Rand was fond of dividing the intellectual world between the Aristotleans and the Platonists, as if everyone, fundamentally, falls into one camp or the other. This division, however, is far too narrow. It leaves out several other basic philosophical positions, such as naturalism and German transcendentalism.
Both Plato and Aristotle arose out of the Socratic philosophical tradition, which holds, in the words of Socrates himself (as transcribed by Plato) "that Reason [is] the disposer of all" and that if anyone "is interesting in finding out the cause or generation or destruction or existence of anything, he must find out what [is] best for that thing." Things are to be understood by their uses and purposes, not by their elements or antecedents. Hence we find Plato arguing that the eyes, nose, and mouth are in front of the head, because the front is the nobler side! And in somewhat the same manner, we find Aristotle coming up with the concept of final causes, which expresses this methodology at its most abstract.
Aristotle tried his best to integrate this Socratic-Platonist philosophy with common sense, which accounts for Aristotle's popularity with thinkers like Rand. But the methodology of the tradition—a methodology satirized by George Santayana when he compared it to the chorus Moliere's Le Malade Imaginaire, which sings that opium puts people to sleep because it has dormitive virtue—runs deep in Aristotle, as it does, unfortunately, in Rand herself. The whole idea of founding a metaphysics on axioms, is inspired by this dubious Socratic methodology. And the notion of reality as logical—this too, Rand owes to Aristotle and the Socratic tradition.
The naturalist tradition, on the other hand, eschews this sort of nonsense. According to naturalism, matters of fact cannot be determined by logical constructions, for the simple reason that the external world is not a logical system. It doesn't not conform to "reason," whatever that may mean! Naturalism instead asserts two very simple but, from a Socratic point of view, heretical principles: (1) "Nothing arises in the world that we may use it, but what arises brings forth its use." Here we have "that discarding of final causes on which all progress in science depends," comments Santayana. And (2) "Nature is her own standard and must be accepted on her own terms, not on ours." Or in other words, the world does not exist for the convenience of our intellects! Nature must be accepted as she is, not as how we, or our philosophical principles, wish her to be.
Now although Objectivism frequently gives lipservice to naturalistic ideas, fundamentally Objectivism is not a naturalistic philosophy. It belongs to a different philosophical tradition. Hence Peikoff's assertion that philosophy has a veto right over all other disciplines, including physics. Or consider the statement "contradictions cannot exist in reality." How do Objectivists know that? Have they run empirical tests? No, of course, not. They "deduce" it from their axioms. Well this is really no different than Plato deducing the circularity of the planets' orbits from the divininity of the circle as a form. It's the classic metaphysical gesture of determining matters of fact from logical, moral or rhetorical constructions. It is not the way scientists work in empirical disciplines like physics and chemistry, where conjectures are inferred from facts and then corroborated through extensive empirical testing. Nature is its own standard, and to find that standard we must consult nature!