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

by Peter Pesic (Author) "In 1824, a young Norwegian named Niels Henrik Abel published a small pamphlet announcing a new mathematical proof for an old problem..." (more)
Key Phrases: abelian equations, irrational magnitudes, quintic equation, Fundamental Theorem of Algebra, Évariste Galois, Luca Pacioli (more...)

Editorial Reviews

Review
"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 Viète 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

"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 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

Product Description
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.

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Product Details

• Hardcover: 221 pages
• Publisher: The MIT Press; annotated edition edition (May 1, 2003)
• Language: English
• ISBN-10: 0262162164
• ISBN-13: 978-0262162166
• Product Dimensions: 8 x 5.5 x 0.8 inches
• Shipping Weight: 13.4 ounces
• Average Customer Review: 4.4 out of 5 stars See all reviews (9 customer reviews)
• Amazon.com Sales Rank: #1,563,171 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

76 of 80 people found the following review helpful:

5.0 out of 5 stars Unsolvable yet quite graspable, November 14, 2003

By	James (Portland, OR) - See all my reviews

To me, Abel's Proof successfully bridges the difficult gap that separates math books from fun books. Being one who appreciates the history and development of ideas and who is not afraid of a few equations, my needs as a reader were tastefully satisfied. If you, like me, find yourself enticed by some of the more subtle problems in math and science, while at the same time, have not the recourse to explore each one to their fullest, this book will be a welcome guide. Pesic uses Niels Abel's proof (1824) regarding the general insolvability by radicals of fifth degree equations as the central trunk of a robust tree whose branches contain delightful episodes of mathematical examples, human dramas, twists of fate, and historical parades. As much a biography as anything else, I could feel the personalities of the mathematicians evinced through their contributions to the question of solvability. From the near misses of Ruffini and Gauss to the final QEDs of Abel and Galois, one sees the human elements of struggle, triumph, anger, and success, set thoughtfully alongside the mathematical details. Carefully arranged mathematical sidebars allow this book to be read with as much technical intent as one chooses to bring; the math is there for the taking (little goes beyond a basic familiarity with algebra). In short, this book offers a delightful way to see some intriguing math and the characters who made it happen.

33 of 36 people found the following review helpful:

4.0 out of 5 stars Nice mixture of history and popular explanation, August 8, 2004

By	Bukkene Bruse (Iowa) - See all my reviews

This review is from: Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability (Paperback)

Pesic tells a very deep and broad story in about 150 pages of core text. In the first sixty or so pages, Pesic does a great job of covering the history of what people understood to be a solution of an algebraic equation, and hence the evolution of the notion of number. Starting with how the Greeks moved from understanding whole numbers and rational numbers to discovering the irrational roots, he moves gracefully to the understanding of imaginary, and then complex numbers in the 1600's.

The flow of the book is rougher for the next 25 pages or so, as the mathematics becomes less elegant, really quite a zoo. Attempts here to give a verbal explanation of the mathematics confuse more than they enlighten. The last half of the book is the meat of the work and is also the best done. Beginning with Abel's tragic personal story and interweaving the lives and work of other mathematicians of the time, in particular the other famous tragedy of Galois, Pesic then moves on to a very lucid description of elementary group theory. Also touched upon are transcendental numbers and matrices. The last chapters on what it all means for science and human understanding summed up the message of the book quite nicely.

I recommend the book for anyone looking to understand a bit more about pure mathematics. It is short, easy to read, and extremely well written and reasoned in the main.

One gripe: Pesic refers to two Persian mathematicians, Omar Khayyam and al-Khwarizimi, as Arabs. Both are from historic Khorasan province which is now in either northeastern Iran or in Uzbekistan and spoke Farsi or a Farsi variant, not Arabic, as their native language (http://en.wikipedia.org/wiki/Al-Khawarizmi, http://en.wikipedia.org/wiki/Omar_Khayyam). Persians are not Arabs, and al-Khwarizimi writing his math in Arabic doesn't make him so. Pesic does manage to tell the Europeans apart, and did somehow figure out that Abel was Norwegian even though he never wrote a math paper in Dano-Norwegian or Swedish.

10 of 10 people found the following review helpful:

5.0 out of 5 stars a nice little gem, January 23, 2006

By	O. Burak Okan (Cambridge, MA USA) - See all my reviews (REAL NAME)

This review is from: Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability (Paperback)

"Abel`s Proof" is a nice little book which tackles with the unsolvability issue in mathematics within the context of Niels Henrik Abel`s proof of the unsolvability of quintic equations with radicals. The text is an enjoyable account of a rather important subject in the whole history of mathematics in some 200 pages, and the quality of writing is laudable. The mathematical details and clarifications are given in boxes along the way, and the book in general is blended with numerous mathematical figures and portraits.

A firm high-school background in basic algebra should suffice to grasp the whole material, yet it has a real potential of teaching a noticeable chunk of mathematics to almost anyone along with valuable comments on its subject-matter.

I recommend this book wholeheartedly to anyone who has some genuine interest in going one step beyond the conventional popular science writing. And the price is right of course.