Uniform probability in cosmology (2308.12229v1)

Abstract: Problems with uniform probabilities on an infinite support show up in contemporary cosmology. This paper focuses on the context of inflation theory, where it complicates the assignment of a probability measure over pocket universes. The measure problem in cosmology, whereby it seems impossible to pick out a uniquely well-motivated measure, is associated with a paradox that occurs in standard probability theory and crucially involves uniformity on an infinite sample space. This problem has been discussed by physicists, albeit without reference to earlier work on this topic. The aim of this article is both to introduce philosophers of probability to these recent discussions in cosmology and to familiarize physicists and philosophers working on cosmology with relevant foundational work by Kolmogorov, de Finetti, Jaynes, and other probabilists. As such, the main goal is not to solve the measure problem, but to clarify the exact origin of some of the current obstacles. The analysis of the assumptions going into the paradox indicates that there exist multiple ways of dealing consistently with uniform probabilities on infinite sample spaces. Taking a pluralist stance towards the mathematical methods used in cosmology shows there is some room for progress with assigning probabilities in cosmological theories.

• The paper demonstrates how alternative probabilistic frameworks, including finitely additive and non-Archimedean methods, overcome inconsistencies in applying uniform probability to infinite cosmological models.
• It compares traditional Kolmogorov axioms with revised methodologies to effectively address the normalization and additivity issues in eternal inflation scenarios.
• The analysis advocates a pluralistic, interdisciplinary approach that bridges historical debates and contemporary insights, guiding future research in cosmological probability measures.

An Essay on "Uniform Probability in Cosmology"

Sylvia Wenmackers' paper, "Uniform Probability in Cosmology," addresses the substantive challenges and theoretical implications of applying uniform probability measures in cosmological contexts, specifically within inflation theory. This paper endeavors to bridge gaps between foundational probability theory and its applications in cosmological measure problems, invoking historical and contemporary discussions from both philosophical and physical perspectives.

The principal concern in the paper revolves around the "measure problem" in cosmology. This issue manifests due to difficulties in assigning probabilities to events in the infinite sample spaces characteristic of cosmological models. A central paradox emerges when one attempts to apply uniform probability measures across countably infinite sample spaces, such as the potential infinite number of universes in inflationary cosmology. The standard probability framework, dominated by Kolmogorov’s axioms, faces inherent inconsistencies when dealing with such infinite and uniform constructs.

Core Issues and Theoretical Insights

The paper begins by highlighting the foundational dilemma known as the "infinite fair lottery paradox"—an incompatibility between uniform probability distributions on infinite sets (like the natural numbers) and the axioms of standard probability theory. The key issues arise from the normalization and additivity constraints within Kolmogorov's framework, leading to paradoxical or undefined outcomes when dealing infinitely.

Wenmackers reviews the historical discourse on this paradox through the lenses of notable theorists like Kolmogorov, de Finetti, and Jaynes, providing a backdrop for understanding diverging approaches to resolving these inconsistencies. She elaborates that a pluralistic approach in mathematical methods may offer a pathway forward, accommodating multiple probabilistic formalisms that include, but are not limited to, finitely additive probability theory and non-standard analysis employing hyperreal numbers.

Alternative Formalisms

The paper investigates four distinct methodologies that address these foundational issues by circumventing standard probability theory's limitations:

1. Finitely Additive Probability (FAP): This approach, primarily stemming from de Finetti’s work, relaxes the additivity constraints, allowing broader application to infinite scenarios.
2. Non-Archimedean Probability (NAP): Emphasized by Benci et al., this framework utilizes hyperreal numbers to incorporate infinitesimals, thus permitting a more nuanced treatment of infinite sums.
3. Infinite Lottery Logic (ILL): Norton's proposal for non-probabilistic reasoning suits scenarios where probabilistic frameworks fall short, though it is not without its limitations when applied practically in cosmology.
4. Non-Normalizable Quasi-Probability: Employed informally by some physicists, this allows for measures that do not necessarily sum to one, thus dodging strict normalization requirements.

Implications for Cosmology

The paper proceeds to apply these theoretical insights to cosmological models, particularly those involving eternal inflation. Here, the universe’s very fabric presents infinite characteristics, necessitating non-standard probabilistic approaches. Measures in cosmology, especially those predicting properties like the cosmological constant across a multiverse, encounter these daunting measure problems.

Wenmackers explores how each proposed formalism provides different insights or potential solutions to these problems within cosmology. She emphasizes the need for constructing probability measures that are not only mathematically coherent but also physically justifiable in the context of infinite cosmological structures.

Future Directions and Conclusion

The analysis ultimately advocates for a pluralistic and interdisciplinary approach, leveraging various probabilistic frameworks to tackle different facets of the measure problem in cosmology. Wenmackers indicates that such diversity in methodological approaches allows for a richer understanding and potentially more robust modeling of the universe’s probabilistic characteristics.

In summary, the paper is a vital contribution to the ongoing discourse on the intersection of probability theory and cosmology. By rigorously exploring alternative mathematical frameworks, Wenmackers provides a balanced and informed perspective on navigating the complexities of infinite probabilities in cosmology, encouraging further theoretical developments and interdisciplinary dialogues in this challenging research area.