A Plea for the Mathematician; Nature, Vol. 1, p. 261.
[725]. The conception of correspondence plays a great part in modern mathematics. It is the fundamental notion in the science of order as distinguished from the science of magnitude. If the older mathematics were mostly dominated by the needs of mensuration, modern mathematics are dominated by the conception of order and arrangement. It may be that this tendency of thought or direction of reasoning goes hand in hand with the modern discovery in physics, that the changes in nature depend not only or not so much on the quantity of mass and energy as on their distribution or arrangement.—Merz, J. T.
History of European Thought in the Nineteenth Century (Edinburgh and London, 1903), p. 736.
[726]. Now this establishment of correspondence between two aggregates and investigation of the propositions that are carried over by the correspondence may be called the central idea of modern mathematics.—Clifford, W. K.
Philosophy of the Pure Sciences; Lectures and Essays (London, 1901), Vol. 1, p. 402.
[727]. In our century the conceptions substitution and substitution group, transformation and transformation group, operation and operation group, invariant, differential invariant and differential parameter, appear more and more clearly as the most important conceptions of mathematics.—Lie, Sophus.
Leipziger Berichte, No. 47 (1895), p. 261.
[728]. Generality of points of view and of methods, precision and elegance in presentation, have become, since Lagrange, the common property of all who would lay claim to the rank of scientific mathematicians. And, even if this generality leads at times to abstruseness at the expense of intuition and applicability, so that general theorems are formulated which fail to apply to a single special case, if furthermore precision at times degenerates into a studied brevity which makes it more difficult to read an article than it was to write it; if, finally, elegance of form has well-nigh become in our day the criterion of the worth or worthlessness of a proposition,—yet are these conditions of the highest importance to a wholesome development, in that they keep the scientific material within the limits which are necessary both intrinsically and extrinsically if mathematics is not to spend itself in trivialities or smother in profusion.—Hankel, Hermann.
Die Entwickelung der Mathematik in den letzten Jahrhunderten (Tübingen, 1884), pp. 14-15.