Ana Historia Mathematica Bernhard Riemann, 1826–1866: Turning Points in the Conception of Mathematics: By Detlef Laugwitz....
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Reviews / Historia Mathematica 30 (2003) 217–226 223 Neuenschwander, E., 1980. Riemann und das “Weierstraßsche” Prinzip der analytischen Fortsetzung durch Potenzreihen. Jahresber. Deutsch. Math. Verein. 82, 1–11. Neuenschwander, E., 1981. Über die Wechselwirkungen zwischen der französischen Schule, Riemann und Weierstraß. Eine Übersicht mit zwei Quellenstudien. Archive Hist. Exact Sciences 24, 221–255; English transl.: Studies in the history of complex function theory. II: Interactions among the French school, Riemann, and Weierstrass. Bull. Am. Math. Soc. (N.S.) 5 (1981), 87–105. Neuenschwander, E., 1987. Riemanns Vorlesungen zur Funktionentheorie, allgemeiner Teil. Preprint No. 1086. Technische Hochschule Darmstadt, Fachbereich Mathematik. Riemann, B., 1990. Gesammelte mathematische Werke, wissenschaftlicher Nachlaß und Nachträge. Collected Papers, Weber, H., Dedekind, R., Narasimhan, R. (Eds.). Springer-Verlag, Berlin and B.G. Teubner Verlagsgesellschaft, Leipzig. Peter Ullrich Fachbereich Mathematik, Universität Siegen D-57068 Siegen, Germany E-mail address: email@example.com 10.1016/S0315-0860(02)00031-9 Bernhard Riemann, 1826–1866: Turning Points in the Conception of Mathematics By Detlef Laugwitz. Tr. Abe Shenitzer with the editorial assistance of the author, Hardy Grant, and Sarah Shenitzer. Boston (Birkhäuser). 1999. xvii + 357 pp. ISBN 0-8176-4040-1, 3-7643-4040-1 For years we have waited for a serious, book-length biography of Riemann, so this book is especially welcome. Biographer Detlef Laugwitz has chosen to concentrate on Riemann’s conceptual innovations in mathematics, placing them in the context of the work of his predecessors, his contemporaries, and his successors. To do so, he has structured the biography as follows. In an introductory chapter (Chapter 0), he discusses Riemann’s life and times and an overview of the background to Riemann’s analysis. He devotes the next three chapters to complex analysis (Chapter 1); real analysis (Chapter 2); and geometry, physics, and phil; osophy (Chapter 3). In his final chapter he discusses “turning points in the conception of mathematics” (Chapter 4). Riemann’s dissertation of 1851 introduced his theory of complex functions, based upon the ideas of mapping and Riemann surface. His habilitation paper of 1853 discussed trigonometric series and introduced his definition of the integral. The habilitation lecture of 1854 began the subject of Riemannian geometry, while making allusions to issues in physics and philosophy. Thus, the central three chapters of this book trace Riemann’s principal investigations in mathematics in the order in which he wrote his qualifying papers as a student. Organizing the biography in this way permits the author to disentangle mathematical themes that occupied Riemann through overlapping periods of several years. The introductory chapter sets the historical stage, and the final chapter offers a view and assessment of Riemann’s mathematical work as a whole. In each of the three central chapters, Laugwitz’s discussion of Riemann’s work includes analysis of sections of his first work on the subject. This method of exposition makes us feel close to Riemann’s thought. It works best in the longest chapter, on complex analysis, the heart of which consists of the discussion of several paragraphs from Riemann’s dissertation on complex analysis. By his analysis, Laugwitz gives substance to such programmatic statements as this (from Article 20 of the dissertation, quoted on p. 101): 224 Reviews / Historia Mathematica 30 (2003) 217–226 The previous methods of treating such functions always set down as a definition an expression for the function, whereby its value was given for every value of its argument; our investigation shows that, as a result of the general character of a function of a variable complex quantity, in such a definition some of the data are a consequence of the remaining ones, namely, the proportion of data has been reduced to those indispensable for the determination. In explanation, Laugwitz tells us that most of Riemann’s predecessors concentrated on a power series expansion rather than the function that it represents. By shifting emphasis to the latter, Riemann could eliminate superfluous information, determining a complex function from its singularities. Here we see an example of the biographer’s theme: Riemann’s work used simple concepts in place of the lengthy and sometimes obscure computations typical of his predecessors and contemporaries. He refers to Riemann’s lecture notes to show “the steady decrease in the amount of attention Riemann seems to have paid to power series between 1856 and 1861” (p. 80), thereby suggesting how Riemann’s thought matured, shifting further away from computation. He illustrates how, even when using his great computational abilities, Riemann still focused upon concepts rather than the computation itself. For example, Riemann constructed a function that has simple zeros at z = 0, 1, 2, . . . and is “finite for all finite z.” Laugwitz remarks (p. 87) that The road to this g(z) is heuristic, but this is of no consequence to Riemann. All he wants is to find some function with the prescribed zeros. By contrast, Weierstrass always aims to obtain formula representations of given functions. Such comments help us to understand Riemann’s mathematics as more than just the next step (obvious or not) on the road to today’s mathematics. Where Laugwitz reveals nuances of mathematical purpose and preferred mathematical methods, the book is at its best. Furthermore, Laugwitz tries to give us clues to the origins of Riemann’s ideas, as in his account of a letter (first reported by E. Neuenschwander) to Felix Klein from F.E. Prym, a former student of Riemann. Riemann noted (says Laugwitz) that relations obtained from series expansions of functions retain their validity outside. . . [their] regions of convergence. . . . “What actually continues functions from region to region”? He arrived at the insight that it was the partial differential equation. [pp. 112–113] Throughout the book, Laugwitz draws heavily upon his own earlier historical work (for example, on Euler’s analysis) and on the work of several recent scholars, including U. Bottazzini, R. Narasimhan, E. Scholz, and especially E. Neuenschwander. The book already serves a purpose by bringing together the expositions and conclusions of these scholars, but its greatest value comes from Laugwitz’s careful exegeses of passages from Riemann. The author has interesting things to say on a variety of subjects, not all closely related to Riemann. Unfortunately, sometimes his remarks seem incomplete or confusing. For example, he suggests (p. 204) that “Riemann may have arrived at his notion of an integral” in answer to the question of whether the Fourier coefficients of a given function tend to 0 (as n goes to infinity). Yet on the next page he characterizes Riemann’s introduction of his integral as ad hoc and remarks, History would have been different if one had asked the question: what kind of integral implies the equality lim ab fn = ab f , where fn is a monotonically increasing sequence of integrable functions that converge pointwise to the limit f ? [p. 305] But why should one have asked that question at that time, and what was wrong with Riemann’s question? If there is more to this critical remark than hindsight, further details would be welcome. Reviews / Historia Mathematica 30 (2003) 217–226 225 Again, in discussing Riemann’s geometry, we read that “linear algebra was a trivial matter” (p. 223) for Riemann. How are we to reconcile this with Laugwitz’s statement (p. 242) that the early developments of Riemannian geometry were “prolix and opaque” because “the development of linear algebra failed for a long time to keep pace with the progress of analysis”? Many of Laugwitz’s remarks offer real insight, however. He remarks upon the superfluity of the concept of the integral for functional analysis (p. 321), the dangers of a classification theory for the development of mathematics (p. 249), the historical significance of rearrangement of series (pp. 300–301), algebraic versus analytic treatments of a theory (p. 160), and many other interesting points. He explains how the expression “Riemannian geometry” came to be used for the geometry of spaces of constant positive curvature. Apparently the usage is a legacy from Felix Klein, whose own transformational view of geometry left no place for the full Riemannian scheme. Now that we know this, let us eschew this misleading nomenclature, which suggests both interest in non-Euclidean geometry that Riemann never demonstrated and superficiality totally at odds with his actual work. Laugwitz develops one comment at greater length in “A self-contained chapter: Gauss, Riemann, and the Göttingen atmosphere” (Chapt. 2, Sect. 5). Reacting to Felix Klein, who spoke of a “mystical, undeniable and yet not clearly understandable influence of the general atmosphere” upon Riemann, Laugwitz says, “The present conception of the history of science makes it impossible for us to be satisfied with Klein’s necromancy” (p. 214). Laugwitz’s clear thinking is welcome. In sum, this book is an excellent introduction to the mathematical work of Riemann that contains many additional interesting points. The author has not attempted to write a complete biography, however. Indeed, the narrative of Riemann’s life occupies merely 43 pages, including illustrations. Although some other external details emerge in later chapters, the reader who hopes for a three-dimensional picture of Riemann will go away disappointed. The author makes scant use of Riemann’s surviving correspondence. He treats even Riemann’s physics only briefly. For example, despite the fact that Riemann was considered an important physicist as late as a century ago, the author dismisses his papers on shock waves and on the motion of a homogeneous liquid ellipsoid. A short time ago, some experts . . . praised Riemann’s work in these two areas. This being so, we can dispense with a discussion of the details in these papers that are not very relevant to the image of Riemann as a mathematician. [pp. 254–255] We certainly fail to get a rounded view even of Riemann’s intellectual life. Of course, the author has the right to narrow his focus as he sees fit, to make the book a study rather than a full-scale biography. However, his decision cuts us off from some sources of his mathematical inspiration. For example, the author notes (p. 222) Speiser’s conjecture that Riemann’s geometric ideas arose from his concurrent search for an ethereal theory of gravitation, electromagnetism, heat, and light, which led him to the introduction of a family of metrics in three-dimensional space. “Thus it is conceivable that Weyl . . . was wrong to claim . . . that Riemann’s attempts to find a connection between light, electricity, magnetism, and gravitation . . . contemporaneous with the preparation of his [habilitation] lecture, not only were unrelated but actually interfered with one another. . . .” But having given this rather obscure hint, the author does not elaborate, although U. Bottazzini and R. Tazzioli, and independently, the reviewer gave detailed evidence of Riemann’s probable path from his investigations of ether to his definition of curvature. The discussion of Riemann’s geometry gives the author one of several opportunities to return to his main theme, that Riemann’s mathematics replaced computations by concepts. He characterizes the developments in the years immediately following. 226 Reviews / Historia Mathematica 30 (2003) 217–226 We witness here a development similar to that in complex analysis. Algorithmic thinking was rampant until well into the 20th century and seemed to have supplanted Riemann’s thinking in terms of manifolds with differentiable or complex structure. But in the second half of the 20th century Riemann’s thinking again came to occupy a leading position. [pp. 244–245] Perhaps this is what he means when he says, “Riemann’s reduction of geometry to analysis could not go unchallenged” (p. 232). At any rate, might one not with equal justice say that Riemann reduced complex analysis to geometry? Laugwitz’s sympathies appear to lie more with analysis than with geometry, which may explain why, in describing mathematics subsequent to Riemann, he devotes more attention to transfinite induction and nonstandard analysis than to algebraic topology, and why we find illustrations of Cantor and Hilbert but not of Poincaré. The author’s apparent preference for analysis over geometry may affect how he treats his main theme. In the final chapter he argues more fully that Riemann’s emphasis on concepts rather than computations represents a turning point in mathematics. After his detailed discussion of Riemann’s mathematics, his characterization of it as conceptual seems to be an advance on Klein’s vague remarks about intuition— remarks with which Laugwitz shows some impatience (see, for example, p. 150). Hilbert, on the other hand, saw very clearly that Riemann turned mathematics to “proofs impelled by thought alone and not by computation” (p. 302). But does Hilbert’s observation fully characterize Riemann’s style? Is it useless to talk of intuition? Should we ignore, in addition to Klein, both Hadamard and Poincaré, who also spoke of Riemann’s intuitive approach to mathematics? Would it not have been worthwhile to mention the latter, the inheritor of so much of both the substance and the style of Riemann’s mathematics? In our own time, Michael Atiyah has championed a view of intuition similar to Poincaré’s. To the question, “What is geometry?” Atiyah answers “that geometry is not so much a branch of mathematics as a way of thinking that permeates all branches.” So perhaps Laugwitz will not have the last word in characterizing Riemann’s style. Nevertheless, in the words of translator A. Shenitzer, “I don’t always agree with the author, but I find him stimulating and enlightening” (p. xvii). Biographer Detlef Laugwitz has given us an informative and provocative study of Riemann. We are beholden to the author, and also to Abe Shenitzer and his team for this English translation. [Ed. Note: We regret the passing both of Detlef Laugwitz and of a member of the translating team, Sarah Schenitzer.] Paul R. Wolfson Department of Mathematics, West Chester University, West Chester, PA 19383, USA 10.1016/S0315-0860(02)00011-3