Necessity, Contingency and Counterfactuals in Mathematics

一些事实是偶然的：尽管它们实际上是真的，但它们可能是假的。其他事实似乎是必要的：他们的虚假在客观上是不可能的。这一区别从一开始就在系统哲学中占据了突出的地位。至少有一些数学事实看起来一定是真的。例如，三加三必须等于六，而不可能等于七。哲学家们长期以来一直认为，事实上，所有的数学真理都是绝对必要的。这个项目考虑公认的智慧是否正确，以及在多大程度上是正确的。在这样做的过程中，它的目的是澄清必然性、偶然性和反事实推理在数学实践中的作用。第一部分，“公理的必然性”，涉及Zermelo-Fraenkel集合论和选择(ZFC)的公理，该公理长期以来一直是数学的官方基础。我认为，ZFC公理并不一定都是正确的。该项目的第二部分“巧合、偶合和几乎错误的定理”处理的是数学家认为勉强为真的数学事实现象。我认为这样的定理是偶然数学真理的很好的候选者，并且这种洞见对我们关于偶然性和解释的思考有重要的影响。该项目的第三部分“数学实践中的反事实”探索了数学中反事实推理的性质和认知目标。我把重点放在解析数论中的“西格尔零点”的例子上：那些被强烈认为不存在的物体，但它们是广泛理论的主题。总而言之，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。

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.