E. O. Wilson just had his say on mathematics in biology. But R. A. Fisher had his say back in 1930. Here’s Fisher writing on pages viii-ix in his Preface to The Genetical Theory of Natural Selection. And if you’re expecting an anti-empirical rant, or an homage to mathematics that ignores or mischaracterizes other routes to biological insight, well, read on and be surprised.
The types of mind which result from training in mathematics and in biology certainly differ profoundly; but the difference does not seem to lie in the intellectual faculty. It would certainly be a mistake to say that the manipulation of mathematical symbols requires more intellect than original thought in biology ; on the contrary, it seems much more comparable to the manipulation of the microscope and its appurtenances of stains and fixatives ; whilst original thought in both spheres represents very similar activities of an identical faculty. This accords with the view that the intelligence, properly speaking, is little influenced by the effects of training. What is profoundly susceptible of training is the imagination, and mathematicians and biologists seem to differ enormously in the manner in which their imaginations are employed. Most biologists will probably feel that this advantage is all on their side. They are introduced early to the immense variety of living things ; their first dissections, even if only of the frog or dog fish, open up vistas of amazing complexity and interest, at the time when the mathematician seems to be dealing only with the barest abstractions, with lines and points, infinitely thin laminae, and masses concentrated at ideal centres of gravity. Perhaps I can best make clear that the mathematician’s imagination also has been trained to some advantage, by quoting a remark dropped casually by Eddington in a recent book–
*We need scarcely add that the contemplation in natural science of a wider domain than the actual leads to a far better understanding of the actual.* (p. 267, The Nature of the Physical World.)
For a mathematician the statement is almost a truism. From a biologist, speaking of his own subject, it would suggest an extra- ordinarily wide outlook. No practical biologist interested in sexual reproduction would be led to work out the detailed consequences experienced by organisms having three or more sexes; yet what else should he do if he wishes to understand why the sexes are, in fact, always two ? The ordinary mathematical procedure in dealing with any actual problem is, after abstracting what are believed to be the essential elements of the problem, to consider it as one of a system of possibilities infinitely wider than the actual, the essential relations of which may be apprehended by generalized reasoning, and subsumed in general formulae, which may be applied at will to any particular case considered. Even the word possibilities in this statement unduly limits the scope of the practical procedures in which he is trained ; for he is early made familiar with the advantages of imaginary solutions, and can most readily think of a wave, or an alternating current, in terms of the square root of minus one. The most serious difficulty to intellectual co-operation would seem to be removed if it were clearly and universally recognized that the essential difference lies, not in intellectual methods, and still less in intellectual ability, but in an enormous and specialized extension of the imaginative faculty, which each has experienced in relation to the needs of his special subject. I can imagine no more beneficial change in scientific education than that which would allow each to appreciate something of the imaginative grandeur of the realms of thought explored by the other.