Enhanced descriptions from Syndetics:

The name of Bernard Riemann is well known to mathematicians and physicists around the world. His name is indelibly stamped on the literature of mathematics and physics. 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.

Table of contents provided by Syndetics

• Preface (p. ix)
• Note to the Reader (p. xiii)
• List of Illustrations (p. xv)
• Translator's Remarks (p. xvii)
• 0 Introduction (p. 1)
• 0.1 Bernhard Riemann in his time (p. 1)
• 0.1.1 His life and the development of his personality (p. 1)
• 0.1.2 The political and economic situation (p. 3)
• 0.1.3 Upbringing and education (p. 5)
• 0.1.4 Riemann's homeland (p. 9)
• 0.1.5 Göttingen and Berlin as centers of study (p. 17)
• 0.1.6 Riemann as full professor: 1859-1866 (p. 22)
• 0.2 The golden 1850s in Göttingen: From Gauss and Dirichlet to Riemann and Dedekind (p. 24)
• 0.2.1 Riemann and Dedekind: Personal circumstances (p. 24)
• 0.2.2 Towards change in mathematics (p. 33)
• 0.2.3 Snapshots by an English observer (p. 36)
• 0.3 Influences in the final years: Riemann between Germany and Italy (p. 39)
• 0.4 Competing Conceptions of analysis before Riemann (p. 43)
• 0.4.1 Riemann in the historical evolution of analysis: An overview (p. 43)
• 0.4.2 Algebraic analysis (p. 46)
• 0.4.3 Infinitesimal analysis (p. 52)
• 0.4.4 Geometric deliberations: Fourier (p. 55)
• 0.4.5 The limit conception: Newton (p. 56)
• 0.4.6 Towards epsilontics: Cauchy and Dirichlet (p. 57)
• 1 Complex Analysis (p. 64)
• 1.1 The genesis of complex analysis up to Riemann's time (p. 64)
• 1.1.1 Preliminary remarks (p. 64)
• 1.1.2 The complex numbers (p. 65)
• 1.1.3 Complex functions and their derivatives (p. 69)
• 1.1.4 Integration (p. 73)
• 1.1.5 Power series (p. 77)
• 1.1.6 Further applications (p. 84)
• 1.1.7 Multivalued functions and Riemann surfaces (p. 88)
• 1.1.8 Doubly periodic functions (p. 92)
• 1.2 The dissertation of 1851 (p. 96)
• 1.2.1 Riemann's view of the motives for the paper: Article 20 of the dissertation, Part I (p. 96)
• 1.2.2 A short account of the contents of the dissertation (p. 99)
• 1.2.3 Riemann's summary of the dissertation. The program: Article 20, and Part II, Article 22 (p. 101)
• 1.2.4 On the prehistory of the dissertation (p. 107)
• 1.2.5 The effect of the dissertation (p. 118)
• 1.3 The elaborations (p. 124)
• 1.3.1 Ordinary differential equations (p. 124)
• 1.3.2 Analysis as the source of topology (p. 130)
• 1.3.3 Abel's theorem (p. 133)
• 1.3.4 Algebraic curves (p. 139)
• 1.3.5 Minimal surfaces (p. 142)
• 1.3.6 Riemann's students and their notes on function theory (p. 144)
• 1.3.7 Later evaluations (p. 147)
• 1.3.8 Dedekind and the algebraization of complex function theory (p. 152)
• 1.4 The zeta function and the distribution of primes (p. 162)
• 1.4.1 Preliminary remarks (p. 162)
• 1.4.2 An approach (p. 163)
• 1.4.3 The functional equation (p. 169)
• 1.4.4 Riemann's explicit formula for the prime number function (p. 173)
• 1.4.5 The zeros and the Riemann hypothesis (p. 175)
• 1.4.6 The Nachlass (p. 176)
• 1.4.7 The evaluations (p. 178)
• 2 Real Analysis (p. 181)
• Preliminary remarks (p. 181)
• 2.1 Foundations of real analysis (p. 181)
• 2.1.1 The concept of an integral (p. 181)
• 2.1.2 Rigor in analysis (p. 185)
• 2.1.3 The new status of special cases: Examples and counterexamples (p. 187)
• 2.2 Trigonometric series before Riemann (p. 191)
• 2.2.1 Preliminary remarks (p. 191)
• 2.2.2 From Euler to Fourier (p. 193)
• 2.2.3 On the development of function concepts (p. 196)
• 2.2.4 From Fourier to Dirichlet (p. 199)
• 2.3 Riemann's results (p. 204)
• 2.3.1 Application of the concept of the integral to the Fourier coefficients (p. 204)
• 2.3.2 Riemann's associated function F(x) (p. 206)
• 2.4 Trigonometric series after Riemann (p. 209)
• 2.4.1 From trigonometric series to set theory (p. 209)
• 2.4.2 On the further development of trigonometric series: The arithmetization of functions and their emancipation in functional analysis (p. 211)
• 2.5 A self-contained chapter: Gauss, Riemann, and the Göttingen atmosphere (p. 213)
• 3 Geometry; Physics: Philosophy (p. 219)
• Preliminary remarks: The central role of the habilitation lecture of 1854 (p. 219)
• 3.1 Geometry (p. 223)
• 3.1.1 From Euclid to Descartes and to non-Euclidean geometry (p. 223)
• 3.1.2 Gauss' theory of surfaces (p. 226)
• 3.1.3 The n-fold extended manifold (p. 230)
• 3.1.4 The metric determinations (p. 234)
• 3.1.5 Curvature (p. 236)
• 3.1.6 Effects on geometry and physics in the first fifty years after Riemann's death (p. 239)
• 3.1.7 The algorithmic developments (p. 242)
• 3.1.8 The influence of Felix Klein (p. 246)
• 3.1.9 Dedekind: Analytic investigations related to Bernhard Riemann's paper on the hypotheses which lie at the foundations of geometry (p. 252)
• 3.2 Physics (p. 254)
• 3.2.1 The interest in physics (p. 254)
• 3.2.2 Physics as a field theory (p. 257)
• 3.2.3 Mathematical methods for physics (p. 263)
• 3.2.4 Riemann's electrodynamics from the viewpoint of the physicists (p. 269)
• 3.2.5 Riemannian geometry in the physics of the twentieth century: Einstein and Weyl (p. 272)
• 3.3 On philosophy (p. 277)
• 3.3.1 Preliminary remarks (p. 277)
• 3.3.2 The spiritual atmosphere in 1853/54: The debate over materialism (p. 279)
• 3.3.3 New mathematical principles of natural philosophy (p. 281)
• 3.3.4 The role of Herbart's philosophy (p. 287)
• 4 Turning Points in the Conception of Mathematics (p. 293)
• 4.1 The Historians' search for revolutions in mathematics (p. 293)
• 4.2 Turning point in the conception of the infinite in mathematics (p. 296)
• 4.3 Turning point in the method: Thinking instead of computing (p. 302)
• 4.4 Turning point in the ontology: Mathematics as thinking in concepts (p. 304)
• 4.4.1 General concepts and the modes of their determination (p. 304)
• 4.4.2 The primacy of the continuous over the discrete in Riemann's mathematics (p. 307)
• 4.4.3 Riemann's concept of a manifold in the philosophical tradition (p. 309)
• 4.4.4 Thinking in mathematical concepts before Riemann (p. 311)
• 4.5 The ontology and methodology of mathematics after Riemann (p. 313)
• 4.5.1 The primacy of number in the case of Dedekind (p. 313)
• 4.5.2 From arithmetization to axiomatization: Hilbert 1897/1899 (p. 318)
• 4.5.3 The role of Georg Cantor (p. 322)
• 4.5.4 The Berlin tradition (p. 325)
• 4.6 Concluding remarks (p. 329)
• Bibliography (p. 341)
• Name Index (p. 351)