Dirk van Dalen’s biography studies the fascinating life of the famous Dutch mathematician and philosopher Luitzen Egbertus Jan Brouwer. Brouwer belonged to a special class of genius; complex and often controversial and gifted with a deep intuition, he had an unparalleled access to the secrets and intricacies of mathematics. Most mathematicians remember L.E.J. Brouwer from his scientific breakthroughs in the young subject of topology and for the famous Brouwer fixed point theorem. Brouwer’s main interest, (...) however, was in the foundation of mathematics which led him to introduce, and then consolidate, constructive methods under the name ‘intuitionism’. This made him one of the main protagonists in the ‘foundation crisis’ of mathematics. As a confirmed internationalist, he also got entangled in the interbellum struggle for the ending of the boycott of German and Austrian scientists. This time during the twentieth century was turbulent; nationalist resentment and friction between formalism and intuitionism led to the Mathematische Annalen conflict. It was here that Brouwer played a pivotal role. The present biography is an updated revision of the earlier two volume biography in one single book. It appeals to mathematicians and anybody interested in the history of mathematics in the first half of the twentieth century. (shrink)

We present a new English translation of L.E.J. Brouwer's paper ‘De onbetrouwbaarheid der logische principes’ of 1908, together with a philosophical and historical introduction. In this paper Brouwer for the first time objected to the idea that the Principle of the Excluded Middle is valid. We discuss the circumstances under which the manuscript was submitted and accepted, Brouwer's ideas on the principle of the excluded middle, its consistency and partial validity, and his argument against the possibility of (...) absolutely undecidable propositions. We note that principled objections to the general excluded middle similar to Brouwer's had been advanced in print by Jules Molk two years before. Finally, we discuss the influence on George Griss' negationless mathematics. (shrink)

The ArgumentL. E. J. Brouwer and David Hubert, two titans of twentieth-century mathematics, clashed dramatically in the 1920s. Though they were both Kantian constructivists, their notoriousGrundlagenstreitcentered on sharp differences about the foundations of mathematics: Brouwer was prepared to revise the content and methods of mathematics (his “Intuitionism” did just that radically), while Hilbert's Program was designed to preserve and constructively secure all of classical mathematics.Hilbert's interests and polemics at the time led to at least three misconstruals of intuitionism, (...) misconstruals which last to our own time: Current literature often portrays popular views of intuitionism as the product of Brouwer's idiosyncratic subjectivism; modern logicians view intuitionism as simply applying a non-standard formal logic to mathematics; and contemporary philosophers see that logic as based upon a pure assertabilist theory of meaning. These pictures stem from the way Hilbert structured the controversy.Even though Brouwer's own work and behavior occasionally reinforce these pictures, they are nevertheless inaccurate accounts of his approach to mathematics. However, the framework provided by the Brouwer-Hilbert debate itself does not supply an adequate correction of these inaccuracies. For, even if we eliminate these mistakes within that framework, Brouwer's position would still appear fragmented and internally inconsistent. I propose a Kantian framework — not from Kant's philosophy of mathematics but from his general metaphysics — which does show the coherence and consistency of Brouwer's views. I also suggest that expanding the context of the controversy in this way will illuminate Hilbert's views as well and will even shed light upon Kant's philosophy. (shrink)

"This comprehensive volume - containing 27 chapters and contributions from six continents - presents and discusses key principles, perspectives, and practices of social learning in the context of sustainability. Social learning is explored from a range of fields challenged by sustainability including: organizational learning, environmental management and corporate social responsibility; multi-stakeholder governance; education, learning and educational psychology; multiple land-use and integrated rural development; and consumerism and critical consumer education. An entire section of the book is devoted to a number of (...) reflective case studies of people, organizations and communities using forms of social learning in moving towards sustainability. 'This book brings together a range of ideas, stories, and discussions about purposeful learning in communities aimed at creating a world that is more sustainable than the one currently in prospect....The book is designed to expand the network of conversations through which our society can confront various perspectives, discover emerging patterns, and apply learning to a variety of emotional and social contexts.' From the Foreword by Fritjof Capra, co-founder of the Center of Ecoliteracy. 'Joining what is so clear and refreshing in this book with the larger movements toward a critically democratic and activist education that is worthy of its name, is but one step in the struggle for sustainability. But it is an essential step if we are to use the insights that are included in this book.' From the Afterword by Michael Apple, author of 'Educating the ""Right"" Way: Markets, Standards, God, and Inequality'.". (shrink)

We live in turbulent times, our world is changing at accelerating speed. Information is everywhere, but wisdom appears in short supply when trying to address key inter-related challenges of our time such as; runaway climate change, the loss of biodiversity, the depletion of natural resources, the on-going homogenization of culture, and rising inequity. Living in such times has implications for education and learning. This book explores the possibilities of designing and facilitating learning-based change and transitions towards sustainability. In 31 chapters (...) contributors from across the world discuss (re)emerging forms of learning that not only assist in breaking down unsustainable routines, forms of governance, production and consumption, but also can help create ones that are more sustainable. The book has been divided into three parts: re-orienting science and society, re-connecting people and planet and re-imagining education and learning. This is essential reading for educators, educational designers, change agents, researchers, students, policymakers and entrepreneurs alike, who are concerned about the well-being of the planet and convinced of our ability to do better. The content and related issues can be discussed on the blog by editor Arjen Wals: Transformative learning. 'We are living in times of incertitude, complexity, and contestation, but also of connectivity, responsibility, and new opportunities. This book analyses the consequences of these times for learning in formal, non-formal, and informal education. It explores the possibilities offered by the concept of sustainability as a central category of a holistic paradigm which harmonizes human beings with Earth. To change people and to change the world are interdependent processes - this book contributes to both.' Moacir Gadotti, Director of Paulo Freire Institute, São Paulo, Brazil 'I hope you share my excitement about the innovations for sustainability that this book catalogues and analyses. While the ecological news is grim, the human news is not. Even in a time of accelerating change, people are showing their enormous capacities to learn, adapt, restore and protect.' From the Foreword by Juliet Schor, author of ‘True Wealth: how and why millions of Americans are creating a time-rich, ecologically-light, small-scale high-satisfaction economy’ 'This book implies a ‘culture of critical commitment’ in educational thinking and practice - engaged enough to make a real difference to social-ecological resilience and sustainability but reflexively critical enough to learn constantly from experience and to keep options open in working for a sustainability transformation.' From the Afterword by Stephen Sterling, Professor of Sustainability Education, Centre for Sustainable Futures, Plymouth University, United Kingdom. (shrink)

This theory accords to volitions the status of basic mental actions, maintaining that these are spontaneous exercises of the will--a "two-way" power which ...

L. E. J. Brouwer, the founder of mathematical intuitionism, believed that mathematics and its objects must be humanly graspable. He initiated a program rebuilding modern mathematics according to that principle. This book introduces the reader to the mathematical core of intuitionism – from elementary number theory through to Brouwer's uniform continuity theorem – and to the two central topics of 'formalized intuitionism': formal intuitionistic logic, and formal systems for intuitionistic analysis. Building on that, the book proposes a systematic, (...) philosophical foundation for intuitionism that weaves together doctrines about human grasp, mathematical objects and mathematical truth. (shrink)

The paper examines Andrei A. Markov’s critical attitude towards L.E.J. Brouwer’s intuitionism, as is expressed in his endnotes to the Russian translation of Heyting’s Intuitionism, published in Moscow in 1965. It is argued that Markov’s algorithmic approach was shaped under the impact of the mathematical style and values prevailing in the Petersburg mathematical school, which is characterized by the proclaimed primacy of applications and the search for rigor and effective solutions.