Philosophy

philosophy

# Philosophy

## Scope

In the philosophical part of $n$Lab we discuss higher algebra, homotopy theory, type theory, category theory, and 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.

## Nonscope

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.

## Idea of relevance of higher mathematical structures

Philosophical interest in higher mathematical structures 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).

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?

### Research programs in mathematics

• Thomas Kuhn?

• Imre Lakatos?

• paradigm?

• paradigm shift?

• research program?

This entire subject about past research programs, paradigms in mathematics and paradigm shifts could be expanded on in the nLab. Examples include the shift from Euclidean geometry to non-Euclidean geometries in the 19th century, and the dominance of the material set theory paradigm in the 20th century and its failure with higher structures, the evolution of analytic concepts such as the differential, the integral, the real numbers, over the course of the 20th century, but there are surely others out there.

## Philosophical positions

### Books

• 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)

### Articles

Some philosophical aspects of the role of category theory are touched upon in some parts of the introductory paper

### Talks

category: philosophy

Last revised on May 16, 2022 at 23:29:23. See the history of this page for a list of all contributions to it.