×

Es steht alles schon bei Dedekind: aspects of the influence of Dedekind’s work on Italian mathematics. (Italian. English summary) Zbl 1405.01016

This paper draws its title from a sentence written by Emmy Noether, according to which “Everything is already present in Dedekind”. This means that the most important ideas and results in modern algebra were due to Dedekind. The basic idea of this work is to show the influence exerted by Dedekind on the development of algebraic studies in Italy between the publication of his celebrate Supplement to Dirichlet’s Vorlesungen über Zahlentheorie (second edition 1871) and the thirties of the 20th century.
As a matter of fact, the author goes beyond the declared aim of his research because he presents a good outline of Dedekind’s main ideas, methods and results in algebra as well as their reception in Germany and in other countries, though the focus is on Italy.
One of the most significant purposes of this work is to show that the ideas of Dedekind were scarcely known in Italy at least until the 1920s.
The author begins pointing out the most relevant novelties introduced by Dedekind in the Supplement: ring theory and theory of ideals. Though the reference picture was offered by Kummer’s work, Dedekind framed the problem in a completely different manner: by working on the idea of abstract algebraic structure. This created many problems to the mathematician and many years were necessary until Dedekind’s ideas were fully grasped (p. 18).
The author explains clearly how Dedekind introduced the concept of corpus – at the beginning referring to the complex numbers – as well as those of algebraic integer and modulus. Two important pages (p. 20 and p. 21) are dedicated to expound Dedekind’s conception of ideal. A very brief comparison with Kronecker is also outlined.
After having expounded the main features of Dedekind’s researches, the author analyses the Italian situation in the last thirty years of the 19th century: Enrico Betti was probably the only one to grasp the importance of Dedekind’s ideas. However, the first mathematician who gave important contributions to modern algebra, inspired by Dedekind, was Luigi Bianchi. He entered the Scuola Normale in 1873. In the meantime, something important happened: an Italian mathematician, Aureliano Faifofer, translated Dirichlet’s Vorlesungen from German into Italian, including Dedekind’s Supplement. Faifofer was not an academician, he taught in a Liceo, but his personality was important in the context of Italian mathematics in the 19th century. He was in epistolary contact with Dedekind who followed the Italian translation of his Supplement. Dirichlet’s Vorlesungen plus Dedekind’s supplements were published into Italian in 1881 (pp. 23–25).
The author explains that Padova – and in part Pisa thanks to Bianchi – were, in fact, the only two Italian centres where number theory was studied with the modern technics derived from algebra. Paolo Gazzaniga and Umberto Scarpis were the protagonists of this season. In the mid 1880s, things change partially because, around Giuseppe Battaglini, a group of young researchers interested in algebra was born in Rome (p. 26).
After having traced this picture, the author examines the main features and results of Bianchi’s research in number theory and algebra (pp. 26–29).
In the further course of his paper, the author expounds the main properties of the fundamental work by Dedekind on algebraic functions, which is, obviously, connected to the basic concepts of the modern algebra he had introduced. The seminal text on algebraic functions is the celebrated “Theorie der algebraischen Functionen einer Veränderlichen”, published by Dedekind in collaboration with H. Weber [J. Reine Angew. Math. 92, 181–291 (1882; JFM 14.0352.01)]. The author gives a brief but precise and informative picture of the most important results and concepts developed by Dedekind as well as their connection with Riemann’s ideas. So, rational functions as meromorphic functions of the Riemann sphere are introduced as well as the field of algebraic functions. The author clarifies that Dedekind goes in the opposite direction of Riemann’s: Dedekind tried to obtain, by a purely algebraic method, Riemann’s surfaces, which, hence, are not anymore “primitive” objects. The author refers to the main step of Dedekind’s algebraization of geometry and number theory (pp. 30–31).
After that, he focuses, once again, on the reception of Dedekind and Weber’s work in Italy. He points out that the great Italian school of algebraic geometry (Segre and Castelnuovo and, after few years, Enriques) developed its methods and results independently of Dedekind’s researches. On the other hand, the author remarks that Segre – apart from Bianchi – was probably the only Italian mathematician who had a profound knowledge of Dedekind’s works. Nonetheless, “For the rest, the Italian algebraic geometers, who were engaged in the great adventure concerning the study of the surfaces, do not appear to be interested in the increasing abstraction of the arithmetical methods” (p. 33). In contrast to this, between the end of the 19th and the beginning of the 20th century, Dedekind’s methods and order of ideas began to become fundamental for the development of algebra and number theory. In this respect, the author outlines the influence exerted by Dedekind’s work on Kurt Hensel, Abraham Fraenkel and Emanuel Lasker. An interesting subsection is also devoted to Francis Sowerby Macaulay, the most important of the not-German mathematicians who, in that period, dealt with the ideals in the ring of polynomials.
Two brief sections concerning the state of the art on algebraic functions between the 19th and the 20th century and Dedekind’s theory of algebras follow.
In the conclusions, the author claims that, at the beginning of the 1920s, the situation in Italy changed: at the Scuola Normale in Pisa a school of algebra and number theory was born. The author mentions several of the important mathematicians who contributed to such development. Only to remind some of the greatest, let us mention: Gaetano Scorza, Luigi Fantappiè, Michele Cipolla and Giovanni Ricci. Luigi Bianchi himself renewed his interest in algebraic number theory. He also wrote significant treatises on these subjects, which played an important role in the education of the mathematicians mentioned above. It was about a very research and didactical project (p. 39). In this context, the character of the already mentioned Gaetano Scorza played a significant role: He first wrote – in 1921 – a text on the theory of algebras.
However, in the course of about ten years, the situation changed and the scholars who had dedicated their research to algebra either changed their field of research or were marginalized. So that, at the beginning of the 1920s, the Italian situation was comparable to that of other countries – apart from Germany – in the middle of the 1930s, Italy was a marginal country at least with regard to modern algebra and number theory. Furthermore, the arithmetical theory of algebraic geometry did not establish itself in Italy, which is disappointing, given the important contributions to algebraic geometry by numerous Italian mathematicians in the previous years.
The paper under review is well conceived and clear. The outline of the technical aspects is well interconnected with the historical development of modern algebra, in particular focusing on the Italian situation. This work is very informative. Probably, a more in-depth look at the causes that determined the crisis of the Italian algebraic school during the 1930s would have been interesting.

MSC:

01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
11-03 History of number theory
14-03 History of algebraic geometry

Citations:

JFM 14.0352.01
PDFBibTeX XMLCite