[1221]. An all-inclusive geometrical symbolism, such as Hamilton and Grassmann conceived of, is impossible.—Burkhardt, H.
Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 5, p. 52.
[1222]. The language of analysis, most perfect of all, being in itself a powerful instrument of discoveries, its notations, especially when they are necessary and happily conceived, are so many germs of new calculi.—Laplace.
Oeuvres, t. 7 (Paris, 1896), p. xl.
CHAPTER XIII
MATHEMATICS AND LOGIC
[1301]. Mathematics belongs to every inquiry, moral as well as physical. Even the rules of logic, by which it is rigidly bound, could not be deduced without its aid. The laws of argument admit of simple statement, but they must be curiously transposed before they can be applied to the living speech and verified by observation. In its pure and simple form the syllogism cannot be directly compared with all experience, or it would not have required an Aristotle to discover it. It must be transmuted into all the possible shapes in which reasoning loves to clothe itself. The transmutation is the mathematical process in the establishment of the law.—Peirce, Benjamin.
Linear Associative Algebra; American Journal of Mathematics, Vol. 4 (1881), p. 97.
[1302]. In mathematics we see the conscious logical activity of our mind in its purest and most perfect form; here is made manifest to us all the labor and the great care with which it progresses, the precision which is necessary to determine exactly the source of the established general theorems, and the difficulty with which we form and comprehend abstract conceptions; but we also learn here to have confidence in the certainty, breadth, and fruitfulness of such intellectual labor.—Helmholtz, H.