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

## TL;DR

This paper extends Descartes' geometric foundational approach to include differential algebra and tractional motion, proposing a new balance between symbolic and analog computation for a constructive foundation of infinitesimal calculus.

## Contribution

It introduces a framework combining analog and symbolic computation through differential universality, enabling a constructive foundation of infinitesimal calculus without infinity.

## Key findings

- Defined constructive limits of tractional motion.
- Proved a differential universality theorem.
- Established a new balance between symbolic and analog computation.

## Abstract

In La G\'eom\'etrie, 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.