Natural Philosophy

by Wilhelm Ostwald

In the course of these observations we have learned how co-ordination can be used for obtaining a number of fundamental and multifariously applied principles. From this alone the great importance of co-ordination is evident, and later we shall see that its significance is even more far-reaching. The entire methodology of all the sciences is based upon the most manifold and many-sided application of the process of co-ordination, and we shall have occasion to make use of it repeatedly. Its significance may be briefly characterized by stating that it is the most general means of bringing connection into the aggregate of our experiences.

29. Counting.

The group of integral numbers, because of its fundamental simplicity and regularity, is by far the best basis of co-ordination. For while arithmetic and the theory of numbers give us a most thorough acquaintance with the peculiarities of this group, we secure by the process of co-ordination the right to presuppose these peculiarities and the possibility of finding them again in every other group which we have co-ordinated with the numerical group. The carrying out of such co-ordination is called counting, and from the premises made it follows that we can count all things in so far as we disregard their differences.

We count when we co-ordinate in turn one member of a group after another with the members of the number series that succeed one another until the group to be counted is exhausted. The last number required for the co-ordination is called the sum of the members of the counted group. Since the number series continues indefinitely, every given group can be counted.

Numerals have been co-ordinated with names as well as with signs. The former are different in the different languages, the latter are international, that is, they have the same form in all languages. From this proceeds the remarkable fact that the written numbers are understood by all educated men, while the spoken numbers are intelligible only within the various languages.

The purpose of counting is extremely manifold. Its most frequent and most important application lies in the fact that the amount affords a measure for the effectiveness or the value of the corresponding group, both increasing and decreasing simultaneously. A further number serves as a basis for divisions and arrangements of all kinds to be carried out within the group, whereby liberal use is made of the principle that everything that can be effected in the given number group can also be effected in the co-ordinated counted group.

30. Signs and Names.

The co-ordination of names and signs with numbers calls for a few general remarks on co-ordination of this nature.

The possibility of carrying out the formal operations effected in one of the groups upon the co-ordinated group itself facilitates to an extraordinary extent the practical shaping of the reality for definite purposes. If by counting we have ascertained that a group of people numbers sixty, we can infer without actually executing the steps that it is possible to form these men in six rows of ten, or in five rows of twelve, or in four rows of fifteen, but that we cannot obtain complete rows if we try to arrange them in sevens or elevens. These and numberless other peculiarities we can learn of the group of men from its amount, that is, from its co-ordination with the numerical group of sixty. In co-ordination, therefore, we have a means of acquainting ourselves with facts without having to deal directly with the corresponding realities.

It is clear that men will very soon notice and avail themselves of so enormous an advantage for the mastery and shaping of life. Thus, we see the process of co-ordination in general use among the most primitive men. Even the higher animals know how to utilize co-ordination consciously. When the dog learns to answer to his name, when the horse responds to the "Whoa" and the "Gee" of his driver there is in each case a co-ordination of a definite action or series of actions, that is, of a concept with a sign, or, in other words, of a concept with a member of another group; and in this there need not be the least similarity between the things co-ordinated with each other. The only requirement is that on the one hand the co-ordinated sign should be easily and definitely expressed and be to the point, and that, on the other hand, it should be easily "understood," that is, comprehended by the senses and unmistakably differentiated from other signs co-ordinated with other things.