New Directions in Reverse Mathematics and Applied Computability Theory

逆向数学和应用可计算性理论的新方向

• 批准号: 1400267
• 负责人: Damir Dzhafarov
• 金额: $ 15万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1400267&HistoricalAwards=false
• 关键词: New; Directions; Reverse; Mathematics; Applied

Mathematics today benefits from having "firm foundations", by which we usually mean a system of axioms sufficient to prove the various theorems we care about. But given a particular theorem, can we specify precisely which axioms are needed to derive it? This is a natural question, and also an ancient one: over 2000 years ago, the Greek mathematicians were asking it about Euclid's geometry. Reverse mathematics is an area of mathematical logic that offers a modern approach to this kind of question, by classifying mathematical theorems according to their logical strength. This offers a deeper insight into the fundamental ideas and methods needed to prove a given theorem. More precisely, reverse mathematics provides a framework in which to compare and contrast results from disparate areas of mathematics, which helps elucidate the underpinnings of various branches of the mathematical sciences, and thereby leads to a better understanding of mathematics and its applications.A striking fact repeatedly demonstrated in this area is that the vast majority of mathematical propositions can be classified into one of a small number of categories. But for some very important and fundamental theorems this is not the case. Dzhafarov's research focuses on theorems of this ?irregular" type, including Ramsey's theorem, various equivalents of the axiom of choice, and principles arising from certain problems in cognitive science. In this project, Dzhafarov will work to achieve a greater understanding of the complexities of these "irregular" theorems, to find new examples of such theorems from previously unexplored areas of mathematics, and to apply the reverse mathematics analysis to questions from outside of mathematics. This will be facilitated by the application of methods from computability theory and proof theory, and by the addition of ideas from various collaborations across a number of areas of pure and applied mathematics, as well as interactions with members of the multidisciplinary University of Connecticut logic group.

今天的数学受益于拥有“牢固的基础”，我们通常是指足以证明我们关心的各种定理的公理系统。但是，鉴于特定的定理，我们可以精确地指定得出哪些公理来得出它？这是一个自然的问题，也是一个古老的问题：2000年前，希腊数学家询问了欧几里得的几何形状。反向数学是数学逻辑领域，它通过根据其逻辑强度对数学定理进行分类，为这种问题提供了现代方法。这为证明给定定理所需的基本思想和方法提供了更深入的了解。更确切地说，反向数学提供了一个框架，在该框架中，与数学的不同领域进行比较和对比，这有助于阐明数学科学的各个分支的基础，从而使对数学及其应用的更好理解及其应用中的一个巨大的概述，从而使数学的主要范围逐渐划分了数学，因此可以更好地理解数学及其应用。但是对于某些非常重要和基本的定理而言，情况并非如此。 Dzhafarov的研究重点是这种不规则的“类型”，包括拉姆西定理，选择的公理的各种等效以及由认知科学中的某些问题引起的原则。将反向数学分析应用于数学之外的问题。