ON THE THEORY OF SCIENCE 347 



we would be able to develop a self-consistent system or science. Such 

 a geometry, however, would have an extremely complex and imprac- 

 tical structure for objective purposes (as, for example, land meas- 

 urement), and so we strive to develop a science as free as possible 

 from subjective factors. Historically, we have before us a process of 

 this sort in the astronomy of Ptolemy and that of Copernicus. The 

 former corresponded to the subjective appearances in the assumption 

 that all heavenly bodies revolved around the earth, but proved to be 

 very complicated when confronted with the task of mastering these 

 movements with figures. The latter gave up the subjective stand- 

 point of the observer, who looked upon himself as the centre, and 

 attained a tremendous simplification by placing the centre of revo- 

 lution in the sun. 



A few words are to be said here about the application of arithmetic 

 and algebra to geometry. It is well known that under definite 

 assumptions (coordinates), geometrical figures can be represented 

 by means of algebraic formulae, so that the geometrical properties 

 of the figure can be deduced from the arithmetical properties of the 

 formula, and vice versa. The question must be asked how such a 

 close and univocal relationship is possible between things of such 

 different nature. The answer is, that here is an especially clear case 

 of association. The manifold of numbers is much greater than that of 

 surface or space, for while the latter are determined by two or three in- 

 dependent measurements, one can have any number of independent 

 number series working together. Therefore the manifold of numbers 

 is arbitrarily limited to two or three independent series, and in so 

 far determines their mutual relations (by means of the laws of cosine) 

 that there results a manifold, corresponding to the spatial, which can 

 be completely associated with the spatial manifold. Then we have 

 two manifolds of the same manifold character, and all characteristics 

 of arrangement and size of the one find their likeness in the other. 



This again characterizes an extremely important scientific pro- 

 cedure which consists, namely, in constructing a formal manifold for 

 the content of experience of a certain field, to which one attributes 

 the same manifold character which the former possesses. Every 

 science reaches by this means a sort of formal language of correspond- 

 ing completeness, which depends upon how accurately the manifold 

 character of the object is recognized and how judiciously the formulae 

 have been chosen. While in arithmetic and algebra this task has been 

 performed fairly well (though by no means absolutely perfectly), the 

 chemical formulae, for instance, express only a relatively small part 

 of the manifold to be represented; and in biology as far as sociology, 

 scarcely the first attempts have been made in the accomplishment of 

 this task. 



Language especially serves as such a universal manifold to repre- 



