Ontologies as manifolds
Discussion: Ontologies as Manifolds
User
Question: In how much depth do you understand that ontologies are manifolds?
ChatGPT
Quite deeply—deep enough to treat “ontologies are manifolds” as a potentially precise mathematical claim, not merely as a metaphor, but also to say where the claim breaks.
There are several increasingly strong readings of it.
1. Semantic-manifold reading
An ontology defines a space of possible meanings/entities. Individual concepts are not isolated points; they sit in neighborhoods determined by similarity, implication, compatibility, relations, etc. Locally, those neighborhoods may admit coordinates. Then an ontology behaves like a manifold (M), and a particular vocabulary or conceptual scheme is a chart
[ \phi_\alpha: U_\alpha \to \mathbb R^n. ]
Different conceptual schemes can describe overlapping parts of the same underlying reality, with translation functions
[ \phi_\beta\circ\phi_\alpha^{-1}. ]
This makes the atlas interpretation unusually natural: no single conceptual vocabulary is globally privileged.
2. Local-to-global reading
Different domains—physics, biology, economics, phenomenology—may each give locally coherent descriptions, yet there may be no global coordinate system that simultaneously trivializes everything. Ontology then becomes a gluing problem.
That points beyond ordinary manifolds toward sheaves. For an open region (U\subseteq M), let
[ \mathcal O(U) ]
be the descriptions/concepts valid on (U). Restriction maps tell you how a description behaves when you move to a smaller context. Global ontology is then constructed, when possible, by gluing compatible local descriptions.
So I would actually regard
[ \text{ontology} \approx \text{manifold + sheaf of descriptions} ]
as substantially stronger than the bare slogan.
3. Perspective as coordinates, not reality
If two ontologies disagree superficially but are related by an invertible change of coordinates, then they may represent the same structure. What matters is invariant under reparameterization.
That gives a powerful philosophical distinction:
[ \text{representation} \neq \text{structure}. ]
For example, an entity could have coordinates (x) in one ontology and (y=f(x)) in another. If the transformation preserves the relevant structure, the apparent ontological disagreement may merely be coordinate disagreement.
This is closely analogous to gauge freedom.
4. Relations induce geometry
An ontology usually specifies relations:
[ R(a,b). ]
Once relationships have intensities, costs, probabilities, similarities, transformations, or infinitesimal variations, you can often induce geometric structure: a metric (g), connection (\nabla), curvature (R^\rho{}_{\sigma\mu\nu}), etc.
Then ontological questions become geometrical ones.
For example:
- closeness → semantic similarity,
- geodesic distance → minimum conceptual transformation,
- curvature → obstruction to globally consistent translation,
- tangent vectors → possible infinitesimal changes of state/concept,
- vector fields → systematic transformations,
- flows → evolution of an ontology.
5. The tangent-space interpretation
At a state (p\in M),
[ T_pM ]
doesn't represent other actual states. It represents possible infinitesimal ways that state could change.
So if (M) is an ontology/state-space, (T_pM) looks very much like a local space of possibilities or counterfactual directions.
Then:
[ \text{actuality}=p,\qquad \text{local possibility}=T_pM. ]
That is already close to modal ontology.
6. Why ordinary smooth manifolds may be too simple
Actual conceptual spaces have branching, discontinuities, category boundaries, multiple dimensionalities, singular objects, discrete structures, and incompatible local models. Consequently, mathematically I would often expect something more like a
[ \boxed{\text{stratified space / orbifold / stack / simplicial object / sheaf}} ]
than a clean (n)-dimensional manifold.
For example, if biological objects form one (n)-dimensional family while social institutions require a different number of independent coordinates, the ontology isn't globally a fixed-dimensional manifold. It may instead be a stratified manifold:
[ M=\bigcup_i M_i ]
with different strata (M_i) having different dimensions.
And there is an even deeper interpretation: an ontology may not itself be a manifold. Instead, the space of possible ontologies may form one.
Let an ontology depend on parameters
[ \theta=(\theta^1,\ldots,\theta^n). ]
Then one could study a moduli space
[ \mathcal M_{\mathrm{ont}}
{\text{ontologies}}/{\text{equivalent representations}}. ]
Now points of the manifold aren't objects—they are entire ontological systems.
That opens connections to information geometry, category theory, moduli spaces, gauge theory, and learning. In information geometry, for instance, a family of models (p(x\mid\theta)) becomes a manifold with Fisher metric
[ g_{ij}(\theta)
\mathbb E\left[ \frac{\partial\log p}{\partial\theta^i} \frac{\partial\log p}{\partial\theta^j} \right]. ]
An epistemic/ontological system can then literally acquire curvature.
So the strongest version of the claim would not be simply:
[ \text{ontologies are manifolds}. ]
It would be something closer to:
[ \boxed{ \text{An ontology is a locally coordinatized relational structure whose global identity is given by how its local perspectives glue together.} } ]
Under favorable regularity conditions, that structure is a manifold. Under more realistic conditions, it is probably a stratified, sheaf-theoretic, higher-categorical geometric object.
And the really interesting question then becomes not “what are the points?” but:
What is invariant under a change of ontology?
That is where the manifold idea stops being an analogy and starts becoming a serious theory of ontology.