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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 Thinkingand 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.