Aspects of Pluralism in the Foundations of Mathematics

这一建议是为了发展国际和平研究所正在进行的对数学基础中的多元化的非平凡例子的调查，涉及似乎不相容但可行的方法。这种多样性的两个主要轨迹是(1)在分析、微分几何等的表述中在经典逻辑和构造性逻辑之间的选择(放弃某些经典原则)，以及(2)不同的本体论框架，如在集合论的单一固定宇宙中，与范式论的多宇宙方法(CT)和相关的结构主义变种形成对比。在(1)中特别有趣的是“光滑无穷小分析”(SIA)，它发展了一个非点状的连续体概念，其中无穷小的平方为0。必须使用构造性逻辑来避免不一致，但其结果是经典分析的惊人简化，并证明了微积分中的早期证明是正确的，长期以来，这些证明被认为是不可信的。然而，由于某些涉及身份的经典推论被禁止，因此出现了解释问题。这种明显的冲突是真实的，还是仅仅是表面的，在后一种情况下，如何实现和解，是本项目将处理并试图解决的重要问题。关于(2)，尽管具有固定宇宙的集合论传统上被认为是发展数学理论的背景框架，但范畴理论提供了一个可选的、更一般的和适应性更强的框架的主张值得认真关注。本项目试图澄清作为自主选择的CT的地位，以及它与其他结构主义方法(例如，PI的情态-结构主义)的关系，这些方法也像CT一样，提供了数学话语的“多个世界”。在这些问题上的进展有助于形成对数学本质以及它与其他科学之间关系的总体理解，这些科学表现出类似的“多元化”。考虑到这项研究的更广泛影响，它应该促进一种概念，即数学是一个比科学家和公众普遍认为的更多方面的知识体系。这一点可以反映在各种背景下的数学教育中，最直接的是在大学层面上。PI的工作说明了容忍、甚至促进多种看似相互冲突的方法的重要性，这些方法不仅有用，甚至可能对于理解丰富而复杂的主题是必要的。与此同时，正如这里重点讨论的特定数学理论所充分说明的那样，在追求这些多重方法的过程中，不需要牺牲严谨或其他相关的智力标准。该项目有助于促进对多元化的容忍，同时在各种不同的背景下，在学术内外保持适当的逻辑和证据标准。

This proposal is to develop the PI's ongoing investigation into non-trivial examples of pluralism in the foundations of mathematics involving seemingly incompatible but viable approaches. Two main loci of such multiplicity are (1) the choice between classical and constructive logic (renouncing certain classical principles) in the formulation of analysis, differential geometry, etc., and (2) differing ontological frameworks as in the single fixed universe of set theory as contrasted with the multiple universe approach of category theory (CT) and related varieties of structuralism. Of particular interest under (1) is "smooth infinitesimal analysis" (SIA) which develops a non-punctiform conception of the continuum, with infinitesimals whose square is 0. Constructive logic must be used to avoid inconsistency, but the results are striking simplifications of classical analysis and a vindication of early proofs in calculus, long thought to be discredited. Problems of interpretation, however, arise due to the banning of certain classical inferences involving identity. Whether such apparent conflicts are genuine or merely apparent, and, in the latter case, how reconciliation is to be achieved, are important questions that this project will address and try to resolve. Concerning (2), although set theory with a fixed universe is traditionally taken as the background framework for developing mathematical theories, category theory's claims to provide an alternative, more general and adaptable framework deserve serious attention.This project seeks to clarify the status of CT as an autonomous alternative, as well as its relation to other structuralist approaches (e.g. the PI's modal-structuralism) that, like CT, also provide for "multiple universes" of mathematical discourse. Progress on these questions can help shape the general understanding of the nature of mathematics as well as its relation to other sciences, which exhibit analogous kinds of "pluralism."Concerning the broader impacts of this research, it should promote a conception of mathematics as a much more multi-faceted body of knowledge than is widely thought, both among scientists and the public. This can be reflected in mathematical education in a wide variety of contexts, most immediately at the college level. The PI's work illustrates the importance of tolerating, and even promoting, a multiplicity of seemingly conflicting approaches, which are not only useful but may even be necessary for comprehending a rich and complex subject matter. At the same time, as the particular mathematical theories focused on here well illustrate, there need be no sacrifice of rigor or other relevant intellectual standards in the pursuit of these multiple approaches. This project can help promote tolerance of pluralism, while maintaining appropriate standards of logic and evidence, in a wide variety of contexts, within and beyond the academic.