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Foundational Theories of Classical and Constructive Mathematics

Part of the The Western Ontario Series in Philosophy of Science Book Series (volume 76)

Editors: Sommaruga, Giovanni

DOI: 10.1007/978-94-007-0431-2

ISBN: 978-94-007-0430-5

Publisher: Springer Netherlands - Springer Journals

Feb 28, 2011

1 - 10 of 15 Chapters

Available

Foundational Frameworks

Hellman*, Geoffrey

Pages: 53 –69

[A good place to begin is with a delineation of the sorts of questions that foundational frameworks for mathematics, as we conceive them, can usefully address and which motivate their development in the first place. This will in turn guide us in framing some important desiderata that such...
Available

The Problem of Mathematical Objects

Hale, Bob

Pages: 71 –84

[In seeking a foundation for mathematics, one may be looking for what may be called a foundation in the logical sense: a single, unified set of principles—perhaps unified by their jointly constituting an acceptable axiomatization of some concept or concepts plausibly taken as fundamental—from...
Available

Set Theory as a Foundation

Maddy, Penelope

Pages: 85 –96

[The view of set theory as a foundation for mathematics emerged early in the thinking of the originators of the theory and is now a pillar of contemporary orthodoxy. As such, it is enshrined in the opening pages of most recent textbooks; to take a few illustrative examples: All branches of...
Available

Foundations: Structures, Sets, and Categories

Shapiro, Stewart

Pages: 97 –110

[Recent years have seen a wealth of discussion on the topic of the foundations of mathematics, and the extent to which category theory, set theory, or some other framework serves, or can serve, as a foundation, or the foundation of some, most, or all of mathematics. Of course, adjudications of...
Available

From Sets to Types, to Categories, to Sets

Awodey, Steve

Pages: 113 –125

[Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of...
Available

Enriched Stratified Systems for the Foundations of Category Theory

Feferman, Solomon

Pages: 127 –143

[This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than 35 years. The first three were “Set-theoretical foundations of category theory” (1969), “Categorical foundations and foundations of category theory” (1977), and much more...
Available

Recent Debate over Categorical Foundations

McLarty, Colin

Pages: 145 –154

[The proposal by Shapiro (2009, p. 76) “to sharpen the battle lines a little” around categorical philosophy and foundations for mathematics suggests also extending the lines to include the original publications by mathematicians William Lawvere (1963, 1964, 1966) and Saunders Mac Lane (1986, 1998).]
Available

The Axiom of Choice in the Foundations of Mathematics

Bell, John L.

Pages: 157 –169

[The principle of set theory known as the Axiom of Choice ( AC) has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s axiom of parallels which was introduced more than two thousand years ago”. It...
Available

Reflections on the Categorical Foundations of Mathematics

Lambek, Joachim; Scott, Philip J.

Pages: 171 –186

[Most practicing mathematicians see no need for the foundations of their subject. But those who wish to place it on a solid ground usually pick set theory, an axiomatic treatment of the membership relation expressed in first order logic. Some of us feel that higher order logic is more appropriate...
Available

Local Constructive Set Theory and Inductive Definitions

Aczel, Peter

Pages: 189 –207

[Local Constructive Set Theory (LCST) is intended to be a local version of constructive set theory (CST). Constructive Set Theory is an open-ended set theoretical setting for constructive mathematics that is not committed to any particular brand of constructive mathematics and, by avoiding any...

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