Umberto Bottazzini | Università degli Studi di Milano - State University of Milan (Italy) (original) (raw)

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Papers by Umberto Bottazzini

Revue D Histoire Des Sciences, 1989

Historia Mathematica Toronto, 1980

In the 1850s Weierstrass succeeded in solving the Jacobi inversion problem for the hyper-elliptic... more 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 Weierstrass' unpublished correspondence with his former student Schwarz provides strong evidence of this.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.

Archives Internationales D Histoire Des Sciences, 1992

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 2014

Karl Weierstraß (1815–1897), 2015

Social History of Nineteenth Century Mathematics, 1981

Mathematicians in Bologna 1861–1960, 2011

Mathematical Lives, 2010

In an isosceles triangle, if the ratio between the base angle and the vertex angle is an algebrai... more In an isosceles triangle, if the ratio between the base angle and the vertex angle is an algebraic but irrational number, is the ratio between the base and the side always transcendent?

Revue d'histoire des sciences, 1989

Mathematics and Computers in Simulation, 1987

... These ideas often have complex connections and causes both within the theory itself and withi... more ... These ideas often have complex connections and causes both within the theory itself and within the wider ... in the nineteenth century consisted of the theories of elliptic and Abelian functions, which today are generally taken to be parts of the history of algebraic geometry and ...

This paper sets out to examine some of Riemann's papers and notes left byhim, in the light of... more This paper sets out to examine some of Riemann's papers and notes left byhim, in the light of the "philosophical" standpoint expounded in his writings on Naturphilosophie. There is some evidence that manyof Riemann's works, including his Habilitationsvortrag of 1854 on the foundations of geometry, may have sprung from his attempts to find a unified explanation for natural phenomena, on

Historia Mathematica, 2000

Hidden Harmony—Geometric Fantasies, 2013