Leibniz and Peano. From binary number systems to machines|Leibniz e Peano Dalla numerazione binaria alle macchine

This paper presents some reflections on binary arithmetic and its applications by Gottfried Wilhelm Leibniz (1646-1716) and Giuseppe Peano (1858-1932). The topic is relevant both for the intrinsic historical, mathematical, philosophical, theological, and linguistic interests motivating its development, and for the social implications derived by the design and the realization of machines. In the light of Leibniz's texts on dyadic between 1663 and 1705, we highlight the results that he obtained as well as the aims he did not achieve, both in the theoretical and in the applicative field. We also underline the ability of the German philosopher and mathematician to involve high-level experts, teachers, and academics from various countries in his projects on dyadic. With regard to Peano, we analyse the sources that between 1898 and 1903 brought him to deal with the history of the binary number system in the wake of Leibniz. Based on the examination of some unpublished manuscripts, we reconstruct his research and show how it culminates in the construction of a binary shorthand machine, in which mathematics, linguistics and technology merged together. For both authors, research in different cultural fields and their historical status constituted an important step in identifying and achieving their objectives.