nLab Practical Foundations of Mathematics

The book

  • Paul Taylor:

    Practical Foundations of Mathematics

    Cambridge Studies in Advanced Mathematics 59

    Cambridge University Press (1999)

    ISBN 0-521-63107-6


is a description of how (in Taylor's opinion) the foundations of mathematics should really be done, with an eye towards matching how mathematics is done in practice (with the consequence that the system is no stronger than necessary).

The result is actually a series of foundations, most constructive, suitable for different sorts of mathematics. Ultimately, these are described as logic in categories defined by sketches and equipped with distinguished pullback-stable classes of display morphisms.

The book includes a self-contained, though dense, introduction to category theory. Before the three chapters on category theory comes a chapter “Posets and Lattices”, which “does for posets everything that is later done for categories” (per Taylor’s summary); compare category theory vs order theory.


The text is available online in a somewhat unreadable format.

There is also a summary in a readable format. This is basically an expanded table of contents together with an abbreviated introduction, with a link into the above-mentioned online text for each section.

A useful survey of some of the topics discussed there is also in

which is an exposition of Taylor’s Abstract Stone Duality.

category: reference

Last revised on August 24, 2023 at 10:08:07. See the history of this page for a list of all contributions to it.