XVII] THE COMPARISON OF RELATED FORMS 1031 



experiences, which we make as very little children and of which no adult has 

 any recollection. The fact that from this basis men of genius have evolved 

 wonderful methods of dealing with numerical relations should not blind us 

 to another fact, namely, that the observational basis of mathematics is, 

 psychologically speaking, very minute compared with the observational basis 

 of even a single minor branch of biology. . . . While therefore here and there 

 the mathematical methods may aid us, we need a kind and degree of accuracy 

 of which mathematics is absolutely incapable. . . . With human minds constituted 

 as they actually are, we cannot anticipate that there will ever be a mathe- 

 matical expression for any organ or even a single cell, although formulae will 

 continue to be useful for dealing now and then with isolated details..." 

 {op. cit. p. 19, 1911). It were easy to discuss and criticise these sweeping 

 assertions, which perhaps had their origin and parentage in an obiter dictum 

 of Huxley's, to the effect that "Mathematics is that study which knows nothing 

 of observation, nothing of experiment, nothing of induction, nothing of 

 causation" {cit. Cajori, Hist, of Elem. Mathematics, p. 283). But Gauss, 

 "rex mathematicorum," called mathematics "a science of the eye"; and 

 Sylvester assures us that "most, if not all, of the great ideas of modem 

 mathematics have had their origin in observation" {Brit. Ass. Address, 1869, 

 and Laws of Verse, p. 120, 1870. 



Reaumur said the same thing two hundred years ago {Mem. i, p. 49, 1734). 

 Maupertuis, he said, was both naturalist and mathematician; and all his 

 mathematics "n'ont en rien affaibli son gout pour les insectes, personne 

 peut-etre n'a plus d' amour pour eux." He goes on to say: "L' esprit 

 d'observation qu'on regarde comme le ' caractere d' esprit essentiel aux 

 naturalistes, est egalement necessaire pour faire des progres en quelque science 

 que ce soit. C'est I'esprit d'observation qui fait appercevoir ce qui a 

 echappe aux autres, qui fait saisir des rapports qui sont entre des choses 

 qui semblent differentes, ou qui fait trouver les differences qui sont entre 

 celles qui paroissent semblables. On ne resoud les problemes les plus epineux 

 de Geometric qu'apres avoir S9u observer des rapports, qui ne se decouvrent 

 qu'a un esprit penetrant, et extremement attentif. Ce sont des observations 

 qui mettent en etat de resoudre les problemes de physique comme ceux 

 d'histoire naturelle, car I'histoire natureUe a ses problemes a resoudre, et 

 elle n'en a meme que trop qui ne sont pas resolus." It is in a deeper sense 

 than this, however, that the modem physicist looks on mathematics as an 

 "empirical" science, and no longer a matter of pure intuition, or "reine 

 Anschauung." Cf. Max Bom, on Some philosophical aspects of modem 

 physics, Proc. R.S.E. Lvn, pp. 1-18, 1936.. 



For one reason or another there are very many organic forms which 

 we cannot describe, still less define, in mathematical terms: just as 

 there are problems even in physical science beyond the mathematics 

 of our age. We never even seek for a formula to define this fish or 

 that, or this or that vertebrate skull. But we may already use 

 mathematical language to describe, even to define in general terms, 



