Customer Reviews

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

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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.

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.

"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.

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.

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.

The actual proof of Abel is in the appendix at the end of the book and the book outlines a history on the life of the mathematicians that worked on the problem. So the book is in fact a history book instead of a real
mathematics book. Therefore I was a little bit disappointed because there is little explanation on the actual proof.

I think this book is great. It provides a very readable history and talks about the high level ideas behind the proofs. It nicely provides short boxes that clarify details in the story.

For those who want to understand the mathematics in more detail, it provides additional information in appendices. It also includes an English translation of Niels Abel's 6 page paper that established his famous proof.

The discussion on Paolo Ruffini and Francois Viete are especially worthwhile. There's information here that's not generally well known.

This book is a model of how a math book should be written. I am very glad to highly recommend this book. Most of the books on this topic focus primarily on Evariste Galois so it's very nice to see Niels Abel also get his due. :-)

Math, it seems, has been a long string of embarassments. The classical Greek discovery of irrational numbers, for example, so horrified the mathematicians of the day that (according to myth), the discoverer was thrown overboard at sea. Zero had its detractors, for a while, who claimed that it represented nihilism and the denial of god. Negative numbers lay hidden for centuries, while mathematicians desperately pushed terms back and forth across their equations to keep everything positive. Then there were the complex numbers, still known as imaginary. Each embarassment came when some bedrock article of mathematical faith crumbled. One such article was the algorithmic solvability of general polynomials, the idea that any equation written only in constants and positive powers of X could be solved according to some fixed, reliable recipe. Although the recipes became increasingly complex, they existed for polynomials of degrees 2, 3, and 4. The recipe for degree 5 seemed inevitable.

That's why Abel took it on. Like others, he beat his head against the obdurate quintic, never giving up. In time though, every approach to the general solution fell apart. In a brilliant stroke of mathematical frustration, Abel turned to proving that there could be no fixed solution for the general quintic. There, he succeeded. There can be no fixed, finite recipe for solving arbitrary polynomials of degrees five and up. There are special, soluble cases for any polynomials of any degree, as well as techniques for approximating solutions. Still the laws of math form a fortress defending the general case against straightforward attack. With that statement, another firm but unproven certainty simply vanished. As happened with all earlier such losses, however, deeper beauty lay behind the simplistic but false facade.

Pesic does a fair job of summarizing the history of polynomial solutions, including the brutal competitions and rivalries that rose between Renaissance solvers. He also does reasonably well at outlining the logic that demonstrated the quintic's adamant nature. In any book like this, however, the author must balance the math against the human story behind it, or risk losing the popular audience. Although the earlier parts of the book keep the math at a widely approachable level, it becomes increasingly dense as Pesic approaches the core truths. In appendices, Pesic walks through the actual proof with helpful annotations. By that point, however, only readers with college math fresh in mind are still with him. Too bad - like Galois, Abel could have been cast as a much more exciting figure.

-- wiredweird

This book is intelligently written and does not assume advanced mathematics. Look for it to become a classic. It includes, as an appendix, Pesic's translation of Abel's 1824 paper "Memoir on algebraic equations, in which is demonstrated the impossibility of solving the general equation of the fifth degree", along with a brief commentary. Two additional appendices further help in understanding Abel's paper. In his notes, Pesic offers a wealth of suggestions for reading beyond his book.

The story is of mathematicians' pursuit of solving polynomial equations. Pesic tells this story with concision, clarity and brilliant elegance. Technical details are boxed and separate from the main text. Not only does this allow readers to bypass such details if they choose but, in being highlighted, the details can be easily located later.

To this reviwer who claims the author made a mistake be referring to Al-Khwarizimi and Khayyam as Arab Mathematicians, the author did not make any mistake nor did any of all the authors who wrote on math history. Those two Arab Mathematicians as well as many more did live in an Arab Empire. Al Khwarizmi, his name is Muhammad bin Mosa - arabic for english moses - was born and lived in Baghdad. He was a close friend to the great Calif Al-Maamoon. Al-Maamoon used to pay the jewish translators the weight of their translation - from greek into arabic- in gold. The arab mathematicians preserved and added to the greek mathmatics. Later at the beginning of european renaissance, a latin scholar had to pretend himself a muslim to translate from arabic in todays Morroco the books of Euclid into latin because the greek original was lost. - Ref. Non-Euclidean Geometry by Roberto Bonola Dover- I think as mathematician we should transcend above such bigotry.