# Abel's Proof:

# An Essay on the Sources and Meaning of Mathematical Unsolvability

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 fiancée.

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.

"Pesic's book is a good place to begin to learn about this important piece of intellectual history." , Fernando Q. Gouvea, American Scientist

"Peter Pesic's tale of how maths came to be is as exciting as any fiction."—Economist

"A unique book. Peter Pesic's chronicle of the long road mathematicians traveled toward understanding when an equation can be solved—and when it can't—is enjoyable, lucid, and user-friendly. The author takes pains to credit less familiar names such as Vi'te and Ruffini and requires of his readers no more than basic algebra—and most of that placed conveniently apart from the main text."

—Tony Rothman, Department of Physics, Bryn Mawr College

"Peter Pesic writes about Abel's work with enthusiasm and sensitivity, beautifully evoking this marvelous moment in the development of algebra."

—Barry Mazur, Gerhard Gade University Professor, Harvard University

"Readers of Pesic's fascinating little book will be led to an inescapable verdict: Niels Abel was guilty of ingenuity in the fifth degree."

—William Dunham, Muhlenberg College, and author of Journey through Genius: The Great Theorems of Mathematics

"This book is a splendid essay on Abel's proof that the general quintic cannot be solved by radicals. The author does an excellent job of providing the historical and mathematical background so that the reader can understand why this question is so compelling. The vivid nontechnical style of the text captures the intricate dance of mathematics and the passionate lives of the people involved."

—David A. Cox, Department of Mathematics and Computer Science, Amherst College

Also available in German translation *Abels Beweis* (Heidelberg: Springer Verlag, 2005); Italian translation *La prova di Abel: Saggio sulle fonti e sul significato della irrisolvibilità in matematica* (Torino: Bollati Boringhieri, 2005, reissued 2018 by Corriere della Sera); Japanese translation **ア****ー****ベルの****証****明****―****解けない方程式を解く**(Tokyo: Nippon Hyoronsha, 2005); Norwegian translation *Abels bevis: Å lose det uløselige* (Oslo: Athena forlag, 2005).