Abstract
A mathematical lecture without formulas, a geometric treatise without pictures or illustrations, a manuscript of only 16 pages that just came into being by chance, but a text that has shaped mathematics like few others works, which were all significantly longer, considerably more detailed, and much more carefully worked out. In this regard, we might mention the “Methodus inveniendi” by Leonhard Euler, which founded the calculus of variations; Carl Friedrich Gauss’ “Disquisitiones arithmeticae” that established mathematics as an independent discipline; Georg Cantor’s set theory, which introduced the modern conception of the infinite in mathematics; the theory of transformation groups of Sophus Lie, that is, the systematic study of symmetries that forms the mathematical basis for quantum mechanics; the programmatic writings of David Hilbert on the axiomatic foundation of various mathematical disciplines; or more recently the work of Alexander Grothendieck on the systematic unification of algebraic geometry and arithmetic. We are talking here of Bernhard Riemann’s “Ueber die Hypothesen, welche der Geometrie zugrunde liegen” (“On the hypotheses which lie at the bases of geometry”), and this short script, written in 1854, but only published in 1868 after Riemann’s death, whose wide ranging effects even take it beyond these works. This is because its position is at the intersection of mathematics, physics and philosophy, and it not only founds and establishes a central mathematical discipline, but also paves the way for the physics of the twentieth century and at the same time represents a timeless refutation of certain philosophical conceptions of space. In the present volume, this key text of mathematics will be edited, positioned in the controversies of its time, and its effects on the development of mathematics will be analyzed and compared to those of its opponents. Riemann’s “Ueber die Hypothesen, welche der Geometrie zugrunde liegen” has shaped and transformed mathematics in a manner very different from, say, Euclid’s Elements, the writings of Leibniz and Newton on the creation of the infinitesimalsimal calculus or the above-mentioned works. It has, in a manner no less fundamental than those, influenced the development of mathematics as a science. Moreover, this text is essential for Einstein’s theory of General Relativity. More recently, it also provided the mathematical structure underlying quantum field theory and its developments in theoretical Elementary Particle Physics (superstring theory, quantum gravity etc.).
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Notes
- 1.
I am also preparing an edition of this text for the current series.
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In the sense that it develops its problems autonomously and intrinsically, instead of obtaining them from physics or other sciences.
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Helmholtz says that he had developed the essential elements of his consideration before learning of Riemann’s work (which had been published with a four 10-year delay), but certainly later than Riemann, who had his lecture delivered and script written in 1854.
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It might appear natural to contrast this work here with Riemann’s text. After giving this option serious thought, however, in the end I refrained from it, because this work of Helmholtz did not achieve the same depth and elegance as Riemann’s. Moreover, among the various writings of Helmholtz on epistemological issues, this particular text is not the best and the clearest, and so, by the choice of that particular work, the important physiologist and physicist Helmholtz would have appeared in a wrong light. So if we had wanted to represent Helmholtz’ theory here by one of his writings, then we should have selected another of his writings, namely “Über den Ursprung und die Bedeutung der geometrischen Axiome” (On the origin and the importance of the geometrical axioms) or his inaugural address as Rektor (president) of the University of Bonn “Die Tatsachen in der Wahrnehmung” (The facts in perception), but then we would have lost the close relationship with Riemann’s habilitation address.
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Of course, the occasion of Riemann’s text should be taken into account. It was a colloquium before the Faculty of Arts, and Riemann certainly wanted to pay respect to the lack of mathematical knowledge of most of the people in his audience. Among these, besides Gauss, who was not a professor of mathematics, but a professor of astronomy and director of the observatory, mathematics was represented only by the two Professors Ulrich (1798–1879) and Stern (1807–1894). However, other such lectures or writings, like Klein’s Erlangen program with which he introduced himself to the faculty in Erlangen, certainly could be much more formalized, and if the faculty had chosen one of the other topics suggested by Riemann, the presentation would probably have been developed in mathematical formulas.
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Gauss was born in Brunswick in modest circumstances. Since his outstanding mathematical talent was recognized early on, he was, however, generously supported by the Brunswick Duke. Already at a young age he made significant mathematical discoveries, such as on the question of the constructability of regular polygons. His Disquisitiones Arithmeticae, published in 1801, but written already some years earlier, are considered as the work that founded modern mathematics as an autonomous science. A spectacular success of his mathematical methods of error calculation was the rediscovery of the minor planet (asteroid) Ceres in the same year. This minor planet had been discovered by astronomers, but then again lost sight of until the Gaussian methods of path calculation would permit prediction of its position with high enough precision so that the astronomers knew to which position in the sky they had to turn their telescopes to find it. Since 1807, Gauss was a professor in Göttingen and the director of the observatory. Gauss is considered the greatest mathematician of all time, and he has influenced almost all areas of modern mathematics and even founded many of them. Together with the physicist Wilhelm Weber (1804–1891) he constructed the first telegraph. The mathematical methods developed by him are fundamental for astronomy and geodesy. Especially in old age, Gauss was difficult to approach, undoubtedly also due to a not very happy family life, and the shy Riemann could not establish direct personal contact with him. Riemann therefore acquired the mathematical theories and discoveries of Gauss by self-study. A recent biography of Gauss is Walter Kaufmann Bhler, Gauss. A biographical study, Berlin etc., Springer, 1981.
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Dirichlet was born in Düren in the Rhineland as a son of the local postmaster, whose father had immigrated from the Walloon region in present-day Belgium, where the Romanesque name comes from. During a stay in Paris from 1822 to 1827, as a foreigner, however, he was not allowed to attend the courses of the then leading French mathematician Augustin Louis Cauchy (1789–1857) at the Ecole Polytechnique. fortunately, he succeeded in gaining access to the circles of Jean-Baptiste Louis Fourier (1768–1830), who, starting from physical problems of thermodynamics, had introduced the famous series representations for periodic functions. Dirichlet proves a basic result about these series expansions. Alexander von Humboldt (1769–1859), who after his famous expeditions initially stayed in Paris and then held influential positions in Berlin, is impressed by him and supports and encourages him and brings him as a professor to Prussia, first to Breslau and then in 1829 to Berlin. Dirichlet and his friend and colleague Carl Gustav Jacob Jacobi (1804–1851) turn the University of Berlin, which had been founded in 1810 by Wilhelm von Humboldt (1767–1835) in the course of the reforms motivated and necessitated by the Napoleonic aggression against Prussia, into a center of mathematical research. Dirichlet’s wife Rebecca was a granddaughter of the philosopher Moses Mendelssohn (1729–1786), a niece of the author Dorothea (von) Schlegel (1764–1839), who in turn was the wife of the writer and theorist of Romanticism Friedrich (von) Schlegel (1772–1829), and a sister of the composer Felix Mendelssohn Bartholdy (1809–1847), who as head of the Leipzig Gewandhaus Orchestra, initiated the rediscovery and renaissance of the baroque music of Bach and Händel. In this way, Dirichlet’s life was intertwined with those of many other prominent personalities. Dirichlet was friendly and open towards Riemann, and Riemann could learn a lot from him. Dirichlet made in particular important contributions in number theory, and he founded the analytic direction of number theory. A historically oriented introduction can be found in W. Scharlau, H. Opolka, From Fermat to Minkowski. Lectures on the theory of numbers and its historical development, New York, Springer,22010 (translated from the German). The principles applied by Dirichlet in the calculation of variations later played a central role in Riemann’s studies on function theory and Riemann surfaces.
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On Dedekind see Winfried Scharlau (ed.), Richard Dedekind. 1831—1981, Braunschweig/Wiesbaden, Vieweg, 1981. The letters printed there also contain biographical material on Riemann, which can complement the picture in Dedekind’s biography of Riemann in the latter’s collected works.
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In Riemann’s collected works edited by Heinrich Weber and Richard Dedekind, there is a 20 page biography of Riemann, written by his friend and colleague Dedekind. Hans Freudenthal wrote a short biography for the Dictionary of Scientific Biography. In addition to other short biographical sketches, there are the scientific biographies of Michael Monastyrsky and Detlef Laugwitz in which the development and impact of Riemann’s scientific work is placed in the context of the circumstances of his life. The scientific biography of Laugwitz has been of great help to me at various places, and it also contains a detailed description of Riemann’s life that is accessible to a general readership. Various other such analyses can be found in the reissue of Riemann’s collected works edited by Raghavan Narasimhan. A systematic research of the unpublished scientific notes and sketches and the available biographical material on Riemann has been started by Erwin Neuenschwander. For some results, see Erwin Neuenschwander, Riemanns Einfhrung in die Funktionentheorie. Eine quellenkritische Edition seiner Vorlesungen und einer Bibliographie zur Wirkungsgeschichte der Riemannschen Funktionentheorie. Abhandlungen der Akademie der Wissenschaften zu Gttingen, Math.”=Phys. Klasse, Bd. 44, 1996. Various mathematical historical studies discuss the development of geometry before, through and after Riemann, but usually not from a biographical perspective. Sources can be found in the bibliography at the end of this book.
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Jost, J. (2016). Introduction. In: Jost, J. (eds) On the Hypotheses Which Lie at the Bases of Geometry. Classic Texts in the Sciences. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-26042-6_1
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