396 



SCIENCE 



[N. S. Vol. XXXII. No. 821 



ieal precepts, as given directly in unan- 

 alyzed experience, are wholly unfitted to 

 form the basis of an exact science. More- 

 over, physical intuition fails altogether to 

 afford anj^ trustworthy guidance in connec- 

 tion with the concept of the iniinite, which, 

 as we have seen, is in some form indis- 

 pensable in the formation of a coherent 

 system of mathematical analysis. The 

 hasty and uncritical extension to the region 

 of the infinite, of results which are true 

 and often obvious in the region of the 

 finite, has been a fruitful source of error 

 in the past, and remains as a pitfall for the 

 unwary student in the present. The no- 

 tions derived from physical intuition must 

 be transformed into a scheme of exact defi- 

 nitions and axioms before they are avail- 

 able for the mathematician, the necessary 

 precision being contributed by the mind 

 itself. A very remarkable fact in connec- 

 tion with this process of refinement of the 

 rough data of experience is that it contains 

 an element of arbitrariness, so that the re- 

 sult of the process is not necessarily unique. 

 The most striking example of this want of 

 uniqueness in the conceptual scheme so ob- 

 tained is the case of geometry, in which it 

 has been shown to be possible to set up 

 various sets of axioms, each set self-con- 

 sistent, but inconsistent with any other of 

 the sets, and yet such that each set of 

 axioms, at least under suitable limitations, 

 leads to results consistent with our percep- 

 tion of actual space-relations. Allusion is 

 here made, in particular, to the well-known 

 geometries of Lobatehewsky and of Rie- 

 mann, which differ from the geometry of 

 Euclid in respect of the axiom of parallels, 

 in place of which axioms inconsistent with 

 that of Euclid and with one another are 

 substituted. It is a matter of demonstra- 

 tion that any inconsistency which might 

 be supposed to exist in the scheme known 

 as hyperbolic geometry, or in that known 



as elliptic geometry, would necessarily en- 

 tail the existence of a corresponding incon- 

 sistency in Euclid's set of axioms. The 

 three geometries, therefore, from the logical 

 point of view, are completely on a par with 

 one another. An interesting mathematical 

 result is that all efforts to prove Euclid's 

 axiom of parallels, i. e., to deduce it from 

 his other axioms, are doomed to necessary 

 failure ; this is of importance in view of the 

 many efforts that have been made to obtain 

 the proof referred to. When the question 

 is raised which of these geometries is the 

 true one, the kind of answer that will be 

 given depends a good deal on the view 

 taken of the relation of conceptual schemes 

 in general to actual experience. It is 

 maintained by M. Poincare, for example, 

 that the question which is the true scheme 

 has no meaning ; that it is, in fact, entirely 

 a matter of convention and convenience 

 which of these geometries is actually em- 

 ployed in connection with spatial measure- 

 ments. To decide between them by a cru- 

 cial test is impossible, because our space 

 perceptions are not sufficiently exact in 

 the mathematical sense to enable us to de- 

 cide between the various axioms of par- 

 allels. Whatever views are taken as to the 

 difficult questions that arise in this connec- 

 tion, the contemplation and study of 

 schemes of geometry wider than that of 

 Euclid, and some of them including 

 Euclid's geometry as a special case, is of 

 great interest not only from the purely 

 mathematical point of view, but also in 

 relation to the general theory of knowledge, 

 in that, owing to the results of this study, 

 some change is necessitated in the views 

 which have been held by philosophers as 

 to what is known as Kant's space-problem. 

 The school of thought which has most 

 emphasized the purely logical aspect of 

 mathematics is that which is represented in 

 this country by Mr. Bertrand Russell and 



