This page is to record the reference:
Practical Foundations of Mathematics
Cambridge Studies in Advanced Mathematics 59
Cambridge University Press (1999)
on a formulation of the foundations of mathematics, with an eye towards the actual practice of mathematics (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 on the author’s website (www.PaulTaylor.EU/Practical-Foundations/index.html) though with many mathematical symbols and formulas failing to render.
There is also a summary (www.paultaylor.eu/~pt/prafm/summary.html) with better typesetting. 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.
Robert Harper: Practical Foundations for Programming Languages, Cambridge University Press (2016) [webpage]
William Lawvere, Robert Rosebrugh: Sets for Mathematics, Cambridge University Press (2003) [book homepage, GoogleBooks, pdf]
Last revised on February 23, 2026 at 05:17:38. See the history of this page for a list of all contributions to it.