On Transformations (1900-1949)

"The power of differential calculus is that it linearizes all problems by going back to the 'infinitesimally small', but this process can be used only on smooth manifolds. Thus our distinction between the two senses of rotation on a smooth manifold rests on the fact that a continuously differentiable coordinate transformation leaving the origin fixed can be approximated by a linear transformation at О and one separates the (nondegenerate) homogeneous linear transformations into positive and negative according to the sign of their determinants. Also the invariance of the dimension for a smooth manifold follows simply from the fact that a linear substitution which has an inverse preserves the number of variables." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"It can, you see, be said, with the same approximation to truth, that the whole of science, including mathematics, consists in the study of transformations or in the study of relations." (Cassius J Keyser. "Mathematical Philosophy: A Study of Fate and Freedom", 1922)

"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 sequence of different positions of the same particle at different times forms a one-dimensional continuum in the four-dimensional space-time, which is called the world-line of the particle. All that physical experiments or observations can teach us refers to intersections of world-lines of different material particles, light-pulsations, etc., and how the course of the world-line is between these points of intersection is entirely irrelevant and outside the domain of physics. The system of intersecting world-lines can thus be twisted about at will, so long as no points of intersection are destroyed or created, and their order is not changed. It follows that the equations expressing the physical laws must be invariant for arbitrary transformations." (Willem de Sitter, "Kosmos", 1932)

"A mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformations of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs." (Vannevar Bush, "As We May Think", 1945)

"It should be observed first that the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation […]. The idea of a category is required only by the precept that every function should have a definite class as domain and a definite class as range, for the categories are provided as the domains and ranges of functors. Thus one could drop the category concept altogether […]" (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"We have assumed that the laws of nature must be capable of expression in a form which is invariant for all possible transformations of the space-time co-ordinates." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)