The Mathematics of Will: Modeling Self-Determined Action

Author: Denis Avetisyan

A new mathematical framework proposes how agency can arise from physical systems without violating deterministic laws.

This review formalizes a ‘dual-laws’ model employing supervenient causation and self-referential feedback to address the philosophical problem of agent determinism.

Defining agency remains a fundamental challenge, particularly reconciling intentional action with the deterministic framework of physics. In ‘A Mathematical Formalization of Self-Determining Agency’, we address this paradox by formalizing a model wherein higher-level dynamics-governed by independent laws-can exert causal influence on lower-level physical processes without violating determinism. This is achieved through a novel concept of ‘supervenient causation’, enabling a dual-laws system where agent-level determination is not merely epiphenomenal. Could this framework offer a mathematically rigorous foundation for understanding self-determination and responsibility in both natural and artificial agents?

The Limits of Conventional Causation

Conventional causal models frequently encounter limitations when applied to systems exhibiting top-down causation, where emergent properties at higher levels of organization exert demonstrable influence on components at lower levels. These models typically prioritize a bottom-up approach, tracing effects from initial conditions to outcomes, and struggle to adequately represent scenarios where overarching constraints or intentional states shape fundamental processes. For instance, the cognitive state of an organism – a high-level phenomenon – can directly alter neural firing patterns – a lower-level process – demonstrating that causation isn’t always a linear progression from base elements. This disconnect proves particularly problematic in fields like biology, where organismal goals impact gene expression, and in social sciences, where cultural norms shape individual behavior, highlighting the need for causal frameworks that accommodate influences originating from system-wide properties rather than solely from constituent parts.

Conventional models of causation frequently dissect processes into linear sequences of events – one action directly triggering another – yet this simplification overlooks the significant influence of agency and intentionality. These traditional approaches struggle to accommodate scenarios where purpose or conscious direction actively shapes outcomes, particularly in systems exhibiting goal-oriented behavior. Consider a biological organism; its actions aren’t merely a chain reaction of biochemical events, but are directed by complex regulatory networks responding to internal goals and external stimuli. Similarly, in social systems, the intentions of individuals and groups demonstrably influence collective behaviors, a dynamic absent from purely mechanistic explanations. By prioritizing simple event chains, these models risk portraying complex systems as passive and deterministic, failing to capture the crucial role of proactive forces in driving change and shaping reality.

A fundamental challenge in understanding intricate systems arises from the limitations of strictly bottom-up causal explanations. While conventional models excel at detailing how individual components interact to produce emergent properties, they often fail to adequately address instances of top-down causation, where higher-level phenomena exert genuine influence on lower-level processes. This is particularly evident in biological organisms, where an organism’s goals and overall state can directly impact gene expression and cellular behavior, and within social structures, where cultural norms and collective intentions shape individual actions. The inability to fully integrate these reciprocal influences creates a significant gap in the comprehensive modeling of complex systems, hindering advancements in fields ranging from medicine and ecology to economics and sociology. Recognizing and accounting for top-down causation is therefore crucial for developing more accurate and predictive frameworks for understanding the world.

Supervenience and Dual-Laws Systems: A Necessary Shift

Supervenience describes a hierarchical relationship between properties where a higher-level property is dependent upon, but not reducible to, lower-level properties. This means changes at the lower level can potentially cause changes at the higher level, but changes at the higher level do not necessarily entail specific changes at the lower level; the same lower-level state can, in principle, support multiple higher-level states. Crucially, supervenience does not imply that the higher-level property is simply a logical consequence of the lower-level properties, nor that it is merely an illusion; instead, it establishes a dependency relation where the higher-level properties are realized in, or emerge from, the lower-level structure. A change at the higher level necessitates some change at the lower level, but the exact nature of that lower-level change is not uniquely determined by the higher-level change.

A Dual-Laws Model posits that systems are subject to laws operating at both the level of subvenience – governing the underlying physical mechanisms – and the level of supervenience – governing the higher-level emergent phenomena. This contrasts with reductionist approaches by allowing for genuine top-down causation, where supervenient laws can influence the behavior of subvenient components. The mathematical formalization presented in this paper demonstrates this bi-directional causality through a system of coupled equations, explicitly modeling how variables at the supervenient level constrain and modify the dynamics of the subvenient level, and vice versa. This allows for the possibility of constraints imposed by higher-level rules on lower-level physical processes, rather than solely bottom-up determination.

The Dual-Laws Model rejects the notion of epiphenomenalism, where higher-level properties are considered mere byproducts of lower-level processes without causal efficacy. Instead, this framework posits that supervenient phenomena-those arising from but not reducible to lower-level components-exert genuine downward causation. This means that higher-level states and properties can directly influence the behavior of the underlying components that give rise to them. This interaction isn’t simply descriptive; the model formally allows for alterations in lower-level dynamics predicated on the state of the higher-level system, indicating a bidirectional causal relationship and a departure from purely reductionist views of complex systems.

Feedback Control and Self-Modification: The Architecture of Adaptation

Feedback control mechanisms establish a directed, hierarchical relationship where signals representing system outcomes are transmitted to control centers operating at a higher level of abstraction. These control centers then modulate the system’s parameters or processes, effectively altering its future behavior. This process relies on quantifiable metrics representing the desired state or performance; deviations from these metrics generate error signals. The magnitude and polarity of these error signals dictate the corrective action applied to the lower-level components. Consequently, the system adjusts its operation to minimize the error, demonstrating a capacity to adapt based on observed results and maintain stability or achieve specific goals. This implementation of causal influence is distinct from simple stimulus-response interactions, as it involves evaluation of outcomes and iterative refinement of internal states.

Self-referential feedback control extends standard feedback mechanisms by enabling a system to alter its own internal operational parameters based on observed outcomes. This differs from typical feedback where adjustments affect only output; self-referential control directly modifies the system’s governing rules or internal structure. Consequently, the system doesn’t simply react to external stimuli, but actively changes how it responds, leading to a demonstrable degree of autonomy in its behavior. This internal modification can involve altering weighting factors, connection strengths, or even the algorithms used for processing information, allowing the system to adapt and optimize its performance over time without external intervention.

Index sequence dynamics, as implemented through algebraic expressions, provide a formalized method for modeling internal adjustments within self-modifying systems. These dynamics utilize indexed variables within equations to represent system states and their evolution; the values of these variables are not static but are updated based on the outcomes of previous states, effectively creating a feedback loop. $x_{t+1} = f(x_t, p)$ represents this process, where $x_t$ is the state at time t, p represents adjustable parameters, and f is a function defining the state transition. By manipulating the parameters p based on observed system performance – itself represented within the algebraic framework – the system achieves self-regulation, altering its behavior without external intervention. This mathematical representation allows for precise analysis of system stability, responsiveness, and the potential for complex adaptive behavior.

Agency, Intentionality, and Causal Power: Reconciling Levels of Explanation

The concept of supervenient causation offers a compelling account of how agency functions within a physically determined universe. It posits that higher-level mental states – intentions, beliefs, and desires – don’t simply correlate with lower-level physical processes, but can genuinely exert causal influence upon them. This isn’t a matter of mental states overriding physical laws, but rather a nuanced relationship where the mental, existing at a supervenient level of description, constrains and shapes the realization of physical events. For instance, a decision to raise one’s arm isn’t merely explained by neuronal firings; the intentional state of wanting to raise an arm plays a causal role in how those firings occur, selecting from a range of physically possible outcomes. This framework allows for meaningful agency, acknowledging physical determinism while preserving the idea that an agent’s mental states can be genuine forces in bringing about action and change.

Intentional causation proposes that an agent’s desires and beliefs aren’t simply correlated with actions, but actually constitute their cause. This framework diverges from purely physical explanations by asserting that an individual’s mental states-what they want and what they believe to be true-can genuinely bring about changes in the world. Rather than reducing action to a chain of neurological events, intentional causation posits that these mental states function as reasons for the action, providing a level of explanation distinct from, and potentially irreducible to, physical mechanisms. This doesn’t negate the role of physical processes; instead, it suggests that these processes are often mediated by intentional states, allowing for a richer understanding of agency and the capacity to act for reasons. The power to act based on beliefs and desires distinguishes intentional systems from objects moved solely by external forces, offering a compelling account of how agents shape their environments.

Agent determinism proposes that even within a framework of universal laws – those governing all physical systems – agents possess a unique capacity for self-determination. This isn’t a rejection of causality, but rather a refinement of it; an agent’s actions are still caused, but those causes originate from internal states – desires, beliefs, and intentions – rather than solely from external forces. This internal origination is the crucial distinction from purely mechanical systems, like a falling domino, where every movement is wholly dictated by prior physical events. The framework suggests that while an agent’s choices are determined, they are determined by the agent’s own psychological states, allowing for a meaningful sense of autonomy and responsibility that wouldn’t exist if actions were simply the inevitable result of prior physical configurations. This nuanced view navigates the apparent conflict between determinism and the intuitive feeling of free will, proposing that the two are not mutually exclusive.

Towards a Mechanistic Understanding of Causation: A Path Forward

A comprehensive approach to understanding causation hinges on identifying the specific pathways through which one variable influences another – a process facilitated by Causal Mechanism Models. These models move beyond simply noting that two variables correlate, instead detailing how a change in one leads to a change in the other, outlining the intermediate steps and underlying processes. When coupled with Statistical Relevance Models, this framework gains further power, allowing researchers to quantify the strength of these causal pathways within complex systems. This combination doesn’t just establish that a relationship exists, but assesses its relative importance compared to other potential influences, offering a nuanced understanding of interconnectedness. By explicitly mapping these mechanisms and assigning statistical weight to each, scientists can move closer to predicting system behavior and intervening effectively – a critical step in fields ranging from epidemiology to engineering.

Traditional scientific inquiry often identifies correlations – that two variables change together – but establishing whether one causes the other requires a more rigorous approach. This methodology moves beyond such limitations by dissecting complex systems to reveal the underlying mechanisms driving observed relationships. Instead of simply noting that A and B occur together, this framework seeks to demonstrate how changes in A systematically lead to changes in B, potentially through a chain of intermediate variables operating across multiple levels of organization – from molecular interactions to behavioral patterns. This allows researchers to differentiate spurious correlations from genuine causal links, offering a more accurate and predictive understanding of how systems function and respond to interventions, ultimately moving beyond description towards explanation.

The integration of Causal Mechanism Models and Statistical Relevance Models, as detailed in this work, offers a pathway towards resolving longstanding ambiguities in defining causation within complex systems. This combined approach doesn’t merely identify that a relationship exists, but elucidates how a causal effect propagates through a system, mapping the specific mechanisms and relevant statistical factors involved. By moving beyond correlational analysis, researchers can begin to dissect the intricate web of interactions governing phenomena across diverse fields-from biological processes and social dynamics to technological networks-and develop a more robust and nuanced understanding of agency itself. The formalized framework presented allows for the quantification of causal pathways, providing a means to predict system behavior, intervene effectively, and ultimately, gain deeper insights into the fundamental nature of complexity.

The pursuit of formally defining agency, as demonstrated in this work, necessitates a compromise between the elegance of fundamental physical laws and the convenient, if messy, reality of emergent causation. The authors propose a dual-laws model, attempting to reconcile deterministic lower-level processes with the apparent freedom of action at a higher, supervenient level. As Albert Einstein observed, “The most incomprehensible thing about the world is that it is comprehensible.” This sentiment resonates with the challenge undertaken here: to build a mathematical framework that captures the subjective experience of agency within the constraints of a fundamentally deterministic universe. The model, while complex, attempts to map the contours of this comprehension, acknowledging that any formalization will inevitably be an approximation of the phenomenon it seeks to explain – a simplification for the sake of understanding, rather than a perfect reflection of reality.

What Remains to be Seen?

The formalization presented here doesn’t, of course, solve the problem of agency. It merely relocates it – from the murky domain of philosophical debate to the slightly more tractable, if equally frustrating, realm of mathematical consistency. The model’s dependence on ‘coarse-graining’ parameters, for instance, feels less like a fundamental insight and more like a confession. It admits that the very scale at which agency manifests is, conveniently, where the mathematics becomes manageable. One suspects that any sufficiently complex system, given enough adjustable knobs, could appear self-determining.

Future work will inevitably focus on the empirical implications – or, more likely, the lack thereof. Establishing whether these ‘supervenient’ laws leave any detectable signature on lower-level physical processes is a challenge bordering on the metaphysical. It isn’t a search for evidence of agency, precisely, but a test of whether a particular mathematical abstraction maps, even loosely, to reality. The truly interesting failures, however, may lie in identifying precisely where the model breaks down.

Perhaps the most humbling prospect is the realization that this entire endeavor might be a sophisticated exercise in rationalization. Not of variance, but of the enduring human impulse to believe in something more than deterministic clockwork. The persistence of the illusion, after all, doesn’t require a mechanism – only a sufficiently convincing story.

Original article: https://arxiv.org/pdf/2601.02885.pdf

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