The Philosophical Review/Volume 1/Review: Husserl - Philosophie der Arithmetik

by Frank Thilly

Philosophie der Arithmetik. Psychologische und logische Untersuchungen. Von Dr. E. S. Husserl, Privatdocent der Philosophie an der Universität zu Halle. Erster Band. Halle-Saale. C. E. M. Pfeffer (Robert Strieker), 1891. — pp. xvi, 324.

We have here the first volume of what promises to be a very thorough and detailed account of the philosophy of arithmetic. The present instalment is a valuable contribution to the understanding of the fundamental concepts underlying the science 'of number. The undertaking is a significant one, if only in the sense that it marks a new departure in logic. Hitherto the modern tendency to specialization has not appeared in this department, works on logic contenting themselves with a meagre account of the philosophy of the sciences. Dr. Husserl describes the field of his research as a circle within many circles, and directs his attention to the principles of a single science.

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 relation to the science of arithmetic. If arithmetic operated with the actual ideas of number, we should have to regard addition and division as its fundamental operations. But this is not the case. Logicians have overlooked the fact that all ideas of number beyond the first few are symbolical. If we could have real ideas of all numbers, arithmetic would be superfluous. Only an infinite understanding, however, could possess such powers of abstraction. Arithmetic is merely an artificial means of overcoming the imperfections of a finite intellect. The most we can do is to cognize concrete pluralities composed of twelve elements. When we present to ourselves a real idea of plurality, every member of the group is conceived in connection with all the rest. If we were restricted to this act, no conception of a multitude (Menge) would be possible. A hasty glance at a crowd of persons at once gives us the idea that it is a multitude. This is due not to a "collective combination," but to sensible quasi-qualities of the multitude itself, viz. to figural elements (row, heap, group), to the sensible contrasts existing between the members themselves, or between them and their background, to movements, etc. (pp. 227-240). The psychological process, occurring in the formation of such a symbolical idea of multitude, is partly like that in the actual formation: there is psychical activity as regards some of the elements, and this serves as a sign that the process may be continued. Now symbolical numbers rest on the symbolical notion of multitude. Symbolically we may, therefore, speak of numbers whose actual ideation transcends the limits of human powers. Signs or names are employed to designate groups that can be collectively combined. The sign remains as the fixed framework of the group; by means of it the latter may be reconstructed in thought. But a systematic principle is required for the formation of symbolical number-forms. If the advance from given numbers to new numbers results from the application of a transparent, simple principle, this only need be remembered. If the designations are appropriate, the signs will indicate the whole process. The following scheme, in which x represents the ground-number, embodies the principle underlying the logical formation of number:

	⁠⁠	1 $\cdot$ 2 $\cdot$ 3	⁠.⁠.⁠.⁠	x - 1
	⁠⁠	1x $\cdot$ 2x $\cdot$ 3x	⁠.⁠.⁠.⁠	(x – 1)x
	⁠⁠	1x2 $\cdot$ 2x2 $\cdot$ 3x2	⁠.⁠.⁠.⁠	(x – 1)x2
	⁠⁠	1x3 $\cdot$ 2x3 $\cdot$ 3x3	⁠.⁠.⁠.⁠	(x – 1)x3, etc.

The same system is expressed in the formation of sensible signs. Concepts are the sources from which the rules of all arithmetical operations spring, but the sensible signs only are taken account of in practice.

With a chapter on the logical sources of arithmetic Dr. Husserl ends his first volume.

The method of sensible signs is the logical method of arithmetic. In the solution of a problem, the thought from which we proceed is first translated into signs, we operate with these signs according to the laws governing the system, and then translate the resulting signs back again into ideas. Hence, the task of arithmetic is to find general rules for the reduction of different forms to certain normal forms. Arithmetical operations will then signify no more than the methods of performing this reduction. With an examination of the processes of addition, multiplication, subtraction, and division the volume closes.

Frank Thilly.