[1533]. Now there are several scores, upon which skill in mathematicks may be useful to the experimental philosopher. For there are some general advantages, which mathematicks may bring to the minds of men, to whatever study they apply themselves, and consequently to the student of natural philosophy; namely, that these disciplines are wont to make men accurate, and very attentive to the employment that they are about, keeping their thoughts from wandering, and inuring them to patience in going through with tedious and intricate demonstrations; besides, that they much improve reason, by accustoming the mind to deduce successive consequences, and judge of them without easily acquiescing in anything but demonstration.—Boyle, Robert.
Works (London, 1772), Vol. 3, p. 426.
[1534]. It is not easy to anatomize the constitution and the operations of a mind [like Newton’s] which makes such an advance in knowledge. Yet we may observe that there must exist in it, in an eminent degree, the elements which compose the mathematical talent. It must possess distinctness of intuition, tenacity and facility in tracing logical connection, fertility of invention, and a strong tendency to generalization.—Whewell, W.
History of the Inductive Sciences (New York, 1894), Vol. 1, p. 416.
[1535]. The domain of physics is no proper field for mathematical pastimes. The best security would be in giving a geometrical training to physicists, who need not then have recourse to mathematicians, whose tendency is to despise experimental science. By this method will that union between the abstract and the concrete be effected which will perfect the uses of mathematical, while extending the positive value of physical science. Meantime, the [use] of analysis in physics is clear enough. Without it we should have no precision, and no co-ordination; and what account could we give of our study of heat, weight, light, etc.? We should have merely series of unconnected facts, in which we could foresee nothing but by constant recourse to experiment; whereas, they now have a character of rationality which fits them for purposes of prevision.—Comte, A.
Positive Philosophy [Martineau], Bk. 3, chap. 1.
[1536]. It must ever be remembered that the true positive spirit first came forth from the pure sources of mathematical science; and it is only the mind that has imbibed it there, and which has been face to face with the lucid truths of geometry and mechanics, that can bring into full action its natural positivity, and apply it in bringing the most complex studies into the reality of demonstration. No other discipline can fitly prepare the intellectual organ.—Comte, A.
Positive Philosophy [Martineau], Bk. 3, chap. 1.
[1537]. During the last two centuries and a half, physical knowledge has been gradually made to rest upon a basis which it had not before. It has become mathematical. The question now is, not whether this or that hypothesis is better or worse to the pure thought, but whether it accords with observed phenomena in those consequences which can be shown necessarily to follow from it, if it be true. Even in those sciences which are not yet under the dominion of mathematics, and perhaps never will be, a working copy of the mathematical process has been made. This is not known to the followers of those sciences who are not themselves mathematicians, and who very often exalt their horns against the mathematics in consequence. They might as well be squaring the circle, for any sense they show in this particular.—De Morgan, A.