Research Plan: Enduring Paradoxes of Infinitesimals, Zero, and Infinity I. Research Goal: To comprehensively analyze the persistent conceptual, philosophical, and pedagogical challenges associated with infinitesimals, zero, and infinity in mathematics, beyond the formalization provided by modern calculus. II. Research Questions: * Historical & Philosophical Evolution: How have the understanding and conceptualization of infinitesimals, zero, and infinity evolved from ancient times through the development of calculus and into contemporary mathematics? What were the key philosophical objections and justifications at each stage? * Conceptual Gaps & Paradoxes: Despite the rigor of modern calculus (epsilon-delta limits), what specific conceptual gaps or "paradoxes" (e.g., in intuitive understanding, notation) still arise from the interplay of infinitesimals, zero, and infinity? How do these manifest in different mathematical contexts (e.g., analysis, set theory)? * The Role of Non-Standard Analysis (NSA): How does Non-Standard Analysis, through hyperreal numbers, provide a rigorous framework for infinitesimals, and to what extent does it resolve or reframe the historical paradoxes? What are its pedagogical implications and limitations for broader adoption? * Pedagogical Challenges: What are the common misunderstandings and difficulties students encounter when learning about limits, derivatives (especially dy/dx), integrals, and infinite series, stemming from an intuitive or incomplete grasp of infinitesimals, zero, and infinity? How do textbooks and curricula address (or fail to address) these nuances? * Modern Interpretations & Applications: How are infinitesimals (or their conceptual descendants) used in contemporary mathematics, physics, and engineering beyond standard calculus (e.g., functional analysis, quantum field theory, numerical methods)? Do these applications introduce new conceptual challenges? * Future Directions & Foundational Debates: What ongoing foundational debates in mathematics pertain to these concepts? Are there new theoretical frameworks being developed that offer alternative perspectives or resolutions to these enduring issues?