As you may know, Aristotle was an immanent realist, not a Platonic realist. He regarded mathematical objects as an aspect of the world that could be investigated (albeit in a more abstract sense), not as existing apart from it (in the sense of Plato's Forms which he rejected).

The best way to conduct an investigation in every case is to take that which does not exist in separation and consider it separately; which is just what the arithmetician or the geometrician does. — Aristot. Met. 13.1078a — Andrew M

As I understand it, the essence or universal of circularity is in the circular object, because for Aristotle, concrete objects demonstrate mathematical properties (weight, volume, extension, etc.) The essence of circularity is not floating around in a Platonic heaven somewhere.

I think it is correct to say that Aristotle believed we could understand mathematics in a more abstract sense, as mathematics and logic are derived from being and particular objects. He also mentions in the Posterior Analytics that the mind is so constituted that we can apprehend and understand these more abstract principles. The above quote from Aristotle's Metaphysics seems to indicate that he didn't think mathematics exists in the same way other things exist (which I think is intuitively correct). However, does that make Aristotle a conceptualist or nominalist? (I know conceptualism and nominalism are later philosophical phenomena. However, I had a professor tell me that Aristotle laid the intellectual foundation for nominalism and I'm trying to figure out for myself if that is really true.)

Thanks again for all your insight and help. I know I have a lot more to learn about this!

Yes, that makes sense. Thanks!

Thank you! That is very interesting and enlightening.

Excellent. Thank you so much for all the helpful comments. I really appreciate it.

Another thought ... Perhaps a Platonic objection but I was wondering what you thought. From an Aristotelian perspective, if I could destroy all the circular objects in the world, would I have successfully destroyed the essence of circularity? What might an Aristotelian response be? (I've read Aristotle but I can't remember if he addresses this question.)

Thanks again for the help!

Thank you for sharing that. I found the article very helpful. Some Platonists accuse Aristotle of laying the groundwork for nominalism. I don't think it is fair to accuse Aristotle of nominalism. Are there other good Aristotelian responses to nominalism?

I think that is correct. Aristotle believed that mathematical properties are immanent within concrete objects. I'm wondering how he would account for the laws of logic and the principles of mathematics that make geometry and math possible in the first place? I'm guessing they are just abstractions of concrete objects or fundamental principles of being?

Thanks. I think you are completely correct in your analysis. :)

This was just a hypothetical question in the sense of "What if we could teleport Aristotle into a room with some of the great 18th-century idealists such as Kant, Fichte, Hegel, or Schelling"? I'm sure he would think idealism was Platonism gone amuck but how would he argue that? :)