philosophy (Rev #21) in nLab

Philosophy

Scope

In philosophical part of nnLab we discuss the higher category theory and its repercussions in philosophy. More widely, the future 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.

Idea

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.

• Illustrative of mathematics as intellectual enquiry: Such a reconstitution of the fundamental language of mathematics reveals much about mathematics as a tradition of enquiry stretching back several millennia, for instance, the continued willingness to reconsider basic concepts.

• Higher category theory provides a new foundation for mathematics - logical and philosophical.

• Higher category theory refines the notion of sameness to allow more subtle variants. It advocates the avoidance of evil.

• 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. Mike Shulman’s project: 2-categorical logic.

• 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.

Enquiry

“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?

Some literature

• foundations and logic
• wikipedia: philosophy of mathematics; wikipedia.ru Философия математики

Blog and forum discussions

• Mathematical reality: I, II
• Gavin wrote about “The dangers of category theory”, the query is now at nForum here

Books

• Michael D. Potter, Set theory and its philosophy: a critical introduction, Oxford Univ. Press 2004

• Fernando Zalamea, Filosofía sintética de las matemáticas contemporáneas, (Spanish) [Synthetic philosophy of contemporary mathematics] Obra Selecta. Editorial Universidad Nacional de Colombia, Bogotá, 2009. 231 pp. MR2599170, ISBN: 978-958-719-206-3, pdf

• 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

Articles

• Glenn G. Parsons, James Robert Brown, Platonism, metaphor, and mathematics, Dialogue 43 (2004), no. 1, 47–66, MR2004k:00004