Mathematics and physics employ different conceptual frameworks to describe the physical universe.
Mathematics and physics utilize distinct abstract structures and conceptual frameworks, with mathematics often serving as the formal language used to model physical reality.
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The Nexus of Reality: A Synthesis of Computation, Mathematics, and Physics. 2025. https://doi.org/10.5281/zenodo.15771017
The Nexus of Reality: A Synthesis of Computation, Mathematics, and Physics by Dean Kulik Part I: The Discrete and the Continuous — Frameworks for Modeling Reality The translation of the continuous laws of physics and the abstract structures of mathematics into a form amenable to computation is one ofthe central challenges and triumphs of modern science. This process of discretization, of replacing the infinitesimal with the finite, is not merely a practical necessity but a profound conceptual shift that reveals deep connections between seemingly disparate fields. The foundational concepts of this translation—finite differences, discrete geometry, and cellular automata—form a shared language for modeling the dynamics of reality, whether in the physical world or in the abstract realm of numbers. Section 1: The Language of Discretization At the heart of computational modeling lies the need to represent continuous systems within a discrete framework. This requires a set of mathematical and algorithmic tools that can approximate continuous processes with finite, computable steps. 1.1 Finite Difference Operators: Approximating the Infinitesimal The most fundamental tool for discretizing differential equations is the finite difference operator, which approximates derivatives by combining function values at nearby points. This approach forms the basis of the Finite Difference Method (FDM), a cornerstone of numerical analysis for solving differential equations by converting them into
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Roles of mathematics in physics education: A systematic review. 2025. https://doi.org/10.1103/wwww-gwp8
Mathematics plays many roles in physics and physics education. While these roles have previously been extensively discussed in the physics education research community, no systematic picture of the multifaceted considerations has yet been formed. To gain a comprehensive overview of the previous studies on the topic, we conducted a systematic literature review on 122 journal articles published between 2000 and 2023 that examine the role of mathematics in physics and physics learning. In the reviewed articles, we employed qualitative content analysis, coded each article for its characteristics, and used network maps for visualization. We identified eight thematic article categories, highlighting the complex integration of mathematics in, for example, physical reasoning, problem solving, modeling, and experiments. Additionally, the review examines theoretical frameworks and contexts, revealing an overemphasis on problem solving in mechanics and a limited exploration of advanced physics topics like quantum mechanics. A detailed inductive analysis further identified six overarching roles that mathematics plays in physics and physics education: (i) supporting learning and achievement, (ii) enabling mathematical manipulations, (iii) guiding reasoning and sensemaking, (iv) facilitating experiments and modeling, (v) serving as a language, and (vi) providing a structural foundation for physics as a science. Based on the analysis, we discuss how the roles assigned to mathematics have been conceptualized in the reviewed articles and provide an overall picture of the types and features of the reviewed corpus. Our findings suggest opportunities for future research, including deeper explorations of underrepresented physics contexts and targeted investigations into specific roles of mathematics in teaching and learning.
The Nature of Reality: The Quantum Narrative Matrix Hypothesis. 2026. https://doi.org/10.5281/zenodo.18259687
This work presents a complete theoretical framework that attempts to unify quantum mechanics, the holographic principle, and cosmology, providing scientifically reproducible approaches to fundamental questions including the unreasonable effectiveness of mathematics in physics, foundational physics philosophy, and the origin of the universe. The framework is computable and reproducible. Guidance and feedback are welcome. If there are any errors or inconsistencies, please provide feedback! Thank you! Version January 16, 2026 The latest inspection reveals that there are still minimal empirical coefficients remaining. I am currently working on their complete elimination and will attach the elimination records along with the next version update, providing updated and more accurate data. Code Refinement and Theoretical Purity Verification.This update implements a rigorous “first-principles” code audit. We have refined the derivation logic for the dark energy equation of state (w0w0) to remove any residual dependency on H0H0, ensuring 100% theoretical purity. Integrity Audit: Included a comprehensive Academic Integrity Audit Report verifying the derivation process. Result Robustness: The core results remain highly consistent with the previous version (H0≈74.3H0≈74.3 km/s/Mpc), demonstrating the model’s exceptional numerical stability under rigorous constraints. Version January 15, 2026 certain parameters (specifically the H₀-dependent correction term in w₀ calculation with coefficie
Blending of Conceptual Physics and Mathematical Signs. 2019. https://doi.org/10.48550/arxiv.1909.11618
Mathematics is the language of science. Fluent and productive use of mathematics requires one to understand the meaning embodied in mathematical symbols, operators, syntax, etc., which can be a difficult task. For instance, in algebraic symbolization, the negative and positive signs carry multiple meanings depending on contexts. In the context of electromagnetism, we use conceptual blending theory to demonstrate that different physical meanings, such as directionality and location, could associate to the positive and negative signs. With these blends, we analyze the struggles of upper-division students as they work with an introductory level problem where the students must employ multiple signs with different meanings in one mathematical expression. We attribute their struggles to the complexity of choosing blends with an appropriate meaning for each sign, which gives us insight into students' algebraic thinking and reasoning.
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