L'infini numérique dans l'Arénaire d'Archimède (Q2541212)

The problem which Archimedes deals with in the Ψαμμίτης (Psammites) is that of showing ``by means of geometrical proofs, which you will be able to follow, that some of the numbers named by me and given in the work I sent to Zeuxippus, exceed not only the number of the mass of sand equal in magnitude to the earth filled up in the way described, but also that of a mass equal in magnitude to the universe''. The purpose of our work has been to show that for Archimedes and probably for the Alexandrine mathematical milieu, the problem of the numerical infinite had been solved in the sense of the potential infinity of the set of natural numbers. However, this concept lacked a suitable numerical notation: as is known with the Greek notation one could not express numbers greater than \(10^8\). In his Ψαμμίτης Archimedes provides an example of how the numerical system of octads can enable one to count not only numbers greater than \(10^8\), but also numbers apparently inaccessible such as the number of the grains of sand contained in the volume of the sphere of the fixed stars. The Arenarius, therefore, seems to prove beyond doubt that in the third century B. C. Greek mathematics had realized the mathematical importance of the sequel of natural numbers. The work also contains a comparison and an analysis of the Aristotelian theory on the actual and potential infinity (in relation also to mathematical objects such as space and the set of natural numbers) and ends with an analysis of the so-called postulate of Eudoxos-Archimedes. Thus the work constitutes a first element for the study of the concept of mathematical infinity in the Archimedean corpus.