CONTRIBUTION TO THE THEORY OF SCIENCE 231 



according to their richness. Further, inasmuch as we advanced by a 

 single member, that is, we have made the smallest step possible, we are 

 certain to have omitted no possible group that is poorer than the 

 richest to which we have advanced our operation. 



This whole procedure is well known. It yields the entire series of 

 positive numbers — the cardinal numbers. It is to be noted that the 

 concept of magnitude does not appear as yet. What we have obtained 

 is merely the concept of number. The individual members may be 

 chosen quite arbitrarily. They need in no way be equal. Each num- 

 ber represents a quantity type; and it is the sphere of arithmetic to 

 examine these different types in respect to subdivision and combina- 

 tion. If this be done without considering the amount of the number, 

 we call the corresponding science algebra. On the other hand, the 

 extension of formal rules beyond their original application has led to 

 one development of numbers after the other. Thus counting backwards 

 leads to zero and the negative numbers, the square root of the latter to 

 the imaginary numbers. The quantity-type of all the positive num- 

 bers is, to be sure, the simplest, though by no means the only possible 

 one. For the purpose of representing other arrangements such as 

 occur among our experiences these new types have proved very useful. 



At the same time the numerical series yields a most useful type of 

 arrangement. From its very origin it is arranged in an orderly 

 fashion and it is therefore employed for the purpose of arranging other 

 quantities. Thus we are accustomed to apply the signs of the numer- 

 ical series to any objects which we desire to use in a definite order, such 

 as the pages of a book, the seats in a theater, as well as countless other 

 groups. We, however, tacitly make the assumption that the arranged 

 groups are to be used in the same sequence in which the natural num- 

 bers follow one another. These sequence-numbers represent no magni- 

 tudes nor do they represent the only type of arrangement possible. 

 They are, however, the very simplest. 



We do not reach the concept of magnitude until we reach the 

 science of time and space. A science of time has not been developed 

 separately. On the contrary, what there is to say about time usually 

 appears for the first time in mechanics. However, it is possible for 

 us to state the fundamental characteristics of time here, so that the 

 want of a distinct science of time will not be felt. 



The first and most important property of time (and also of space) 

 is that it is continuous. In other words, any portion of time may be 

 divided at any point. In the numerical series this is not the case; it 

 may be divided only between numbers. The series one to ten has nine 

 places of division, and only nine. A minute or a second, on the other 

 hand, has an unlimited number of possible points of division. In 

 other words, there is nothing in the passage of time preventing us at 



