A Projective Algebra for Ansatz: Resolving Wigner's Puzzle and the Existence of External Realms

TL;DR

This paper introduces a projective algebra framework to model the role of Ansatz in physics, resolving Wigner's puzzle and arguing for the necessity of an external abstract realm to contain all mathematical entities, supporting Mathematical Realism.

Contribution

It develops a formal algebraic approach to Ansatz, providing a logical argument for the existence of an external realm of abstract entities in science and philosophy.

Findings

Resolves Wigner's puzzle via cardinality equivalence.

Shows no universe can contain all mathematical abstractions.

Provides logical evidence for an external abstract realm.

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.