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Krantz takes the reader on a journey around the globe and through centuries of history , exploring the many transformations that mathematical proof has undergone from its inception at the time of Euclid and Pythagoras to its versatile, present-day use . The author elaborates on the beauty, challenges and metamorphisms of thought that have accompanied the search for truth through proof.
The first two chapters examine the early beginnings of concept of proof and the creation of its elegant structure and language, touching on some of the logic and philosophy behind these developments. The history then unfolds as the author explains the changing face of proofs. The more well-known proofs , the mathematicians behind them, and the world that surrounded them are all highlighted . Each story has its own unique past; there was often a philosophical, sociological, technological or competitive edge that restricted or promoted progress. But the author's commentary and insights create a seamless thread throughout the many vignettes.
Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to date. This is shown in noting some of the more prominent discussions currently underway, such as Gorenstein's effort to classify finance groups, Thomas Hale's resolution of the Kepler sphere-packing problem, and other modern tales. Most of the proofs are discussed in detail with figures and
some equations accompanying them, allowing both the professional mathematician and those less familiar with mathematics to derive the same joy from reading this book.
The first two chapters examine the early beginnings of concept of proof and the creation of its elegant structure and language, touching on some of the logic and philosophy behind these developments. The history then unfolds as the author explains the changing face of proofs. The more well-known proofs , the mathematicians behind them, and the world that surrounded them are all highlighted . Each story has its own unique past; there was often a philosophical, sociological, technological or competitive edge that restricted or promoted progress. But the author's commentary and insights create a seamless thread throughout the many vignettes.
Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to date. This is shown in noting some of the more prominent discussions currently underway, such as Gorenstein's effort to classify finance groups, Thomas Hale's resolution of the Kepler sphere-packing problem, and other modern tales. Most of the proofs are discussed in detail with figures and
some equations accompanying them, allowing both the professional mathematician and those less familiar with mathematics to derive the same joy from reading this book.
From the reviews:
"The author traces the development of the idea of proof from Euclid through computer-aided and computer-generated proofs, pointing out the way some current social trends in mathematics affect the construction of nonstandard proofs. ... This work provides good outside class reading for students--and not just mathematics majors; one could easily imagine this as a supplement to courses on the history or philosophy of science. Summing Up: Recommended. Lower-division undergraduates through researchers/faculty; general readers."
(D. Robbins, Choice, Vol. 49 (2), October, 2011)
"The author traces the development of the idea of proof from Euclid through computer-aided and computer-generated proofs, pointing out the way some current social trends in mathematics affect the construction of nonstandard proofs. ... This work provides good outside class reading for students--and not just mathematics majors; one could easily imagine this as a supplement to courses on the history or philosophy of science. Summing Up: Recommended. Lower-division undergraduates through researchers/faculty; general readers."
(D. Robbins, Choice, Vol. 49 (2), October, 2011)
"This book is to describe the essence, nature, and methodology of mathematical proof, with a strong emphasis on the change of these concepts in time. ... It is written in a very clear and suggestive manner that makes the reading pleasant and rewarding ... . Any reader will notice that the author has reached this goal in very convincing way, and the outcome is a brilliant work which should be found in every math library and department office." (Jürgen Appell, zbMATH 1318.00005, 2015)
"In this book Steven Krantz undertakes the Miltonic task of justifying the ways of (pure) mathematicians to the world at large. ... The concept of mathematical proof is at the heart of Krantz's book. ... mathematicians should find the book interesting, illuminating, provocative ... ." (J. M. Plotkin, Mathematical Reviews, Issue 2012 b)
"The author traces the development of the idea of proof from Euclid through computer-aided and computer-generated proofs, pointing out the way some current social trends in mathematics affect the construction of nonstandard proofs. ... This work provides good outside class reading for students--and not just mathematics majors; one could easily imagine this as a supplement to courses on the history or philosophy of science. Summing Up: Recommended. Lower-division
undergraduates through researchers/faculty; general readers." (D. Robbins, Choice, Vol. 49 (2), October, 2011)
"Krantz's book is entertaining, can be read by the early undergraduate and puts forward some serious issues. There are few math books that are useful and valuable reading for all mathematicians, but this is one of them." (Charles Ashbacher, The Mathematical Association of America, June, 2011)
"In this book Steven Krantz undertakes the Miltonic task of justifying the ways of (pure) mathematicians to the world at large. ... The concept of mathematical proof is at the heart of Krantz's book. ... mathematicians should find the book interesting, illuminating, provocative ... ." (J. M. Plotkin, Mathematical Reviews, Issue 2012 b)
"The author traces the development of the idea of proof from Euclid through computer-aided and computer-generated proofs, pointing out the way some current social trends in mathematics affect the construction of nonstandard proofs. ... This work provides good outside class reading for students--and not just mathematics majors; one could easily imagine this as a supplement to courses on the history or philosophy of science. Summing Up: Recommended. Lower-division
undergraduates through researchers/faculty; general readers." (D. Robbins, Choice, Vol. 49 (2), October, 2011)
"Krantz's book is entertaining, can be read by the early undergraduate and puts forward some serious issues. There are few math books that are useful and valuable reading for all mathematicians, but this is one of them." (Charles Ashbacher, The Mathematical Association of America, June, 2011)