1. Aristotle introduces his intensional account of universals in Z, 10 (to be extended to dunamis in book H, and to mathematical entities in M, 3). In book M he compares his take to the Pythagorean one - that numbers (or other mathematical entities) are parts of sensible things and therefore themselves to be counted among sensible things - and the Platonic one - that numbers (as ideas and mathematical entities) are separate from sensible things. Aristotle proposes that mathematical entities are, like universals, aspects of things - in fact aspects of substances and, as such, enjoy a mode of being (tropon) which depend on what is substantial. Aristotle held that ousia protai to onton (substance has priority over other modes of being). Aspects are associated to his intensional turn in ontology, so to speak. He's explicit about that in M, 3: mathematical entities are aspects of sensible things just as, for geometry, it is an accident that a circle is white but not that it is circular while if we study the whiteness, the shape is to be treated like an accident. Aspects are out there and this is why we can have a perfect knowledge of them. In fact, this is maybe the hidden inspiration for McDowell's idea of de re senses and for his response dependence account of the content of perceptual experience - requiring conceptual capacities to have contents. Aspects are in the world - things harbor affordances.
2. Book M clarifies further Aristotle's idea of substances, especially of sensible substances when he discusses the Pythagorean and the Platonic alternatives to his intensional view of abstracta. Incidentally, these two alternatives are the ones commonly mentioned now-a-days as typical forms of realism about mathematics if we accept the similarities between the Pythagorean view and Quinean or Millian naturalisms. In both cases, abstracta are somehow among sensible things. But going back to Aristotle's substance-oriented ontology: substances are the units of reality. They are not like objects in OOO because their parts are not necessarily substances and there are things that are not as prior as substances but do exist - his ontology is not flat. Otherwise, there are similarities. Substances are everywhere and the fundamental relations are between are the important ones. It is pluralist - there are many substances and many types of substance. They have priority (ousia protai to onton) over all other modes of being and they are units that (supposedly) carve the world in its joints.
3. M, 4 presents explicitly an attempt to debunk Plato's doctrine of ideas in terms of its Heraclitean roots. Aristotle says that Plato's theory of ideas comes from his acceptance of a Heraclitean doctrine of the sensible according to which every sensible thing is in a perennial flow. No knowledge or science is possible without the aid of a separate intelligible realm with immutable realities. It is an interesting diagnosis. In fact, one can then compare Plato's move with Hume's conception of the sensible as unpredictable actuality - and the next steps are to criticize metaphysics (as the study of anything necessary within the sensible) and to appeal to our efforts for second creation to provide stability, predictability and some kind of knowledge. Aristotle's answer to Plato can then be put like this: we cut the evil by its roots by proposing a substance-oriented account of the sensible.
2. Book M clarifies further Aristotle's idea of substances, especially of sensible substances when he discusses the Pythagorean and the Platonic alternatives to his intensional view of abstracta. Incidentally, these two alternatives are the ones commonly mentioned now-a-days as typical forms of realism about mathematics if we accept the similarities between the Pythagorean view and Quinean or Millian naturalisms. In both cases, abstracta are somehow among sensible things. But going back to Aristotle's substance-oriented ontology: substances are the units of reality. They are not like objects in OOO because their parts are not necessarily substances and there are things that are not as prior as substances but do exist - his ontology is not flat. Otherwise, there are similarities. Substances are everywhere and the fundamental relations are between are the important ones. It is pluralist - there are many substances and many types of substance. They have priority (ousia protai to onton) over all other modes of being and they are units that (supposedly) carve the world in its joints.
3. M, 4 presents explicitly an attempt to debunk Plato's doctrine of ideas in terms of its Heraclitean roots. Aristotle says that Plato's theory of ideas comes from his acceptance of a Heraclitean doctrine of the sensible according to which every sensible thing is in a perennial flow. No knowledge or science is possible without the aid of a separate intelligible realm with immutable realities. It is an interesting diagnosis. In fact, one can then compare Plato's move with Hume's conception of the sensible as unpredictable actuality - and the next steps are to criticize metaphysics (as the study of anything necessary within the sensible) and to appeal to our efforts for second creation to provide stability, predictability and some kind of knowledge. Aristotle's answer to Plato can then be put like this: we cut the evil by its roots by proposing a substance-oriented account of the sensible.
Book Nu makes it clear that Aristotle's existential pluralism is no flat ontology. His emphasis on the priority of substances (ouisai protai ton onton, in book Lambda) over all other mode of being is combined with his claim (in Nu, 1) that relations are themselves the mode of being that comes last. Nothing has less being than relations. It is interesting to compare this point with Leibniz on substances and relations. Leibniz makes relations crucial for substances and, therefore, entertaining all priority substances enjoy. Aristotle, on the other hand, takes relations to be non-substantial (i.e. always external).
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