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 2 3 | ... | x - 1 | ||
| 1x 2x 3x | ... | (x – 1)x | ||
| 1x2 2x2 3x2 | ... | (x – 1)x2 | ||
| 1x3 2x3 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.
