Teaching with infinitesimals

Abstract

Why has it always been “impossible” to teach Analysis, ever since the infinitesimal calculus appeared, three centuries ago ? For the first two centuries, it was because the tools of the infinitesimal calculus, though effective and splendid, were impossible to justify by consistent discursive reasoning. Thereafter, once we knew how to talk rigourously about limits and how to define the topological structure of E, it was because of the great technical and conceptual difficulties which proved so disturbing for beginners. Our own experience leads us to think that the emergence of nonstandard analysis makes the task of teaching analysis so much more possible. In practice, we suggest that the beginnings of analysis be approached in two stages:

• - The first level (here called “level 0”) is based on the numerical manipulation of orders of magnitude so leading to familiarity with small and large numbers and with their behaviour with respect to elementary operations and functions. This allows one to grasp the fact that some numbers are negligible in comparison with others and to encounter notions of closeness and of asymptotic or local behaviour which leads, in turn, to the algebra of limits, the continuity of functions ,the study of sequences and Taylor (asymptotic) expansions.

• - The second level (here called “level 1”) is geared more towards the introduction of elementary topological notions and, particularly, towards definitions and results related to the completeness of R Thus we get down to problems whose difficulty we ought not to hide and whose ontological nature the title of Dedekind’s fundamental article aptly conveys: Was sind und was sollen die Zahlen? We believe that the shadow concept now becomes very useful didactically and should be introduced geometrically