History and Philosophy of Non-Archimedean Mathematics

非阿基米德数学的历史和哲学

• 批准号: 9223839
• 负责人: Philip Ehrlich
• 金额: $ 5万
• 依托单位: Individual Award
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-04-15 至 1995-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9223839&HistoricalAwards=false
• 关键词: History; Philosophy; Non; Archimedean; Mathematics

Between the 1870's and the appearance of A. Robinson's extraordinary work, "Non-standard Analysis" in 1961, there emerged a large, diverse, technically deep and philosophically pregnant body of consistent (non-Archimedean) mathematics of the infinitely large and infinitely small. Unlike non-standard analysis, which is concerned with providing a treatment of the calculus making use of infinitesimals, most of the former work either focused on geometry and the concepts of number and magnitude, or grew out of the natural evolution of such discussions. Moreover, whereas non- standard analysis derives its arithmetic content from an exceptionally narrow, albeit an extremely important, subclass of the class of real-closed ordered number systems, the former work deals with arbitrary ordered number fields as well as with ordered general conception of numbers great and small, as well as with ordered number systems having failures of communitivity, associativity, etc. This so-called non-Archimedean mathematics thereby reveals a more general conception of numbers great and small, as well as a richer theory of the infinitely large and infinitely small. This body of work, published under the rubric The Theory of Non-Archimedean Geometric and (Totally) Ordered Algebraic Systems has continued to grow in recent decades producing a number of major accomplishments including J.H. Conway's work on Surreal Numbers. In this project, Dr. Ehrlich aims to provide a detailed sketch of the historical development of this largely ignored but philosophically important subject, and to develop a number of the philosophical implications of its results for the concepts of number, the infinite, the infinitesimal, and the continuum, as well as for a number of issues in the philosophy of measurement and the philosophy of space and time.

从19世纪70年代到A.罗宾逊的 非凡的工作，“非标准分析”在1961年，出现了 一个巨大的，多样的，技术上有深度的， 无穷大的相容（非阿基米德）数学体 大的和无限小的。 与非标准分析不同， 提供一种治疗结石的方法 无穷小，大多数以前的工作要么集中在几何 以及数和大小的概念，或者说是从 这种讨论的自然演变。 此外，非- 标准分析的算术内容来自 非常狭窄，尽管非常重要， 一类实闭有序数系统，前者的工作 处理任意有序数域和有序数域 数字的一般概念，大和小，以及与 具有通信故障的有序数系统， 结合性等等。这个所谓的非阿基米德数学 从而揭示了一个更普遍的概念的数字大， 小，以及一个更丰富的理论的无限大， 无穷小。 这部著作以 非阿基米德几何与（全）序理论 近几十年来，代数系统不断发展， 一些重大成就，包括J. H。康威的工作 超现实数字 在这个项目中，埃利希博士的目标是提供一个详细的草图， 这一历史发展在很大程度上被忽视， 这是一个非常重要的问题，并制定了一些 哲学的影响，其结果的概念 数、无穷、无穷小和连续统 至于测量哲学中的一些问题， 空间和时间的哲学。