philosophy

# Philosophy

## Scope

In philosophical part of $n$Lab 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.

## 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

## 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?

### Books

• Hegel, Wissenschaft der Logik ( Science of Logic )

• 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

### Blog and forum discussions, conferences

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

category: philosophy

Revised on March 3, 2014 04:04:11 by Urs Schreiber (89.204.155.195)