Logic of Partitions Reveals Duality Linking Classical Physics and Information Theory

The foundations of logic and information theory are being re-examined through a surprising connection to the core principles of physics, as David Ellerman, an independent researcher, and the late Gian-Carlo Rota explore a previously unfinished line of inquiry. Their work traces an intellectual path, beginning with Rota’s investigations into the logic of equivalence relations and partitions, revealing a fundamental duality that underpins much of the exact sciences. The researchers demonstrate how this duality positions classical physics on one side and modern physics on the other, offering a novel framework for understanding mechanics through the lens of partitions and, crucially, resolving long-standing mysteries surrounding quantum phenomena like the two-slit experiment. By applying Rota’s approach to enumerative combinatorics, this research provides a fresh perspective on the very foundations of how we understand definiteness, indefiniteness, and statistical analysis

Partition Logic, Equivalence, and Quantum Foundations

This document explores the connections between quantum mechanics, combinatorics, and logic, building upon the work of Gian-Carlo Rota, a highly influential mathematician known for his contributions to combinatorial analysis and its philosophical implications. It proposes a framework based on partition logic and equivalence relations to understand the foundations of quantum theory, arguing for objective indeterminacy, where quantum states are genuinely indefinite until measured, rather than simply reflecting a lack of knowledge on the part of the observer. Partition logic, the logic of dividing a whole into parts, is positioned as fundamental to understanding both combinatorial structures, arrangements of objects, and the inherent uncertainty present in quantum systems. This framework proposes a reality comprising definite classical and indefinite quantum realms, drawing heavily on Rota’s emphasis on the importance of combinatorial structures as a basis for mathematical and physical reasoning.

A compelling parallel is drawn between Leibniz’s principle of indiscernibility, the philosophical assertion that two distinct objects must differ in at least one property, and the Pauli exclusion principle, a fundamental tenet of quantum mechanics stating that no two identical fermions can occupy the same quantum state simultaneously. This connection suggests a unifying principle governing both classical and quantum reality, implying that the very act of individuation, of distinguishing one entity from another, operates under similar logical constraints in both realms. The author emphasizes that partition logic is more foundational than traditional Boolean logic, as it addresses the basic operation of dividing a whole into parts, a pre-logical operation upon which logical inferences are built. Equivalence relations, which group elements based on shared properties, serve as a mathematical model for both combinatorial structures, such as grouping objects with the same colour or shape, and the relationships between quantum states, where states are considered equivalent if they yield the same probabilities for measurement outcomes. The work rejects the notion that quantum indeterminacy arises solely from limited knowledge, arguing for genuine, objective indeterminacy inherent in quantum states until measurement, supported by concepts like density matrices, mathematical objects representing the probabilistic state of a quantum system, and Gleason’s theorem, which establishes a connection between quantum states and probability measures.

The Born rule, a central postulate of quantum mechanics determining the probabilities of measurement outcomes, is presented as a natural consequence of the superposition of quantum states, where a quantum system can exist in a combination of multiple states simultaneously. This is not merely a mathematical convenience, but a fundamental aspect of quantum reality, and the Born rule provides the mechanism for collapsing this superposition into a definite outcome upon measurement. The author utilizes lattices, partially ordered sets where elements have a defined relationship of ‘less than or equal to’, to model the structure of quantum reality, representing relationships between quantum states and the possible transitions between them., A lattice structure allows for a hierarchical understanding of quantum states, where states can be ordered based on their energy, momentum, or other relevant properties. This approach draws on the work of philosophers such as Immanuel Kant, whose concept of ‘categories of understanding’ influenced the structuring of experience, Werner Heisenberg, who articulated the uncertainty principle, and Abner Shimony, a prominent philosopher of physics who explored the foundations of quantum mechanics, to strengthen this perspective. The document unfolds as a sustained argument, outlining core themes and introducing key concepts before delving into the mathematical foundations of partition logic, equivalence relations, and lattice theory. It then applies these concepts to the interpretation of quantum mechanics, focusing on objective indeterminacy, and concludes by summarizing the arguments and highlighting the interconnectedness of logic, combinatorics, and quantum mechanics.

The author advocates for a holistic view of reality, where logic, mathematics, and physics are deeply intertwined, suggesting that fundamental principles governing one domain may also apply to others. This perspective challenges the traditional separation between these disciplines, proposing that a unified framework is necessary to fully understand the nature of reality. The interpretation of quantum mechanics as involving objective indeterminacy is central to this framework, meaning that quantum properties are not predetermined but are only defined upon measurement, and this is not due to a lack of information but is an inherent feature of the quantum world. Partition logic is presented as a powerful tool for understanding both combinatorial structures and the foundations of quantum mechanics, providing a formal language for describing and reasoning about both classical and quantum systems. This work stands as a testament to the legacy of Gian-Carlo Rota and his emphasis on combinatorial structures as a unifying principle in mathematics and beyond. Ultimately, this document represents an ambitious attempt to provide a unified framework for understanding the foundations of reality, offering a unique and thought-provoking perspective on fundamental questions in science and philosophy, and potentially opening new avenues for research in both theoretical physics and the foundations of mathematics.