Mathematics through diagrams: Microscopes in non-standard and smooth analysis

Abstract

Diagrams play an important role in the construction of mathematical concepts, mainly in (some) "limit" situations, like in the case of the mental representation of geometric tangent lines. They have many properties and can be viewed as particular epistemic mediators. Further, they are able to provide a better understanding of some mathematical concepts because they can be manipulated. In this paper we investigate how a particular kind of diagram (microscope) can serve to obtain two different and interesting visual representations of how a real function appears in small neighborhoods of its points. © 2007 Springer-Verlag Berlin Heidelberg.