserl examines the concepts, plurality, unity, and sum, which latter forms the fundamental notion of number. After investigating the time-succession theory, Lange's thesis that the synthesis upon which number is founded is a synthesis of space-intuitions, and the views of Baumann, Sigwart, Jevons, and Schuppe, he finds the origin of the concepts, plurality, and sum, to be due to the "collective combination" of the mind, which cognizes every member of a sum by itself and in connection with all the rest. The concrete phenomena, which serve as the basis for this abstraction, may be either physical or psychical. This explanation seems to me to be far more satisfactory than the superficial reasoning of Mill, who, like Bain, advocates the theory of physical abstraction. Of course, no concept can be conceived without being based on a concrete intuition, but the special nature of the particular object is of no account whatsoever. The notion of plurality ultimately rests upon that of the somewhat (Etwas), a concept which cannot be further analyzed, nor even explained in the way in which Dr. Husserl explains the other concepts. It seems to be a category in the Kantian sense of "function" or "form" of the intellect, a fact which the author does not, in my opinion, sufficiently appreciate.
Part II proposes to explain, pyschologically and logically, the art of reckoning based on the notions hitherto analyzed, and to investigate its
