Whatever its source, mathematics has come down to the present by the t — Unification in science and mathematics

"It appears... that the elastic theories of light, if Kelvins gyrostatic adynamic ether be admitted, have not been wholly routed. Nevertheless the great electromagnetic theory of light propounded by Maxwell (1864, Treatise, 1873) has been singularly apt not only in explaining all the phenomena reached by the older theories and in predicting entirely novel results, but in harmoniously uniting as parts of a unique doctrine, both the electric or photographic light vector of Fresnel and Cauchy and the magnetic vector of Neumann and MacCullagh. Its predictions have, moreover, been astonishingly verified by the work of Hertz (1890), and it is to-day acquiring added power in the convection theories of Lorentz (1895) and others."

"All knowledge... is unification of the multiple."

"The age-old conflict between our notions of continuity and the scientific concept of number ended in a decisive victory for that latter. This victory was brought about by the necessity of vindicating, of legitimizing... a procedure which ever since the days of Fermat and Descartes had been an indispensable tool of analysis. ...analytic geometry ...this discipline which was born of the endeavors to subject problems of geometry to arithmetical analysis, ended by becoming the vehicle through which the abstract properties of number are transmitted to the mind. It furnished analysis with a rich, picturesque language and directed it into channels of generalization hitherto unthought of. Now, the tacit assumption on which analytic geometry operated was that it was possible to represent the points on a line, and therefore points in a plane and in space, by means of numbers. ...The great success of analytic geometry... gave this assumption an irresistible pragmatic force. ...Under such circumstances mathematics proceeds by fiat. It bridges the chasm between intuition and reason by a convenient postulate. On the one hand, there was the logically consistent concept of real number and its aggregate, the arithmetic continuum; on the other, the vague notions of the point and its aggregate, the linear continuum. All that was necessary was to declare the identity of the two, or, what amounted to the same thing, to assert that: It is possible to assign to any point on a line a unique real number, and, conversely, any real number can be represented in a unique manner by a point on a line. This is the famous Dedekind-Cantor axiom."

"Although it is true that it is the goal of science to discover rules which permit the association and foretelling of facts, this is not its only aim. It also seeks to reduce the connections discovered to the smallest possible number of mutually independent conceptual elements. It is in this striving after the rational unification of the manifold that it encounters its greatest successes, even though it is precisely this attempt which causes it to run the greatest risk of falling a prey to illusions."

"The remarkable insight that characterized Klimpts later work was contemporaneous with Freuds psychological studies and presaged the inward turn that would pervade all fields of inquiry in Vienna in 1900. This period... was characterized by the attempt to make a sharp break with the past and to explore new forms of expression in art, architecture, psychology, literature, and music. It spawned an ongoing pursuit to link these disciplines. ...Viennese life at the turn of the century provided opportunities in salons and coffeehouses for scientists, writers, and artists to come together in an atmosphere that was at once inspiring, optimistic, and politically engaged. ...science was no longer the narrow and restrictive province of scientists but had become an integral part of Viennese culture. ...a paradigm for how an open dialogue can be achieved."

"At first the mathematical disciplines were not sharply defined. As knowledge increased, individual subjects split off from the parent mass and became autonomous. Later, some were overtaken and reabsorbed in vaster generalizations of the mass from which they sprang. Thus trigonometry issued from surveying, astronomy, and geometry only to be absorbed, centuries later, in the analysis which had generalized geometry. This recurrent escape and recapture has inspired some to dream of a final, unified mathematics which shall embrace all. Early in the twentieth century it was believed by some for a time that the desired unification had been achieved in mathematical logic. But mathematics, too irrepressibly creative to be restrained by any formalism, escaped."