Review by Choice Review
This is a truly exceptional work. In an almost gripping tour de force, Byers (Concordia Univ., Montreal) examines the creative impulse of mathematics, which to him is the notion of ambiguity, understood to "involve a single idea that is perceived in two self-consistent but mutually incompatible frames of reference." Paradoxes, proofs by contradiction, and infinity are taken to be instances of ambiguity. Assuming of the reader no more than an understanding of basic analysis, the author delves into these and many other topics in depth, with precision and uncanny clarity. The proof of the uncountability of real numbers is as transparent as an outline of Godel's proof of the Incompleteness Theorem. Throughout, Byers focuses on what is really going on, exposing the ideas behind the formalities of proofs. Endnotes are terse, the bibliography extensive, the index complete. Despite the occasional wordiness, this is a must-read book for every mathematics student and professor. To the former, it is a sorely needed complement to often-formulaic textbooks; to the latter, it may open vistas of mathematics they never would see. An incredible book, for libraries to announce to all. Summing Up: Essential. Lower-division undergraduates through faculty. J. Mayer emeritus, Lebanon Valley College
Copyright American Library Association, used with permission.
Review by Library Journal Review
This first book by Byers (mathematics, Concordia Univ.) is a compelling discussion of the intersection of mathematical thinking, psychology, and philosophy. The author focuses on how new ideas in mathematics are developed from ambiguous concepts such as infinity, chaos, and randomness. Specifically, Byers concentrates on the cognitive approaches mathematicians use to reconcile opposing objectives, a contradictory idea (paradox) to create proofs to formalize the behavior of these ideas. He incorporates a great deal of mathematically oriented discussion on contradictory ideas and how mathematicians have applied them creatively to make new discoveries. A psychological description of how mathematicians work and the functions of cognitive processes during problem-solving might have strengthened the author's premise, as this title is classed in psychology. Other similar works on mathematical thinking, e.g., Robert J. Sternberg and Talia Ben-Zeev's The Nature of Mathematical Thinking and Brian Butterworth's The Mathematical Brain, focus on mathematical learning and comprehension in student populations rather than creative problem solving among researchers, but both titles nicely supplement this text. Strongly recommended for academic libraries and specialized math and psychology collections.-Elizabeth Brown, Binghamton Univ. Libs., NY (c) Copyright 2010. Library Journals LLC, a wholly owned subsidiary of Media Source, Inc. No redistribution permitted.
(c) Copyright Library Journals LLC, a wholly owned subsidiary of Media Source, Inc. No redistribution permitted.
