It is well known that the construction of an identity for the Italian School of Algebraic Geometry directed by C. Segre was the result of a complex dynamic of scientific exchanges with the international mathematical community. In particular, Felix Klein was a reference interlocutor for Segre, Fano, Enriques and Castelnuovo, and he exerted a notable influence on mathematical studies, on the teaching of mathematics, on publishing activity, and on the organization of cultural and academic life. In this paper, in light of the correspondence that Segre carried on with Italian and foreign colleagues, we will illustrate the international relationships of the Italian School of Algebraic Geometry in the period of formation (1883–1891). Further, through the reconstruction of Segre’s partnerships with American scholars, we will show how he took on the role of a ‘Maestro a distanza’ or ‘distance Maestro’, for foreign geometers up to the first years of 20th century. From these letters and other archival sources there will emerge the cosmopolite features of the research activities pursued by the Segre’s team, the diffusion of Italian geometric methods and results abroad, as well as some aspects of the biography of Segre, related to his institutional, political and editorial committment.

From Turin to Göttingen: dialogues and correspondence (1879-1923)

Corrado Segre and his Disciples. The Construction of an International Identity for the Italian School of Algebraic Geometry

David Hilbert. David Hilbert's lectures on the foundations of geometry, 1891–1902. Michael Hallett and Ulrich Majer, eds. David Hilbert's Foundational Lectures; 1. Berlin: Springer-Verlag, 2004. ISBN 3-540-64373-7. Pp. xxviii + 661

In 1886 Corrado Segre wrote to Felix Klein about his intention to study ‘geometrie projective pure’, completing and developing the work of von Staudt. He would continue this research project throughout the whole of his scientific life. In 1889, following a suggestion of Segre, Mario Pieri published his translation of the Geometrie der Lage, and from 1889 to 1890 Segre published four important papers, “Un nuovo campo di ricerche geometriche”, in which he completely developed complex projective geometry, considering new mathematical objects such as antiprojectivities and studying the Hermitian forms from a geometrical point of view with the related ‘hyperalgebraic varieties’. Segre developed the same ideas in 1892 in a new paper published in the Matematische Annalen, in which he also considers the so-called “bicomplex numbers” and provided the first example of a projective geometry on an algebra with zero-divisors. In 1891, during one of his celebrated courses in higher geometry, Segre asked his students to find a system of independent axioms for projective hyperspatial geometry. Fano, Enriques, Pieri and Amodeo, who wrote important papers, followed this proposal.

Segre and the Foundations of Geometry: From Complex Projective Geometry to Dual Numbers

Up to the present, the research on the collections of models in Italy has been limited to examining and cataloguing the collections existing at various universities, with a great deal of work being carried out mostly by Franco and Nicla Palladino,1 but an in-depth study of their use in university and pre-university teaching does not yet exist. In my paper I will deal with this problem, considering the period running from the mid-nineteenth century to the early decades of the twentieth century. In particular I will focus on the following points: the lack of interest in designing and making models for teaching at the university level at the turn of the twentieth century in Italy, and its causes; the one exception to this, i.e., Beltrami’s cardboard model of the pseudospherical surface; models in pre-university schools in the 1800s; Corrado Segre and the use of models at the University of Turin; the role of models in the “laboratory school” (Vailati, Marcolongo, Montessori, Emma Castelnuovo); the reconstruction of university collections of models in the 1950s. Finally I will try to draw some conclusions from this first historical analysis of the question. 1. The lack of interest in designing and making models for teaching at the university level at the turn of the twentieth century, and its causes The use of models for research and teaching of mathematics began to spread in the second half of the nineteenth Century and saw mathematicians of the highest scientific calibre engaged in this effort. The most important initiatives developed in France (Paris), in the United Kingdom (London, Manchester) and especially in Germany (Munich, Darmstadt, Karlsruhe, Göttingen).2 Although some first collections of models date from the early decades of the nineteenth Century, the mass production begun in the seventies mainly in Munich when Felix Klein came to teach at the local Technische Hochschule and began his collaboration with Alexander 1 See the Bibliography. 2 See for example N.& F. Palladino 2009. Brill.3 In the following years various exhibitions4 and the publication of catalogues5 favoured the spreading of the use of models in teaching at international level. Italy remained marginal in the activity of conceiving and making geometric models, in spite of the many young mathematicians who went to Germany for post-graduate study.6 Actually there was an attempt by Giuseppe Veronese (1854-1917) to set up a national laboratory for the production of models. In 1883 Veronese, who did postgraduate work in 1880-1881 in Leipzig where at the time Klein was teaching, in a report to the minister of education wrote:7 “I believe that for increasingly substantial progress of mathematical studies in Italy, an Atelier such as that in Munich would benefit us greatly, because our schools would be made independent in this respect too from foreigners and could procure their collections at less expense”.8 In fact he thought that “Intuition in Geometry consists in our representing in our minds the figures in space, in such a way that our thoughts can go into them, alternately uniting them and separating them, and discovering the intimate nexus that permeates them. This is the intuition of space that must be developed in the minds of young people from their most tender years, and to that end it is useful to accompany each geometric proof, as far as possible, with drawings and models by which the young person can better comprehend and intuit the geometric properties of bodies without many mental efforts”.9 In spite of the support of important Italian mathematicians F. Brioschi, E. D’Ovidio, R. De Paolis, U. Dini and E. Bertini the initiative was unsuccessful.

background

Models in mathematical teaching in Italy (1850-1950)

One of the greatest architects of the “geometric Risorgimento” (Coolidge 1927, 352) in Italy, Corrado Segre provides a shining example of the role of mentor in the history of mathematics. His university courses were a veritable forge for future researchers. The years between 1891 and the beginning of the 20th century witnessed the launch in Turin, under his guidance, of the Italian School of algebraic geometry, which in a short time would assume an internationally recognised role. Undoubtedly decisive in the formation of the School was the fact that Segre had fostered the onset and consolidation of the following lines of research: hyperspatial projective geometry; research in the foundations of the hyperspatial projective geometry; birational algebraic geometry; enumerative geometry; projective differential geometry; and projective geometry in the complex domain. However, also significant was the role of his university teaching, a valuable record of which is conserved in the forty handwritten notebooks of his university lectures. These not only make it possible to reconstruct the genesis and developments of his scientific research, but also allow us to understand how his own research stimulated and closely interacted with that of his students, thus showing the importance of his teaching in the flourishing of the Italian School. Here, after a brief overview of Segre’s scientific contributions, we intend to show the close interaction between teaching and research, in order to bring out the influence of his university courses on his students and to pinpoint the moment when his role as leader began to be acknowledged; and make evident the enthusiasm, the nature of the collective work of his group of researchers, at least up to the early years of the twentieth century. We also intend to show how his vision of mathematics education—closely linked to his idea of the objectives of mathematics—was transmitted to the future teachers who attended his courses at the Scuola di Magistero (Teachers College) of the University of Turin. Finally we will try to identify the most salient features of his scientific leadership.

From Turin to Göttingen: dialogues and correspondence (1879-1923)

Citations

Corrado Segre and his Disciples. The Construction of an International Identity for the Italian School of Algebraic Geometry

David Hilbert. David Hilbert's lectures on the foundations of geometry, 1891–1902. Michael Hallett and Ulrich Majer, eds. David Hilbert's Foundational Lectures; 1. Berlin: Springer-Verlag, 2004. ISBN 3-540-64373-7. Pp. xxviii + 661

Segre and the Foundations of Geometry: From Complex Projective Geometry to Dual Numbers

Models in mathematical teaching in Italy (1850-1950)

Segre’s University Courses and the Blossoming of the Italian School of Algebraic Geometry

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