One difference is in where their primary interests lie. The natural scientist studying astronomy is interested in mathematics because it helps him model and predict the motions of the heavenly bodies that are his primary interest. The mathematician is interested in astronomy because the paths of the heavenly bodies trace geometric forms that are his primary interest. I suppose the natural scientist would tend to lean more on empiricism while the mathematician would tend to favor rationalism. Aristotle, so far as I can tell, seems to favor a balance between the extremes. He’s just finished arguing that form is more nature than matter, and geometry could be regarded as the study of pure form, but he doesn’t strike me as all that interested in the study of mathematics for its own sake. He is happy to use either observation or deductive reasoning and mathematics if they help him understand the natures of light and sound, etc.
A good example of how the two approaches can complement one another might be the Greeks’ discovery a generation or two before Socrates that Hesperus, the evening star, and Phosphorus, the morning star, were in fact the same heavenly body we now know as Venus. (This is often attributed to Pythagoras, but there are other traditions.) Because Venus orbits the Sun closer than the Earth, it will rise and set in the evening for a time, then it will no longer be visible while its rising and setting take place during daylight hours, and then it will later reappear in the morning sky. The cycle then repeats in reverse. Since no one can see Venus with the naked eye during the day, it required math and reason to work out that the apparently separate planets were actually one. The mathematicians could not have accomplished this, though, without careful observation of the planet’s movements while it was visible. The discovery had a profound impact on Greek philosophers and provided ample fodder for their poets on the subjects of beginnings and endings, death and new life. There’s an epigram attributed to Plato that Shelley translates thus:
Thou wert the morning star among the living,
Ere thy fair light had fled;
Now, having died, thou art as Hesperus, giving
New splendour to the dead.
Well. that tangent was a bit longer than I intended, and I feel like there’s a lot of nuance in this passage that I’m missing, but I hope that if my answer isn’t exactly complete, it will at least add something interesting.
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