Classicism, Constructivism, and Scientific Indispensability Arguments in Mathematics

9310667 Hellman现代数学的工具中有多少是真正需要其在当代科学中的应用，特别是物理学？ 当代数学的一些较小的子集（其中一些选择是有限主义，建构主义，直觉主义或谓词主义）。 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工作的问题的科学initiativity古典无穷（即“非建设性”）数学和问题的意义，这种initiativity的哲学数学。 他的项目的第一部分侧重于建构主义程序，经典数学的替代品之一，以及它们恢复物理科学中数学的经验重要部分的能力，这些数学似乎构成了特殊的挑战，特别是变分法和时空物理学的奇点定理（霍金和彭罗斯）。 第二部分关注的是介于建构主义和完全古典系统之间的有趣的中介系统，特别是被称为谓词主义的准建构主义（起源于庞加莱和魏尔）和某些唯名论系统。 他正在研究科学和数学哲学的挑战，以评估谓词主义的可行性作为数学的基础框架的目的。 最后，在他的研究的第三部分，赫尔曼教授开始评估的整体范围和力量的“不可或缺”的论点（作为理由抽象数学），根据他的早期工作的结果，也根据最近的挑战，有效性和说服力的这些论点。 ***

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. ***