All exact knowledge supposes the mind to be able to apply, steadily and clearly, not only the processes of reasoning, but also certain fundamental ideas; and it is one main office of a liberal education to fix and develope these ideas. The ideas of Space and of Number are the subject matter of Geometry, of Arithmetic, and of Algebra in its character of Universal Arithmetic: and since all our knowledge, relative to the external world, must be subject to the conditions of space and number, the elementary portions of mathematics just mentioned are, rightly and necessarily, made the basis of all intellectual education. If we advance further in mathematical study, with the view of its thus serving as an intellectual discipline, what other ideas do we thus bring to activity and use? I reply, that the main general ideas which we have next to introduce, and which consequently should be the governing principles of the second stage of a liberal education, are the following:--the mechanical ideas of Force and Body, with their various modifications; the idea of the Symmetry of symbolical expressions;--the idea of the Universal Interpretation of symbols, including as an important branch of this, the Application of Algebra to Geometry;--and the idea of a Limit.
[William Whewell, The Doctrine of Limits, page viii.]
I hope to start a series on Whewell later this week, and, of course, the final section of the series on ideas for a video game based on philosophical movement-building, and also something on natural law theory, but I am not feeling well today, so they'll have to wait.