Richard Dedekind and the Structuralist Transformation of Modern Mathematics

本数学史研究项目由美国国家科学基金会科学、技术与社会计划资助，并由SBE和MPS理事会共同以“数学、物理科学与社会研究”的“致同事信”的形式资助。科学哲学的主要任务之一是研究革命性科学发展的意义，如古希腊理论数学的诞生，近代早期实验自然科学的兴起，以及20世纪相对论和量子力学的出现。另一个恰当的例子是发生在19世纪和20世纪初的数学的根本转变，从对“量”的研究转变为对“关系结构”（群、环、场、几何流形、拓扑空间等）的更普遍的研究。上面提到的数学转型的中心人物是理查德·戴德金德，他被公认为19世纪最有影响力的数学家之一，也是代数、数论和数学基础最重要的贡献者之一。然而，关于戴德金德和他的作品，并没有一本书的篇幅。这个项目的主要目标是从事研究，最终将产生这样一本书。这本书将特别关注“结构主义”的数学方法，戴德金比其他任何人都更有助于开创这种方法。这项拟议的研究将以几位数学历史学家和哲学家的成果为基础。它将综合已经取得的见解，并将相应的研究扩展到新的数学子领域。此外，该项目将评估戴德金作品中常用的概念性工具，并改进这些工具。最后，由于它们的中心地位和持续的相关性仍然被低估，另一个目标将是通过将它们直接与当前的数学哲学联系起来，使戴德金的思想结出硕果。

This research project in the history of mathematics is supported by the Science, Technology and Society program at NSF and jointly by the SBE and MPS directorates by way of a Dear Colleague Letter, Research on Mathematical and Physical Sciences and Society. One of main tasks in the philosophy of science is to study the significance of revolutionary scientific developments such as the birth of theoretical mathematics in Ancient Greece, the rise of experimental natural science in the early modern period, and the emer¬gence of relativity theory and quantum mechanics in the twentieth century. Another case in point is the radical transformation of mathematics that took place in the nineteenth and early twentieth cen¬turies, a shift from the study of "quantity" as it had been under¬stood for millennia to the much more general study of "relational structures" (groups, rings, fields, geometric manifolds, topological spaces, etc.). A central figure in the transformation in mathematics noted above was Richard Dedekind, who is generally recognized as one of the most influential mathematicians of the nineteenth century, as well as one of the most important contributors to algebra, number theory, and the foundations of mathematics of all time. Yet there is no book-length treatment of Dedekind and his work. The main goal of this project is to engage in research that will eventually result in such a book. The book will have a special focus on the "structuralist" approach to mathe¬matics that Dedekind, more than anyone else, helped to inaugurate.The proposed research will build on the results of several historians and philosophers of mathe¬matics. It will be to synthesize the insights already achieved, and it will be to expand the corresponding investigations into new sub-fields of mathematics. In addition, the project will assess the conceptual tools commonly used in charac¬terizing Dedekind's works and sharpen those tools. Finally, because the centrality and the continuing relevance of them remain underestimated, another goal will be to make Dedekind's ideas fruit¬ful by relating them directly to current philo¬sophy of mathe¬mat¬ics.