UP Mathematics,Philosophy

Sources on:

Philosophy of maths

Several good collections of philosophical papers are available, e.g. [Benacerraf83], [Hart96] and [Tymoczko98].

For material on logicism see the writings of Gottlöb Frege and those of Bertrand Russell.

Some good sources on intuitionism and constructive mathematics are Errett Bishop, Douglas Bridges and Michael Beeson. For philosophical material relating to intuitionism Dummett must be mentioned (not withstanding his conspicuous absence from my bibliography).

The "anti-foundationalist" heresy may be found in the works of Imre Lakatos, Reuben Hersh and Phillip Davis and is copiously represented in [Tymoczko98]. (see Are Foundations Necessary? for my affirmation of orthodoxy, though aimed a computer scientists rather than anti-foundationalists)

On the more technical side From Frege to Gödel provides a collection of important papers published in the first three decades of the 20th century. While predominantly by logicians this collection is a valuable resource for philosophers of mathematics. Each paper is prefaced by a contemporary overview and evaluation.

Mathematics,Foundations Foundationsof maths
There is a lot of material on the foundations of mathematics in the references above.

Introduction to the Foundations of Mathematics by Wilder is an informal text on the Foundations of Mathematics (though, according to [Tymoczko98] Wilder was not appreciated by philosophers until they began to question foundations). A comprehensive formal account of The Logical Foundations of Mathematics is provided by William Hatcher, including a discussion of Category Theoretic approaches. An extensive account of constructive foundation systems may be found in Foundations of Constructive Mathematics by Michael Beeson.


Books on Categorical Foundations

The following list of books was culled from the f.o.m. mailing list. Many of them can't really be said to be about the foundations of mathematics, but are nevertheless thought by some to be at least relevant to the foundations of mathematics.
Johnstone, "Topos Theory", Academic Press, 1977.

Goldblatt, "Topoi: the categorical analysis of logic", North-Holland,

Makkai and Reyes, "First Order Categorical Logic", Springer-Verlag, 1977.

Barr and Wells, "Toposes, Triples, and Theories," Springer-Verlag, 1985.

Lambek and Scott, "Introduction to Higher-Order Categorical Logic,"
Cambridge, 1986.

Bell, "Toposes and Local Set Theories," Oxford, 1988.

Makkai and Pare, "The Foundations of Categorical Model Theory", AMS
CM104, 1989.

Freyd and Scedrov, "Categories, Allegories", North-Holland, 1990.

Moerdijk and Reyes, "Models for Smooth Infinitesimal Analysis",
Springer-Verlag, 1991.

McLarty, "Elementary Categories, Elementary Toposes," Oxford, 1992.

Mac Lane and Moerdijk, "Sheaves in Geometry and Logic", Springer-Verlag,

Borceux, "Handbook of Categorical Algebra" (three volumes), Cambridge,

Lawvere and Schanuel, "Conceptual Mathematics", 1997.

A. Joyal and I. Moerdijk, Algebraic Set Theory, LMS Lecture
Note Series 220, Cambridge University Press, 1995.

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