On Differential Equations (1900-1924)

"The equations of Newton's mechanics exhibit a two-fold invariance. Their form remains unaltered, firstly, if we subject the underlying system of spatial coordinates to any arbitrary change of position ; secondly, if we change its state of motion, namely, by imparting to it any uniform translatory motion ; furthermore, the zero point of time is given no part to play. We are accustomed to look upon the axioms of geometry as finished with, when we feel ripe for the axioms of mechanics, and for that reason the two invariances are probably rarely mentioned in the same breath. Each of them by itself signifies, for the differential equations of mechanics, a certain group of transformations. The existence of the first group is looked upon as a fundamental characteristic of space. The second group is preferably treated with disdain, so that we with un-troubled minds may overcome the difficulty of never being able to decide, from physical phenomena, whether space, which is supposed to be stationary, may not be after all in a state of uniform translation. Thus the two groups, side by side, lead their lives entirely apart. Their utterly heterogeneous character may have discouraged any attempt to compound them. But it is precisely when they are compounded that the complete group, as a whole, gives us to think." (Hermann Minkowski, "Space and Time" ["Raum und Zeit"], [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

"It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations." (Felix Klein, "Lectures on Mathematics", 1911)

"The difficulty involved in the proper and adequate means of describing changes in continuous deformable bodies is the method of differential equations. […] They express mathematically the physical concept of contiguous action." (Max Born, "Einstein's Theory of Relativity", 1920)

"The works of the highest faculty of man, judgment, is always directed toward the constant limiting of the infinite, toward the breaking up of the infinite into comfortably digestible portions, differentials." (Yevgeny Zamyatin, "We", 1921)

"The conception of tensors is possible owing to the circumstance that the transition from one co-ordinate system to another expresses itself as a linear transformation in the differentials. One here uses the exceedingly fruitful mathematical device of making a problem "linear" by reverting to infinitely small quantities." (Hermann Weyl, "Space - Time - Matter", 1922)

"The difficulty involved is that the proper and adequate means of describing changes in continuous deformable bodies is the method of differential equations. […] They express mathematically the physical concept of contiguous action." (Max Born, "Einstein’s Theory of Relativity", 1922)

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