A differential extension of Descartes' foundational approach: a new balance between symbolic and analog computation

In La Géométrie, Descartes proposed a balance between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In modern terms, that is a balance between analog and symbolic computation. Descartes' geometric foundational approach (analysis without infinitary objects and synthesis with diagrammatic constructions) has been extended beyond the limits of algebraic polynomials in two different periods: by late 17th century tractional motion and by early 20th century differential algebra. This paper proves that, adopting these extensions, it is possible to define a new convergence of machines (analog computation), algebra (symbolic manipulations) and a well determined class of mathematical objects that gives scope for a constructive foundation of (a part of) infinitesimal calculus without the conceptual need of infinity. To establish this balance, a clear definition of the constructive limits of tractional motion is provided by a differential universality theorem.