Natural Philosophy

by Wilhelm Ostwald

Now, according to a previous observation ([p. 64]), not only does the order bring every number into relation with the preceding one, but since this last for its part already possesses a great number of relations to all preceding, these relations exert their influence also upon the new relation. This fact gives rise to extraordinarily manifold relations between the various numbers and to manifold laws governing these relations. The elucidation of them forms the subject of an extensive science.

26. Arithmetic, Algebra, and the Theory of Numbers.

From this regular form of the number series numerous special characteristics can be established. The investigations leading to the discovery of these characteristics are purely scientific, that is, they have no special technical aim. But they have the uncommonly great practical significance that they provide for all possible arrangements and divisions of numbered things, and so have instruments at hand ready for application to each special case as it arises. I have already pointed out that in this lies the positive importance of the theoretical sciences. For practical reasons the study of them must be as general as possible. This science is called arithmetic.

Arithmetic undergoes an important generalization if the individual numbers in a calculation are disregarded and abstract signs standing for any number at all are used in their place. At first glance this seems superfluous, since in every real numerical calculation the numbers must be reintroduced. The advantage lies in this, that in calculations of the same form, the required steps are formally disposed of once for all, so that the numerical values need be introduced only at the conclusion and need not be calculated at each step. Moreover, the general laws of numerical combination appear much more clearly if the signs are kept, since the result is immediately seen to be composed of the participating members. Thus, algebra, that is, calculation with abstract or general quantities, has developed as an extensive and important field of general mathematics.

By the theory of numbers we understand the most general part of arithmetic which treats of the properties of the "numerical bodies" formed in some regular way.

27. Co-ordination.

So far our discussion has confined itself to the individual groups and to the properties which each one of them exhibits by itself. We shall now investigate the relations which exist between two or more groups, both with regard to their several members and to their aggregate.

If at first we have two groups the members of which are all differentiated from one another, then any one member of the one group can be co-ordinated with any one member of the other group. This means that we determine that the same should be done with every member of the second group as is done with the corresponding member of the first group. That such a rule may be carried out we must be able to do with the members of all the groups whatever we do with the members of one group. In other words, no properties peculiar to individual members may be utilized, but only the properties that each member possesses as a member of a group. As we have seen, these are the properties of association.

First, the co-ordination is mutual, that is, it is immaterial to which of the two groups the processes are applied. The relation of the two groups is reciprocal or symmetrical.