Geometric Calculus and Geometry Foundations in Peano

Abstract

First, Peano’s geometrical calculus theory is a general theory which is of intrinsic mathematical interest and which is also applied to mechanics and to physics. Peano’s contributions, which come from an elaboration of Grassmann’s ideas, consist in an Euclidean interpretation of relative concepts. Moreover, in this context, Peano proves important fundamental theorems of projective geometry. For this reason, Peano’s geometrical calculus has an implicit foundational interest. In our opinion, the protophysical role of Euclidean geometry in Peano’s works is essential and decisive. He distinguishes position geometry from Euclidean geometry, and from a theoretical point of view, it is appropriate. In his ‘Sui fondamenti della geometria’ the congruence theory is well determined and regulated. Classical geometry constitutes the crucial model for the study of the foundations of geometry. Even Hilbert, deep down, takes Euclid into account²⁰. During this period, we have many proposals of systems with different essential or primitive notions and axioms. Hence, we can observe “equivalent theories” for the foundation of elementary geometry, and in this way we have a “theoretical relativism” regarding the choice of primitive elements and fundamental axioms. This is epistemologically and historiographically²¹ very important²².