Note 643 — Chapter XVII · ON THE THEORY OF TRANSFORMATIONS, OR THE COMPARISON OF RELATED FORMS*

In this there lies a certain justification for a saying of Minot’s, of the greater part of which, nevertheless, I am heartily inclined to disapprove. “We biologists,” he says, “cannot deplore too frequently or too emphatically the great math­e­mat­i­cal delusion by which men often of great if limited ability have been misled into becoming advocates of an erroneous conception of accuracy. The delusion is that no science is accurate until its results can be expressed math­e­mat­i­cally. The error comes from the assumption that mathematics can express complex relations. Unfortunately mathematics have a very limited scope, and are based upon a few extremely rudimentary 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 math­e­mat­i­cal 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 called mathematics “a science of the eye”; and Sylvester assures us that “most, if not all, of the great ideas of modern mathematics have had their origin in observation” (Brit. Ass. Address, 1869, and Laws of Verse, p. 120, 1870).