—Coleridge, S. T.
A Mathematical Problem.
[1420]. Mathematics connect themselves on the one side with common life and physical science; on the other side with philosophy in regard to our notions of space and time, and in the questions which have arisen as to the universality and necessity of the truths of mathematics and the foundation of our knowledge of them.—Cayley, Arthur.
British Association Address (1888); Collected Mathematical Papers, Vol. 11, p. 430.
[1421]. Mathematical teaching ... trains the mind to capacities, which ... are of the closest kin to those of the greatest metaphysician and philosopher. There is some color of truth for the opposite doctrine in the case of elementary algebra. The resolution of a common equation can be reduced to almost as mechanical a process as the working of a sum in arithmetic. The reduction of the question to an equation, however, is no mechanical operation, but one which, according to the degree of its difficulty, requires nearly every possible grade of ingenuity: not to speak of the new, and in the present state of the science insoluble, equations, which start up at every fresh step attempted in the application of mathematics to other branches of knowledge.—Mill, J. S.
An Examination of Sir William Hamilton’s Philosophy (London, 1878), p. 615.
[1422]. The value of mathematical instruction as a preparation for those more difficult investigations, consists in the applicability not of its doctrines, but of its methods. Mathematics will ever remain the most perfect type of the Deductive Method in general; and the applications of mathematics to the simpler branches of physics, furnish the only school in which philosophers can effectually learn the most difficult and important portion of their art, the employment of the laws of the simpler phenomena for explaining and predicting those of the more complex. These grounds are quite sufficient for deeming mathematical training an indispensable basis of real scientific education, and regarding, with Plato, one who is ἀγεωμέτρητος, as wanting in one of the most essential qualifications for the successful cultivation of the higher branches of philosophy.—Mill, J. S.
System of Logic, Bk. 3, chap. 24, sect. 9.
[1423]. In metaphysical reasoning, the process is always short. The conclusion is but a step or two, seldom more, from the first principles or axioms on which it is grounded, and the different conclusions depend not one upon another.
It is otherwise in mathematical reasoning. Here the field has no limits. One proposition leads on to another, that to a third, and so on without end. If it should be asked, why demonstrative reasoning has so wide a field in mathematics, while, in other abstract subjects, it is confined within very narrow limits, I conceive this is chiefly owing to the nature of quantity, ... mathematical quantities being made up of parts without number, can touch in innumerable points, and be compared in innumerable different ways.—Reid, Thomas.