Indeterminism in Classical Mechanics: Re-evaluating the Role of Real Numbers
The paper "Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?" by Nicolas Gisin addresses a fundamental assumption in classical mechanics regarding the use of real numbers to represent initial conditions. This assumption has profound implications for determinism in physics and the interpretation of both classical and quantum mechanics.
Overview and Central Argument
The paper questions the conventional use of real numbers in physics to define initial conditions for dynamical systems. Real numbers possess an infinite amount of information, which is not physically representable within a finite space. Therefore, Gisin suggests that real numbers should not be considered genuinely real in the physical sense. Instead, he proposes that the term "random numbers" might better describe such numbers due to their infinite and random bit sequences.
Gisin introduces an alternative form of classical mechanics that utilizes finite-information numbers, as opposed to real numbers. This alternative theory maintains the same predictive power as conventional classical mechanics but is inherently non-deterministic. The argument rests on the premise that since a finite volume cannot sustain infinite information, the randomness observed in chaotic classical systems is intrinsic, not merely apparent.
Implications for Determinism and Alternative Theories
The implications of Gisin's arguments challenge the deterministic view traditionally held in physics, particularly in classical mechanics. Classical mechanics, when expressed with infinite-information real numbers, is deterministic in its equations and predictions. However, the proposed alternative using finite-information numbers suggests that indeterminism is more naturally aligned with physical reality, as it limits the system's description to quantities that actually exist within the constraints of finite information.
Gisin also explores the conceptual parallels between deterministic interpretations of classical mechanics and quantum mechanics, notably through Bohmian mechanics. In both classical and quantum domains, determinism can be restored by introducing supplementary variables—such as real numbers for initial conditions in classical systems or hidden variables in Bohmian mechanics. The distinction he emphasizes is that the choice to adopt these supplementary variables is an ontological addition rather than an empirical necessity.
Future Directions and Theoretical Considerations
The paper posits future investigations into the nature of information and its physical limitations as a field promising rich conceptual and practical developments. It proposes that classical and quantum theories could be comprehensively framed within a non-deterministic paradigm, where the emphasis on embedding supplementary variables to ensure determinism is re-evaluated.
Gisin's proposition that the completeness of classical mechanics does not necessitate determinism opens further dialogue on the roles of computation and simulation in physical theory. Moreover, the exploration of finite-information frameworks could lead to a re-examination of how predictions in chaotic systems are approached, potentially influencing fields such as statistical mechanics and information theory.
Conclusion
Gisin's challenging views on the physical irrelevance of real numbers in classical mechanics question the philosophical and methodological approaches to determinism in physics. As he argues, the reconsideration of infinite information in the structure of classical and quantum theories might provide new insights into the understanding of randomness and the distinction between potentiality and actuality. This paper invites experts in the field to re-evaluate deeply-held assumptions about the fabric of physical reality and the tools used to describe it, paving the way for novel interpretations and methodologies in theoretical physics.