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

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MIT Press, Feb 27, 2004 - Mathematics - 213 pages

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 fiancÚ.

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.

 

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User Review  - FPdC - LibraryThing

The formula for the solution of the general equation of the third degree by the italians Del Ferro, Tartaglia, and Cardano, in the 16th Century, was one of the triumphs of Renaissance mathematics, and ... Read full review

LibraryThing Review

User Review  - gam3 - LibraryThing

If you enjoy math history you should enjoy this book. Abel had a rather difficult life, not quit as sad as Galois', but not a happy one by any means. The math in this book is easy to follow, and might be more assessable than most as the proof covers something that most learn in High School Algebra. Read full review

Contents

The Scandal of the Irrational
5
Controversy and Coefficients
23
Impossibilities and Imaginaries
47
Spirals and Seashores
59
Premonitions and Permutations
73
Abels Proof
85
Abel and Galois
95
Seeing Symmetries
111
Solving the Unsolvable
145
Abels 1824 Paper
155
Abel on the General Form of an Algebraic Solution
171
Cauchys Theorem on Permutations
175
Notes
181
Acknowledgments
203
Index
205
Copyright

The Order of Things
131

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About the author (2004)

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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