Category Theory in Philosophy of Science

本文的主题是应用范畴论方法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存在的感兴趣的话题：哲学家们一直对两个理论何时等价等问题感兴趣，即，当他们说同样的事情关于世界;当两个理论站在一个关系的还原，即，当一个理论可以在另一个理论中"代表“时;当一个理论是另一个理论的极限时，即，当一个理论可以被视为在某些物理制度中对另一个的近似时。这些问题不仅本身是令人感兴趣的，而且直接关系到科学哲学中的问题，甚至关系到科学实践本身。例如，理解等价的条件对于确定一个人可以在多大程度上拥有与所有相同的经验证据相一致的不同（即不等价）的理论是很重要的，也就是说，在多大程度上现实主义者应该关注数据对理论的不充分决定。或者，再举一个例子，理解一个理论的内容在多大程度上被保留在一个理论中，它是一个限制，这与评估科学知识在理论变化中是否稳健直接相关，因为人们经常发现，一个早期的理论是作为其后继者的限制案例出现的。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，我们就很难弄清楚新理论的新颖之处，或者不同的研究方案之间的差异。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，范畴方法涉及根据它们之间的结构保持映射（例如群同态、群同态或线性变换）来刻画某些类别的数学结构（例如群、光滑流形或向量空间）。最近人们开始认识到，把一个理论看作是这样的结构（理论的模型或解）的集合，再加上一个适当的态射类，可以使人们对理论的结构有很深的了解。这个项目将吸引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？正确分析理论需要什么样的范畴理论资源？范畴理论工具能为理解约简和极限带来什么样的见解？

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?