philosophy (Rev #60) in nLab

Philosophy

Scope

In philosophical part of nnLab we discuss 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.

Nonscope

There are many articles which are not philosophy, but rather essays on general mathematics, and so on, often opinion pieces on what is important and so on. That is not philosophy per se, but it may be relevant thoughts and we could link them rather at related pages, like opinions on development of mathematics.

Idea of relevance of higher categories

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.

metaphysics

• 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.

• Homotopy type theory may be thought of as a vertical categorification of logic to (∞,1)(\infinity,1).

• Higher category theory may provide the right tools to take physics forward. A Prehistory of n-Categorical Physics See also physics.

• higher category theory and physics, geometry of 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?

• category (philosophy), modality

• logos (in philosophy), Nous

• metaphysics

• hermeticism, mysticism, gnosticism

• logic, reasoning

• Science of Logic

  • unity of opposites

    • becoming :\colon nothing ⊣\dashv being

• Platonism

• idealism

  • objective idealism

  • subjective idealism

• positivism

• analytic philosophy

• philosophy of mathematics

• philosophy of physics

• Converging Realities -- Toward a common philosophy of physics and mathematics

References and links

• foundations and logic, history of mathematics, structuralism, opinions on development of mathematics

General

• wikipedia: philosophy of mathematics; wikipedia.ru Философия математики

• Stanford Encyclopaedia of Philosophy online, contents

• Stewart Shapiro, The Oxford handbook of philosophy of mathematics and logic, Oxford University Press 2005

• free-philosophy-ebooks at openculture

Books

• Hegel, Wissenschaft der Logik ( Science of Logic )

• Ludwig Wittgenstein, Tractatus Logico-Philosophicus

• 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

Articles

• Glenn G. Parsons, James Robert Brown, Platonism, metaphor, and mathematics, Dialogue 43 (2004), no. 1, 47–66, MR2004k:00004
• John Baldwin, Model theoretic perspectives on the philosophy of mathematics, pdf
• Yu. I. Manin, Mathematical knowledge: internal, social and cultural aspects, arXiv:math.HO/0703427; Georg Cantor and his heritage, arxiv/math.AG/0209244; Truth as value and duty: lessons of mathematics, arxiv/0805.4057
• M. G. Katz, E. Leichtnam, Commuting and noncommuting infinitesimals, Amer. Math. Monthly 120 (2013), no. 7, 631-641 arxiv/1304.0583
• M. G. Katz, Thomas Mormann, Infinitesimals as an issue in neo-Kantian philosophy of science, arxiv/1304.1027
• Mikhail Gromov, Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2., pdf; Structures, Learning and Ergosystems: Chapters 1-4, 6 (2011) pdf
• William Lawvere, Cohesive Toposes and Cantor's "lauter Einsen"

Blog and forum discussions, conferences

• nnCafé on Mathematical reality: I, II
• Gavin wrote about “The dangers of category theory”, see nnForum here
• Simplicityonference: Ideals of Practice in Mathematics & the Arts