A History Of Mathematical Impossibility

Author : Jesper Lützen
Publisher :
ISBN 13 : 9780192693020
Total Pages : 0 pages
Book Rating : 4.26/5 ( download)

DOWNLOAD NOW!

Download or read book A History of Mathematical Impossibility written by Jesper Lützen and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book tells the history of impossibility theorems starting with the ancient Greek proof of the incommensurability of the side and the diagonal in a square.

Author : David S. Richeson
Publisher : Princeton University Press
ISBN 13 : 0691218722
Total Pages : 450 pages
Book Rating : 4.24/5 ( download)

DOWNLOAD NOW!

Download or read book Tales of Impossibility written by David S. Richeson and published by Princeton University Press. This book was released on 2021-11-02 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive look at four of the most famous problems in mathematics Tales of Impossibility recounts the intriguing story of the renowned problems of antiquity, four of the most famous and studied questions in the history of mathematics. First posed by the ancient Greeks, these compass and straightedge problems—squaring the circle, trisecting an angle, doubling the cube, and inscribing regular polygons in a circle—have served as ever-present muses for mathematicians for more than two millennia. David Richeson follows the trail of these problems to show that ultimately their proofs—which demonstrated the impossibility of solving them using only a compass and straightedge—depended on and resulted in the growth of mathematics. Richeson investigates how celebrated luminaries, including Euclid, Archimedes, Viète, Descartes, Newton, and Gauss, labored to understand these problems and how many major mathematical discoveries were related to their explorations. Although the problems were based in geometry, their resolutions were not, and had to wait until the nineteenth century, when mathematicians had developed the theory of real and complex numbers, analytic geometry, algebra, and calculus. Pierre Wantzel, a little-known mathematician, and Ferdinand von Lindemann, through his work on pi, finally determined the problems were impossible to solve. Along the way, Richeson provides entertaining anecdotes connected to the problems, such as how the Indiana state legislature passed a bill setting an incorrect value for pi and how Leonardo da Vinci made elegant contributions in his own study of these problems. Taking readers from the classical period to the present, Tales of Impossibility chronicles how four unsolvable problems have captivated mathematical thinking for centuries.

Author : Jesper Lützen
Publisher : Oxford University Press
ISBN 13 : 0192693034
Total Pages : 305 pages
Book Rating : 4.37/5 ( download)

DOWNLOAD NOW!

Download or read book A History of Mathematical Impossibility written by Jesper Lützen and published by Oxford University Press. This book was released on 2022-12-30 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many of the most famous results in mathematics are impossibility theorems stating that something cannot be done. Good examples include the quadrature of the circle by ruler and compass, the solution of the quintic equation by radicals, Fermat's last theorem, and the impossibility of proving the parallel postulate from the other axioms of Euclidean geometry. This book tells the history of these and many other impossibility theorems starting with the ancient Greek proof of the incommensurability of the side and the diagonal in a square. Lützen argues that the role of impossibility results have changed over time. At first, they were considered rather unimportant meta-statements concerning mathematics but gradually they obtained the role of important proper mathematical results that can and should be proved. While mathematical impossibility proofs are more rigorous than impossibility arguments in other areas of life, mathematicians have employed great ingenuity to circumvent impossibilities by changing the rules of the game. For example, complex numbers were invented in order to make impossible equations solvable. In this way, impossibilities have been a strong creative force in the development of mathematics, mathematical physics, and social science.

Author : John Stillwell
Publisher : CRC Press
ISBN 13 : 1439865779
Total Pages : 249 pages
Book Rating : 4.74/5 ( download)

DOWNLOAD NOW!

Download or read book Yearning for the Impossible written by John Stillwell and published by CRC Press. This book was released on 2006-05-23 with total page 249 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores the history of mathematics from the perspective of the creative tension between common sense and the "impossible" as the author follows the discovery or invention of new concepts that have marked mathematical progress: - Irrational and Imaginary Numbers - The Fourth Dimension - Curved Space - Infinity and others The author puts t

Author : Jesper Lützen
Publisher : Oxford University Press
ISBN 13 : 0192867393
Total Pages : 305 pages
Book Rating : 4.91/5 ( download)

DOWNLOAD NOW!

Download or read book A History of Mathematical Impossibility written by Jesper Lützen and published by Oxford University Press. This book was released on 2023-01-26 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many of the most famous results in mathematics are impossibility theorems stating that something cannot be done. Good examples include the quadrature of the circle by ruler and compass, the solution of the quintic equation by radicals, Fermat's last theorem, and the impossibility of proving the parallel postulate from the other axioms of Euclidean geometry. This book tells the history of these and many other impossibility theorems starting with the ancient Greek proof of the incommensurability of the side and the diagonal in a square. Lützen argues that the role of impossibility results have changed over time. At first, they were considered rather unimportant meta-statements concerning mathematics but gradually they obtained the role of important proper mathematical results that can and should be proved. While mathematical impossibility proofs are more rigorous than impossibility arguments in other areas of life, mathematicians have employed great ingenuity to circumvent impossibilities by changing the rules of the game. For example, complex numbers were invented in order to make impossible equations solvable. In this way, impossibilities have been a strong creative force in the development of mathematics, mathematical physics, and social science.

Author : John Stillwell
Publisher : CRC Press
ISBN 13 : 0429998031
Total Pages : 312 pages
Book Rating : 4.34/5 ( download)

DOWNLOAD NOW!

Download or read book Yearning for the Impossible written by John Stillwell and published by CRC Press. This book was released on 2018-04-27 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: Yearning for the Impossible: The Surprising Truth of Mathematics, Second Edition explores the history of mathematics from the perspective of the creative tension between common sense and the "impossible" as the author follows the discovery or invention of new concepts that have marked mathematical progress. The author puts these creations into a broader context involving related "impossibilities" from art, literature, philosophy, and physics. This new edition contains many new exercises and commentaries, clearly discussing a wide range of challenging subjects.

Author : Jeff Suzuki
Publisher : MAA
ISBN 13 : 9780883855706
Total Pages : 432 pages
Book Rating : 4.04/5 ( download)

DOWNLOAD NOW!

Download or read book Mathematics in Historical Context written by Jeff Suzuki and published by MAA. This book was released on 2009-08-27 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: An exploration of the interaction between mathematics, mathematicians and society. What would Newton see if he looked out his window?

Author : Eric Maskin
Publisher : Columbia University Press
ISBN 13 : 0231153287
Total Pages : 164 pages
Book Rating : 4.87/5 ( download)

DOWNLOAD NOW!

Download or read book The Arrow Impossibility Theorem written by Eric Maskin and published by Columbia University Press. This book was released on 2014-07-08 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: Kenneth Arrow's pathbreaking Òimpossibility theoremÓ was a watershed in the history of welfare economics, voting theory, and collective choice, demonstrating that there is no voting rule that satisfies the four desirable axioms of decisiveness, consensus, nondictatorship, and independence. In this book, Amartya Sen and Eric Maskin explore the implications of ArrowÕs theorem. Sen considers its ongoing utility, exploring the theoremÕs value and limitations in relation to recent research on social reasoning, while Maskin discusses how to design a voting rule that gets us closer to the idealÑgiven that achieving the ideal is impossible. The volume also contains a contextual introduction by social choice scholar Prasanta K. Pattanaik and commentaries from Joseph E. Stiglitz and Kenneth Arrow himself, as well as essays by Sen and Maskin outlining the mathematical proof and framework behind their assertions.

Author : Davide Crippa
Publisher : Springer
ISBN 13 : 3030016382
Total Pages : 184 pages
Book Rating : 4.88/5 ( download)

DOWNLOAD NOW!

Download or read book The Impossibility of Squaring the Circle in the 17th Century written by Davide Crippa and published by Springer. This book was released on 2019-03-06 with total page 184 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is about James Gregory’s attempt to prove that the quadrature of the circle, the ellipse and the hyperbola cannot be found algebraically. Additonally, the subsequent debates that ensued between Gregory, Christiaan Huygens and G.W. Leibniz are presented and analyzed. These debates eventually culminated with the impossibility result that Leibniz appended to his unpublished treatise on the arithmetical quadrature of the circle. The author shows how the controversy around the possibility of solving the quadrature of the circle by certain means (algebraic curves) pointed to metamathematical issues, particularly to the completeness of algebra with respect to geometry. In other words, the question underlying the debate on the solvability of the circle-squaring problem may be thus phrased: can finite polynomial equations describe any geometrical quantity? As the study reveals, this question was central in the early days of calculus, when transcendental quantities and operations entered the stage. Undergraduate and graduate students in the history of science, in philosophy and in mathematics will find this book appealing as well as mathematicians and historians with broad interests in the history of mathematics.

Author : David S. Richeson
Publisher : Princeton University Press
ISBN 13 : 0691191999
Total Pages : 336 pages
Book Rating : 4.97/5 ( download)

DOWNLOAD NOW!

Download or read book Euler's Gem written by David S. Richeson and published by Princeton University Press. This book was released on 2019-07-23 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: How a simple equation reshaped mathematics Leonhard Euler’s polyhedron formula describes the structure of many objects—from soccer balls and gemstones to Buckminster Fuller’s buildings and giant all-carbon molecules. Yet Euler’s theorem is so simple it can be explained to a child. From ancient Greek geometry to today’s cutting-edge research, Euler’s Gem celebrates the discovery of Euler’s beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. Using wonderful examples and numerous illustrations, David Richeson presents this mathematical idea’s many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map. Filled with a who’s who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem’s development, Euler’s Gem will fascinate every mathematics enthusiast. This paperback edition contains a new preface by the author.