This chapter looks at the mathematization of the study of nature by focusing on how practical mathematicians from the sixteenth century onward understood mathematics as primarily devoted to solving problems through mathematical construction. This constructive understanding of the nature of mathematics is then related to the double movement of physicalizing mathematics (giving physical interpretations to mathematical constructions) and mathematizing physics (understanding physics as basically involving the solution of problems). The work of seventeenth-century thinkers like Galileo, Descartes, and Mersenne is used to further illustrate these ideas, which led to the establishment of mathematical physics as characterized by its problem-solving nature.

Sidereus Nuncius was a seminal text of the Scientific Revolution. It reported celestial observations whose implications upended the geocentric cosmology few had ever doubted. But Galileo’s treatise also combined topics and methodologies that traditionally had been assigned separately to mixed mathematics and natural philosophy. Whereas the bounds between these disciplines had been weakened by earlier controversies, particularly about the regressus method and about the certainty of mathematics (i.e., the Quaestio de Certitudine), Sidereus Nuncius broke them down altogether, to the delight and dismay of readers. Galileo’s application of mathematical methods—empiricism and quantification—to natural philosophy framed the ensuing discussions, such that even those who disagreed with his conclusions responded on those grounds. Thus, the book was pivotal in the process of disciplinary admixture that reorganized the study of nature into modern science.

Newton’s remarkable achievements in planetary theory and dynamics were followed by a century of equally remarkable advances in the generalized science of forces acting on bodies in motion. Far from merelyformalizing the Newtonian framework, these advances aimed at solving deep problems left in Newton’s wake. First, Newtonian theory focused on centripetal forces, which are not characteristic of motion under arbitrary constraints or bodily deformation due to pressure and stress. Second, Newton’s attempts at fluid mechanics made clear that media could not be adequately analyzed using strategies developed for point-particles. Both considerations suggested that different forms of body required different analytical and conceptual tools. They required the formulation of generalized principles flexible enough to treat heterogeneous material configurations, but stringent enough to preserve the unity of mechanics as a study of matter and motion. Ultimately, Newton’s successors established a new discipline, analytical mechanics. Its primary object of investigation was the functional representation of the invariant relations behind dynamic phenomena, not the geometric representation of trajectories. The philosophical suppositions behind the new mechanics constituted a new mechanical philosophy, but one that hearkened back to, and completed the project of, the mechanical philosophy of the mid-seventeenth century.