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The Legacy of Bernhard Riemann After One Hundred and Fifty Years, Volume 1

Lizhen Ji, Frans Oort, and Shing-Tung Yau, editors
International Press
Publication Date: 
Number of Pages: 
Advanced Lectures in Mathematics 35.1
[Reviewed by
Steven Deckelman
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In the preface to the second edition of his famous work The World as Will and Representation, the sometimes acerbic Arthur Schopenhauer wrote on the topic of reading the original writings of the masters:

Only from their creators themselves can we receive philosophical thoughts. Therefore the man who feels himself drawn to philosophy must himself seek out its immortal teachers in the quiet sanctuary of their works. The principal chapters of any one of these genuine philosophers will furnish a hundred times more insight into their doctrines than the cumbersome and distorted accounts of them produced by commonplace minds…

Perhaps Schopenhauer’s stricture seems unreasonably harsh. Commentaries and other compendia can be immeasurably helpful to understanding not only original writings but also their subsequent developments and ramifications. Nonetheless Riemann’s original writings even today remain imbued with life and inspiration for the 21st century, and reading Riemann’s original works is still worthwhile. Niels Henrick Abel made a similar observation in mathematics through his famous saying “It appears to me that if one wants to make progress in mathematics one should study the masters and not their pupils.”

This two volume book is a collection of twenty three articles by different authors on the mathematical legacy of Bernhard Riemann. The articles deal to varying extents with different aspects of Riemann’s life, his work, and the subsequent evolution and impact on mathematics during the one hundred and fifty years following his death in 1866. The articles generally touch on the history of mathematics, philosophy of mathematics, Riemann’s original papers and contemporary currents of mathematical thought they gave rise to.

It should be pointed out that the collection does not include any of Riemann’s own works, though many references are given to the myriad publications going back to the first German editions in 1876 and 1892 along with later French, Russian and English editions. These various editions are themselves a valuable source of surveys and commentary on the impact of Riemann’s work at various points in time. In English, a particulary comprehensive one was published by Springer with a long preface by Narasimhan in 1990. The original German language papers are available online. The collected works of Riemann have also been published by Dover in paperback. An English translation by Christensen and Baker was published by the Kendrick Press in 2004, though at the time of this review it seems to be out of print and it seems difficult to find an affordable copy.

The articles vary quite a bit in length. The two longest articles are the ones by Brian Conrey, titled “Riemann’s Hypothesis”, and James Milne, titled “The Riemann Hypothesis over Finite Fields: From Weil to the Present Day”, each in excess of 80 pages. By contrast, Sir Michael F. Aityah’s “Riemann’s Influence in Geometry, Analysis and Number theory” is only 12 pages.

A question raised by the editors and contributors is how to place the work of Bernhard Riemann in the pantheon of mathematicians. This question is in essence a philosophical one; it relates to how we define what mathematics is, what mathematicians are, and what constitutes knowledge and progress in mathematics. In particular, what is it that makes mathematics good and great? The very first article in the collection, “What One Should Know About Riemann But May Not Know?” by Lizhen Ji and Shing-Tung Yau address these question at length. They draw upon the views of a number of distinguished mathematicians including Arnold, Atiyah, Borel,Browder,Dehn Dyson Gelfand, Gowers, Grothendieck, Halmos, Klein, Mac Lane Shafarevich, Thurston, Von Neumann, J.C. Fields. Indeed a plausible argument can be made is that by some standards, Riemann might be regarded as the greatest mathematician in history.

This first article also contains a short history of Riemann’s life and overview of his work and mathematical legacy and serves as an overview of the twenty-two articles that follow. Some of Riemann’s work is of course familiar. These include subjects like Riemannian Geometry, Number Theory, Complex Analysis and Real Analysis; all of these are treated in depth in the many articles that follow. What readers will find interesting is that Riemann also made deep contributions to physics and was deeply influenced by philosophy, in particular the post-Kantian philosopher Friedrich Herbart. His famous 1854 Habilitationsschrift on the foundations of geometry has an unmistakable philosophical tone. This lead Hans Freudenthal to later write that Riemann would have been recognized as a great philosopher had he lived longer.

Another question the first article addresses is what we can learn from reading the original work of a great mathematician like Riemann. Perhaps in essence it is merely the inspiration that can be had from reading the masters own words and seeing great ideas in their seminal form. Riemann, however, is not easy to read, and not merely due to the depth of his ideas but also to his Germanic writing style, which Hans Freudenthal described as exhibiting some of the worst aspects of German syntax, presumably influenced by his philosophical reading. Nonetheless, professors Ji and Yau give this marvelous metaphor in terms of the Bible:

There are many commentaries on the Bible, and people also need to go to sermons, to help them understand the spirit in the Bible. But if they really want to understand the spirit of the Bible, people still need to, and it is essential, to read the Bible, and there is no replacement or shortcut to the demanding job of reading the Bible, and even in the original language.

In addition to the first articles, the articles by Lizhen Ji, “The Historical Roots of the Concept of Riemann Surfaces” and “The Story of Riemann’s Moduli Space”, as well as Jurgen Jost’s “Riemann and the Modern Concept of Space”, also contain a good deal of history. The second article by Sir Michael Atiyah, “Riemann’s Influence in Geometry, Analysis and Number Theory”, gives a very concise but readable account of the Riemann Zeta Function, Riemann Surfaces and the Riemannian metric. The remaining articles are more specialized and treat various modern outgrowths of Riemann’s work.

Each article contains its own bibliography and is rich in references into the literature. This collection will be of interest to those interested in the history and philosophy of mathematics as well as to anyone seeking a central source of papers on mathematical topics having a connection to Riemann’s work.

Steven Deckelman is a professor of mathematics at the University of Wisconsin-Stout, where he has been since 1997. He received his Ph.D from the University of Wisconsin-Madison in 1994 for a thesis in several complex variables written under Patrick Ahern. Some of his interests include complex analysis, mathematical biology and the history of mathematics.

The table of contents is not available.