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

Rating:
Summary: Nice mixture of history and popular explanation
Review: 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.

Rating:
Summary: Unsolvable yet quite graspable
Review: 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.

Rating:
Summary: Unsolvable yet quite graspable
Review: 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.

Rating:
Summary: Reply to a Reviewer
Review: 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.