Necessity, Contingency and Counterfactuals in Mathematics

有些事实是偶然的：虽然实际上是真的，但它们也可能是假的。其他事实似乎是必要的：它们是错误的客观上是不可能的。这种区别从一开始就在系统哲学中占有突出地位。至少有些数学事实似乎必然是正确的。例如，3加3一定等于6，而不可能等于7。哲学家们长期以来一直认为，事实上，所有数学真理都是绝对必要的。这个项目考虑是否和在多大程度上接受的智慧是正确的。在这样做的过程中，它的目的是澄清必要性，偶然性和反事实推理在数学实践中的作用。该项目有三个组成部分。第一部分，“公理的必要性”，是关于Zermelo-Fraenkel集合论（ZFC）的公理，它长期以来一直是数学的官方基础。我认为ZFC公理不一定都是正确的。该项目的第二部分，“巧合、偶然性和几乎是假的定理”，处理数学家们认为仅仅是勉强正确的数学事实现象。我认为，这些定理是偶然性数学真理的良好候选，这种见解对我们思考偶然性和解释具有重要影响。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本质和认知目标。我关注的是解析数论中的“西格尔零”：人们强烈认为不存在的对象，但它们是广泛理论化的主题。总而言之，这个项目的各个部分对长期以来关于数学必要性的观点构成了重大挑战。

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.