Abstract
The book gives an informal but thorough introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics has been completely revised for this second edition. Brouwer’s proof of the Bar Theorem has been reworked, the account of valuation systems simplified, and the treatment of generalized Beth Trees and the completeness of intuitionistic first-order logic rewritten. The work explores the philosophical foundations of constructive mathematics, examining how the rejection of the law of excluded middle leads to a fundamentally different approach to mathematical reasoning. Dummett presents intuitionism as both a mathematical program initiated by Brouwer and a broader philosophical position about the nature of mathematical truth and existence.
Outline
Chapter 1: Preliminaries
- Constructive proofs
- The meaning of logical constants
- Sample of logical principles
- Functional Completeness
Chapter 2: Elementary Intuitionistic Mathematics
- Heyting Arithmetic and basic number theory
- Real numbers in constructive mathematics
- Order relations and their properties
- The axiom of choice in intuitionistic context
Chapter 3: Choice Sequences and Spreads
- The concept of choice sequences
- Infinity in constructive mathematics
- The fan theorem
- Bar induction principles
- The continuity principle
- Brouwer’s Bar Theorem and its proof
- Continuous functionals and their representation
- The uniform continuity theorem
Chapter 4: The Formalization of Intuitionistic Logic
- Natural deduction systems
- The sequent calculus
- Cut-elimination procedures
- Decidability of intuitionistic sentential logic
- Normalization theorems
Chapter 5: Semantics for Intuitionistic Logic
- Valuation systems and their properties
- Lattices and finite model property
- Topological semantics
- Beth trees and their applications
- Semantics for intuitionistic predicate logic
- Completeness of intuitionistic predicate logic
- Generalized Beth trees
- Compactness theorems
Chapter 6: Some Further Topics
- Intuitionistic formal systems
- Realizability interpretations
- The Creative Subject as idealized mathematician
- Thought-experiments for motivating axioms
Chapter 7: Philosophical Remarks
- Philosophical foundations of constructive mathematics
- The notion of proof in intuitionism
- Partial functions and their role
- Logical constants as represented on Beth trees
- The notion of choice sequences and their philosophical significance
Themes
The work addresses the fundamental tension between classical and intuitionistic approaches to mathematics, exploring how the rejection of the principle of bivalence leads to a reconstruction of mathematical reasoning. Dummett connects Brouwer’s mathematical intuitionism to broader questions in the philosophy of language and anti-realism, arguing that mathematical statements should be understood in terms of what we can construct or prove rather than in terms of objective truth conditions.