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arXiv:2610.23355

Randomness from initial conditions: The challenge of incompleteness

Klaas Landsman

Published Oct 16, 2026• 0 citations
arXivHTMLPDF

Abstract

Certain prima facie deterministic theories, notably Bohmian mechanics and kinetic theory, require ``random'' initial conditions in order to achieve their goals. I analyze recent proposals that this idea should be implemented using algorithmic randomness, seen as a mathematical sharpening of the typicality arguments originally used to understand these theories. On the one hand my analysis includes a critique of these arguments and hence a defense of this sharpening. On the other hand, I argue that Chaitin's incompleteness theorems for random sequences make it impossible for a dynamical theory to either process given random initial conditions beyond some finite (but not arbitrary) number of their digits, or else to state such conditions with arbitrary precision without compromising their randomness (these are two sides of the same coin). This challenges the ability of such theories to make any other predictions than those that hold typically in the usual sense of the word (like macroscopic statements), and moves their alleged determinism high up the metaphysical ladder.

DOI
10.48550/arxiv.2610.23355

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Certain prima facie deterministic theories, notably Bohmian mechanics and kinetic theory, require ``random'' initial conditions in order to achieve their goals. I analyze recent proposals that this idea should be implemented using algorithmic randomness, seen as a mathematical sharpening of the typicality arguments originally used to understand these theories. On the one hand my analysis includes a critique of these arguments and hence a defense of this sharpening. On the other hand, I argue that Chaitin's incompleteness theorems for random sequences make it impossible for a dynamical theory to either process given random initial conditions beyond some finite (but not arbitrary) number of their digits, or else to state such conditions with arbitrary precision without compromising their randomness (these are two sides of the same coin). This challenges the ability of such theories to make any other predictions than those that hold typically in the usual sense of the word (like macroscopic statements), and moves their alleged determinism high up the metaphysical ladder.
Abstract I argue that Bohmian mechanics (or any similar pilot-wave theory) cannot reasonably be claimed to be a deterministic theory. If one assumes the “quantum equilibrium distribution” provided by the wave function of the universe, Bohmian mechanics requires an external random oracle in order to describe the (Kolmogorov–Levin–Chaitin) algorithmic randomness properties of typical outcome sequences of long runs of repeated identical experiments (which provably follow from the Born rule). This oracle lies beyond the scope of Bohmian mechanics (or any deterministic extension thereof), including the impossibility of explaining the randomness property in question from “random” initial conditions. Thus the advantages of Bohmian mechanics over other interpretations of quantum mechanics, if any, must lie at an ontological level, and in its potential to derive the quantum equilibrium distribution and hence the Born rule.
Appeals to typicality are common in physics, but it is often unclear what it means for a physical state to be typical relative to a probability measure, and correspondingly unclear what a law that appeals to typicality asserts. Here we consider how one might characterize physical typicality using ideas from the theory of algorithmic randomness. As a concrete example, we show how taking a physical state to be typical relative to a computable measure when it is Martin-Löf random allows one to formulate the distribution postulate in Bohmian mechanics as a statistical constraining law of the theory. Using a toy model, we show how this constraint guarantees the standard Born statistics for computable experimental protocols. Algorithmic Bohmian mechanics (aBM) thus illustrates how algorithmic randomness may be used to provide precise content to a statistical law.
In order to make the right empirical predictions Bohmian mechanics requires a special statistical boundary condition -- the distribution postulate -- but it is unclear how best to understand this condition. We show how one might use the theory of algorithmic randomness to formulate the distribution postulate as an objective constraining law. The framework requires us to say something about admissible quantum-mechanical states and measurements. In return, algorithmic Bohmian mechanics (aBM) guarantees the standard Born statistics for a collection of canonical quantum experiments in the limit, not just with high probability. The algorithmic distribution postulate provides a sharp typicality condition, clarifies the status of quantum probabilities in the deterministic theory, and provides a concrete example of how notions provided by the theory of algorithmic randomness can aid in specifying the content of a physical law.
This paper deals with the notion of objective randomness in classical deterministic theories. After the introduction, section 2 establishes an important distinction between a strictly metaphysical thesis of determinism (as characterized in the Montague-Earman definition, for example), and the doctrine of determinism, which can be roughly characterized as a methodological set of principles. The doctrine of determinism is associated with the idea that probability assignments can only reflect our ignorance of facts, and it also grounds the (ontological) thesis of separability: A system or process can be characterized completely in terms of the properties that a system has when in a given state, independently of the properties of other systems, The key notions of "completeness" and "independence" are only briefly discussed, as they are examined more in depth elsewhere. Section three examines attempts to characterize a notion of objective randomness in ergodic theory. The characterization can be seen to be equivalent to the formulation of a notion of a "physically impossible process". One way of expressing this idea is grounded on the thesis of the "coarse graining" of our measuring instruments. This leads to the problem of distinguishing "objective" from "apparent" randomness. The problem seems to be intrinsic to any attempt of characterizing the required notion of physical impossibility (and thus of objective randomness) in terms of an ideal observer. The alternative of trying to characterize physical impossibility in terms of a theory of algorithms suffers from the difficulty that it is not clear what would be the required notion of (non-epistemic) computability. The "coarse graining" approach, as well as the alternative in terms of a theory of algorithms seem to share the usual confusion between a strictly metaphysical thesis of determinism and the (methodological) doctrine of determinism. In section four an alternative approach is suggested. It is noticed that a denial of the thesis of separability is compatible with a strict deterministic theory, and thus that at least some classical systems (the "statistical" ones) can be described in terms of non-separable states. We do not have to think that the impossibility of preparing a classical state is too only possible explanation for physical (objective) randomness in deterministic theories. This proposal is an elaboration of an idea of Blatt (1959), although Blatt was still trying to understand objective randomness within an ideal-observer framework.[S.M.]
Abstract It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality.
A long sequence of tosses of a classical coin produces an apparently random bit string, but classical randomness is an illusion: the algorithmic information content of a classically-generated bit string lies almost entirely in the description of initial conditions. This letter presents a simple argument that, by contrast, a sequence of bits produced by tossing a quantum coin is, almost certainly, genuinely (algorithmically) random. This result can be interpreted as a strengthening of Bell's no-hidden-variables theorem, and relies on causality and quantum entanglement in a manner similar to Bell's original argument.