Category Theory in Philosophy of Science

提出的主题是应用范畴论方法来分析科学理论之间的关系。理解这种关系是科学哲学中一个长期感兴趣的话题：哲学家们一直对以下问题感兴趣：当两种理论是等价的，即当它们对世界说同样的事情时；当两种理论处于还原关系中，即当一种理论可以在另一种理论中“表现”时；当一种理论是另一种理论的极限时，也就是说，当一种理论在某些物理条件下可以看作是另一种理论的近似值时。这些问题不仅本身令人感兴趣，而且直接关系到科学哲学问题，甚至关系到科学实践本身。例如，理解等效的条件对于确定一个人能够拥有与所有相同经验证据一致的不同（即非等效）理论的程度是很重要的，也就是说，现实主义者应该在多大程度上关注数据对理论的不确定。或者，再举一个例子，了解一个理论的内容在多大程度上保留在一个理论中，它是一个极限，这与评估科学知识是否在理论变化中是稳健的直接相关，因为人们经常发现，一个早期的理论作为其继任者的极限情况出现。这些问题直接关系到如何评价新理论，以及我们应该如何解释实验结果：如果没有适当的工具来比较理论之间的差异，我们就很难描述新理论的新颖之处，或者其他研究项目之间的差异。最近，这些领域的研究人员已经意识到，分类方法为解决这些问题提供了一个强大的工具。粗略地说，范畴方法涉及到用它们之间的保结构映射（如群同态、微分同态或线性变换）来描述某些数学结构类（如群、光滑流形或向量空间）。最近，人们开始认识到，将一个理论视为这些结构（理论的模型或解）的集合，加上适当的态射类，可以对理论的结构有很大的了解。这个项目将聚集一群对这些工具的应用感兴趣的研究人员，并研究三个具体问题：如何使用范畴理论方法来判断理论之间的等价性？正确的理论分析需要什么样的范畴论资源？范畴论工具在理解约简和极限方面能产生什么样的见解？

The proposed topic is the application of category-theoretic methods to the analysis of relations between scientific theories. Understanding such relations is a longstanding topic of interest in the philosophy of science: philosophers have been interested in questions such as when two theories are equivalent, i.e., when they say the same thing about the world; when two theories stand in a relation of reduction, i.e., when it is that one theory can be ``represented'' within another; and when one theory is a limit of another, i.e., when one theory may be regarded as an approximation to the other in certain physical regimes. These questions are not only of interest in themselves, but also have direct bearing on issues in philosophy of science, and indeed on scientific practice itself. For instance, understanding conditions for equivalence is important for determining the extent to which one could have distinct (i.e. non-equivalent) theories which are consistent with all the same empirical evidence—that is, on the extent to which realists should be concerned about the underdetermination of a theory by data. Or, for another example, understanding the extent to which the content of a theory is preserved in a theory of which it is a limit is directly relevant to assessing whether scientific knowledge is robust across theory change, since one often finds that an earlier theory emerges as a limiting case of its successor. These questions bear directly on how to evaluate new theories, and how we should interpret the results of experiments: without appropriate tools for comparing theories to one another, we can struggle to characterise what is novel about new theories, or how alternative research programs differ from one another.Recently, researchers in these areas have come to realise that categorical methods offer a powerful tool for addressing such questions. Roughly speaking, categorical methods involve characterising certain classes of mathematical structures (e.g. groups, smooth manifolds, or vector spaces) in terms of the structure-preserving maps between them (e.g. group homomorphisms, diffeomorphisms, or linear transformations). It has recently come to be appreciated that treating a theory as a collection of such structures (the models or solutions of the theory), together with an appropriate class of morphisms, gives a great deal of insight into the structure of the theory. This project will draw together a community of researchers interested in the application of these tools, and examine three specific issues: how can category-theoretic methods be used to make judgments of equivalence between theories? What kinds of category-theoretic resources are required for a proper analysis of theories? And what insights can category-theoretic tools yield into understanding reduction and limits?