In 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.
He is also willing to allow that even when Bolzano is profound he may nothave been saying what later mathematicians came to say. The book is exemplary in a number of ways. One is that Russ has been at pains to say the minimum necessary tointroduce Bolzano, and as a result we have a very generous amount of material here.
His comments are judicious andinformative; one might have asked for more. Another is that the selection not only covers the better known early worksthat would have to be in any one-volume translation, but brings up later works that show how Bolzano scholarshipcontinues to advance.
As already noted, the translations read fluently while remaining faithful to Bolzanos not alwaysoptimal choice of notation.
Lastly, Oxford University Press is to be congratulated on producing a handsome book thatmatches the significance of its subject. The Semantic Tradition from Kant to Carnap. From Kant to Hilbert: A Source Book in the Foundations of Mathematics.
A translation of Bolzanos paper on the Intermediate Value Theorem.
Historia Mathematica 7, ISBNpp. Al-though the problem of finding solutions for the quintic had been weighing on the minds of great mathematicians forcenturies, the proof received little attention at the time of its publication; even Gauss could not be bothered to read it.
It was not until after Abels death that the proof began to receive the attention it deserved. It is easy to forget that each mathematical concept or piece of notation we employ has a rich history. In Abels Proofby Peter Pesic, we are reminded that the mathematical tools we apply without question were once new and strange.
Pesics book is not just an explanation of a year-old proof, it is a story about the acceptance of new mathematicalideas, ideas that may fly in the face of beliefs held for hundreds of years. The atmosphere of controversy that surrounded the question of the solvability of the quintic in radicals is felt fromthe first chapter of Pesics book.
Although Pesic waits until much later to discuss Abels proof, in the first chapterhe provides the reader with a glimpse of the mathematical turmoil to come with his account of the scandal of theirrational. He takes us back to ancient Greece and the myths that surround the Pythagorean brotherhood, includingthat the arrival of the irrational caused such upset that it may have resulted in the drowning of its discoverer.
Next, Pesic describes the gradual appearance of algebra as we now know it.
We see that although we are stillsolving some of the same problems that were studied by the Babylonians for example, quadratic equationsthemethod we use to formulate these problems and provide solutions has changed drastically.
Interestingly, the generalsolutions for quadratic, cubic, and quartic equations all preceded the introduction of an algebraic notation for variablesand coefficients by Franois Vite in the 16th century.
The solution of these equations was followed by another tumultuous period for new mathematical ideas. Negatives were thought of as absurd and false and Cardano remarked that an imaginarynumber was as subtle as it is useless.
In the fourth chapter of Pesics book, he gives sketches of the ingenious proofs of two important results. The firstis Newtons argument to show that all simple closed curves have areas that cannot be described by finite algebraicequations and the second is Gauss proof of the Fundamental Theorem of Algebra.
Pesic then develops the stories of Lagrange and Ruffini. Even after determining that the methods used for solvingthe cubic and quartic would not work for the quintic, Lagrange continued to believe that it was solvable in radicals.
However, Gauss stated in his Disquisitiones Arithmeticae that he believed that the quintic had no general radicalsolution, and Ruffini went so far as to propose six versions of a proof of its unsolvability, although none was completelyaccepted by the mathematical community.
Finally, in the sixth chapter, we meet Abel and are given a summary of his proof.Abel's proof: an essay on the sources and meaning of mathematical unsolvability / Peter Pesic. Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability.
di Peter Pesic Recensione di David Graves, dal sito della MAA, Mathematical Association of America. This fascinating book owes its existence to the author's desire for a "basic insight" into the general unsolvability of polynomial equations of degree five or higher.
Buy Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability (The MIT Press) on attheheels.com FREE SHIPPING on qualified orders/5(14). Get this from a library! Abel's proof: an essay on the sources and meaning of mathematical unsolvability.
[Peter Pesic] -- "In 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. Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability by Peter Pesic starting at $ Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability has 2 available editions to buy at Alibris.
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