Classicism vs. Constructivism: On the Indispensability of Abstract Mathematics

在当代数学哲学中，数学中的“古典主义”被理解为以古典逻辑推理为特征的标准数学实践。这种做法通常与“柏拉图主义”的哲学立场相联系，根据这种立场，公理和定理被视为关于独立于头脑的数学现实的客观真理，该现实由数学话语似乎量化的那种抽象对象组成。另一方面，“建构主义”涵盖了那些认为古典实践中不合理的中心组成部分而寻求发展基于建构主义原则的替代实践的少数派学校。在这个项目中被考察的特定的建构主义学派拒绝接受作为经典逻辑基础的客观数学真理的经典概念，他们拒绝在数学背景下的普通的“存在”的概念。取而代之的是建构性对象原则和建构性证据原则。在这个项目中，赫尔曼教授正在研究古典数学实践和领先的建构主义学派之间的长期对立，强调数学的科学应用的重要性。首先，他正在根据最近的“情态结构解释”来审视建构主义挑战的哲学基础，支持古典主义，但避免标准柏拉图主义的某些问题。然后，他正在研究抽象数学对科学来说“不可或缺”的论点，并评估这些论点的范围和力量，特别是考虑到最近的数学-逻辑工作(谓词还原、逆数学)似乎严重限制了这些论点的影响。最后，他在古典主义-建构主义之争中运用了这样的论点。建构主义用于科学目的的可行性正在根据最近发现的适用障碍进行评估，这些障碍通常是在经验不确定的情况下，特别是在量子物理领域。

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.