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American Journal of Physics

Probability and Determinism

Miriam Lipschütz-Yevick

Published Nov 1, 1957• 3 citations
Publisher

Abstract

Various physical examples are displayed in which the probability distribution can be traced back to its causal origin. It is argued that probability is never irreducible, but is, in principle, always derivable from a deterministic process which determines its numerical value. It is suggested that deterministic and probabilistic theories are interlocked, each presenting a different aspect of reality, yet that no description of the laws of nature in terms of either of these concepts will ever be ultimate.

DOI
10.1119/1.1934551

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Various physical examples are displayed in which the probability distribution can be traced back to its causal origin. It is argued that probability is never irreducible, but is, in principle, always derivable from a deterministic process which determines its numerical value. It is suggested that deterministic and probabilistic theories are interlocked, each presenting a different aspect of reality, yet that no description of the laws of nature in terms of either of these concepts will ever be ultimate.
In this chapter, we consider the way quantum theory represents deterministic, statistical, and probabilistic aspects of Nature. After a discussion in Section 3.1 of uncertainty in general, we discuss in Sections 3.2-3.4 several formal notions of classical probability, their relation to the probability concept used in applied statistics, and their dependence on the description used. Section 3.5 briefly discusses various examples, where probabilistic features emerge from fully deterministic situations-a theme to be taken up in much more detail in Chapter 11. Section 3.6 then shows how the statistical aspects of the quantum formalism naturally follow from the weak law of large numbers. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined in Section 3.7. Historically, the concept of classical probability (including its use in stochastic processes) was given an undisputed formal mathematical foundation in 1933 in terms of the measure-theoretic setting of Kolmogorov [164]. Apart from this traditional axiomatic foundation of classical probability theory, there is a less well-known equivalent axiomatic treatment by Whittle [302] in terms of expectations. Here probabilities appear as the expectations of statements, {0, 1}-valued random variables. von Plato [240] discusses the history of the concept of probability. Krüger et al. [169] discuss the history of probability in the various fields of application. Sklar [275] discusses the philosophical problems of the probability concept, with an emphasis on statistical mechanics.
This paper argues that such strictly deterministic formulations rely on idealizations that obscure an essential physical feature: the objective growth of information associated with irreversible processes. Drawing on thermodynamics, information theory, and the foundations of quantum mechanics, the paper is based on the notion of ontic entropy as a measure of the objective informational content instantiated by physical systems. This paper proposes a classification of determinism based on the informational characteristics of physical evolution. By distinguishing between strict causal determinism, probabilistic determinism, and chaotic determinism, it becomes possible to analyze causal structure without conflating necessity with lawfulness or probability with indeterminism.
On a very intuitive way of thinking, if it is already determined that some event will happen, then there is no non-trivial chance of it failing to happen, and if it is already determined that some event will not happen, then there is no non-trivial chance of it happening. A different sort of worry that arises for an actual frequentist analysis of deterministic chance is that such an analysis seems unsuited to theories like Boltzmannian statistical mechanics or Bohmian mechanics insofar as those theories are supposed to describe the evolution of the Universe as a whole. It should be clear, then, that there are options available for understanding deterministic chance that are in keeping with several of the leading metaphysical analyses of chance. It is widely accepted that whatever chances are, they are to be contrasted with individual credences . According to classical statistical mechanics, the fundamental laws are Newtonian and thus deterministic.
Time and reality are central concepts in both philosophy and physics. While classical mechanics describes the universe as deterministic, where the present emerges from past conditions, quantum theory introduces probabilistic interpretations suggesting inherent uncertainty at microscopic scales. These two perspectives have led to an ongoing debate regarding whether reality is fundamentally deterministic, probabilistic, or a combination of both. The Zia Theory of Temporal Reality (ZTTR) integrates these frameworks by proposing that the present state of reality is determined by accumulated past causes, while the future evolves probabilistically under uncertainty. To formalize this concept, the Zia Unified Continuity Equation (ZUCE) is expressed as a stochastic model where the state of reality represents a combination of initial conditions, integrated causal influences, and stochastic fluctuations represented by a Wiener process. To enhance generality and address limitations of fixed functional assumptions, the uncertainty term is further extended as a generalized function, allowing different forms such as diffusion-like, logarithmic, or nonlinear growth depending on system dynamics. In this framework, deterministic causal accumulation represents the structured continuity of past influences, while the stochastic component reflects uncertainty arising from complexity, interaction, and incomplete knowledge of system variables. This interpretation aligns with developments in statistical mechanics, information theory, and complex systems research. The applicability of the model is illustrated through conceptual and numerical examples drawn from physical processes, biological systems, and learning dynamics. These examples demonstrate that deterministic historical structures and stochastic variations can coexist within a single temporal framework. The Zia Theory of Temporal Reality thus provides a generalized and mathematically consistent perspective on temporal evolution and contributes to interdisciplinary research in physics, complexity science, and artificial intelligence.
It is argued that quantum mechanics follows naturally from the assumptions that there are no fundamental causal laws but only probabilities for physical processes that are constrained by symmetries, and reality is relational in the sense that an object is real only in relation to another object that it is interacting with. The first assumption makes it natural to include in the action for a gauge theory all terms that are allowed by the symmetries, enabling cancellation of infinities, with only the terms in the standard model observable at the energies at which we presently do our experiments. In this approach, it is also natural to have an infinite number of fundamental interactions.