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Abstract:
This essay explores the power and fertility of mathematical imagination, as they are manifest in the thought of a nineteenth-century German mathematician Bernhard Riemann (1826-1866), one of the greatest and most imaginative mathematicians who ever lived. Riemann introduced radically new ideas in every main field of modern mathematics: algebra, analysis, geometry, and topology. These ideas transformed each of these fields and played major roles in making mathematics into what it is now. The essay considers in particular two interrelated aspects of Riemann's work: the first is his concept of "manifold(ness)," which transformed our mathematical, physical, and philosophical understanding of spatiality; and the second is the conceptual character of Riemann's mathematical thinking as responsible for the radical nature of his ideas, such as those concerning spatiality and/as manifoldness. The essay also addresses, in closing, some of the implications of Riemann's ideas for modern physics, most especially for Albert Einstein's general relativity-his non-Newtonian theory of gravitation.
Introduction
Richard Feynman once said that "a new idea is extremely difficult to think of. It takes a fantastic imagination."1 It is, accordingly, all the more remarkable that Bernhard Riemann (1826-1866), a nineteenth- century German mathematician-and one of the greatest and most imaginative mathematicians who ever lived-thought of radically new ideas in every main field of modern mathematics: algebra, analysis, geometry, and topology. His ideas transformed each of these fields and, in part through establishing new connections among them (one of the hallmarks of Riemann's thought), played major roles in making each field and mathematics as a whole into what they are now. As Detlef Laugwitz observes in his biography of Riemann, Bernhard Riemann: Turnings Points in the Conception of Mathematics: "It is an amazing fact that fundamental parts of modern mathematics have their origins in Riemann's analysis."2 Laugwitz's subtitle is worth noting for its plural of "turning points," and for its suggestion that these were the points at which not only mathematics itself, but also our conception of it changed. Given my limits here, I shall focus primarily on two interrelated aspects of Riemann's work. The first is the power and fertility of his thought and imagination as manifest in his concept of "manifold," or "manifoldness," one of his great inventions, which transformed our mathematical, physical, and philosophical...