On a Peculiar Lacuna in the Historiography of Non-Archimedean Mathematics: The Case of Giuseppe Veronese (1854-1917)

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• Saša Popović (University of Rijeka Centre for Logic and Decision Theory (UniRi CLDT))
• Wednesday 22 January 2025, 11:45-12:45
• Seminar Room 1, Newton Institute.

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MHMW01 - Modern history of mathematics: emerging themes

Many modern histories of non-Archimedean, i.e., infinitesimal mathematics typically tend to focus on the development of the calculus and mathematical analysis, and less so on the role of infinitesimals in geometry or the so-called theory of magnitudes (Grössenlehre, teoria delle grandezze). Moreover, there are also authors working on the history of non-Archimedean mathematics who make leaps from, e.g. late 17th-century Leibnizian infinitesimal analysis to 20th-century advances such as A. Robinson’s Non-Standard Analysis from the 1960s or W. F. Lawvere’s Smooth Infinitesimal Analysis from the 1970s. Such leaps are oftentimes corroborated by claims that only these more recent developments allow us to work with infinitesimals consistently and with sufficient exactness and precision. I claim that such views are necessarily lacunar insomuch as they omit what I take to be the crucial formative phase in the historical development of non-Archimedean mathematics, i.e., the fin-de-siècle period. During this period, there emerged a large and thematically diverse, technically sophisticated, and, from a foundational and philosophical point of view, quite profound body of consistent non-Archimedean mathematics that laid the groundwork for much of the 20th-century developments in the theory of non- Archimedean ordered (algebraic) systems. My talk may be seen as a case study and an overview of the contributions of one of the pioneers of this late 19th-century work on infinitesimals, the Italian mathematician Giuseppe Veronese (1854-1917), who was dubbed “the chief modern champion of the infinitely small” by F. Cajori already in 1915. Veronese introduced and developed non-Archimedean geometry in a series of seminal publications from 1889 to 1909, culminating in his Fondamenti di Geometria a più dimensioni e a più specie di unità rettilinee, esposti in forma elementare (1891). Initially, Veronese’s results were widely discussed by leading mathematicians (e.g. T. Levi-Civita, G. Peano, G. Cantor, D. Hilbert, M. Dehn, L. E. J. Brouwer, C. S. Peirce, H. Poincaré, H. Hahn, etc.), as well as in major reference journals (e.g. the Jahrbuch), mathematical encyclopaedias and lexicons of the time. His work also quickly garnered the attention of philosophers working on the foundations of mathematics (e.g. the Marburg neo-Kantians). However, shortly after this initial burst of enthusiasm for Veronese’s new geometry, already at the beginning of the 1920s we can see that discussions in both mathematical and philosophical circles started shifting to other mathematical subjects and, consequently, to other authors, resulting in a somewhat peculiar situation that Veronese’s results seem to be almost entirely unknown by contemporary mathematicians, philosophers and historians of mathematics alike. This striking asymmetry in the initial and final stages of the reception of Veronese’s theory will be at the heart of my talk. I will indicate what were the key factors which negatively impacted further dissemination and development of Veronese’s ideas, as well as who were the “main culprits” for what may be considered a damnatio memoriae of Giuseppe Veronese in the actual practice of contemporary (non-Archimedean) mathematics, as well as in contemporary historiography. In this context, I will especially focus on the roles of G. Cantor, G. Peano, B. Russell, and H. Poincaré in Veronese’s Nachleben. In 1990, H. Mehrtens proposed an analysis of the modernization process of fin-de-siècle mathematics in terms of a conflict between “the moderns” and the “counter-moderns”. We shall see how Veronese’s work fits into the broader modernist geometrical foundational programmes, especially Hilbert’s, and, what is more, I will try to show that Veronese’s Fondamenti di geometria (1891) not only represents a clear example of the “perfect pre-Hilbert style” of doing mathematics, but that there is a continuous developmental path from Veronese’s geometrical investigations from the 1890s all the way up to late 20th-century non-Archimedean theories such as, e.g., J. H. Conway’s theory of surreal numbers from the 1970s.

This talk is part of the Isaac Newton Institute Seminar Series series.

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