In this paper, we bring a new solution to two unusual questions in Computer Science relative to recursive Program Synthesis (PS). To clarify our ideas we introduce the concepts of Newtonian and Cartesian paradigms to scientific creativity when related to PS. The main contribution of the paper is a thorough discussion on the difference between disruptive Cartesian creation and classical Newtonian construction of a theorem prover devoted to PS. We illustrate these ideas by an analysis of Peano's axioms defining the set of non negative integers, from the point of view of creativity and we explain why Newtonian systemic creativity is not suited for conceiving this simple recursive system. This analysis is then applied to a more complex case of the general framework for our own 'Constructive Matching Methodology' (CMM) as a Cartesian paradigm to the creation of an autonomous theorem prover for PS. This methodology illustrates that Cartesian Intuitionism can be viewed as a 'generator of new ideas'.