A finite and geometric foundation of differential equations (historical and computational insights)
What is it about?
Descartes' 1637 Géométrie proposed a “balance” between geometric constructions and symbolic manipulation, introducing suitable ideal machines. In modern terms, that is a balance between analog and symbolic computation. This 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. Adopting these extensions, it is possible to define a new convergence of machines (analog computation), algebra (symbolic manipulations), and a 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 tractional motion's constructive limits is provided by a differential universality theorem.
Why is it important?
This paper starts from the historical analysis and uses computational tools to prove a result that provides a new perspective for future developments in the foundation of mathematics.