[Paper Review] Four Departures in Mathematics and Physics

Elemér E Rosinger|arXiv (Cornell University)|Mar 1, 2010

Mathematical and Theoretical Analysis12 references3 citations

TL;DR

This paper challenges four foundational constraints in mathematics and physics—no contradictions, no self-reference, the Archimedean Axiom, and the neglect of matrix module structures—arguing that each can be overcome using non-standard mathematical frameworks such as inconsistent logical systems, non-Archimedean algebras, and reduced power algebras. The key contribution is demonstrating that modern physics, especially quantum mechanics and cosmology, demands mathematical structures beyond the real numbers and classical logic, offering consistent alternatives that resolve long-standing infinities and conceptual paradoxes.

ABSTRACT

Much of Mathematics, and therefore Physics as well, have been limited by four rather consequential restrictions. Two of them are ancient taboos, one is an ancient and no longer felt as such bondage, and the fourth is a surprising omission in Algebra. The paper brings to the attention of those interested these four restrictions, as well as the fact that each of them has by now ways, even if hardly yet known ones, to overcome them.

Motivation & Objective

• To identify and critique four long-standing, restrictive assumptions in mathematics and physics that hinder progress in foundational theories.
• To argue that the rejection of contradictions and self-reference is not logically necessary, but historically and culturally entrenched.
• To demonstrate that the Archimedean Axiom is not a necessary truth but a limiting assumption that excludes richer mathematical structures.
• To show that the n-dimensional Euclidean space ℝⁿ has a deeper algebraic structure as a module over the non-commutative algebra ℳⁿ of n×n real matrices, which is systematically overlooked.
• To advocate for the adoption of non-Archimedean mathematics, particularly reduced power algebras, as a more suitable foundation for modern physics, including quantum field theory and cosmology.

Proposed method

• Introduces inconsistent logical systems, such as those underlying digital computers, where M+1=M for a large integer M, showing practical consistency despite formal inconsistency.
• Analyzes self-reference through historical and philosophical lenses, citing the Liar Paradox and Russell’s Paradox, and argues for its legitimacy in foundational systems.
• Replaces the Archimedean Axiom with non-Archimedean structures, using reduced power algebras to construct ordered fields with infinitesimals and infinite elements.
• Constructs tensor products over different rings and homomorphisms, showing that the tensor product structure depends on the underlying ring, leading to distinct algebraic behaviors.
• Demonstrates that ℝⁿ is not just a vector space over ℝ but also a module over the non-commutative algebra ℳⁿ, revealing a richer algebraic structure often ignored.
• Uses algebraic decomposition theorems for modules over principal ideal domains (PIDs), such as R/(Rp_i^{α_ij}), to illustrate structural richness in module theory.

Experimental results

Research questions

• RQ1 Can mathematical physics be advanced by relaxing the taboo against logical contradictions, given that digital computers operate consistently under inconsistent axioms?
• RQ2 Is self-reference a logical flaw or a necessary feature of foundational systems, especially in light of its presence in divine self-identification and modern logic?
• RQ3 Does the Archimedean Axiom uniquely determine the real line, or are there viable, mathematically consistent alternatives that better model physical space-time?
• RQ4 Can the structure of ℝⁿ be enriched by considering it as a module over the non-commutative algebra ℳⁿ of n×n matrices, and what are the implications for physics?
• RQ5 Do modern physical theories like quantum field theory and cosmology require non-Archimedean mathematical frameworks to resolve issues of infinity and multiverse state counting?

Key findings

• Digital computers operate under an inconsistent arithmetic system where M+1=M for a large integer M, yet function reliably, proving that inconsistency need not imply incoherence.
• The uniqueness of ℝ as the only complete ordered field is not a deep mathematical truth but a direct consequence of the Archimedean Axiom, which can be removed to yield richer mathematical structures.
• Reduced power algebras provide a consistent, non-Archimedean framework that includes infinitesimals and infinite numbers, offering a viable alternative to ℝ for modeling space-time.
• The n-dimensional space ℝⁿ possesses a natural structure as a module over the non-commutative algebra ℳⁿ of n×n real matrices, a fact that is systematically overlooked in standard treatments.
• The tensor product over a ring R depends on R, and changing the underlying ring—from R to a subring S or via ring homomorphisms f and g—leads to distinct tensor product structures, showing that the choice of ring is mathematically significant.
• The existence of quantum algorithms requiring more than 10^500 states, while the observable universe contains fewer than 10^80 atoms, suggests a need for mathematical structures beyond standard real-number-based models, which non-Archimedean algebras can provide.