EXAMINATION OF THE PRINCIPLES 159 



in the nature of the abstraction which is made in the two sciences. 

 The laws of economics are protected by an 'other things being 

 equal,' where there is by no means a definite conception as to 

 what these other things may possibly include. In mechanics 

 there is no 'other things being equal.' The antecedent of each 

 formula purports, at least, to set forth the precise conditions 

 under which the consequent must follow. Aside from this we 

 can only say that mechanical laws represent a far higher grade of 

 universality and precision than economic laws have attained, or, 

 very possibly, will ever attain. 



The case of geometry and that of mechanics hang closely to- 

 gether. It is known that the principles of the two sciences are 

 so related that considerable alterations can be made in either 

 and sufficiently compensated by corresponding alterations in the 

 other. A non-Euclidean geometry, coupled with its appropriate 

 non-Newtonian mechanics, can describe our world as exactly 

 as the Euclidean can do. In short, geometry is recognizedly a 

 branch of applied mathematics an experimental science which has 

 long since reached the deductive stage. If mathematicians some- 

 times appear to take it otherwise, that is because they have re- 

 defined the term. It then no longer professes to treat of the 

 space-relations of our experience, but is, as the phrase goes, a 

 science of 'cross-classification.' 



There remains only pure mathematics that is to say, formal 

 logic and the sciences of number and order deducible from formal 

 logic as a possible obstacle to an evolutionary view of scientific 

 validity. We are inclined to the belief that this also is no insuper- 

 able obstacle, that logic, like geometry and mechanics, repre- 

 sents a stage in the development of scientific universality, not the 

 ideal consummation. The numerical formulas (such as Kant's no- 

 torious 7 + 5 = 12), upon whose a priori certainty so much stress 

 was formerly laid, are in themselves, as has been definitely shown, 

 analytical propositions and, indeed, absolute identities: the defi- 

 nitions of the two members of the equality can always be reduced 

 to an identical form. 1 The vital question is whether the under- 



J Cf. L. Couturat, Les Principes des Mathematiques, p. 255, n. 3. 



