Beyond Determinism: How Models Shape Our View of Reality

Author: Denis Avetisyan

A new analysis suggests the line between deterministic and indeterministic physics isn’t a property of the universe itself, but a consequence of how we choose to model it.

This review argues that ontological commitments should be based on model-invariant structures rather than the apparent determinism or indeterminism of specific representations.

The persistent debate surrounding determinism versus indeterminism in physics often obscures the role of modelling choices themselves. This paper, ‘Determinism and Indeterminism as Model Artefacts: Toward a Model-Invariant Ontology of Physics’, argues that this duality frequently arises as a representational artefact, not an ontological commitment. By prioritizing model-invariant structures – features stable across empirically equivalent formulations – a fallibilist structural realism emerges, grounding ontological commitment in robust, observable relations. Could this framework ultimately reconcile seemingly intractable problems like the measurement problem and the relationship between determinism and free will by distinguishing genuine physical structure from mere modelling convention?

The Illusion of Predictability: A Quantum Reassessment

For centuries, the prevailing understanding of the physical world rested on the principle of determinism. This concept posits that, given complete knowledge of a system’s present state – encompassing position, velocity, and all relevant forces – its future evolution is, in principle, entirely predictable. Essentially, the universe operates like a complex clockwork mechanism, where each gear’s movement is dictated by the preceding one, and the entire sequence unfolds with absolute necessity. This isn’t merely a philosophical stance; it’s deeply embedded within the equations of classical physics, such as Newton’s laws of motion and Maxwell’s equations of electromagnetism, which offer precise, deterministic predictions. Consequently, any apparent randomness was historically attributed to a lack of complete information, rather than inherent unpredictability within the universe itself.

Quantum mechanics fundamentally disrupts the classical notion of a predictable universe by positing that indeterminacy isn’t merely a limitation of measurement, but an intrinsic property of reality itself. Unlike classical physics, where precise initial conditions theoretically allow for perfect future prediction, quantum systems are governed by probability waves described by the Ψ function. This means that even with complete knowledge of a particle’s present state, only the probability of finding it in a particular future state can be known. This isn’t due to incomplete information, but because certain properties, like position and momentum, exist as superpositions – multiple possibilities simultaneously – until measured, at which point the wave function collapses into a single, definite outcome. This inherent randomness, demonstrated through phenomena like radioactive decay and quantum tunneling, suggests the universe isn’t a clockwork mechanism unfolding with absolute certainty, but rather a probabilistic landscape where chance plays a fundamental role.

The persistent discord between deterministic and indeterministic perspectives compels a re-evaluation of how physical systems are represented and understood. Traditionally, models aimed to predict future states with absolute certainty, assuming complete knowledge of present conditions – a cornerstone of classical physics. However, quantum mechanics introduces an inherent probabilistic nature, suggesting that even with perfect knowledge of the present, only the probability of future outcomes can be known. This isn’t merely a matter of incomplete information; it’s a fundamental property of reality at the quantum level. Consequently, scientists are increasingly focused on developing models that incorporate probabilistic predictions, shifting from seeking definitive answers to quantifying uncertainties and exploring the range of possible behaviors. This transition impacts not only theoretical frameworks but also practical applications, from weather forecasting to materials science, demanding a more nuanced and statistically-grounded approach to understanding the physical world.

Empirical Equivalence: The Limits of Ontological Certainty

Empirical equivalence describes the scenario where multiple models, potentially founded on distinct ontological assumptions regarding the underlying nature of reality, yield identical predictions when tested against observable data. This means that despite differences in how these models describe a system – for example, whether they posit deterministic or probabilistic mechanisms – they produce the same results when making predictions about measurable quantities. The phenomenon highlights a fundamental limitation in using empirical observation alone to definitively validate or invalidate specific ontological commitments, as differing theoretical frameworks can be observationally indistinguishable. This isn’t a statement about the ‘truth’ of either model, but rather a demonstration that empirical data can be insufficient to uniquely determine the underlying causal structure of a system.

Research conducted by Werndl, building upon earlier work by Ornstein and Weiss, has established that deterministic and stochastic dynamical systems can yield indistinguishable empirical results. Specifically, these studies demonstrate that under certain conditions, the probabilistic behavior generated by a stochastic model can be perfectly replicated by a deterministic system with sufficient complexity, and vice-versa. This is achieved by mapping random variables in the stochastic model to hidden variables in the deterministic one, effectively mirroring the statistical properties of the stochastic process. Consequently, observational data alone cannot definitively determine whether the underlying dynamics are fundamentally deterministic or inherently stochastic, highlighting a key challenge in model selection and inference.

The observation of empirical equivalence between models with differing ontological commitments presents a significant challenge to model selection. If multiple models accurately predict observable phenomena but posit fundamentally different underlying mechanisms – for example, one deterministic and another stochastic – the justification for preferring one over the other cannot rest solely on predictive power. Standard criteria such as parsimony or computational efficiency may be insufficient, as empirically indistinguishable models can exhibit differing levels of complexity or require varying computational resources. This necessitates consideration of factors beyond empirical adequacy, potentially including theoretical considerations, prior beliefs, or the broader context of the scientific framework within which the models are situated, to rationally justify ontological preference.

Gauge freedom, a concept originating in physics, describes the ability to perform transformations on a system’s parameters without affecting its predicted outcomes. Specifically, these transformations involve redundancies in the mathematical description of a system; altering these redundant elements does not change the observable predictions of the model. This principle extends beyond physics, suggesting that multiple distinct mathematical or conceptual descriptions can accurately represent the same empirical reality. The existence of gauge freedom therefore supports the notion that ontological commitments – the specific entities and relationships posited by a model – are not uniquely determined by observational data; multiple, equally valid, descriptions are possible, differing only in their underlying ontological assumptions.

Mathematical Tools for Describing Dynamical Systems

Hamiltonian Dynamics employs a mathematical formalism based on the total energy of a system, expressed as a function of generalized coordinates and momenta. This approach utilizes $H(q, p, t)$ , the Hamiltonian, to define the time evolution of the system. Central to this framework is Phase Space, a multidimensional space where each point represents a unique state of the system; for a system with $N$ degrees of freedom, Phase Space is $2N$ -dimensional, comprising $N$ position coordinates and $N$ corresponding momenta. The equations of motion, known as Hamilton’s Equations, dictate the trajectory of a point in Phase Space, effectively describing the system’s evolution over time. This provides a complete and deterministic description of the system, given initial conditions.

A Poincaré section is a technique used to analyze the qualitative behavior of continuous dynamical systems. This is achieved by observing the intersections of a trajectory with a chosen hypersurface in the system’s phase space. By restricting the continuous flow to discrete points of intersection, the analysis is simplified, transforming the problem from studying a continuous flow to studying a discrete map. The resulting pattern of points – the Poincaré plot – reveals important properties of the system, such as periodicity, quasi-periodicity, and chaos; for example, a single closed curve in the Poincaré plot indicates periodic motion, while a filled region suggests chaotic behavior. The choice of hypersurface is crucial and often selected to exploit symmetries within the system, further simplifying the analysis and visualization of dynamics.

The Liouville measure, denoted as $d\mu$ , is a fundamental concept in Hamiltonian dynamics that defines a probability measure on phase space. Specifically, it represents the density of states, indicating the proportion of trajectories concentrated in a given region. A key property is its preservation under Hamiltonian flow; meaning that the volume of phase space occupied by a collection of trajectories remains constant over time, even as the shape of the region distorts. Mathematically, this is expressed by the preservation of the integral of a smooth function $f$ over any region of phase space: $\frac{d}{dt} \in t f \, d\mu = 0$ . This conservation is not a statement about the conservation of energy or momentum, but rather a property of the transformation itself, and is crucial for understanding the statistical mechanics of Hamiltonian systems.

The Bernoulli map, defined by the equation $x_{n+1} = r x_n (1 - x_n)$ , demonstrates that chaotic behavior can arise from purely deterministic systems. Even with a fixed parameter $r$ and a single initial condition $x_0$ , the resulting sequence $x_n$ can exhibit extreme sensitivity to that initial condition; infinitesimally small differences in $x_0$ lead to exponentially diverging trajectories. This sensitivity manifests as unpredictable long-term behavior, despite the map being fully defined by a simple, non-random equation. Specifically, for values of $r$ greater than approximately 3.57, the map enters a chaotic regime characterized by seemingly random behavior and positive Lyapunov exponents, quantifying the rate of divergence of nearby trajectories.

Beyond Determinism and Indeterminism: A New Criterion for Understanding

Statistical mechanics, as powerfully demonstrated by Sklar, provides a framework for comprehending physical systems even when those systems inherently exhibit stochastic, or seemingly random, behavior. Rather than demanding a pinpoint, deterministic explanation for every particle’s trajectory, this approach focuses on the collective behavior of vast ensembles, revealing predictable patterns and probabilities. This is particularly crucial when dealing with systems containing an immense number of particles-like gases or liquids-where tracking individual components is not only impractical but fundamentally misses the emergent properties defining the system’s macroscopic state. Sklar’s work highlights that the laws of statistical mechanics don’t necessarily require underlying randomness; they simply offer the most efficient and accurate method for describing systems where complete, deterministic knowledge is inaccessible or irrelevant, effectively shifting the focus from knowing exactly what will happen to predicting how likely different outcomes are.

The rigorous mathematical framework for understanding systems evolving with inherent randomness largely stems from the pioneering work of Van Kampen and Kolmogorov. Kolmogorov, in the 1930s, established a foundational theory for stochastic processes, providing the mathematical tools to describe and analyze phenomena where future states are not entirely determined by past ones – instead governed by probability distributions. Building upon this, Nicholas van Kampen, in the mid-20th century, developed the Kramers-Moyal expansion and the system size expansion, offering powerful techniques to approximate the behavior of stochastic processes, particularly those arising in physics and chemistry. These contributions weren’t simply about describing randomness; they provided methods for modeling it, enabling scientists to predict the statistical properties of systems – like Brownian motion or chemical reaction rates – even when precise deterministic predictions were impossible. Their combined efforts shifted the focus from seeking absolute certainty to embracing probabilistic descriptions, fundamentally changing how complex systems are analyzed and understood.

This work introduces the Model-Invariance Criterion as a method for determining ontological commitment in physics, moving beyond the traditional debate of determinism versus indeterminism. Rather than seeking to definitively label a system as one or the other, the criterion prioritizes identifying features that persist regardless of how a physical system is modeled-specifically, those remaining consistent across empirically equivalent formulations. This approach suggests that the fundamental reality of a system isn’t necessarily defined by whether its underlying equations are deterministic or stochastic, but by the invariant empirical structure revealed through observation and modeling. By focusing on what remains consistent across different, equally valid descriptions, the criterion offers a pathway to understanding the truly fundamental aspects of physical reality, independent of the specific mathematical tools used to represent it.

The seeming opposition between deterministic and indeterministic accounts of physical systems is often less stark than it appears. Investigations into the Thermal Interpretation, a deterministic approach to statistical mechanics, reveal that its predictive power and mathematical formalism can be entirely equivalent to those of explicitly stochastic models. This equivalence doesn’t signify a mere mathematical coincidence; rather, it suggests that the fundamental physics underlying a phenomenon can be realized through multiple, empirically indistinguishable descriptions. Consequently, attributing ontological significance to whether a process is ‘truly’ deterministic or indeterministic becomes problematic, as the observed behavior remains consistent regardless of the chosen framework. The focus, therefore, shifts from identifying a singular ‘correct’ description of cause and effect to recognizing the inherent flexibility in how physical reality manifests within different, yet equally valid, theoretical representations.

Implications for Modeling, Interpretation, and the Future of Physics

The persistent possibility of empirically equivalent models-multiple descriptions that perfectly align with observed data-fundamentally alters how scientists approach model building. Rather than striving for a single, definitive “correct” representation of a phenomenon, a pragmatic shift prioritizes models based on their utility and suitability for specific purposes. This doesn’t imply a surrender to subjectivity, but rather an acknowledgement that any model is, at its core, an approximation tailored to a particular context and set of questions. Consequently, the emphasis moves from seeking ontological truth to maximizing predictive power and facilitating effective manipulation of the system under study; a model is ‘good’ not because it perfectly mirrors reality, but because it reliably delivers actionable insights within its defined scope and limitations. This perspective encourages the development of a diverse toolkit of models, each optimized for a particular task, and promotes a more flexible, adaptable approach to scientific inquiry.

The principle of Effective Theory demonstrates that physical descriptions aren’t absolute truths, but rather approximations valid within specific energy scales and observational contexts. This framework posits that complex systems can be accurately modeled by focusing on the relevant degrees of freedom at a given scale, effectively ‘integrating out’ high-energy details that are inconsequential at lower energies. Consequently, a theory perfectly adequate for describing phenomena at everyday scales might necessitate modification or replacement when probing extremely high energies or small distances. This isn’t a failure of the theory, but rather a confirmation of its limited domain of applicability, highlighting that physical laws are best understood as emergent properties dependent on the observer’s perspective and the energy regime under investigation – a concept fundamentally shifting the focus from seeking a single, all-encompassing ‘theory of everything’ to embracing a multiplicity of effective descriptions.

The concept of backward causation, while initially counterintuitive, serves as a powerful indicator of the limitations inherent in assuming a strictly linear progression of time. Certain theoretical frameworks, particularly those exploring quantum mechanics and advanced concepts in cosmology, suggest scenarios where future events can, in principle, influence the past. This isn’t necessarily a violation of causality, but rather a challenge to its conventional understanding as an arrow pointing solely from past to future. These models propose that time’s directionality may be emergent, rather than fundamental, and that the universe’s behavior isn’t rigidly constrained by initial conditions. Consequently, the notion of a definitive cause preceding its effect becomes less absolute, prompting researchers to consider more complex, potentially non-linear relationships between events and to refine models that accommodate influences traversing temporal boundaries. This exploration doesn’t necessarily prove time travel is possible, but it underscores the need for flexible theoretical constructs that move beyond simplistic, unidirectional timelines.

The development of robust and reliable models across scientific disciplines hinges on a careful consideration of determinism and indeterminism. While classical physics often operates under deterministic principles – where future states are fully determined by present conditions – many complex systems exhibit inherent uncertainty. Ignoring this indeterminacy can lead to models that fail to accurately predict real-world behavior, particularly in fields like quantum mechanics, climate science, and epidemiology. Conversely, embracing randomness without acknowledging underlying deterministic constraints can result in models lacking predictive power or failing to identify meaningful patterns. A truly nuanced approach requires identifying the appropriate balance between these forces, incorporating probabilistic methods where necessary, and recognizing that the very act of modeling introduces a level of abstraction that may obscure the full complexity of the system under investigation. This careful calibration is not merely a philosophical exercise; it directly impacts the validity and utility of scientific predictions and informs decision-making in numerous practical applications.

The paper’s exploration of deterministic and indeterministic models as representational choices echoes a profound sentiment: “You cannot teach a man anything; you can only help him discover it for himself.” This assertion, articulated by Galileo Galilei, highlights the limitations of imposing pre-defined frameworks onto reality. Just as one cannot force understanding, the research demonstrates how the perceived dichotomy between determinism and indeterminism arises from the models used to describe physical phenomena, rather than inherent properties of those phenomena themselves. The pursuit of model-invariant structures, as advocated in the article, mirrors the need to allow observation to guide understanding, rather than forcing reality into preconceived notions. An engineer is responsible not only for system function but its consequences; similarly, a physicist must recognize that the choice of model shapes the perceived ontology.

Beyond Determinism’s Shadow

The assertion that the deterministic/indeterministic divide frequently stems from modeling choices, not fundamental reality, presents a peculiar challenge. It suggests the field has been engaged in elaborate debates about the representation of physics, rather than physics itself. This is not a novel observation, but the emphasis on model-invariance as a pathway toward ontological commitment feels less like a resolution and more like a shifting of the goalposts. What exactly is being optimized here? A mathematically elegant description, divorced from the messy contingencies of measurement and interpretation? The pursuit of model-invariance, while logically sound, risks becoming a purely formal exercise, elegantly sidestepping the genuinely difficult questions about the relationship between theory and existence.

Future work must grapple with the practical implications of this representational view. If indeterminism can be ‘coarse-grained’ away, does that diminish its explanatory power, or simply highlight the limitations of a particular modeling approach? The paper rightly points to gauge freedom, but the sheer proliferation of possible representational choices raises a disturbing prospect: an ontology so pliable it becomes effectively meaningless. Algorithmic bias is, after all, a mirror of values; here, the ‘values’ are mathematical convenience and formal consistency.

Transparency, then, is the minimum viable morality. Simply demonstrating model-invariance is insufficient. The choices made in constructing those models – the assumptions baked into the formalism, the parameters deemed significant – require rigorous scrutiny. The next step isn’t simply to find model-invariant structures, but to understand how those structures are constructed, and for whose benefit.

Original article: https://arxiv.org/pdf/2512.22540.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/