Aristotelian “matter” and “material cause” mainly capture notions of circumstance, contingent fact, mediation, and what some 20th century writers called sedimentation.
Early Greek mathematics was not sufficiently advanced to be of much help in understanding natural processes, so Aristotle instead pioneered a logical/semantic approach to our experience of sensible nature. In accordance with this, “matter” for Aristotle is what Brandom would call an expressive metaconcept, rather than being a theory-laden empirical concept like the modern notion. We need to be careful moving between the two.
For Aristotle, matter is not a subsisting thing but a relative concept that has an expressive role. “Matter” and “form” are correlatives, only analytically distinguishable, though the relation is not quite symmetrical (form seems to be more primary). All physical things are conventionally referred to as “composites” of the two, but this should not be taken to mean that either has independent existence. (See also Hylomorphism.)
The relation between matter and form is loosely but definitely not strictly analogous to that between potentiality and actuality. Matter gives concrete embodiment to particulars, whereas potentiality is what provides the space for construction of universals.
Unlike Descartes, Aristotle does not associate matter in any direct way with mathematically analyzable extension. One of his usages for “matter” is as the inferred substrate for sensible properties. But alongside this, a different account is actually more prominent. Some “form” or way of being and doing makes the composite the particular kind of thing it is. In this context, “matter” ends up comprising the concrete circumstances of the actual functioning of a way of being and doing.
Modern people are used to thinking of form as a predicate of some matter. Some in the ancient world thought this way, too, but Aristotle prefers to speak in the opposite way, and to predicate the matter of the form.
Aristotelian matter refers to circumstance, to a body of what is the case about some particular actualized form, more primarily than to a body of stuff. What is of interest with matter is particular matters, or “a” matter, or “the” matter of some particular form — the circumstances of its actualization.
The Greek commentator Alexander of Aphrodisias postulated an ultimate “prime matter” that was distinguished by having no properties at all, which by implication would put all properties whatsoever on the side of form. At the other end of the spectrum, some later commentators spoke of a “proximate matter” (i.e., the matter closest to the form) as highly structured, and as including things like the body of an animal.
Aristotle’s various usages, if reified, would result in a layering of form/matter distinctions. (In the middle ages, there was heated debate over the so-called “unity or plurality of substantial forms”. Theologians were much concerned over the relation of soul to body. Albert the Great, Thomas Aquinas and others took seriously Aristotle’s description of soul as the “form of the body”. Augustinians wanted a dualistic separation of soul and body, more like the pilot in the ship that Aristotle rejected. They therefore argued that the body already had a form of its own, and that the form of the soul is superimposed on it (a “plurality of substantial forms”). The notion of “substantial form” was a late refinement intended to be strictly univocal, where Aristotle’s general usage of “form” had instead been self-consciously overdetermined, overflowing any such conditions in the course of its dialectical development.)
Aristotelian matter is implicitly not just immediate circumstance, but always a result of a sort of layering or accumulation of circumstances over time, including some feedback loops. This is important for the kind of generalization we should expect from Aristotelian science.
Cartesian extensionality is perfectly homogeneous. Aristotelian matter is anything but that. It is a site of differences (as form is also, in a complementary way). The virtual layering or accumulation I have spoken of is a very weak kind of unity. Aristotle’s notions of necessity and universality are also both deliberately weak notions. Something is said to be “necessary” just if there is no known counter-instance. “Universal” means “said of many things”, not “said unconditionally”.
Aristotelian “science” does not aim to codify laws in the modern, univocal sense. It is an open hermeneutic that seeks to understand processes in terms of patterns, while recognizing inherent contingency in matters of fact. Particularly in the biological works, there is a genealogical intertwining of form and accident that makes it impossible to strictly separate the two. Aristotelian natural teleology is not only purely immanent, but also never univocal, as this intertwining makes clear. It gives us tendencies only, never strict predetermination.