The class I am teaching this term covers the emergence of Early Modern philosophy and science. The first five weeks are devoted to a whirlwind tour of Ancient Greek natural philosophy (plus a bit of Renaissance thought), and the last 10 weeks cover 17th century philosophy and the scientific revolution.
We spent half of the past week discussing Ptolemy, and I was struck by something that I had noticed before, but never really appreciated. It is very natural for Ptolemy to use fully 'mathematized' explanations for astronomical phenomena, but not for (most) other physical processes. Why? On Ptolemy's view, astronomical objects share more properties with mathematical objects than they do with terrestrial objects. He thought that astronomical objects are eternal and their properties are unchanging -- like the number 5, but unlike terrestial ones. We give a mathematical treatment of astronomical phenomena because they exhibit properties of mathematical objects.
The application of mathematical methods in Ptolemaic astronomy helps bring into focus the so-called problem of the unreasonable effectiveness of mathematics, which some days appears to me to be an unequivocal pseudo-problem. Ptolemy's application of mathematics to physical phenomena, I think, appears extremely well-justified compared to our own: astronomical phenomena can be mathematized because they share peculiar features with mathematical objects, features that the mundane, material objects in our immediate surroundings lack. During and after the scientific revolution, we preserved and expanded Ptolemy's mathematizing proclivities, but we apparently relinquished his justification for treating the natural world mathematically.
Update (10/02/05): Kenny over at Antimeta just put up an interesting post on the (un)reasonable effectiveness of mathematics too, and it is in (small) part a comment on my post.