Aspects of Pluralism in the Foundations of Mathematics

这个建议是为了发展PI正在进行的调查，以了解数学基础中涉及看似不相容但可行方法的多元主义的重要例子。这种多样性的两个主要方面是：（1）在分析、微分几何等的公式化中，在经典逻辑和构造性逻辑（放弃某些经典原则）之间的选择，（2）不同的本体论框架，如在集合论的单一固定宇宙中，与范畴论（CT）的多宇宙方法和结构主义的相关变体形成对比。在（1）中特别有趣的是“光滑无穷小分析”（SIA），它发展了一种非点状的连续统概念，无穷小的平方为0。必须使用构造性逻辑来避免不一致，但结果是经典分析的惊人简化，并证明了微积分中长期被认为是不可信的早期证明。然而，由于禁止某些涉及同一性的经典推论，解释的问题就出现了。这种明显的冲突是真实的还是仅仅是表面的，以及在后一种情况下，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），尽管传统上把具有固定论域的集合论作为发展数学理论的背景框架，但范畴论声称提供了一种可供选择的、更一般的和适应性更强的框架，这一点值得认真关注。本项目试图澄清范畴论作为一种自主选择的地位，以及它与其他结构主义方法的关系（例如PI的模态结构主义），像CT一样，也提供了数学话语的“多个宇宙”。 在这些问题上的进展可以帮助形成对数学本质的一般理解，以及它与其他科学的关系，这些科学表现出类似的“多元主义”。“关于这项研究的更广泛影响，它应该在科学家和公众中倡导数学作为一种比广泛认为的更加多方面的知识体系的概念。这可以反映在各种各样的数学教育中，最直接的是在大学一级。PI的工作说明了容忍甚至促进看似冲突的方法的多样性的重要性，这些方法不仅有用，而且对于理解丰富而复杂的主题是必要的。与此同时，正如这里所关注的特定数学理论很好地说明的那样，在追求这些多种方法的过程中，不需要牺牲严谨性或其他相关的知识标准。该项目有助于促进对多元主义的容忍，同时在学术内外的各种背景下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.