In the philosophical part of $n$Lab we discuss higher category theory and its repercussions in philosophy. More widely, the entries on philosophy in nLab should also contain philosophy of mathematics in general, and of logic and foundations in particular. As it is usual for philosophy and the study of thought, it is usefully carried on via study of historical thinkers and their ideas, hence some idea-related aspects of the history of mathematics are welcome.
There are many articles which are not directly philosophical, but rather essays on general mathematics, often opinion pieces on what is important and so on. Although mathematicians will often speak of their ‘philosophy’, this is not philosophy per se, but it may be relevant to an understanding of the nature of mathematics through its practice, see, for instance, development and current state of mathematics.
Philosophical interest in n-categories may be characterised as belonging to one of two kinds.
Metaphysical: The formation of a new language which may prove to be as important for philosophy as predicate logic was for Bertrand Russell and the analytic philosophers he inspired (see, e.g., Corfield 20).
Illustrative of mathematics as intellectual enquiry: Such a reconstitution of the fundamental language of mathematics reveals much about the discipline as a tradition of enquiry stretching back several millennia, for instance, the continued willingness to reconsider basic concepts (see, e.g., Corfield 12, Corfield 19).
Higher category theory and its associated type theories, such as homotopy type theory provides a new foundation for mathematics - logical and philosophical.
Higher category theory refines the notion of sameness to allow more subtle variants, adhering to the principle of equivalence.
There ought to be a categorified logic, or 2-logic. There are some suggestions that existing work on modal logic is relevant. Blog discussion: I, II, III, IV, V. Mike Shulman’s project: 2-categorical logic.
Homotopy type theory may be thought of as a vertical categorification of logic to $(\infinity,1)$.
Higher category theory may provide the right tools to take physics forward. A Prehistory of n-Categorical Physics See also physics.
More speculatively, category theory may prove useful in biology.
“Mathematical wisdom, if not forgotten, lives as an invariant of all its (re)presentations in a permanently self–renewing discourse.” (Yuri Manin)
To categorify mathematical constructions properly, one must have understood their essential features. This leads us to consider what it is to get concepts ‘right’. Which kind of ‘realism’ is suitable for mathematics? Which virtues should a mathematical community possess to further its ends: a knowledge of its history, close attention to instruction and the sharing of knowledge, a willingness to admit to what is currently lacking in its programmes?
Hegel, Wissenschaft der Logik ( Science of Logic )
Albert Lautman, Mathematics, ideas and the physical real, 2011 translation by Simon B. Duffy; English edition of Les Mathématiques, les idées et le réel physique, Librairie Philosophique, J. VRIN, 2006
Michael D. Potter, Set theory and its philosophy: a critical introduction, Oxford Univ. Press 2004
Fernando Zalamea, Filosofía sintética de las matemáticas contemporáneas, (Spanish) Obra Selecta. Editorial Universidad Nacional de Colombia, Bogotá, 2009. 231 pp. MR2599170, ISBN: 978-958-719-206-3, pdf. Transl. into English by Zachary Luke Fraser: Synthetic philosophy of contemporary mathematics, Sep. 2011. bookpage. Some excerpts here.
David Corfield, Towards a philosophy of real mathematics, Cambridge University Press, 2003, gBooks
Saunders MacLane, Mathematics, form and function, Springer-Verlag 1986, xi+476 pp. MR87g:00041, wikipedia
George Lakoff, Rafael E. Núñez, Where mathematics comes from, Basic Books 2000, xviii+493 pp. MR2001i:00013
Yuri I. Manin, Mathematics as Metaphor:
Selected Essays of Yuri Manin_, Amer. Math. Soc. 2007
Ralf Krömer, Tool and object: A history and philosophy of category theory, Birkhäuser 2007
Jean-Pierre Marquis, From a geometrical point of view: a study of the history and philosophy of category theory, Springer, 2008
Ian Hacking, Why is there philosophy of mathematics at all?, Cambridge University Press 2014
William Bragg Ewald, From Kant to Hilbert, From Kant to Hilbert: Readings in the Foundations of Mathematics, 2 vols. (original readings in English translation)
Roland Omnès, Converging Realities – Toward a common philosophy of physics and mathematics, Princeton University Press, 2005
David Corfield, Modal homotopy type theory, Oxford University Press 2020 (ISBN: 9780198853404)
Some philosophical aspects of the role of category theory are touched upon in some parts of the introductory paper