发表机构
University of Warwick（华威大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文梳理逆向数学的历史与哲学背景，阐明其通过可计算性理论衡量集合存在原则强度，并关联数学哲学中从奠基纲领到实在论、确定性与应用性的核心议题。
AI 中文摘要
逆向数学是数理逻辑的一个分支，致力于确定推导关于具体结构（如实数线）的普通数学定理所必要且充分的最小集合存在原则。自20世纪70年代中期以来，逆向数学已发展出对数学各领域（从实分析与复分析到无穷组合学）中定理强度的系统分类。本文将从历史与哲学背景中定位逆向数学，并揭示其与数学哲学核心问题的相关性，从希尔伯特和布劳威尔的奠基性纲领到关于实在论、确定性和数学可应用性的当代争论。在此过程中，本文将讨论可计算性理论在衡量集合存在原则强度中的作用，以及当这些原则应用于物理科学和哲学时关于理想化的相关问题。
英文摘要
Reverse mathematics is a branch of mathematical logic dedicated to determining the minimal set existence principles necessary and sufficient to derive ordinary mathematical theorems about concrete structures like the real line. Since the mid-1970s, reverse mathematics has developed a systematic classification of the strength of theorems in areas of mathematics ranging from real and complex analysis to infinitary combinatorics. This essay will place reverse mathematics in its historical and philosophical context, and reveal its relevance to central issues in the philosophy of mathematics, from the foundational programmes of Hilbert and Brouwer to contemporary debates about realism, determinacy, and applicability of mathematics. In doing so, it will discuss the role of computability theory in measuring the strength of set existence principles, as well as related questions about idealisation when these principles are applied in the physical sciences and in philosophy.
Comments22 pages. To appear in the Blackwell Companion to the Philosophy of Mathematics, A. Paseau (ed.)
