Continuity and discontinuity in Italian mathematics after the Unification: From Brioschi to Peano

A period of renewal for Italian mathematics started with the Italian Risorgimento in the middle of the nineteenth century. In the first half of the nineteenth century Italian mathematicians were still essentially linked to French polytechnic models (among them were Ottaviano Fabrizio Mossotti, Antonio Bordoni, Giorgio Bidone, and Giovanni Plana). These models had undergone a crisis in France, too, following the development of mathematical research in the German universities which had opened up new horizons (Steiner, Jacobi, and Möbius). The young Italian mathematicians who measured themselves with the highest scientific level understood that they had to turn to the German schools, above all those of Berlin and Göttingen. Elements of continuity can also be pointed out, but in this case the main element of novelty with respect to the previous situation was a deep change in the direction of mathematical research, with the new reference point of the German schools and their emerging ideas and concepts, which can better explain the birth and the progress of a new school of Italian mathematicians at an international level. The year 1848 brought about a new period in Italy, and in Europe, as well as a new generation of scientists who took an active part in the both the political and cultural activity of the Italian Risorgimento. At the forefront we find Francesco Brioschi, Enrico Betti and Luigi Cremona. Changes in the direction of their mathematical research are examined, as well as the influence of Riemann on Betti and Casorati. The influence of German schools extended to the second generation of Italian mathematicians, among whom the most celebrated and representative have been chosen: Eugenio Beltrami Ulisse Dini, and Giuseppe Peano.