Structuralism, the view that mathematics is the science of structures, can be characterized as a philosophical response to a general structural turn in modern mathematics. Structuralists aim to understand the ontological, epistemological, and semantical implications of this structural approach in mathematics. Theories of structuralism began to develop following the publication of Paul Benacerraf’s paper ‘What numbers could not be’ in 1965. These theories include non-eliminative approaches, formulated in a background ontology of sui generis structures, such as Stewart Shapiro’s ante rem structuralism and Michael Resnik’s pattern structuralism. In contrast, there are also eliminativist accounts of structuralism, such as Geoffrey Hellman’s modal structuralism, which avoids sui generis structures. These research projects have guided a more systematic focus on philosophical topics related to mathematical structuralism, including the identity criteria for objects in structures, dependence relations between objects and structures, and also, more recently, structural abstraction principles. Parallel to these developments are approaches that describe mathematical structure in category-theoretic terms (e.g., in work by Steve Awodey, Elaine Landry, and Colin McLarty). Category-theoretic approaches have been further developed using tools from homotopy type theory. Here we find a strong relationship between mathematical structuralism and the univalent foundations project, an approach to the foundations of mathematics based on higher category theory.

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特刊简介：数学结构主义的基础

结构主义，即数学是结构科学的观点，可以被描述为对现代数学中一般结构转向的哲学反应。结构主义者旨在理解这种结构方法在数学中的本体论，认识论和语义含义。在1965年，保罗·贝纳塞拉夫（Paul Benacerraf）发表论文“不可能有多少”之后，结构主义理论开始发展。这些理论包括非消除性方法，这些方法是在特殊结构的背景本体论中提出的，例如斯图尔特·夏皮罗（Stewart Shapiro）的前结构主义和迈克尔Resnik的模式结构主义。相比之下，也有结构主义的极端主义主义解释，例如杰弗里·赫尔曼（Geoffrey Hellman）的模态结构主义，它避免了特殊的结构。这些研究项目引导人们更加系统地关注与数学结构主义有关的哲学主题，包括结构中对象的标识标准，对象与结构之间的依存关系以及最近的结构抽象原理。与这些发展并行的是以范畴论的术语描述数学结构的方法（例如，史蒂夫·阿沃迪，伊莱恩·兰德里和科林·麦克拉蒂的著作）。使用同伦类型理论的工具进一步发展了分类理论方法。在这里，我们发现数学结构主义与单价基础项目之间存在紧密的关系，后者是基于高级类别理论的数学基础方法。