A quest for exactness: machines, algebra and geometry for tractional constructions of differential equations

The peculiarity of this work lies in the attention to the constructive role of geometry as idealization of machines for foundational purposes. This approach, after the “de-geometrization” of mathematics, is far removed from the mainstream discussions of mathematics, especially regarding foundations. However, though forgotten these days, the problem of defining appropriate canons of construction was very important in the early modern era, and had a lot of influence on the definition of mathematical objects and methods. According to the definition of Bos [2001], these are “exactness problems” for geometry. Such problems about exactness involve philosophical and psychological interpretations, which is why they are usually considered external to mathematics. However, even though lacking any final answer, I propose in conclusion a very primitive algorithmic approach to such problems, which I hope to explore further in future research. From a cognitive perspective, this approach to calculus does not require ifinity and, thanks to idealized machines, can be set with suitable “grounding metaphors” (according to the terminology of Lakoff and Núñez [2000]). This concreteness can have useful fallouts for math education, thanks to the use of both physical and digital artifacts (this part will be treated only marginally).