464 MATHEMATICS 



since Gauss's time by almost all the leading mathematicians who 

 have been conversant with non-Euclidean geometry. 1 Recently, 

 however, Poincare has thrown the weight of his great authority 

 against this view, 2 claiming that the experiments by which it is 

 sought to test the truth of geometric axioms are really not geometrical 

 experiments at all but physical ones, and that any failure of these 

 experiments to agree with the ordinary geometrical axioms could 

 be explained by the inaccuracy of the physical la\vs ordinarily as- 

 sumed. There is undoubtedly an important element of truth here. 

 Every experiment depends for its results not merely on the law we 

 wish to test, but also on other laws which for the moment we assume 

 to be true. But, if we prefer, we may, in many cases, assume as 

 true the law we were before testing arid our experiment will then 

 serve to test some of the remaining laws. If, then, we choose to stick 

 to the ordinary Euclidean axioms of geometry in spite of what any 

 future experiments may possibly show, we can do so, but at the cost, 

 perhaps, of our present simple physical laws, not merely in one 

 branch of physics but in several. Poincare's view 3 is that it will 

 always be expedient to preserve simple geometric laws at all costs.. 

 an opinion for which I fail to see sufficient reason. 



V. Kempe's Definition 



Let us now turn from Peirce's method of defining mathematics to 

 Kempe's, which, however, I shall present to you in a somewhat 

 modified form. 4 The point of view adopted here is to try to define 

 mathematics, as other sciences are defined, by describing the objects 

 with which it deals. The diversity of the objects with which mathe- 

 matics is ordinarily supposed to deal being so great, the first step 

 must be to divest them of what is unessential for the mathematical 

 treatment, and to try in this way to discover their common and 

 characteristic element. 



The first point on which Kempe insists is that the objects of mathe- 

 matical discussion, whether they be the points and lines of geometry, 

 the numbers real or complex of algebra or analysis, the elements of 

 groups or anything else, are always individuals, infinite in number 

 perhaps, but still distinct individuals. In a particular mathematical 

 investigation we may, and usually do, have several different kinds of 

 individuals; as for instance, in elementary plane geometry, points, 

 straight lines, and circles. Furthermore, we have to deal with certain 

 relations of these objects to one an.other. For instance, in the example 



1 Gauss, Riemann, Helmholtz are the names which will carry perhaps the 

 greatest weight. 



2 Cf. La Science et fHypothese. Paris, 1903. 

 1 L. c., chapter v. In particular, p. 93. 



* Kempe has set forth his ideas in rather popular form in the Proceedings of 

 the London Mathematical Society, vol. xxvi (1894), p. 5; and in Nature, vol. XLIII 

 1890), p. 156, where references to his more technical writings will be found. 



