Necessity, Contingency and Counterfactuals in Mathematics

数学中的必然性、偶然性和反事实

• 批准号: EP/Y027957/1
• 负责人: Alexander Paseau
• 金额: $ 25.55万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY027957%2F1
• 关键词: Necessity; Contingency; Counterfactuals; Mathematics

Some facts are contingent: although actually true, they could have been false. Other facts appear to be necessary: their being false is objectively impossible. The distinction has figured prominently in systematic philosophy from its beginnings. At least some mathematical facts appear to be necessarily true. Three plus three must equal six, for instance, and couldn't have equaled seven instead. Philosophers have long held that, in fact, all mathematical truths are absolutely necessary. This project considers whether and to what extent the received wisdom is correct. In doing so, it aims to clarify the roles of necessity, contingency and counterfactual reasoning in mathematical practice.The project has three components. The first part, "The Necessity of the Axioms", is concerned with the axioms of Zermelo-Fraenkel set theory with Choice (ZFC), which has long served as the official foundation for mathematics. I argue that the ZFC axioms are not all necessarily true. The second part of the project, "Coincidence, Contingency and Almost False Theorems", deals with the phenomenon of mathematical facts which mathematicians judge to be only barely true. I argue that such theorems are good candidates for contingent mathematical truths, and that this insight has important consequences for our thinking about contingency and explanation. The third part of the project, "Counterfactuals in Mathematical Practice", explores the nature and epistemic goals of counterfactual reasoning in mathematics. I focus on the case of "Siegel zeros" in analytic number theory: objects which are strongly believed not to exist, but which are the subject of extensive theorizing. Together, the parts of the project represent a major challenge to long-held views on the necessity of mathematics.

有些事实是偶然的：虽然事实上是真的，但它们可能是假的。其他的事实似乎是必要的：它们是假的在客观上是不可能的。这种区别从一开始就在系统哲学中占有突出地位。至少有一些数学事实看起来是必然正确的。例如，三加三一定等于六，而不可能等于七。哲学家们一直认为，事实上，所有的数学真理都是绝对必要的。这个项目考虑是否以及在多大程度上接受的智慧是正确的。在这样做的过程中，它的目的是澄清在数学实践中的必然性，偶然性和反事实推理的作用。第一部分，“公理的必要性”，关注的是Zermelo-Fraenkel选择集理论（ZFC）的公理，它长期以来一直是数学的官方基础。我认为ZFC公理并不一定都是真的。该项目的第二部分，“巧合，偶然性和几乎错误的定理”，涉及数学事实的现象，数学家判断只有勉强正确。我认为，这样的定理是很好的候选人偶然的数学真理，这一见解有重要的后果，我们的思考偶然性和解释。该项目的第三部分，“数学实践中的反事实”，探讨了数学中反事实推理的性质和认识目标。我专注于分析数论中的“西格尔零点”的情况：被强烈认为不存在的对象，但这是广泛的理论化的主题。总的来说，该项目的各个部分对长期以来关于数学必要性的观点提出了重大挑战。