Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability

Overview

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 ...
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Overview

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 fiancee. 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 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.
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Editorial Reviews

From the Publisher

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

The MIT Press

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Product Details

  • ISBN-13: 9780262162166
  • Publisher: MIT Press
  • Publication date: 5/1/2003
  • Pages: 221
  • Product dimensions: 5.64 (w) x 8.36 (h) x 0.68 (d)

Meet the Author

Peter Pesic is Tutor and Musician-in-Residence at St. John's College, Santa Fe. He is the author of Labyrinth: A Search for the Hidden Meaning of Science; Seeing Double: Shared
Identities in Physics, Philosophy, and Literature; Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability
; and Sky in a Bottle, all published by the MIT Press.
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Table of Contents

Introduction 1
1 The Scandal of the Irrational 5
2 Controversy and Coefficients 23
3 Impossibilities and Imaginaries 47
4 Spirals and Seashores 59
5 Premonitions and Permutations 73
6 Abel's Proof 85
7 Abel and Galois 95
8 Seeing Symmetries 111
9 The Order of Things 131
10 Solving the Unsolvable 145
App. A Abel's 1824 Paper 155
App. B Abel on the General Form of an Algebraic Solution 171
App. C Cauchy's Theorem on Permutations 175
Notes 181
Acknowledgments 203
Index 205
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