"Algebraic truths" vs "geometric fantasies": Weierstrass' Response to Riemann

Abstract

In the 1850s Weierstrass succeeded in solving the Jacobi inversion problem for the hyper-elliptic case, and claimed he was able to solve the general problem. At about the same time Riemann successfully applied the geometric methods that he set up in his thesis (1851) to the study of Abelian integrals, and the solution of Jacobi inversion problem. In response to Riemann's achievements, by the early 1860s Weierstrass began to build the theory of analytic functions in a systematic way on arithmetical foundations, and to present it in his lectures. According to Weierstrass, this theory provided the foundations of the whole of both elliptic and Abelian function theory, the latter being the ultimate goal of his mathematical work. Riemann's theory of complex functions seems to have been the background of Weierstrass's work and lectures. Weierstrass' unpublished correspondence with his former student Schwarz provides strong evidence of this. Many of Weierstrass' results, including his example of a continuous non-differentiable function as well as his counter-example to Dirichlet principle, were motivated by his criticism of Riemann's methods, and his distrust in Riemann's ``geometric fantasies''. Instead, he chose the power series approach because of his conviction that the theory of analytic functions had to be founded on simple "algebraic truths". Even though Weierstrass failed to build a satisfactory theory of functions of several complex variables, the contradiction between his and Riemann's geometric approach remained effective until the early decades of the 20$^{th}$ century.