Adam I. Gerard

Ontological Gaps

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.

Forms and Structures

  1. Structures can be (geometric) forms (this is the ancient view) - e.g. as spheres, cubes, or in the Delian Problem solved by Archytas
  2. Today, structures are typically thought of algebraically not geometrically (groups, rings, fields)
  3. 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 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!

Universal and Structures

  1. Ante Rem Structuralism in philosophy math bucks the modern trend and explicitly sees algebraic structures as universals (Platonic Forms).
  2. Platonic Forms - a category of substances or properties that are exemplified in or instantiated by particulars (objects).
  3. 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.
  4. Even Category Theory presupposes algebraic, presumably Set Theoretic, structures.


  1. Are properties?
  2. Special properties like haecceities or quiddities?
  3. But if Redness is a Platonic Form (Universal), on Plato’s view it is both a property that is exemplified and itself.
  4. On his view, Red is Red and so it is of itself that it exemplifies itself.
  5. But, identity is distinct from exemplification.
  6. 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.

  1. But there’s a problem here too. For if there are substances and non-substances as properties what is it that distinguishes these?
  2. What property of substances makes them intrinsically different than other properties?
  3. 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?
  4. 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. 1 and 2 cannot be meaningfully understood independently of the successor relations within which they stand.
  2. Therefore, 1 and 2 have no essence beyond the number line of which they are a part.
  3. This also explains why I and II can be substituted as symbols (Roman, Greek, Arabian numerals, etc.) because the choice of marks is irrelevant.
  4. But this implies that substances are irrelevant as a theory of identity within the philosophy of mathematics.

Relata and Relations

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

Upward Function Composition

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

Downward Function Composition

  1. Objects are just identity relations (identity functions)
  2. But what is a function?
  3. On this view there’s an infinitude of functions - functions of functions ... terminating in identity relations.
  4. 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.