Why did all the ancient philosophers seem so fascinated by astronomy? Their interest in math and "physics" is understandable, as math can be seen as very similar to certain branches of philosophy in that it is not the study of a particular existence, but, rather, the study of "existence," and physics is the study of the seemingly occult laws that govern everything, which is also very similar to philosophy in a sense, but astronomy is just the extrapolation of those two fields on "arbitrarily chosen" pieces of mass. Math, and even physics to a large extent, are "implicit" (for lack of better term) to existence, while astronomy is wholly explicit.

You are completely right to notice the early absorption with astronomy. I have heard people say “Greek philosophy began on May 28, 585 BC, at 6:13 in the evening” – because of astronomy. Thales, who is often called the first Greek philosopher, predicted a solar eclipse that we now know to have taken place on that date.

Not only did Thales thereby establish the credentials of philosophers as “ones who know” by being able to predict a coming natural event; he also thereby proved a point about the natural world that encapsulates early philosophy’s turn away from religion. If astonishing events like solar eclipses are not the capricious actions of mysterious gods but rather quite regular events in the natural world, then the world can be adequately studied through rational methods and without dependence on old stories handed down about divine action.

Mind you, this is only one way to understand the earliest philosophers. My point is that interest in astronomy is part of this picture and even one of the central concerns that those philosophers had, not at all something extraneous to their interests. We’d want to say somewhat different things about what astronomy meant to Anaximander and the Pythagoreans, because they had theories unlike anything ascribed to Thales.

Later in the ancient tradition, i.e. with Plato and Aristotle, astronomy came to take on additional significance. But rather than jumping ahead to them, let me back up to the state of astronomy before the philosophers came along.

First of all it’s highly relevant that astronomy was one of the first success stories in ancient science. The astronomy we encounter in e.g. the Babylonian records is almost entirely observational. People noted each night what phase the moon was in and which constellations were visible. Thales, who must have gotten his data from Babylon, was able to draw on their long history of watching the night sky. Astronomy let the earliest agricultural civilizations organize their calendars, not only observing that weather had begun to cool off in the fall but knowing when to expect it to cool; therefore knowing when the best time to plant and harvest would be. The first calendars began with the observation of recurring patterns in the skies, along with the special problem of coordinating the lunar calendar with a solar year.

It should be obvious why astronomy provides the basis for a calendar. The patterns of the sun, moon, and stars not only fall into patterns, once you start observing them long enough, but are also quite independent of anything that happens on earth. Earthquakes, floods, and fires – to say nothing of merely human events like war, drought, and migration – have no discernible effect on what we see in the stars. Measuring time calls for something that changes in a quantifiable way without being changed by the events one is using the calendar to measure. Nothing else accessible to those ancient civilizations could work as the observable sky could.

This fact also guides us to the second feature of astronomy that would have appealed to philosophers, in addition to the regularity that made the world feel natural and subject to human knowledge. Studying astronomy seemed to bring human beings into contact with relations and events that didn’t mix with lesser natural processes.

We know too much today to think this way. We know that the stars are made of matter like the matter found on earth, and that the observable patterns in our night sky are only the accidental effect of where our little sun is riding around a non-central part of an unimpressive galaxy. But ancient observers who knew none of that perceived what they saw in the sky as close to what we’d call a priori truth.

Ultimately my reply is to reject your assumption about the relative status of mathematics, physics, and astronomy. Astronomy struck a philosopher like Plato as much closer to mathematical truth than anything in the subject that Aristotle called “physics.” It mattered to philosophers (for Plato in particular) precisely because of how close it came to being mathematics.

Plato does distinguish the two subjects, though. In Book 7 of the Republic he has Socrates tell Glaucon that even what we see in the starry sky is visible and hence to some degree subject to the failings of all material objects. Astronomy comes closest of all the sciences to giving us patterns of the abstract truths about geometry and motion, but it still isn’t mathematics. Plato would conclude that although philosophers need to study astronomy as they progress toward higher kinds of knowledge, it is not their final object of inquiry.