Classicism vs. Constructivism: On the Indispensability of Abstract Mathematics

在当代数学哲学中，“古典主义” 在数学中被理解为标准的数学实践 以经典逻辑推理为特征。 这 实践通常与“柏拉图主义者”联系在一起 哲学观点，根据这一观点， 定理被认为是关于心灵的客观真理， 独立的数学实在由抽象的 数学话语似乎量化了 结束了 另一方面，“建构主义”涵盖了 少数民族学校拒绝作为非法的核心组成部分 并寻求发展替代实践 基于建构主义原则。 的特定 在这个项目中正在审查的建构主义学校 反对古典的客观概念，认为它是不合理的 经典逻辑所依据的数学真理， 拒绝“存在”的普通概念， 数学 取而代之的是， 构造性客体和构造性证明。 在这个项目中，赫尔曼教授正在研究 古典数学和现代数学之间 实践和领先的建构主义学校，强调 数学科学应用的重要性。 在 首先，他正在研究的哲学基础， 建构主义的挑战，根据最近的“模态结构” 解释，”支持古典主义，但避免某些 标准柏拉图主义的问题。然后，他正在研究 抽象数学对科学是“不可或缺的”， 评估这些论点的范围和力量，特别是 根据最近法医学逻辑工作（表语 减少，逆向数学），这似乎严重限制了 这些论点的影响。 最后他终于开始 这种关于古典主义和建构主义的争论。 建构主义对于科学目的的可行性是 根据最近发现的障碍进行评估， 适用性，一般情况下，在经验的情况下， 不确定性，特别是在量子物理学领域。

Within contemporary philosophy of mathematics, "classicism" in mathematics is understood as standard mathematical practice as characterized by classical logical reasoning. This practice is usually associated with a "platonist" philosophical standpoint according to which axioms and theorems are viewed to be objective truths about a mind- independent mathematical reality consisting of abstract objects of the sort mathematical discourse appears to quantify over. "Constructivism," on the other hand, covers those minority schools who reject as illegitimate central components of classical practice and seek to develop alternative practice based on constructivist principles. The particular constructivist schools under examination in this project reject as illegitimate the classical conception of objective mathematical truth on which classical logic is based and they reject the ordinary notion of "existence" in context of mathematics. In its place, they substitute principles of constructive object and constructive proof. In this project, Professor Hellman is examining the longstanding opposition between classical mathematical practice and leading constructivist schools, emphasizing the importance of scientific applications of mathematics. At the outset, he is examining the philosophical underpinnings of the constructivist challenge in light of recent "modal-structural interpretations," supporting classicism but avoiding certain problems of standard platonism. He is then examining arguments that abstract mathematics is "indispensable" for science and assessing the scope and force of these arguments, especially in light of recent mathematical-logical work (predicative reductions, reverse mathematics) which seems severely to limit the impact of such arguments. Finally he is bringing to bear such arguments on the classicism-constructivism controversy. The viability of constructivism for scientific purposes is being assessed in light of recently uncovered obstacles to its applicability, both generally, in circumstances of empirical uncertainty, and specially, in the domain of quantum physics.