ON THE THEORY OF SCIENCE 345 



can foretell the future positions of these bodies with a high degree of 

 approximation. 



From the theory of order we come to the theory of number or 

 arithmetic by the systematic arrangement or development of an 

 operation just indicated (page 343). We can arrange any number of 

 groups in such a way that a richer always follows a poorer. But the 

 complex obtained in this manner is always accidental with reference 

 to the number and the richness of its members. A regular and com- 

 plete structure of all possible groups is evidently obtained only if 

 we start from a group of one member or from a simple thing, and by 

 the addition of one member at a time make further groups out of 

 those that we have. Thus we obtain different groups arranged ac- 

 cording to an increasing richness, and since we have advanced one 

 member at a time, that is, made the smallest step which is possible, 

 we are certain that we have left out no possible group which is poorer 

 than the richest to which the operation has been carried. 



This whole process is familiar; it gives the series of the positive 

 whole numbers, that is, the cardinal numbers. It is to be noted that 

 the concept of quantity has not yet been considered; what we have 

 gained is the concept of number. The single things or members in 

 this number are quite arbitrary, and especially they do not need to 

 be alike in any manner. Every number forms a group-type, and 

 arithmetic or the science of numbers has the task of investigating 

 the properties of these different types with reference to their division 

 and combination. If this is done in general form, without attention 

 to the special amount of the number, the corresponding science is 

 called algebra. On the other hand, by the application of formal rules 

 of formation, the number system has had one extension after another 

 beyond the territory of its original validity. Thus counting back- 

 ward led to zero and to the negative numbers; the inversion of 

 involution to the imaginary numbers. For the group-type of the 

 positive whole numbers is the simplest but by no means the only 

 possible one, and for the purpose of representing other manifolds 

 than those which are met with in experience, these new types have 

 proved themselves very useful. 



At the same time the number series gives us an extremely useful 

 type of arrangement. In the process of arising it is already ordered, 

 and we make use of it for the purpose of arranging other groups. 

 Thus, we are accustomed to furnish the pages in a book, the seats in a 

 theatre, and countless other groups which we wish to make use of in 

 any kind of order with the signs of the number series, and thereby 

 we make the tacit assumption that the use of that corresponding 

 group shall take place in the same order as the natural numbers 

 follow each other. The ordinal numbers arising therefrom do not 

 represent quantities, nor do they represent the only possible type 



