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, [Gandon, 2006] and [Rizza, 2009]).
I shall claim that there are two poles in Peano’s construction of geometry. The first pole is the requirement of an empirical foundation of the basic geometrical concepts and propositions. 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. The second pole corresponds to the idea that geometry starts from axioms. 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 are formulated, the specific nature of these concepts becomes irrelevant. I shall claim that Peano’s notion of 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 and his notion of purity, I shall argue that these two poles can be understood as compatible stages of a single process of construction rather than conflicting options.