The paradox of classical reasoning

TL;DR

The paper explores the paradox that classical and quantum formalisms cannot be hierarchically related despite expectations based on their expressive powers, challenging assumptions about quantum-classical correspondence.

Contribution

It demonstrates that classical and quantum formalisms are incomparable in expressiveness, questioning the reductionist view of the physical world.

Findings

Classical and quantum formalisms are not hierarchically related.

Quantum theory's formal language cannot fully encompass classical formalism.

This challenges the assumption of quantum-classical correspondence.

Abstract

Intuitively, the more powerful a theory is, the greater the variety and quantity of ideas can be expressed through its formal language. Therefore, when comparing two theories concerning the same subject, it seems only reasonable to compare the expressive powers of their formal languages. On condition that the quantum mechanical description is universal and so can be applied to macroscopic systems, quantum theory is required to be more powerful than classical mechanics. This implies that the formal language of Hilbert space theory must be more expressive than that of Zermelo-Fraenkel set theory (the language of classical formalism). However, as shown in the paper, such a requirement cannot be met. As a result, classical and quantum formalisms cannot be in a hierarchical relation, that is, include one another. This fact puts in doubt the quantum-classical correspondence and undermines the reductionist approach to the physical world.