Bernhard Riemann, 1826-1866 : turning points in the conception of mathematics - Laugwitz, Detlef (Boston : Birkhäuser, 1999)
This book, originally written in German and presented here in an English-language translation, is the first attempt to examine Riemann's scientific work from a single unifying perspective. Laugwitz describes Riemann's development of a conceptual approach to mathematics at a time when conventional algorithmic thinking dictated that formulas and figures, rigid constructs, and transformations of terms were the only legitimate means of studying mathematical objects.
David Hilbert gave prominence to the Riemannian principle of utilizing thought, not calculation, to achieve proofs. Hermann Weyl interpreted the Riemann principle - for mathematics and physics alike - to be a matter of "understanding the world through its behavior in the infinitely small.".
This remarkable work, rich in insight and scholarship, is addressed to mathematicians, physicists, and philosophers interested in mathematics. It seeks to draw those readers closer to the underlying ideas of Riemann's work and to the development of them in their historical context. This illuminating English-language version of the original German edition will be an important contribution to the literature of the history of mathematics.
0. Introduction -- 1. Complex Analysis. 1.1. The genesis of complex analysis up to Riemann's time. 1.2. The dissertation of 1851. 1.3. The elaborations. 1.4. The zeta function and the distribution of primes -- 2. Real Analysis. 2.1. Foundations of real analysis. 2.2. Trigonometric series before Riemann. 2.3. Riemann's results. 2.4. Trigonometric series after Riemann. 2.5. A self-contained chapter: Gauss, Riemann, and the Gottingen atmosphere -- 3. Geometry; Physics; Philosophy. 3.1. Geometry. 3.2. Physics. 3.3. On philosophy -- 4. Turning Points in the Conception of Mathematics. 4.1. The historians' search for revolutions in mathematics. 4.2. Turning point in the conception of the infinite in mathematics. 4.3. Turning point in the method: Thinking instead of computing. 4.4. Turning point in the ontology: Mathematics as thinking in concepts. 4.5. The ontology and methodology of mathematics after Riemann. 4.6. Concluding remarks.
Includes bibliographical references (p. [341]-349) and index.
Keywords: Riemann, Bernhard, -- 1826-1866.; Mathematicians -- Germany -- Biography; Mathematics -- Germany -- History -- 19th century