Abel’s Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability (The MIT Press) by Peter Pesic (PDF)

Description

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 fiancé.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.

User’s Reviews

Editorial Reviews: Review Peter Pesic’s tale of how maths came to be is as exciting as any fiction.—The Economist—Pesic’s book is a good place to begin to learn about this important piece of intellectual history.—Fernando Q. Gouvea , American Scientist— Review 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 From the Inside Flap “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, Truman Koehler Professor of Mathematics, Muhlenberg College, and author of *Journey Through Genius: The Great Theorems of Mathematics* “A unique book. Peter Pesic’s chronicle of the long road mathematicians traveled towards 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 Viete 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 “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 non-technical 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 “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 About the Author Peter Pesic, writer, pianist, and scholar, is Director of the Science Institute and Musician-in-Residence at St. John’s College, Santa Fe. He is the author of Abel’s Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability; Seeing Double: Shared Identities in Physics, Philosophy, and Literature; Sky in a Bottle; and Music and the Making of Modern Science, all published by the MIT Press. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Abel’s Proof takes an interesting approach to mathematical writing. The author places all mathematical formulas and derivations in boxes that are separate from the flow of the text so you don’t have to do the math if all you want is a knowledge of the history of this era in mathematics. The author does a great job of explaining this history, covering the story of the factoring of polynomials in general but focusing on the unsolvability of the quintic. Abel’s life of poverty is covered in detail. There is a lot of mathematics, if you are interested, including how to factor a cubic polynomial and Abel’s proof of the unsolvability of the quintic. I found the math somewhat hard to follow but worth the effort of doing so. I found Abel’s story to be very sad. The book is worth reading just for his life story alone.

⭐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 ([…] […] 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.

⭐It is a nice introduction to the many issues that surround the unsolvability of the quintic by radicals and also the different approaches by Abel and Galois. It can be read like a good story about a mathematical topic or it can be a starting point to get into more technical details following the appendixes and the references provided. It is up to the reader, the guidance in both cases is excellent.

⭐A good book if you are curious (as I am) of why there is no formula for a polynomial equation greater than the quartic. Be forewarned that it is not easy reading and be current on your knowledge of algebra. I got the gist of what Abel was proving but actually following and understanding the proof is difficult. It is a great insight into how the mathematical mind works and how it is so different from the rest of us. The anecdotes about Cardano, Newton, Cauchy and others are also rewarding.

⭐Peter Pesic has done a wonderful job in explaining the development of Abel’s work. I suspect there are many mathematicians who couldn’t do such a good job. He puts all the relevant building blocks together in their historical context. He gives in a very concise way the “helicopter” view of the substance of the issues that excited some very good minds. Unfortunately many standard texts on Galois theory fail to really develop the motivation for the theory. I commend this to anyone interested in the subject.

⭐While the introductory chapters are a nice summary of early developments in the theory of equations, as the material got more difficult I found I increasingly had to resort to more advanced texts to work out what Pesic was trying to say. Eventually in Appendices A & B where he gets to the heart of Abel’s proof, Pesic becomes incomprehensible and I had to resort entirely to more advanced works on abstract algebra and historical resources off the internet. Buy the book for the history – but do not expect to have Abel’s proof explained to you, there are too many gaps and mistakes in the more advanced sections for that.

⭐Excellent

⭐Great book. I have still to read it properly, it’s not that easy to follow a great man’s work. Like Galois, Abel died young.

⭐Brilliant. A mathematical book readable by mere mortals, telling the story of Abel. It does not go into too much detail about the lives of the protagonists (although there is a fair amount of history). What is rally nice is that the math is in there too. Most of it is explained in the cute “boxes” next to the text. The appendices, however, do contain the full original proof, in a translation by the author. More of this!!!

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