Foundations of mathematics

• Taylor, 1999: Practical Foundations of Mathematics (online )
• Sommaruga, ed., 2011: Foundational Theories of Classical and Constructive Mathematics (doi)
• Awodey, 2011: From sets to types, to categories, to sets (doi, pdf)

Philosophical references are below; for the mathematics, see the homotopy type theory page.

• Ladyman & Presnell, 2014: A primer on HoTT: Part 1: Formal type theory (pdf)
• Ladyman & Presnell, 2018: Does homotopy type theory provide a foundation for mathematics? (doi)

Introductions

• Makkai, 1998: Towards a categorical foundation of mathematics (pdf)
  • Builds on Makkai’s first order logic with dependent sorts (FOLDS)
  • See also: Makkai, 2013: The theory of abstract sets based on first-order logic with dependent types (pdf)

Philosophy surveys

• Landry & Marquis, 2005: Categories in context: Historical, foundational, and philosophical (doi)
• Marquis, 2014, SEP: Category theory, Sec 3: Philosophical significance (online )
  • Useful bibliography on categorical foundations
• Shapiro, IEP: Mathematical structuralism (online )
  • References on connection between structuralism and category theory

Philosophy papers

• Feferman, 1977: Categorical foundations and foundations of category theory (doi)
• Awodey, 1996: Structure in mathematics and logic: a categorical perspective (doi)
• Hellman, 2003: Does category theory provide a framework for mathematical structuralism? (doi, pdf)
• Awodey, 2004: An anwser to Hellman’s question: ’Does category theory provide a framework for mathematical structuralism?’ (doi, pdf)
• McLarty, 2004: Exploring categorical structuralism (doi)
• Shapiro, 2005: Categories, structures, and the Frege-Hilbert controversy: The status of meta-mathematics (doi, pdf)