一个用于假设的投影代数：解决威金的谜题和外部现实的存在
A Projective Algebra for Ansatz: Resolving Wigner's Puzzle and the Existence of External Realms
AI总结：
本文通过投影代数形式化解决威金谜题，并证明外部抽象领域存在的必要性，支持数学实在论。
摘要：
自然哲学将科学观察与抽象框架相结合，通常使用数学假设来推测物理现象。然而，探索其他宇宙的可能性挑战了假设物理定律，如时空几何，超越我们现实的假设。本文认为，数学抽象作为超越物理限制的望远镜，使这种推理成为可能。通过投影代数形式化（第4节），我们建模了假设的机制，抽象地描述物理对象。这为威金的不合理有效性问题提供了通过基数等价性（第5节）的解决方案，并澄清了“证据”和“存在”等术语（第6节）以符合物理学中的惯例。一个受康托启发的悖论表明，没有任何宇宙可以包含所有数学抽象（例如，集合、数字），因为其幂集超过它，需要一个外部抽象领域（第6.4节）。这种逻辑必要性，即使在替代集合理论如新基础理论的背景下也成立，为一个最小的外部宇宙作为抽象领域提供了证据，支持数学实在论。这一结果并非特定于形式化，只要我们接受集合论的原则在数学上是有效的。由于抽象实体无法被实证检测，逻辑证据是恰当的，指导未来的科学和哲学研究，并促进跨学科探索。
英文摘要：
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.
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