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

structureanduniversals.

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.

*Structures*can be (**geometric**) forms (this is the ancient view) - e.g. as spheres, cubes, or in the Delian Problem solved by Archytas- Today,
*structures*are typically thought of**algebraically**not**geometrically**(groups, rings, fields) - Houses, buildings, 3D objects are
**geometric**“structures” in the first sense but not necessarily in the second

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

formsis fairly nuanced and involves a conception of forms asfunctional structures. Suchfunctional structuresare neverthelessgeometricallyinspired (the Ship of Theseus, animals, buildings, and so on.) Again, the wider sense described above takes its inspiration fromalgebraic structures. One key difference is the extremely limited role thatobjects(or,individualsin 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

structuresandfunctional systems(which are fundamentally role-oriented - structures need not berole-orientedin any way they can merely interrelated with no other distinguishingrolesor 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!

*Ante Rem Structuralism*in philosophy math bucks the modern trend and explicitly sees*algebraic structures*as*universals*(Platonic*Forms*).- Platonic
*Forms*- a*category*of*substances*or*properties*that are*exemplified in*or*instantiated by**particulars*(*objects*). *Structures*so-conceived (today) are characterized by*ersatz-objects*or*places-as-objects*(pointed out by Shapiro) and so don't seem to be*structuralist*at all.- Even
**Category Theory**presupposes*algebraic*, presumably**Set Theoretic**,*structures*.

- Are
*properties*? - Special
*properties*like*haecceities*or*quiddities*? - But if Redness is a Platonic
*Form*(*Universal*), on Plato’s view it is both a property that is exemplified and itself. - On his view, Red is Red and so it is of itself that it
*exemplifies*itself. - But,
*identity*is distinct from*exemplification*. - And, wherever Red is it is infinitely-many so (for it
*exemplifies*itself and is of itself*exemplified*).

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*.

- But there’s a problem here too. For if there are
*substances*and non-*substances*as*properties*what is it that distinguishes these? - What property of
*substances*makes them intrinsically different than other*properties*? - But how can a specific kind of
*property*, the very kind of*property*that is what makes something intrinsically distinct, be distinguished from another*property*by way of another*property*? - It’s contradictory.

There are other arguments against *substances* as such. Some involve considerations about the appropriateness of this concept in the first place:

`1`

and`2`

cannot be meaningfully understood independently of the*successor relations*within which they stand.- Therefore,
`1`

and`2`

have no*essence*beyond the number line of which they are a part. - This also explains why
`I`

and`II`

can be substituted as symbols (Roman, Greek, Arabian numerals, etc.) because the choice of marks is irrelevant. - But this implies that
*substances*are irrelevant as a*theory of identity*within the philosophy of mathematics.

- Bradley’s
*regress problem*(against*relations*understood as*universals*) applies here regardless of the view one takes about the nature of*relations*. - It applies equally well against traditional
*Forms*. - F is H - object F exemplifies H but how does exemplification stand in relationship to F and H?

*Functions*are mostly conceived as being built up from*mappings*and*elements*(*objects*,*sets*)- But what are
*mappings*?

*Objects*are just*identity relations*(identity*functions*)- But what is a
*function*? - On this view there’s an infinitude of
*functions*-*functions*of*functions*... terminating in identity relations. - We can define functions using
**Set Theory**which relies on concepts like*domains*,*codomains*,*images*, but all of these depend of*mappings*or*1-1 correspondence*.

These independent considerations also motivate **Connection Theory**.

- Path Deformation
- On Connection Theory
- Ontological Gaps
- Sign, Identity, Relations
- Fun Math Stuff and the Philosopher's Stone

##### post: 11/27/2020

##### update: 7/31/2022