Abstract

I will indicate and address the main lacunas, open questions, misconceptions, and misinterpretations in the current historiography of fin-de-siècle non-Archimedean mathematics, taking the Italian mathematician Giuseppe Veronese (1854–1917) — “the chief modern champion of the infinitely small” (sec. Cajori) — as a particularly illustrative case study. Veronese introduced and developed non-Archimedean geometry in a series of seminal publications from 1889 to 1909, culminating in the Fondamenti di Geometria. Initially, Veronese’s results were widely discussed by leading mathematicians (e.g. Peano, Cantor, Hilbert, Poincaré, etc.), as well as in major reference journals, mathematical encyclopaedias and lexicons of the time. His work also quickly garnered the attention of philosophers working on the foundations of mathematics. 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 that negatively impacted the 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 Veronese in the actual practice of contemporary non-Archimedean mathematics and its historiography. In this context, I will especially focus on the roles of Cantor, Peano, Russell, and Poincaré in Veronese’s Nachleben. I will also provide a critical overview of the few attempts at interpreting some facets of Veronese’s theory that have been advanced in the last 135 years, emphasizing their various shortcomings and omissions. Finally, I will propose a novel interpretative strategy which, in my view, provides us with „un nouvel hommage réparateur“ of Veronese’s insights, and allows us to detect and follow a continuous developmental path from Veronese’s geometrical investigations from the 1890s up to late 20th-century non-Archimedean theories (e.g. Robinsons’s NSA , Vopěnka’s AST , or Conway’s surreal numbers).