"[The] seemingly self-evident possibility of the application of mathematics to physics is quite often almost taken for granted. However ... for those ancient scientists and philosophers who follow the Platonic-Pythagorean programme, such an application is not at all evident and is, in fact, impossible." (p. x)
This is because "for Plato and for the later Neoplatonic thinkers ..., mathematics can give knowledge about those things that cannot be otherwise and therefore has nothing to do with the ever-fluent physical things" (p. 256).
"The central question of this book is thus[:] how and why does it become possible for early modern science to apply mathematics and its methods to the description of rerum natura." (p. x)
The key lies in the notion of construction. In the Platonic tradition---"where, in general, the (already) existing has a higher ontological status than the produced" (p. 259)---the role of constructions in geometry is as a crutch which enables the worldly mind, which is sadly limited by its dependence on sense-perception-based reasoning, to form an impression of the higher world of eternal truths:
"The purpose of the construction of a geometrical figure in the imagination for Proclus is then a sui generis alienation of the figure, when it is objectified ... In this way, the properties are rendered visualizable as imaginable, as if being projected onto the screen of the imagination and into geometrical matter, for better and clearer consideration of and by discursive reason." (p. 260)
Descartes, by contrast, takes constructions to be the epistemological cornerstone of all knowledge:
"Modern philosophy and science ... accept the verum factum principle, according to which an object can be known ... to the extent that, and insofar as, it is and can be produced and constructed. ... In this imaginary recreation of the world, the finite mind easily assumes the role of the demiurg, who knows the world insofar as he himself has created that very world." (p. 259)
The applicability of mathematics to physics therefore becomes a non-issue: geometry simply is physical, since it is only meaningful by virtue of its physical constructions.