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Philosophy of Physics

Non-Conglomerable Probability in Cosmology

Sylvia Wenmackers

Published Jan 1, 2026• 0 citations
Publisher

Abstract

In the search for an explanation of the thermodynamic arrow of time in our universe, cosmologists have proposed central-time models. These recent, speculative models run into a probabilistic paradox that involves a failure of conglomerability. We analyze the non-conglomerability of these models with the help of standard probability theory, as well as three alternative formalisms: finitely additive probability, non-Archimedean probability (NAP), and non-normalized quasi-probability. We find that all formalisms exhibit failures of conglomerability on infinite partitions when applied to cosmological models that invoke non-normalizable measures, such as uniform measures on infinite phase spaces. NAP theory, however, does assure conglomerability on partitions of arbitrary cardinality that meet an additional condition. In any case, failures of conglomerability need not be intrinsically problematic for the development of central-time models.

DOI
10.31389/pop.128

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In the search for an explanation of the thermodynamic arrow of time in our universe, cosmologists have proposed central-time models. These recent, speculative models run into a probabilistic paradox that involves a failure of conglomerability. We analyze the non-conglomerability of these models with the help of standard probability theory, as well as three alternative formalisms: finitely additive probability, non-Archimedean probability (NAP), and non-normalized quasi-probability. We find that all formalisms exhibit failures of conglomerability on infinite partitions when applied to cosmological models that invoke non-normalizable measures, such as uniform measures on infinite phase spaces. NAP theory, however, does assure conglomerability on partitions of arbitrary cardinality that meet an additional condition. In any case, failures of conglomerability need not be intrinsically problematic for the development of central-time models.
Quantum cosmology is the quantum theory of the entire universe. Although strange at first sight, it is appropriate because (1) our world appears to be fundamentally quantum, (2) the classical description of gravity breaks down at singularities it would give at the beginning of the universe, and (3) our universe has many properties that cannot be explained by a classical description of them. A quantum state of the universe should obey certain constraints, such as the Wheeler-DeWitt equations. One approach to interpreting such a state is {\it Sensible Quantum Mechanics} (SQM), in which nothing is probabilistic, except, in a certain frequency sense, conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued {\it awareness operators}. These may be defined even when the quantum state itself is not normalizable, though there still seem to be problems of divergences when one has an infinite amount of inflation, producing infinitely large spatial volumes with presumably infinite measures of perceptions. Ratios of the measures for sets of perceptions (if finite) can be interpreted as frequency-type probabilities for many actually existing sets rather than as propensities for potentialities to be actualized, so there is nothing indeterministic in a specific SQM. One can do a Bayesian analysis to test between different specific SQMs. One can also make statistical predictions of what one might perceive within a specific SQM by invoking the {\it Conditional Aesthemic Principle}: among the set of all conscious perceptions, our perceptions are likely to be typical.
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It is often assumed that the complete physics theory of the universe is computable - in sense that it can provide meaningful theoretical predictions for every phenomenon of the universe. Against this view, it is argued that once quantum mechanics is understood as encompassing a novel concept of probability, thereby resolving the measurement problem in straightforward ways, the interpretation speaks for uncomputability of the complete physics theory. The reasons why a new theory of probability is needed are explored, along the lines of the principle of indifference and the sleeping beauty problem.