I was watching a great conversation from the past between Dr. Martha Nussbaum (University of Chicago, Law) and the legendary Bryan Magee (Parliament PM and TV executive).
Dr. Nussbaum is a foremost Aristotelian and I appreciated her very thorough and succinct description of Aristotle's theory of structure and universals.
This post seeks to briefly demarcate the Aristotelian use of these terms and modern structuralist notions.
Along the way I'd like to identity at least a couple ontological and conceptual gaps in philosophy that are relevant to discussions on this blog.
These metaphors and ontological units do all the work of the theories of which they are a part. But they often remain mysterious and unexplained.
Really a lot of the verbiage surrounding use of “structure” (the word) should be more specific.
Part of modern structuralism is the attempt to overthrow the old notion of Forms (geometric metaphors that they were given the Greek preference for Geometry above all other kinds of math) in favor of algebraic structures.
Dr. Nussbaum has correctly pointed out that Aristotle's treatment of forms is fairly nuanced and involves a conception of forms as functional structures. Such functional structures are nevertheless geometrically inspired (the Ship of Theseus, animals, buildings, and so on.) Again, the wider sense described above takes its inspiration from algebraic structures. One key difference is the extremely limited role that objects (or, individuals in the archaic vernacular) play within structured systems - in many cases, they are eliminated entirely (for example, within certain formulations of Category Theory).
Furthermore, I think we should carefully distinguish between structures and functional systems (which are fundamentally role-oriented - structures need not be role-oriented in any way they can merely interrelated with no other distinguishing roles or characteristics) and explore the idea that structuralism may be an explanation for functionalism (and shouldn't be identified as the same).
This aligns with my criticism that many metaphors in philosophy are misguided. We should leverage modern maths and not be fettered to ancient notions of Geometry!
So, there’s something wrong with conceiving Forms as substances (as has historically been noted).
Substances are understood to be identity properties - the kind of property that makes a specific particular exactly what it is. Its essence.
There are other arguments against substances as such. Some involve considerations about the appropriateness of this concept in the first place:
2cannot be meaningfully understood independently of the successor relations within which they stand.
2have no essence beyond the number line of which they are a part.
IIcan be substituted as symbols (Roman, Greek, Arabian numerals, etc.) because the choice of marks is irrelevant.
These independent considerations also motivate Connection Theory.