Naive Fundamentals of Infinitesimal Methods

Abstract

One of the most widely spread prejudices as regards the infinitely small and the constructions related to them through is the opinion that the apparatus of nonstandard analysis is extremely difficult to master. Moreover, it is usually emphasized that nonstandard analysis is based on the advanced sections of the modern formalized set theory and mathematical logic. Actually, the existence of this relation, although irrefutable, in no way hampers either understanding or handling infinitesimals. The purpose of the present chapter is to corroborate the above statement by way of presenting the methodology of nonstandard analysis at the level of rigour accepted in the modern system of mathematical education which is based on the ideas of the naive set-theoretic stance proposed by G.Cantor. Alongside with elucidating the essence of the concepts of nonstandard set theory and the principles of transfer, idealization and standardization adopted in it, certain attention will be also paid to paralleling quite recent ideas on the objects of elementary mathematical analysis and the approaches of classics. In doing so, we would like to confirm the continuity in the evolution of the ideas of differential and integral calculuses which nonstandard analysis sheds new light upon.