On Continuity XIV (Progress)

"Nothing is accomplished all at once, and it is one of my great maxims, and one of the most completely verified, that Nature makes no leaps: a maxim which I have called the law of continuity." (Gottfried W Leibniz, "New Essays Concerning Human Understanding with an Appealing", 1704)

"Harmonious order governing eternally continuous progress - the web and woof of matter and force interweaving by slow degrees, without a broken thread, that veil which lies between us and the Infinite - that universe which alone we know or can know; such is the picture which science draws of the world, and in proportion as any part of that picture is in unison with the rest, so may we feel sure that it is rightly painted." (Thomas H Huxley, "Darwiniana", 1893–94)

"Mathematics is not a world empire where one man can exert a dominating influence over work along all the lines, but it is continually splitting up into self-determining republics." (Geroge A Miller, "Felix Klein and the History of Modern Mathematics", 1927)

"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. […] it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-Euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation." (Paul A M Dirac, "Quantities singularities in the electromagnetic field", Proceedings of the Royal Society of London, 1931)

"For it is true, generally speaking, that mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers." (David Hilbert, "Anschauliche Geometrie", 1932)

"Science does not mean an idle resting upon a body of certain knowledge; it means unresting endeavor and continually progressing development toward an end which the poetic intuition may apprehend, but which the intellect can never fully grasp." (Max Planck, "The Philosophy of Physics", 1936)

"And yet, on looking into the history of science, one is overwhelmed by evidence that all too often there is no regular procedure, no logical system of discovery, no simple, continuous development. The process of discovery has been as varied as the temperament of the scientist." (Gerald Holton, "Thematic Origins of Scientific Thought: Kepler to Einstein", 1973)