Richard Dedekind and the Structuralist Transformation of Modern Mathematics

这一数学史研究项目得到了美国国家科学基金会科学、技术和社会计划的支持，并由SBE和MPS董事通过一封亲爱的同事信--《数学和物理科学与社会研究》--联合提供支持。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，20世纪相对论和量子力学的兴起。另一个恰当的例子是发生在十九世纪和二十世纪初的数学的根本变革，从几千年来一直被理解的“量”的研究转向对“关系结构”(群、环、场、几何流形、拓扑空间等)的更一般的研究。理查德·迪德金德是上述数学变革的核心人物，他被公认为十九世纪最有影响力的数学家之一，也是有史以来对代数、数论和数学基础做出最重要贡献的人之一。然而，对迪德金德和他的作品并没有一本书那么长的论述。这个项目的主要目标是从事最终将产生这样一本书的研究。这本书将特别关注Dedekin比其他任何人都更多地帮助开创了数学的“结构主义”方法。拟议的研究将建立在几位历史学家和数学哲学家的成果基础上。它将综合已经取得的洞察力，并将相应的研究扩展到数学的新的子领域。此外，该项目还将评估通常用于刻画德德金德作品的概念性工具，并加强这些工具。最后，由于它们的中心性和持续相关性仍然被低估，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，使德德金德的想法取得成果。

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.