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This book contains the papers developing out the presentations given at the International Conference organized by the Torino Academy of Sciences and the Department of Mathematics Giuseppe Peano of the Torino University to celebrate the 150th anniversary of G. Peano's birth - one of the greatest figures in modern mathematics and logic and the most important mathematical logician in Italy - a century after the publication of Formulario Mathematico, a great attempt to systematise Mathematics in symbolic form.



1. Giuseppe Peano and Mathematical Analysis in Italy

If you enter what was latterly the office of the Head of the former Istituto di Analisi Matematica of the University of Torino, you find before you a group of photographs which recall those Professors who held the Chair of Analisi (or Calcolo) in the period from 1811 to 1972; among them, side by side with Giovanni Plana (who occupied the Chair from 1811 to 1864), Angelo Genocchi (from 1864 to 1889), Enrico D’Ovidio (from 1872 to 1918), is the mild countenance of Giuseppe Peano, Professor of Calculus from 1890 to 1931; next to him is his successor Francesco G. Tricomi, Professor of Analysis from 1925 to 1972.
Fulvia Skof

2. Some Contributions of Peano to Analysis in the Light of the Work of Belgian Mathematicians

The period of the main original contributions of Giuseppe Peano (1858–1932) to analysis goes from 1884 till 1900, and his work deals mostly with a critical analysis of the foundations of differential and integral calculus and with the fundamental theory of ordinary differential equations. During this period, the main analysts in Belgium were Louis-Philippe Gilbert (1832–1892) and his successor Charles-Jean de La Vallée Poussin (1866–1962), at the Université Catholique de Louvain, and Paul Mansion (1844–1919) at the Université de Gand. At the Université de Liège, Eugène Catalan (1814–1894) retired in 1884, and his successor was Joseph Neuberg (1840– 1926), an expert in the geometry of the triangle. Analysis at the Université Libre de Bruxelles was still waiting to be awakened by Théophile De Donder (1872–1957)1.
Jean Mawhin

3. Peano, his School and … Numerical Analysis

Giuseppe Peano, an outstanding mathematician of unusual versatility, made fundamental contributions to many branches of mathematics; in numerical analysis, noteworthy results concern representation of linear functionals, quadrature formulas, ordinary differential equations, Taylor’s formula, interpolation, and numerical approximations1. Many results are still of great interest, whereas a few others appear obsolete.
Giampietro Allasia

4. Peano and the Foundations of Arithmetic

At the end of the 1880s two episodes occurred in rapid succession which formed the bases of what we call the foundations of arithmetic: the publication in 1888 of Was sind und was sollen die Zahlen by Richard Dedekind and in 1889 of Arithmetices Principia, nova methodo exposita by Giuseppe Peano. This work was to give Peano lasting fame, in that he had for the first time expounded the axioms for the system of natural numbers; from that time on they were linked to his name, and from the English “Peano Arithmetic” were known by the acronym PA.
Gabriele Lolli

5. Geometric Calculus and Geometry Foundations in Peano

First, Peano’s geometrical calculus theory is a general theory which is of intrinsic mathematical interest and which is also applied to mechanics and to physics. Peano’s contributions, which come from an elaboration of Grassmann’s ideas, consist in an Euclidean interpretation of relative concepts. Moreover, in this context, Peano proves important fundamental theorems of projective geometry. For this reason, Peano’s geometrical calculus has an implicit foundational interest. In our opinion, the protophysical role of Euclidean geometry in Peano’s works is essential and decisive. He distinguishes position geometry from Euclidean geometry, and from a theoretical point of view, it is appropriate. In his ‘Sui fondamenti della geometria’ the congruence theory is well determined and regulated. Classical geometry constitutes the crucial model for the study of the foundations of geometry. Even Hilbert, deep down, takes Euclid into account20. During this period, we have many proposals of systems with different essential or primitive notions and axioms. Hence, we can observe “equivalent theories” for the foundation of elementary geometry, and in this way we have a “theoretical relativism” regarding the choice of primitive elements and fundamental axioms. This is epistemologically and historiographically21 very important22.
Paolo Freguglia

6. The Formulario between Mathematics and History

For almost twenty years, from 1888 to 1908, Peano devoted all his energies to formulating and realising a project, which throughout his life he was to acknowledge as one of the most important results of his mathematical research1. This was the Formulaire de Mathématiques, a huge collection of mathematical propositions expressed in symbols, especially written with his own logic, capable of concentrating in a single volume the knowledge of mathematics of his time. To this end, Peano founded a journal and invited to collaborate on it scholars, assistants, colleagues at the University and at the military Academy, teachers and other mathematicians in Italy and abroad. His total commitment to this enterprise was also accompanied by his voluntary decision to leave his post as Professor of infinitesimal Calculus at the military Academy2, keeping only his University position, and by the purchase of a printing press so that he could set up the text himself, in view of the difficulties that the mathematical symbols created for the publishing houses3.
Clara Silvia Roero

7. Giuseppe Peano: a Revolutionary in Symbolic Logic?

In this paper I consider Peano’s main mathematical concerns in the 1880s, and the relations between them. I shall propose that he had a sort of magical moment that led him to create his mathematical logic, but also that he was obscure, or at least unclear, about one of the major attendant changes in thought. The material covered is summarised historically in Grattan-Guinness (2000, especially chs. 2, 4 and 5), and treated in more detail in various works cited there.
Ivor Grattan-Guinness

8. At the Origins of Metalogic

Among the many aspects of the logical and foundationalistic work of Peano, there is one which, though not ignored, seems not yet to have attracted all the attention it deserves: that which deals with metalogic. Yet the relationship between Peano and this basic theme is decisive both for full appreciation of his work, and for a genuine understanding of its destiny. While he was, indeed, — as we will try to show — one of the genuine founders also of this logical discipline, it was precisely its development that played a decisive role in the decline and even the disappearance from the international scene of his logical tradition.
Ettore Casari

9. Foundations of Geometry in the School of Peano

The question that motivated this paper — why Pieri made an analogy to Peano’s affinities when he introduced segmental transformations — revealed several plausible answers. But perhaps more importantly, seeking to answer the question provided an opportunity to explore the commonalities and differences about the scholars’ views and treatments of projective geometry and its transformations. In this regard, there is one more avenue to explore. What is not evident from his axiomatizations, but is clear from his lectures37 to students, is the evolution of Pieri’s thoughts about projective geometry. In his (1891) notes for a course in projective geometry at the Military Academy — prior to writing his first axiomatization, but after he had translated Staudt (1847) — Pieri took the same approach to projective geometry as had Peano38. But in his notes for the University of Parma (1909–10), after he had written all his foundational papers in projective geometry, Pieri alerted students to the more “desirable” direction of Staudt as opposed to that pursued by J. Poncelet, Möbius, J. Steiner and Chasles, who studied projective geometry as an extension of elementary geometry.
Pieri had learned well from Peano, but was not reluctant to forge his own path. For example, Pieri would adopt Peano’s ideas on point transformations, but took their use to new levels. In (1898b), he demonstrated the possibility of constructing real projective geometry entirely on the basis of point and a projective point transformation that preserves lines. In (1900), he constructed absolute geometry, the theory common to Euclidean and Bolyai-Lobatchevskian geometry, solely on the undefined notions of point and motion39. In that paper, Pieri observed that although the distinction between the synthetic concept of a congruence transformation (motions) from points to points rather than from figure to figure is not, from a logical perspective, significant, the first idea is more “manageable” to the deductive process. Pieri acknowledged Peano (1889d) and Peano (1894c), noting that Peano’s primitives and postulates could be derived from his Pieri (1900a, Prefazione, 174 — Opere 1980, 184).
And the path would come full circle. Peano would be inspired by Pieri’s fertile ideas. For example in 1903 Peano proposed a construction of geometry based on the ideas of point and distance40. His proposal combined Pieri’s plan (announced in Pieri 1901b) to establish elementary geometry on the basis of point and two points equidistant from a third (that would be realized in Pieri 1908), with his own Peano (1898c) construction on point and vector. Using Pieri’s idea of equidistance, Peano was able to define the equivalence of vectors instead of taking it as primitive, as he had previously, and reformulate definitions (include that of vector) on solely on the basis of it. He produced a systemization of geometry founded on three primitives (point, the relation of equidifference between pairs of points, and inner product of two vectors) and nineteen postulates (reducible to seventeen).
It is impossible to exclude the influence of Peano on Pieri’s immersion into the world of foundations. After his (1895–1896) Notes, Pieri would continue to refine his ideas on projective geometry and ultimately produce what Russell (1903)41 called “the best work” on the subject. As I have observed, Peano was involved in a substantial way in propelling Pieri on the path to that achievement. And he shared Russell’s evaluation of it. Peano wrote: “The results reached by Pieri constitute an epoch in the study of foundations of geometry, and all those who later treated the foundations of geometry have made ample use of Pieri’s work and have echoed Russell’s evaluation”. Pieri had made his mentor proud!42
Elena Anne Marchisotto


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