existence; yet that does not entail that the existence of the quiddities found in the external world is something added to them in that external realm, nor that quiddities possess any reality outside the mind beyond their very existence outside the mind.

To clarify the first principle further: the philosophers who follow Aristotle—figures such as Ibn Sīnā and those of lesser rank—do not hold that every non-existent entity, whether an individual or otherwise, has any fixed reality outside the mind.
The claim that it does is made only by certain Muʿtazilites and Shiʿites—and even they are in a better state than the philosophers, for the philosophers themselves split into two divergent opinions.

As for their earlier predecessors—Pythagoras and his adherents, and Plato and his adherents—they were in manifest error on this question, and Aristotle and his followers refuted them, as Ibn Sīnā records in the Healing and elsewhere.
The Pythagoreans imagined that numbers and magnitudes are entities existing outside the mind, independent of the things that are numbered and measured.
Plato’s followers, sensing the flaw in that view, supposed instead that the specific universal natures—such as the essence of “man,” “horse,” and the like—subsist externally apart from the concrete individuals found in the world, and that these natures are eternal and incapable of change. These are the so-called Platonic Forms, or “suspended exemplars.”

Nor did they stop there: they extended the same claim to matter, duration, and place. They posited a bare matter divorced from all forms, existing on its own in the external realm—namely, the primordial hylē, concerning which most rational thinkers, both within their school and outside it, deemed them mistaken. They likewise affirmed a real duration existing apart from bodies and their qualities, and a real vacuum existing outside bodies and their qualities. Aristotle and his circle realized that all of these are merely mental posits, not things that subsist in concrete reality, just as number is with respect to the thing numbered. Yet Aristotle and his followers nevertheless asserted that matter exists externally apart from the visible forms, and that the universal natures are fixed in the external world independently of individual substances.