Determinism and Indeterminism as Model Artefacts: Toward a Model-Invariant Ontology of Physics

The gist: Deterministic–stochastic dualities are available in principle, and arise in a non-contrived way in many scientifically important models. This paper argues that both determinism and indeterminism as commonly understood are representational artefacts of our models rather than ontologically significant features of the world.

Model-Invariant Ontology Criterion

The central claim developed in this paper is that only structural features that remain stable across empirically equivalent formulations, and whose empirically accessible physical effects are preserved under such reformulations, qualify as candidates for realism. This criterion shifts attention away from the specific mathematical form of a model and towards the modal and relational structure that persists through representational variation. This yields a fallibilist form of structural realism grounded in modal robustness rather than in the specifics of any given mathematical representation. Features such as conservation laws, symmetries, and causal or metric structure satisfy this criterion and can be encoded in observable relations in mathematically intelligible ways. By contrast, the localisation of modal selection—whether in initial conditions, stochastic outcomes, or informational collapse mechanisms—is not invariant under empirically equivalent reformulations and is therefore best understood as a gauge choice rather than an ontological feature.

Representational Duality and Model-Equivalence

The paper illustrates how a deterministic dynamical model can be reformulated as a stochastic one in a way that preserves the preparation–measurement correlation structure, rendering the two models empirically indistinguishable for the relevant observables. This representational duality is not confined to the Bernoulli map but is a recurring feature of deterministic dynamical systems exhibiting chaotic behaviour under suitable coarse-grainings. The text notes that any stochastic process can be represented deterministically on an appropriately defined path space, while deterministic models with fine-grained but inaccessible microstates routinely admit stochastic coarse-grained descriptions that capture observable behaviour while suppressing microstate detail. This shows that whether a model is presented in deterministic or indeterministic form depends on representational choices.

Gauge Freedom and Invariant Structure

The paper argues that underdetermination of modal structure is not merely epistemic but reflects a genuine gauge freedom induced by finite empirical resolution. Once fine-grained state spaces are quotiented by observational equivalence classes, multiple inequivalent descriptions—deterministic or stochastic—can generate the same empirically accessible transition structure. What remains invariant across these reformulations is the induced modal and relational structure of the dynamics. This invariance is what I will argue to be the appropriate content of ontological commitment. In classical systems, coarse-graining by a finite partition replaces exact symplectic diffeomorphisms with stochastic dynamics on cells, and the defining invariant content of Hamiltonian mechanics survives as conservation of probability flow.

Quantum Analogy and Measurement Problem

In the quantum case, coarse-graining necessarily leads to stochastic transition rules that may be governed by either deterministic or indeterministic completions of empirically verifiable quantum transition rules. The freedom in how states are represented within a coarse cell is analogous to the gauge freedom relating empirically indistinguishable microstates in the classical case. The resulting description consists of unitary evolution interspersed with irreducibly probabilistic transitions associated with observation, which reclassifies the measurement problem as a question about how different representational frameworks encode the same invariant empirical transition structure.

Conclusion on Ontological Commitment

The opposition between determinism and indeterminism is not a metaphysical divide in nature but a difference in modelling practice that reflects how invariant empirical structure is encoded. The task of ontology, on this view, is not to choose between such representations, but to identify and articulate the structural features that remain stable across them. Where model-invariant structure thus leaves open whether decisive modal transitions are stochastic or deterministic in origin, it is more accurate to say that this distinction is simply not fixed by the empirically accessible dynamical structure itself.

General Applications

The analysis suggests that empirical access constrains only a finite-resolution dynamical structure while leaving open multiple, representationally distinct ways of realizing that structure at sub-empirical scales. The most permissive level is an empirical gauge, under which only those features fixed by observed transition statistics are held invariant. This empirical probability current itself is what remains invariant across all representations.

Final Summary of Findings

The paper establishes a model-invariance criterion for ontological commitment where a structural feature qualifies only if it is fixed across all empirically equivalent representations. This framework explains how long-standing problems in the foundations of physics arise from the reification of representational artefacts. The distinction between determinism and indeterminism is thus reframed as a difference in modelling practice rather than a metaphysical divide in nature. This approach allows for the possibility that the decisive modalities modulating between extremes are not fixed by the current theory.

Acknowledgements

I would like to thank Nicholas Parkin for conversations, and for drawing my attention to some relevant physics work, and Arnold Neumeier for comments on the first draft of this paper.

References

The references listed at the end of the paper support the claims made throughout these sections. The specific references cited are [1] Barandes, J.A., 2018, [6] Kolmogorov, A.N., 1950, and others listed in the bibliography.

1. This system can distinguish between model-invariant structure and modelling conventions. It will prioritize features such as conservation laws, symmetries, and causal or metric structure that remain stable across empirically equivalent representations.

2. The AI can adopt a fallibilist form of structural realism where it treats only model-invariant structure as the primary locus of ontological commitment while classifying features such as determinism and indeterminism as representational artefacts.

3. It will treat "the localization of modal selection—whether in initial conditions, stochastic outcomes, or informational collapse mechanisms—is not invariant under empirically equivalent reformulations and is therefore best understood as a gauge choice rather than an ontological feature."