This original anthology assembles eleven accessible essays by a giant of modern mathematics. Hermann Weyl (1885-1955) made lasting contributions to number theory as well as theoretical physics, and he was associated with Princeton's Institute for Advanced Study, the University of Gottingen, and ETH Zurich. Spanning the 1930s-50s, these articles offer insights into logic and relativity theory in addition to reflections on the work of Weyl's mentor, David Hilbert, and his friend Emmy Noether.
Subjects include "Topology and Abstract Algebra as Two Roads of Mathematical Comprehension," "The Mathematical Way of Thinking," "Relativity Theory as a Stimulus in Mathematical Research," and "Why is the World Four-Dimensional?" Historians of mathematics, advanced undergraduates, and graduate students will appreciate these writings, many of which have been long unavailable to English-language readers.

1824 gelang einem jungen Norweger namens Niels Henrik Abel der endgultige Beweis, dass algebraische Gleichungen funften Grades im allgemeinen nicht durch Wurzeln auflosbar sind. In diesem Buch zeigt Peter Pesic auf, welche Bedeutung diesem Ereignis in der Geschichte des Denkens zukommt. Es ist aber auch eine bemerkenswerte menschliche Geschichte, denn Abel war einundzwanzig, als er seinen Beweis auf eigene Kosten veroffentlichte, und funf Jahre spater starb er, verarmt und deprimiert, kurz bevor sein Beweis begann, weite Anerkennung zu finden. Abels Versuche, die mathematische Elite seiner Zeit zu erreichen, erlebten eine verachtliche Abweisung; es war ihm nicht moglich, eine Anstellung zu finden, die es ihm erlaubte, in Ruhe zu arbeiten und seine Verlobte zu heiraten.

Aber Pesics Geschichte beginnt lange vor Abels Zeit und setzt sich bis zum heutigen Tage fort, denn Abels Beweis anderte die Art und Weise, wie wir uber Mathematik und ihren Bezug zur "wirklichen" Welt nachdenken.

Beginnend bei den Griechen, bei denen die Idee der mathematischen Beweise entstand, zeigt Pesic, wie die Mathematik ihre Ursprunge im realen Leben nahm (den Formen von Sachen, den Buchfuhrungsbedarf von

Kaufleuten) und dann uber diese Ursprunge hinaus auf etwas Umfassenderes zu zielen. Die Versuche der Pythagoraer, mit irrationalen Grossen umzugehen, kundigen das langsame Entstehen der abstrakten Mathematik an. Pesic konzentriert sich auf die umstrittene Entwicklung der Algebra - der sogar Newton widerstand - und der allmahlichen Anerkennung ihres Nutzens und ihrer Schonheit in der Abstraktrion, die Realitaten in Dimensionen jenseits menschlicher Erfahrung zu beschworen scheint. Pesic erzahlt diese Geschichte hauptsachlich als eine Geschichte der Ideen; mathematische Details werden ausserhalb des Haupttextes ausgefuhrt. Das Buch enthalt auch eine neue, kommentierte Ubersetzung von Abels originalem Beweis. "

Hermann Weyl (1885-1955) was one of the twentieth century's most important mathematicians, as well as a seminal figure in the development of quantum physics and general relativity. He was also an eloquent writer with a lifelong interest in the philosophical implications of the startling new scientific developments with which he was so involved. "Mind and Nature" is a collection of Weyl's most important general writings on philosophy, mathematics, and physics, including pieces that have never before been published in any language or translated into English, or that have long been out of print. Complete with Peter Pesic's introduction, notes, and bibliography, these writings reveal an unjustly neglected dimension of a complex and fascinating thinker. In addition, the book includes more than twenty photographs of Weyl and his family and colleagues, many of which are previously unpublished.

Included here are Weyl's exposition of his important synthesis of electromagnetism and gravitation, which Einstein at first hailed as "a first-class stroke of genius"; two little-known letters by Weyl and Einstein from 1922 that give their contrasting views on the philosophical implications of modern physics; and an essay on time that contains Weyl's argument that the past is never completed and the present is not a point. Also included are two book-length series of lectures, "The Open World" (1932) and "Mind and Nature" (1934), each a masterly exposition of Weyl's views on a range of topics from modern physics and mathematics. Finally, four retrospective essays from Weyl's last decade give his final thoughts on the interrelations among mathematics, philosophy, and physics, intertwined with reflections on the course of his rich life.

The intellectual and human story of a mathematical proof that transformed our ideas about mathematics. In 1824 a young Norwegian named Niels Henrik Abel proved conclusively that algebraic equations of the fifth order are not solvable in radicals. In this book Peter Pesic shows what an important event this was in the history of thought. He also presents it as a remarkable human story. Abel was twenty-one when he self-published his proof, and he died five years later, poor and depressed, just before the proof started to receive wide acclaim. Abel's attempts to reach out to the mathematical elite of the day had been spurned, and he was unable to find a position that would allow him to work in peace and marry his fiance. But Pesic's story begins long before Abel and continues to the present day, for Abel's proof changed how we think about mathematics and its relation to the "real" world. Starting with the Greeks, who invented the idea of mathematical proof, Pesic shows how mathematics found its sources in the real world (the shapes of things, the accounting needs of merchants) and then reached beyond those sources toward something more universal. The Pythagoreans' attempts to deal with irrational numbers foreshadowed the slow emergence of abstract mathematics. Pesic focuses on the contested development of algebra-which even Newton resisted-and the gradual acceptance of the usefulness and perhaps even beauty of abstractions that seem to invoke realities with dimensions outside human experience. Pesic tells this story as a history of ideas, with mathematical details incorporated in boxes. The book also includes a new annotated translation of Abel's original proof.