arXiv subjects

Jeffrey A. Barrett

Publications and source records attributed to Jeffrey A. Barrett.

7 recordsLinked to original sources

arXiv2026

Algorithmic Randomness and Physical Typicality

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.

quant-ph↗

arXiv2026

Algorithmic Randomness, Exchangeability, and the Principal Principle

We introduce a framework uniting algorithmic randomness with exchangeable credences to address foundational questions in philosophy of probability and philosophy of science. To demonstrate its power, we show how one might use the framework to derive the Principal Principle -- the norm that rational credence should match known objective chance -- without circularity. The derivation brings together de Finetti's exchangeability, Martin-Löf randomness, Lewis's and Skyrms's chance-credence norms, and statistical constraining laws (arXiv:2303.01411). Laws that constrain histories to algorithmically random sequences naturally pair with exchangeable credences encoding inductive symmetries. Using the de Finetti representation theorem, we show that this pairing directly entails the Principal Principle of this framework. We extend the proof to partial exchangeability and provide finite-history bounds that vanish in the infinite limit. The Principal Principle thus emerges as a mathematical consequence of the alignment between nomological constraints and inductive learning. This reveals how algorithmic randomness and exchangeability can illuminate foundational questions about chance, frequency, and rational belief.

physics.hist-ph↗

arXiv2026

The Distribution Postulate in Algorithmic Bohmian Mechanics

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.

quant-ph↗

arXiv2025

Algorithmic Randomness and Probabilistic Laws

We apply recent ideas about complexity and randomness to the philosophy of laws and chances. We develop two ways to use algorithmic randomness to characterize probabilistic laws of nature. The first, a generative chance* law, employs a nonstandard notion of chance. The second, a probabilistic* constraining law, impose relative frequency and randomness constraints that every physically possible world must satisfy. The constraining notion removes a major obstacle to a unified governing account of non-Humean laws, on which laws govern by constraining physical possibilities; it also provides independently motivated solutions to familiar problems for the Humean best-system account (the Big Bad Bug and the zero-fit problem). On either approach, probabilistic laws are tied more tightly to corresponding sets of possible worlds: some histories permitted by traditional probabilistic laws are now ruled out as physically impossible. Consequently, the framework avoids one variety of empirical underdetermination while bringing to light others that are typically overlooked.

physics.hist-ph↗

arXiv2019

Typical Worlds

Hugh Everett III presented pure wave mechanics, sometimes referred to as the many-worlds interpretation, as a solution to the quantum measurement problem. While pure wave mechanics is an objectively deterministic physical theory with no probabilities, Everett sought to show how the theory might be understood as making the standard quantum statistical predictions as appearances to observers who were themselves described by the theory. We will consider his argument and how it depends on a particular notion of branch typicality. We will also consider responses to Everett and the relationship between typicality and probability. The suggestion will be that pure wave mechanics requires a number of significant auxiliary assumptions in order to make anything like the standard quantum predictions.

quant-ph↗

arXiv2005

Stability and Paradox in Algorithmic Logic

Type-free systems of logic are designed to consistently handle significant instances of self-reference. Some consistent type-free systems also have the feature of allowing the sort of general abstraction or comprehension principle that infamously leads to paradox in naive set theory. Because type-free systems possess these features, and avoid the hierarchy of types that is felt to be unnatural in some contexts, they have the potential to play an important role in the foundations of mathematics, the theory of classes (producing a richer notion of class than that currently used in set theory and category theory), property theory, natural language semantics, the theory of truth, and theoretical computer science. Clearly, type-free systems must depart from classical logic in some way, but there is little agreement on what kind of type-free system to use, and which departures from classical logic should be allowed. Our approach to type-free logic is to study a naturally occurring type-free system that we believe is in some sense prototypical of systems that will ultimately prove useful. The logic studied in this paper, called algorithmic logic, concerns certain basic statements involving algorithms and the algorithmic rules of inference between such statements. This paper studies the propositional properties of algorithmic logic. A future paper will show that algorithmic logic possesses a general abstraction principle.

math.LO↗

arXiv2000

The Persistence of Memory: Surreal Trajectories in Bohm's Theory

In this paper I describe the history of the surreal trajectories problem and argue that in fact it is not a problem for Bohm's theory. More specifically, I argue that one can take the particle trajectories predicted by Bohm's theory to be the actual trajectories that particles follow and that there is no reason to suppose that good particle detectors are somehow fooled in the context of the surreal trajectories experiments. Rather than showing that Bohm's theory predicts the wrong particle trajectories or that it somehow prevents one from making reliable measurements, such experiments ultimately reveal the special role played by position and the fundamental incompatibility between Bohm's theory and relativity.

quant-ph↗