Indeterminate Ontology: Quantum and Metaphysical Insights

• Indeterminate ontology is a framework that asserts physical properties have inherent indeterminacy, demonstrated by operator non-commutativity and the Kochen–Specker theorem.
• It spans quantum mechanics, quantum gravity, and metaphysics by explaining phenomena such as decoherence, non-accessible mass, and gappy multi-field assignments.
• Models like GRW mass-density, generative quantum theory, and pilot-wave approaches exemplify its role in reconciling measurement paradoxes with clear ontological structure.

Indeterminate ontology encompasses a spectrum of frameworks and interpretations within quantum physics, quantum gravity, metaphysics, and foundational philosophy, all unified by the recognition that certain fundamental aspects of reality are objectively indeterminate rather than merely epistemically unknown or linguistically vague. These frameworks highlight the failure of classical joint value-definiteness for physical properties and frequently identify this indeterminacy as intrinsic—present in the structure of quantum theory, quantum gravity, or in the very fabric of physical modeling. The notion is distinguished from semantic or subjective ignorance and is often formalized via operator non-commutativity, determinable/determinate property hierarchies, and model-theoretic invariance.

1. Formal and Conceptual Foundations of Indeterminate Ontology

Indeterminate ontology, as systematized in both foundational physics and analytic metaphysics, asserts that indefiniteness is instantiated in the world itself. Within quantum mechanics (QM), this is articulated by the Eigenstate–Eigenvalue Link (EEL): a system has a definite value of an observable $\mathcal{O}$ if and only if its state is an eigenstate of $\mathcal{O}$. If the state is not an eigenstate, the observable is ontologically indeterminate—not just epistemically unknown (Cinti et al., 2021). The Kochen–Specker theorem precludes the pre-assignment of definite values to all observables in Hilbert spaces of dimension $\geq 3$ without contradiction.

The concept extends beyond standard QM. In quantum gravity (QG), both Loop Quantum Gravity (LQG) and String Theory exemplify indeterminate ontology: geometric operators (area, volume) in LQG and center-of-mass or vibrational modes in String Theory are represented by non-commuting observables, ensuring the impossibility of joint value-definiteness and mandating a non-epistemic, mind-independent indeterminacy of spacetime itself (Cinti et al., 2021).

“Degrees of presence” (Herbut) offer a graded alternative: every projector $E$ is assigned a reality-valued measure $\langle\psi|E|\psi\rangle$, recasting indeterminacy as an ontic “degree of presence” rather than a strict binary (Herbut, 2013).

2. Quantum Theory and Generative Architectures of Indeterminacy

Quantum indeterminacy underpins several modern ontologies of quantum theory, most notably in the Generative Quantum Theory (GQT) framework (Pipa, 2024). GQT models a physical system $S$ as possessing bundles of quantum properties, concretized as a density operator $\rho_S$ and a set of observables $\{\hat{O}_P\}$ whose eigenstates encode candidate properties (e.g., position, spin). A core notion is that value properties (i.e., “$S$ has value $v$ of $\mathcal{O}$0”) become determinate only when the system’s quantum property $\mathcal{O}$1 is stably differentiated—typically through decoherence (suppressing off-diagonal terms in the relevant basis).

GQT identifies seven ontological “knobs”—e.g., property ontology, generative properties, stability conditions, indetermination/determination structures—that, depending on their selection, generate interpretations such as GRW (spontaneous collapse), Many-Worlds (branching via decoherence), Bohmian mechanics (determinacy attached to particle positions), and others. Determinacy becomes a derived or emergent phenomenon; indeterminacy is the default ontological mode, with determination occurring via specific dynamical or environmental criteria.

3. Indeterminacy in Pilot-Wave and Multi-Field Theories

Pilot-wave theories, often considered paradigms of ontological clarity due to their precise primitive ontology (PO) of particles guided by a universal wave function, nevertheless instantiate various forms of metaphysical indeterminacy (Oldofredi, 2024). Oldofredi demonstrates that Bohmian mechanics, Nikolić's relativistic pilot-wave QFT, and Bell-type quantum field theories all leave specific aspects of reality metaphysically indeterminate—even with fully specified ontologies and deterministic (or stochastic, but precisely defined) laws. For instance, in Bohmian mechanics, spin and other non-positional observables remain ontologically indeterminate, while the ontology fails to single out a unique decomposition into subsystem modal trajectories (modal indeterminacy). In Bell-type QFTs, future evolution is not uniquely fixed, resulting in “future-state indeterminacy.”

The multi-field interpretation of the wave function (Romano, 6 Aug 2025) reframes $\mathcal{O}$2 as a “multi-field” in $\mathcal{O}$3-space, assigning values to $\mathcal{O}$4-tuples—determinately for the actual configuration occupied by the particles, but leaving all other $\mathcal{O}$5-tuples gappy or indeterminate. This is characterized as a determinable object-level metaphysical indeterminacy: for each unoccupied tuple, the proposition “$\mathcal{O}$6” simply lacks a truth-value. This gappy structure is intrinsic to the ontology, not a defect or an epistemic artifact.

4. Non-Classical Ontologies: Mass Density, Representation, and Cognition

In the GRW mass–density (GRWm) ontology (Mariani, 2020), indeterminacy appears via the ever-present “tails” of the matter density field, which cannot be made arbitrarily sharp due to the stochastic nature of wave function collapse and the functional relation between the wave function and the mass density field. The metaphysical status of non-accessible mass—mass located outside the centers of mass-density “clumps”—is best understood as an objectively indeterminate state of affairs. Attempts to treat such regions as unreal or anti-realist fail to account for their physical influence and the empirical adequacy of the theory.

Beyond the physical, indeterminate ontology has been invoked in frameworks for understanding the structure of cognition and representation systems (Novikov-Borodin, 2017). Novikov-Borodin distinguishes the “Existing” (what may influence a cognizer $\mathcal{O}$7) from the incognizable, positing that multiple, incompatible but jointly exhaustive representational ontologies co-exist, each indeterminate relative to the others when viewed from the level of basic notions. This model addresses paradoxes and incompatibilities in both the foundations of physics (e.g., cosmology, quantum objects) and the structure of set theory or scientific cognition.

5. Structural Realism, Model Invariance, and Gauge Aspects

Indeterminate ontology also arises when disentangling the status of determinism and indeterminism in physical theory. The model-invariant perspective (Nolland, 27 Dec 2025) argues for a structural realism based on modal robustness: only empirical structures stable across all empirically equivalent models—such as symmetries, conserved quantities, or probability fluxes—are assigned ontological status. Determinism or indeterminism, the degree or location of “collapse,” and even certain supposed ontological features (such as the locus of randomness) are shown to be gauge artifacts—matters of representational convention and not of reality per se. This framework resolves foundational conflicts, such as those between measurement collapse and unitary evolution or determinism and agency, by restricting ontology to invariants under model dualities.

Perspective / Theory	Source	Mode of Indeterminacy / Ontology
Quantum Mechanics / Gravity	(Cinti et al., 2021, Herbut, 2013)	Operator non-commutativity, lack of value-definiteness, degrees of presence, mind-independent indeterminacy (spacetime, matter)
GRWm	(Mariani, 2020)	Non-accessible mass as objectively indeterminate state of affairs
Multi-field (Bohmian)	(Romano, 6 Aug 2025)	Determinable vs. determinate, “gappy” multi-field assignments
GQT	(Pipa, 2024)	Indeterminate quantum properties as default; theory-generated conditions for determination across interpretations
Pilot-wave QFT	(Oldofredi, 2024)	Modal indeterminacy, stochasticity in configuration evolution, lack of determinate observable values
Cognition / Representation	(Novikov-Borodin, 2017)	Indeterminacy via incompatible ontological systems
Structural Realism	(Nolland, 27 Dec 2025)	Model-invariance: only structures invariant under deterministic/stochastic duality are ontological

A plausible implication is that indeterminate ontology, when precisely characterized, offers a powerful unifying lens on interpretative controversies in quantum theory, quantum gravity, and the modeling of physical systems. It restricts ontological commitment to structures that are robust under empirical equivalence and forbids the reification of representational artifacts.

6. Philosophical and Foundational Implications

Indeterminate ontology enforces a separation between ontological clarity in the specification of “what there is” (beables, primitive ontology) and the possibility of metaphysically indefinite facts concerning aspects of the world (e.g., observable values not fixed by the theory, modal partitions of possible histories, inaccessible mass, or value assignments to incompatible observables) (Oldofredi, 2024). This suggests that theories with explicit primitive ontology do not automatically yield a fully determinate world. Instead, the clarity of what is specified (particles, fields) coexists with inherent indeterminacy regarding values or modal evolution.

This view generalizes beyond the quantum domain, motivating fallibilist structural realism (Nolland, 27 Dec 2025) and meta-ontological pluralism (Novikov-Borodin, 2017). Specifying an ontology of only model-invariant features resists metaphysical overreach and clarifies which questions are substantive—those answered by invariant structure—and which are artifacts of particular representations.

Indeterminate ontology thus serves as both a tool for advancing interpretative rigor, especially in the quantum foundations and philosophy of physics, and as a framework for elucidating the boundaries of empirical and metaphysical inquiry.