A Projective Algebra for Ansatz: Resolving Wigner's Puzzle and the Existence of External Realms
Jonathan M. M. Hall
TL;DR
The paper addresses how abstract mathematical ideas via Ansatz relate to physical reality and whether external abstract realms must exist. It introduces a projective algebra combining projection $\mathcal{P}$ and abstraction $\mathcal{A}$, defines $I$-extantness and the relating theorem, and uses Cantor-style arguments to argue for external realms. The main contributions are a formal framework for Ansatz, a resolution of Wigner’s puzzle through cardinality considerations, clarification of the notions of evidence and existence, and a demonstration that external abstract realms are logically necessary. This work supports Mathematical Realism and provides a logic-based, cross-disciplinary framework for future investigations at the boundary between physics and metaphysics.
Abstract
Natural philosophy integrates scientific observation with abstract frameworks, often using a mathematical Ansatz to hypothesise about physical phenomena. Exploring the possibility of other universes, however, challenges assumptions that physical laws, like spacetime geometry, extend beyond our reality. This paper argues that mathematical abstractions, serving as a telescope beyond physical constraints, enable such reasoning. Through a projective algebra formalism (Section 4), we model the mechanism of Ansatz, abstractly describing physical objects. This yields a resolution to Wigner's unreasonable effectiveness via cardinality equivalence (Section 5) and clarifies terms like 'evidence' and 'existence' (Section 6) to align with the conventions used in physics. A Cantor-inspired paradox shows no universe can contain all mathematical abstractions (e.g., sets, numbers), as its power set exceeds it, necessitating an external abstract realm (Section 6.4). This logical necessity, which holds even in the context of alternative set theories like New Foundations, provides evidence for a minimal external universe as an abstract realm, supporting Mathematical Realism. This result is not specific to the formalism, as long as we accept that the principles of set theory are mathematically valid. As abstract entities elude empirical detection, logical evidence is apt, guiding future science and philosophy research and fostering interdisciplinary inquiry.