Gaussian Influences

Gauss was well past his active scientific years when Riemann knew him, which partly explains why they had few if any substantive discussions about scientific questions. What Riemann seems to have known about Gauss's ongoing interests he heard from Wilhelm Weber, with whom he worked closely. The present chapter attempts to show, however, the considerable influence Gaussian ideas exerted on specific aspects of Riemann's work. It begins, however, with a general assessment of what has been called Gaussian precision, a research ethos also found elsewhere in Germany, e.g., in Königsberg, where it was explicitly cultivated in Franz Neumann's physics seminar. Beyond this general influence, Riemann often took note of the manner in which Gauss appealed to the geometry associated with the complex numbers. This viewpoint became so influential that mathematicians today have difficulty recognizing its novelty back then. Indeed, Gauss long hesitated before publicly announcing his views regarding the nature of imaginaries, which Riemann treated not only as legitimate but as equivalent in status with the real numbers. In the case of Riemann's influential paper on hypergeometric functions -- which appeared soon after Gauss's death and was written with knowledge of an unpublished sequel to the original paper -- this was walking directly in Gauss's footsteps, as well as in Kummer's. A more famous, but also far more obscure potential influence concerned the import of non-Euclidean geometry, which first surfaced in the little-known works of Bolyai and Lobachevsky. Since Riemann never made mention of this work, which Gauss knew well, the question arises whether he was aware of it at all. He was, of course, definitely aware of the significance of topological ideas, which can be found in many places in Gauss's (mostly unpublished) works. Indirect evidence suggests that Riemann may have gained insights from Gauss about curve systems that determine the connectivity of a surface, but this was likely based on a misunderstanding that occurred in a conversation between Riemann and Betti.