Peano’s geometry: From empirical foundations to abstract development
Organização:  DLMPST/IUHPST
University of Buenos Aires
27 / 07 / 2023
In Principii di Geometria [1889] and ‘Sui fondamenti della Geometria’ [1894] Peano offers axiomatic presentations of projective geometry. While Peano’s advocacy of the axiomatic method is well known, his view that the basic components of geometry must be founded on intuition is seldom considered (see, however, [Bottazzini, 2001], [Avellone et al, 2002], [Gandon, 2006] and [Rizza, 2009]). There seems to be a tension in Peano’s construction of geometry in these two works. On the one hand, Peano insists that the fundamental geometrical concepts (i.e., the notion of point and the relation of incidence between points) are acquired by experience and the axioms are determined by direct observation. I shall observe that, in a polemic with the Italian mathematician Segre, Peano rejects an abstract foundation of geometry that is disconnected from any intuitive character of the fundamental concepts. On the other hand, geometry starts from axioms, which cannot be attached to a single interpretation. These basic propositions are the result of a specific analysis of the properties of the basic concepts, but they do not properly define them. As Peano suggests, once the axioms areformulated, the specific nature of these concepts becomes irrelevant. I shall claim that Peano’s notionof purity, the focus on the development of a logical apparatus that can regiment geometrical proofs, and his disregard of the specific meaning of the geometrical terms in the demonstration of theorems indicates his endorsement of deductivism.
By studying Peano’s axiomatisation of geometry, I shall argue that the tension can be dissolved if these two seemingly contradictory positions are understood as compatible aspects of a single process of construction, rather than competing options. Specifically, I shall explain that each stance corresponds to a specific phase in Peano’s construction of geometry. I shall describe these two phases, and characterise their relationship by referring to a dispute between Peano and Segre.

Avellone, M.; Brigaglia, A.; Zappulla, C. (2002). The Foundations of Projective Geometry in Italy from De Paolis to Pieri. Archive for History of Exact Sciences, 56(5), pp. 363–425.
Bottazzini, U. (2001). I geometri italiani e il problema dei fondamenti (1889–1899). Bollettino dell’Unione Matematica Italiana, 4A(2), pp. 281–329.
Gandon, S. (2006). La réception des Vorlesungen über neuere Geometrie de Pasch par Peano. Revue d’histoire des mathématiques, 12(2), pp. 249–290.
Peano, G. (1889). Principii di Geometria Logicamente Esposti. Turin: Fratelli Bocca.
Peano, G. (1894). Sui fondamenti della Geometria. Rivista di matematica, 4, pp. 51–90.
Rizza, D. (2009). Abstraction and Intuition in Peano’s Axiomatizations of Geometry. History and Philosophy of Logic, 30(4), pp. 349–368.

| |