A geometrical constructive approach to infinitesimal analysis: epistemological potential and boundaries of tractional motion

Recent foundational approaches to Infinitesimal Analysis are essentially algebraic or computational, whereas the first approaches to such problems were geometrical. From this perspective, we may recall the seventeenth-century investigations of the “inverse tangent problem.” Suggested solutions to this problem involved certain machines, intended as both theoretical and actual instruments, which could construct transcendental curves through so-called tractional motion. The main idea of this work is to further develop tractional motion to investigate if and how, at a very first analysis, these ideal machines (like the ancient straightedge and compass) can constitute the basis of a purely geometrical and finitistic axiomatic foundation (like Euclid’s planar geometry) for a class of differential problems. In particular, after a brief historical introduction, a model of such machines (i.e., the suggested components) is presented. Then, we introduce some preliminary results about generable functions, an example of a “tractional” planar machine embodying the complex exponential function, and, finally, a didactic proposal for this kind of artifact.