On some contributions by Alessandro Padoa and Mario Pieri to the foundations of geometry (Q2837303)

The paper focuses on Mario Pieri's efforts to found geometry on a minimal set of primitive symbols. This culminates in the major result in the foundations of geometry on the side of Peano's school. The idea to search for a minimal set of symbols is typical for Peano and his pupils. At any rate, the author relates two texts by Pieri devoted to foundational aims in geometry, which he calls \textit{point and movement} and \textit{point and sphere}. In the first, written in 1899 [Torino Mem. (2) 49, 173--222 (1899; JFM 30.0426.02)], \textit{M. Pieri} assumed as primitives entities the \textit{point} and the \textit{movement}, from which he derived the concepts of \textit{distance} and \textit{segment}.NEWLINENEWLINE Later, in [Mem. di Mat. e di Fis. Soc. It. Sc. (3) 15, 345--450 (1908; JFM 39.0545.08)], he assumed as primitives concepts those of \textit{point} and \textit{sphere}; from these, he defined the segment and the movement. Now, with `movement' Pieri meant a transformation \(f:(a,b)\to (c,d),\) where \(a,b,c,d\) are points and the couple \((a,b)\) is congruent to \((c,d)\). In other terms, a movement is a function mapping couples of points to congruent couples of points.NEWLINENEWLINE What is the rôle of Alessandro Padoa in this matter? Padoa was a friend of Pieri and addressed a speech on Pieri's foundational efforts at the Paris congress of mathematics in 1900. Furthermore, Pieri was not satisfied with his own work of 1899, hoping for a future simplification. Padoa tried to eliminate the concept of movement in favour of that of sphere, but with no negligible complications. We could say that there was an overlapping between the researches of Pieri and Padoa, both aiming at the same target and both forged at the Peano school. Their work represents the best result on the foundations of geometry in the spirit of Peano. Only a marginal hint to Hilbert's \textit{Vorlesungen} is made (p.~94), while Tarski's categorical foundation of geometry on the concept of sphere is not mentioned at all, not even in the references.