Gnomes in the Fog: The Reception of Brouwer's Intuitionism in the 1920s

ISBN-13: 9783034893947 / Angielski / Miękka / 2012 / 448 str.

Dennis E. Hesseling; Dennis E. Hesseling

Opis

The significance of foundational debate in mathematics that took place in the 1920s seems to have been recognized only in circles of mathematicians and philosophers. A period in the history of mathematics when mathematics and philosophy, usually so far away from each other, seemed to meet. The foundational debate is presented with all its brilliant contributions and its shortcomings, its new ideas and its misunderstandings.

Recenzje

"L.E.J. Brouwer is best known to many mathematicians for his seminal contributions to topology. He is also the founder of mathematical intuitionism, and a key player in the debate on foundations of mathematics that raged for a brief decade in the 1920s, and then subsided. Gnomes in the Fog tells the story of that important and influential episode in the history of mathematics, in fascinating and delicious detail.... [A]nyone with an interest in mathematics and its history and philosophy, should enjoy this book. Mathematicians (especially logicians) may find some surprises in the first chapter, on Brouwer's predecessors; philosophers and science study scholars should especially appreciate the final chapter on the cultural context of the debate.... One of the many treasures to be discovered in reading this book is the rich collection of original quotes in the many languages in which the debate took place, along with the author's translations."

-MAA Online

Spis treści

1 Kronecker, the semi-intuitionists, Poincaré.- 1.1 Introduction.- 1.1.1 Mathematical prerequisites.- 1.2 Kronecker.- 1.2.1 Kronecker’s conflicts.- 1.2.2 Kronecker’s views.- 1.3 The French semi-intuitionists.- 1.3.1 The French semi-intuitionists’ main conflict.- 1.3.2 The French semi-intuitionists’ views.- 1.4 Poincaré.- 1.4.1 Poincaré’s conflicts.- 1.4.2 Poincaré’s views.- 1.5 Conclusion.- 2 The genesis of Brouwer’s intuitionism.- 2.1 Introduction.- 2.2 The early years.- 2.2.1 Brouwer’s youth.- 2.2.2 Brouwer’s profession of faith.- 2.2.3 Mannoury.- 2.2.4 Brouwer’s mysticism.- 2.3 The first act of intuitionism.- 2.3.1 Brouwer’s dissertation.- 2.3.2 The unreliability of the logical principles.- 2.4 Topology.- 2.5 Intuitionism and formalism.- 2.6 The second act of intuitionism.- 2.6.1 Intuitionistic set theory.- 2.6.2 Further development of intuitionistic mathematics.- 2.7 The Brouwer lectures.- 2.7.1 Berlin.- 2.7.2 Amsterdam.- 2.7.3 Vienna.- 2.8 The Mathematische Annalen and afterwards.- 2.9 Brouwer’s personality.- 2.10 Conclusion.- 3 Overview of the foundational debate.- 3.1 Introduction.- 3.2 Quantitative inquiry.- 3.2.1 The Fortschritte.- 3.2.2 ‘All’ public reactions to intuitionism.- 3.3 Qualitative inquiry.- 3.3.1 Themes.- 3.3.2 Tone.- 3.3.3 Currents and schools.- 3.3.4 People.- 3.3.5 Languages and media.- 3.4 Conclusion.- 4 Reactions: existence and constructivity.- 4.1 Introduction.- 4.1.1 Mathematical existence.- 4.1.2 A short history of constructivism.- 4.2 The beginning of the debate.- 4.2.1 Weyl’s Grundlagenkrise.- 4.2.2 Hilbert’s first reactions.- 4.2.3 Becker’s phenomenology.- 4.2.4 Fraenkel’s early commentaries.- 4.2.5 Baldus’ rector’s address.- 4.3 The debate widened.- 4.3.1 Existence in a central position.- 4.3.2 Existence as a minor subject.- 4.4 Later reactions.- 4.4.1 The Königsberg conference.- 4.4.2 Wittgenstein.- 4.4.3 Others.- 4.5 Conclusion.- 5 Reactions: logic and the excluded middle.- 5.1 Introduction.- 5.1.1 A short history of classical logic.- 5.2 The beginning of the debate.- 5.2.1 Weyl’s Grundlagenkrise.- 5.2.2 Hilbert’s first reactions.- 5.2.3 Addresses: Wolff, Finsler and Baldus.- 5.2.4 Fraenkel’s early commentaries.- 5.3 The debate widened.- 5.3.1 The excluded middle in a central position.- 5.3.2 The excluded middle as a minor subject.- 5.4 Later reactions.- 5.4.1 Glivenko, Heyting and Kolmogorov.- 5.4.2 Gödel.- 5.4.3 Barzin and Errera.- 5.5 Conclusion.- 6 The foundational crisis in its context.- 6.1 Introduction.- 6.2 Metaphors.- 6.2.1 Crisis and revolution.- 6.3 Philosophy.- 6.3.1 Lebensphilosophie.- 6.3.2 Mathematical and philosophical intuitionism: a comparison.- 6.3.3 Contemporaries’ remarks.- 6.3.4 Göttingen and Hilbert.- 6.3.5 Spengler.- 6.3.6 Summary.- 6.4 Physics.- 6.4.1 Theory of relativity.- 6.4.2 Quantum mechanics.- 6.5 Art.- 6.5.1 Constructivism.- 6.6 Politics.- 6.6.1 Mathematics and the rise of the Third Reich.- 6.6.2 Bieberbach’s racial interpretation of the foundational debate.- 6.7 Moderne and Gegenmoderne.- 6.8 Conclusion.- Conclusion.- A Chronology of the debate.- B Public reactions to Brouwer’s intuitionism.- C Logical notations.- Dankwoord/ Acknowledgements.