Classicism, Constructivism, and Scientific Indispensability Arguments in Mathematics

9310667赫尔曼：现代数学在当代科学，尤其是物理学中的应用，到底需要多少现代数学工具？当代数学的一些较小的子集(这些替代方案中的一些是有限主义、建构主义、直觉主义或预言主义)。这是赫尔曼教授正在解决的中心问题。他正在进一步发展和扩展他正在进行的关于古典无限(即“非建设性”)数学的科学必然性问题和关于这种必然性对数学哲学的意义的问题的工作。他的项目的第一部分专注于建构主义程序，这是经典数学的替代方案之一，以及它们恢复物理科学中似乎构成特殊挑战的数学中具有经验意义的部分的能力，特别是变分和时空物理学的奇点定理(霍金和彭罗斯)。第二部分涉及哲学上有趣的中介系统，介于(在特殊意义上)建构主义和完全古典系统之间，特别是被称为预测主义(起源于庞加莱和韦尔)的准建构主义和某些唯名论系统。他正在研究对预测主义的科学和数学哲学挑战，目的是评估预测主义作为数学基础框架的可行性。最后，在他的研究的第三部分，赫尔曼教授开始根据他早期工作的结果以及最近对这种论点的有效性和说服力的挑战来评估“不可缺性”论点的总体范围和力量(作为抽象数学的正当理由)。***

9310667 Hellman How much of the apparatus of modern mathematics is really needed for its applications in contemporary science, especially physics? Could some smaller subset of contemporary mathematics (some of these alternatives are finitism, constructivism, intuitionism, or predicativism). This is the central question which Professor Hellman is addressing. He is further developing and extending his ongoing work on the question of the scientific indispensability of classical infinitistic (i.e. "non-constructive") mathematics and on the question of the significance of such indispensability for the philosophy of mathematics. The first part of his project focuses on constructivist programs, one of the groups of alternatives to classical mathematics, and their capacity to recover empirically significant portions of mathematics employed in the physical sciences that appear to pose special challenges, especially the calculus of variations and the singularity theorems (Hawking and Penrose) of space-time physics. The second part concerns philosophically interesting intermediary systems falling "between" (in special senses) constructivist and full classical systems, especially the quasi-constructivism known as predicativism (originating with Poincare and Weyl) and certain nominalistic systems. He is examining both scientific and mathematical- philosophical challenges to predicativism with the aim of assessing predicativism's viability as a foundational framework for mathematics. Finally, in the third part of his study, Professor Hellman is beginning to assess the overall scope and force of "indispensability" arguments (as justification for abstract mathematics) in light of results of his earlier work and also in light of recent challenges to the effectiveness and cogency of such arguments. ***