Let's discuss the idea that there are no such things as properties, just lots of things which are more or less similar

The claim under consideration asserts that without an external, privilege-conferring structure, resemblance nominalism collapses into arbitrariness. The strongest defensible interpretation of this argument is that **similarity relations among concrete particulars are mathematically unconstrained in high-dimensional state spaces and modally fragile across possible worlds, rendering resemblance insufficient on its own to ground objective property boundaries.** This argument does not entail that similarity is useless in cognitive or empirical contexts, but rather that *metaphysical nominalism* cannot use ungrounded resemblance as an ultimate explanatory primitive for objective ontologies. ``` THE ARBITRARINESS DEDUCTION [ Raw Particulars in Infinite-Dimensional State Space ] | v [ Information-Theoretic Equidistance ] (Cover & Thomas) (Infinite projection axes yield identical distances) | v [ Modal Fragility across Possible Worlds ] (Lewis) (Extension of property classes shifts per counterfactual) | v Conclusion: Similarity cannot select objective boundaries without external constraints (Invariants/Pragmatics). ``` --- ### Deductive Structure of the Argument The logical core of the argument proceeds as follows: 1. **Premise 1 (Dimensionality):** Any concrete particular possesses an arbitrary or infinite number of physical, spatio-temporal, and relational features, placing it within an infinite-dimensional state space. 2. **Premise 2 (Information-Theoretic Undetermined Distance):** In a state space with infinite dimensions, the choice of metric or projection axis determines the distance between any two objects. Under unweighted metrics, all distinct objects are equidistant or indefinitely variable in distance [1]. 3. **Premise 3 (Modal Instability):** If a property is defined merely as an extension (a set of resembling particulars), its boundaries are intrinsically bound to the actual contingent population of particulars across modal space [2]. 4. **Premise 4 (Explanatory Circularity):** To select a *privileged* projection axis or metric (i.e., to privilege "charge" over "spatiotemporal proximity to Paris"), the nominalist must either appeal to an intrinsic feature (reintroducing universals or sparse tropes) or to human utility (relativizing reality to psychology). 5. **Conclusion:** Therefore, primitive, ungrounded resemblance cannot ground objective properties without devolving into arbitrary grouping or illicitly reintroducing the very metaphysical structures nominalism sets out to deny. --- ### Primary Evidence and Formal Foundations #### 1. Information-Theoretic Geometry of State Spaces In formal metric space theory and information theory, an object $x$ represented as a sequence of feature bits in a high-dimensional space can be mapped onto an infinite array of orthogonal axes [1]. As established by Thomas M. Cover and Joy A. Thomas, without a predefined weighting vector $W = (w_1, w_2, \dots, w_n)$, the generalized distance metric: $$D(x, y) = \sum_{i=1}^{n} w_i \cdot d(x_i, y_i)$$ is formally underdetermined [1]. For any pair of physical objects—such as a billiard ball and a cherry versus a billiard ball and an electron—there exist infinite projection spaces where the Hamming distance between the former pair is larger than between the latter pair [1]. * **Evidence vs. Analogy:** This is a mathematical reality of high-dimensional geometry, not merely an analogy. Unweighted similarity fails to yield a single, deterministic topology. #### 2. Modal Fragility and Counterpart Theory In David Lewis's *On the Plurality of Worlds*, modal realism and counterpart theory demonstrate that sets of particulars defined purely by intra-world similarity relations suffer from structural instability [2]. If "redness" is identical to the maximal set $S$ of resembling red particulars in the actual world $W_1$, then adding or removing a single object in a counterfactual world $W_2$ fundamentally shifts the boundaries of the class $S$ [2]. Because primitive resemblance lacks an internal "essence" or invariant core to hold the class together across worlds, properties constructed solely out of set-theoretic resemblance are modally fragile and structurally indeterminate across modal space [2]. --- ### Grounding and Resolution Strategies To escape this logical indeterminacy without adopting Platonic universals, contemporary philosophers and physicalists appeal to two constrained framework strategies: | Strategy | Mechanism | Metaphysical Constraint & Trade-off | | :--- | :--- | :--- | | **Physical Invariants** | Ground resemblance in physical symmetries and conserved quantities derived via Noether's theorem (e.g., charge, invariant mass) [3]. | **Constraint:** Grounds objective properties in fundamental physics [3].<br>**Trade-off:** Fails to account for macro-level, biological, or social properties without complex reductionism. | | **Pragmatic / Evolutionary Functionalism** | Similarity metrics are constrained by biological evolution, sensory systems, and niche construction. | **Constraint:** Explains why organisms consistently group objects non-arbitrarily.<br>**Trade-off:** Relinquishes metaphysical realism; turns properties into cognitive projections. | --- ### Crucial Counterevidence and Limitations The most formidable challenge to this argument comes from **Sophisticated Primitive Resemblance Nominalism**, most notably formulated by Gonzalo Rodriguez-Pereyra. Rodriguez-Pereyra argues that resemblance is not an unweighted metric over arbitrary properties, but an irreducible, $n$-adic primitive relation holding directly between concrete particulars as wholes. Under this view: * Nominalists do *not* sum up individual feature dimensions to compute a distance metric. * Rather, the fact that "object $a$ resembles object $b$" is an ultimate, unanalyzable truthmaker. If primitive $n$-adic resemblance is accepted as a foundational metaphysical fact, the information-theoretic charge of "unweighted axes" loses traction because there are no underlying "axes" or "dimensions" being summed; there are only primitive facts of resemblance. --- ### Epistemic Calibration * **Strengths:** The argument forcefully demonstrates that nominalism cannot appeal to "similarity" as a naive, self-evident concept. Similarity without structural or physical constraints is mathematically and modally indeterminate [1, 2]. * **Weaknesses:** It relies on the assumption that nominalists must analyze resemblance in terms of state-space dimensions or feature sets. If a nominalist successfully defends resemblance as a primitive, non-relational truthmaker, the information-theoretic critique functions as a critique of human conceptualization rather than an ontological refutation. **Definitive Proof or Disproof:** The argument would be **strengthened** by empirical or formal proof that microphysical state spaces cannot be mapped onto invariant symmetry groups without invoking abstract relations. It would be **weakened** by a formal proof showing that primitive $n$-adic resemblance can completely avoid both Russell's regress and modal fragility without introducing higher-order set constraints. *** ### References * [1] Cover, T. M., & Thomas, J. A. (2006). *Elements of Information Theory* (2nd ed.). Wiley-Interscience. * [2] Lewis, D. (1986). *On the Plurality of Worlds*. Blackwell. * [3] Brading, K., & Castellani, E. (Eds.). (2003). *Symmetries in Physics: Philosophical Reflections*. Cambridge University Press.