Intuitionistic Analysis
A Constructive Frame of Mind
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Beschreibung
This book introduces the core ideas of L.E.J. Brouwer’s approach to constructivity in mathematics, focusing on analysis, set theory, and topology, while considering his philosophical motivations. Brouwer’s “intuitionism” offers a coherent alternative to classical (nonconstructive) mathematics.
Starting with the rejection of the Principle of the Excluded Middle, the book reconstructs number systems and analysis using Cauchy sequences. It compares constructive and classical methods, highlights where classical theorems fail through “weak counterexamples”, and examines Brouwer’s classical and constructive versions of the Fixed-Point Theorem. Intuitionistic concepts like choice sequences and the Creating Subject lead to surprising results, such as the continuity of all total real functions and the existence of effective but non-recursive functions. Brief but fundamental comparisons are made with the later alternatives of Markov and Bishop.
Intended as an introduction for undergraduates, this book is suitable for mathematics students interested in philosophy as well as philosophers with some mathematical background.
This book introduces the core ideas of L.E.J. Brouwer’s approach to constructivity in mathematics, focusing on analysis, set theory, and topology, while considering his philosophical motivations. Brouwer’s “intuitionism” offers a coherent alternative to classical (nonconstructive) mathematics.
Starting with the rejection of the Principle of the Excluded Middle, the book reconstructs number systems and analysis using Cauchy sequences. It compares constructive and classical methods, highlights where classical theorems fail through “weak counterexamples”, and examines Brouwer’s classical and constructive versions of the Fixed-Point Theorem. Intuitionistic concepts like choice sequences and the Creating Subject lead to surprising results, such as the continuity of all total real functions and the existence of effective but non-recursive functions. Brief but fundamental comparisons are made with the later alternatives of Markov and Bishop.
Intended as an introduction for undergraduates, this book is suitable for mathematics students interested in philosophy as well as philosophers with some mathematical background.
Presents general constructivism with a discussion of Brouwer’s perspective Prioritises mathematical content over logic and formalism Makes systematic use of "weak and strong counterexamples" to demonstrate how classical laws can fail
Autor*in
Dirk van Dalen
Themen in »Intuitionistic Analysis«
Intuitionistic mathematics Constructive mathematics Constructive analysis Philosophy of mathematics Foundations of mathematics Intuitionism in mathematics L.E.J. Brouwer Constructivism in mathematics Intuitionistic logic Constructive logic Weak counterexamples Choice sequences Constructive real numbers Brouwer's Fixed Point Theorem Creating Subject