This paper reflects upon and extends Husserl's analysis of the indirect mathematization of nature by distinguishing three stages thereof: (1) Galileo's extension of classical Euclidean geometry to the study of matter in motion; (2) the development (dependent upon the analytic geometry and the calculus) of the ideal of a fully formal, axiomatic science of nature; and (3) the challenges posed to this view of science by relativity theory, quantum mechanics. The differing mathematical conceptions involved in these challenges also occasion a revised view of how the world presents itself in and to the scientific experience.