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This is an introduction to, and survey of, the constructive approaches to pure mathematics. The authors emphasise the viewpoint of Errett Bishop's school, but intuitionism. Russian constructivism and recursive analysis are also treated, with comparisons between the various approaches included where appropriate. Constructive mathematics is now enjoying a revival, with interest from not only logicans but also category theorists, recursive function theorists and theoretical computer scientists. This account for non-specialists in these and other disciplines.

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• 1. The Foundations of Contructive Mathematics
• 1. Existence and omniscience(p. 1)
• 2. Basic constructions(p. 6)
• 3. Informal intuitionistic logic(p. 11)
• 4. Choice axioms(p. 11)
• 5. Real numbers(p. 12)
• Problems(p. 14)
• Notes(p. 16)
• 2. Constructive Analysis
• 1. Complete metric spaces(p. 18)
• 2. Baire's theorem revisited(p. 21)
• 3. Located subsets(p. 26)
• 4. Totally bounded spaces(p. 28)
• 5. Bounded linear maps(p. 34)
• 6. Compactly generated Banach spaces(p. 41)
• Problems(p. 44)
• Notes(p. 47)
• 3. Russian Constructive Mathematics
• 1. Programming systems and omniscience principles(p. 49)
• 2. Continuity and intermediate values(p. 54)
• 3. Specker's sequence(p. 58)
• 4. The Heine-Borel theorem(p. 60)
• 5. Moduli of continuity and cozero sets(p. 64)
• 6. Ceitin's theorem(p. 67)
• Problems(p. 71)
• Notes(p. 73)
• 4. Constructive Algebra
• 1. General considerations(p. 75)
• 2. Factoring(p. 76)
• 3. Splitting fields(p. 79)
• 4. Uniqueness of splitting fields(p. 82)
• 5. Finitely presented modules(p. 87)
• 6. Noetherian rings(p. 91)
• Problems(p. 97)
• Notes(p. 99)
• 5. Intuitionism
• 1. Sequence spaces(p. 103)
• 2. Continuous choice(p. 106)
• 3. Uniform continuity(p. 110)
• 4. The creating subject and Markov's principle(p. 116)
• Problems(p. 117)
• Notes(p. 119)
• 6. Contrasting Varieties
• 1. The three varieties(p. 120)
• 2. Positive-valued continuous functions(p. 122)
• Problems(p. 129)
• Notes(p. 130)
• 7. Intuitionistic Logic and Topos Theory
• 1. Intuitionistic propositional calculus(p. 131)
• 2. Predicate calculus(p. 134)
• 3. The sheaf model C(X)(p. 138)
• 4. Presheaf topos models(p. 140)
• Problems(p. 143)
• Notes(p. 144)
• Index(p. 146)

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CHOICE Review

Many theorems of classical mathematics demonstrate than an object exists without providing any suggestion of how the object might be constructed; others show that some quantity is finite without indicating how that quality might be computed. In such situations, one might seek a new, more constructive proof of the theorem, or try to analyze the constructive content of the proof one already has. Constructive mathematics takes a more radical approach. It seeks new foundations for mathematics in which only statements that have been proved constructively are considered valid. First one needs to decide what constitutes a construction, a philosophical problem; then one must develop a correspondly appropriate axiomatic system and explore its consequences, a mathematical problem. Bridges and Richman provide a readable introduction to the most important approaches to the mathematical side of constructive mathematics. Constructive mathematicians often use familiar terminology in new ways, leading to what seems, to a classically trained mathematician, like a topsy-turvy world, where familiar facts are invalid and strange things become true. This book emphasizes what are the important differences between the various constructive approaches and conventional mathematics. Highly recommended for all libraries.-D.V. Feldman, University of New Hampshire