CONTRIBUTION TO THE THEORY OF SCIENCE 233 



geometric properties of the figures from the calculatory properties of 

 the formulas, and vice versa. We must inquire how such close and 

 unambiguous relations can exist between things so diverse. The an- 

 swer is that we have to deal in this instance with a particularly obvious 

 case of association. The manifoldness of numbers is far greater than 

 that of planes or space, for, whereas the latter are determined by but 

 two or three independent measurements, any number of independent 

 numerical series may be made to react upon one another. We there- 

 fore arbitrarily limit the manifoldness of the numbers to two or three 

 independent series, and determine (by means of the cosinus law) their 

 mutual relations so that a manifoldness corresponding exactly to that 

 of the space arises to which we are able completely to refer it. We 

 have then two manifoldnesses of identical character; and all properties 

 of arrangement and size of the one are ' depicted ' in the other. In 

 this an extremely important scientific procedure is indicated which 

 consists in giving to the experience-content of a given field a formal 

 manifoldness to which we impart the same manifoldness-character as 

 that possessed by the former. Every science thus develops a formula 

 language of its own, perfect in proportion to the accuracy with which 

 the manifoldness-character of the object has been recognized, and the 

 fitness of the formulas selected. Whereas, in arithmetic and algebra 

 this problem has been solved quite perfectly (though by no means ab- 

 solutely so), chemical formulas, for instance, express only a relatively 

 small part of the characteristics which they ought to express, while in 

 biology and sociology we have hardly progressed beyond the very begin- 

 nings of the solution of this problem. 



One of these universal manifoldnesses designed to express our ex- 

 periences is speech. Inasmuch as it was developed in a primitive 

 civilization it is by no means regular and complete enough to fulfil its 

 purpose satisfactorily. On the contrary, it is quite as unsystematic 

 as were the events in the history of the various peoples. The need to 

 express the infinite variety of events in daily life has been filled by 

 allowing word and concept to correspond only within a wide limit of 

 variation. Therefore all research in the sciences which are forced to 

 employ this means of expression (psychology and sociology or phi- 

 losophy generally) is greatly impeded by the struggle with the in- 

 definiteness and ambiguity of language. An improvement of these 

 conditions is to be attained only by the introduction, as rapidly as the 

 progress of the science warrants it, of symbols to which we refer the 

 manifoldness which experience tells us is peculiar to the concept. 



The sciences which have been classed above as a part of energetics 

 occupy an intermediate position. In addition to the concepts of order, 

 number, magnitude, space and time we meet in this branch of knowl- 

 edge with the new concept of energy, which is applied as universally to 



