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.
Dirk van Dalen is a professor emeritus of philosophy and history of mathematics and logic at Utrecht University. A former student of Arend Heyting, his research interests are intuitionism, constructive mathematics, and their philosophy and history. His publications include the textbook Logic and Structure (5th edition, Springer, 2013) and a biography of L.E.J. Brouwer (Oxford University Press, 1999 and 2005; Springer, 2013).
Mark van Atten is a senior researcher at CNRS, France and a member of the Archives Husserl in Paris. A former student of Dirk van Dalen, his field is the philosophy of mathematics, specifically the intersection of Brouwer's intuitionism and Husserl's phenomenology. Among his publications are On Brouwer (Wadsworth, 2004); Brouwer Meets Husserl (Springer, 2007); and Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer (Springer, 2015).
Craig Smoryński is a mathematician and independent scholar who earned his degree at the University of Illinois in Chicago and spent several years in the Netherlands, where his knowledge of intuitionism matured. He has made contributions to logic, proof theory, arithmetic, and their historiography. He is the author of several texts, including Self-Reference and Modal Logic (Springer, 1985), Logical Number Theory I (Springer, 1991), and Adventures in Formalism (College Publications, 2012).
