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Deep Thinking: What Mathematics Can Teach Us About The Mind

Penned By William Byers
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4.85
98,163 Global Ratings
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Deep Thinking: What Mathematics Can Teach Us About The Mind

Penned By William Byers
4.85 98,163 Ratings
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Editor's Note:
There is more than one way to think. Most people are familiar with the systematic, rule-based thinking that one finds in a mathematical proof or a computer program. But such thinking does not produce breakthroughs in mathematics and science nor is it the kind of thinking that results in significant learning. Deep thinking is a different and more basic way of using the mind. It results in the discontinuous “aha!” experience, which is the essence of creativity. It is at the heart of every paradigm shift or reframing of a problematic situation.The identification of deep thinking as the default state of the mind has the potential to reframe our current approach to technological change, education, and the nature of mathematics and science. For example, there is an unbridgeable gap between deep thinking and computer simulations of thinking. Many people suspect that such a gap exists, but find it difficult to make this intuition precise. This book identifies the way in which the authentic intelligence of deep thinking differs from the artificial intelligence of “big data” and “analytics”.Deep thinking is the essential ingredient in every significant learning experience, which leads to a new way to think about education. It is also essential to the construction of conceptual systems that are at the heart of mathematics and science, and of the technologies that shape the modern world. Deep thinking can be found whenever one conceptual system morphs into another.The sources of this study include the cognitive development of numbers in children, neuropsychology, the study of creativity, and the historical development of mathematics and science. The approach is unusual and original. It comes out of the author's lengthy experience as a mathematician, teacher, and writer of books about mathematics and science, such as How Mathematicians Using Ambiguity, Contradiction, and Paradox to Create Mathematics and The Blind Science and the Crisis of Uncertainty. What is Deep Thinking?Conceptual SystemsDeep Thinking in Mathematics and ScienceDeep Thinking in the Mind and the BrainDeep Thinking and CreativityDeep LearningGood TeachingUndergraduate MathematicsWhat the Mind Can Teach Us About MathematicsWhat Mathematics Can Teach Us About the MindReferences Students, graduate students and researchers with an interest in mathematics, mathematicians, scientists, philosophers, psychologists, and readers who use mathematics in their work.Key In this book, the author, a mathematician, demonstrates the specific difference between creative mathematical thought and the analytic thought of logic and the artificial intelligence of computersThis book shows why learning is a creative activity and demonstrates how teaching and learning must undergo radical changes in this age of rapid technological changeMathematics is a model for how people think and it reveals the essence of intelligenceThe author, a mathematician, demonstrates that creativity is a basic feature of the world. The same phenomenon of creative intelligence underlies the theory of evolution, child development, learning, and scientific and mathematical research. Creativity is so natural that even babies are capable of it but so difficult that adults have great trouble with it. This book explains the reasons behind this apparent paradox

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Released September 22, 2014
Identifier (ISBN) 24063883
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William Byers
The Mind Behind the Work
William Byers

William Byers is a writer and avid reader based in the Pacific Northwest. Before pursuing writing full-time, they spent over a decade working in education. When not working on their next book, William Byers can usually be found hiking with their dog or trying out new coffee shops in their neighborhood.

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Mary Diaz
Mary Diaz March 21, 2024
"Masterpiece from start to finish. The author orchestrated a breathtaking finale. Truly a work of art. I will undoubtedly revisit this masterpiece."
Stephanie Diaz
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"A decent, if standard, experience. Certain acts were fascinating, while others felt like a slight endeavor to complete. Suitable if your reading queue is empty."
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Stephanie Flores
Stephanie Flores June 28, 2025
"Absolutely brilliant. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece."
Sandra Campbell
Sandra Campbell April 18, 2024
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"Honestly one of the most premium reading experiences lately. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. A true literary gem."
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"An acceptable diversion. The prose was competent, yet the characters lacked the depth I usually seek. Perhaps more aligned with a different audience's tastes."
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"Masterpiece from start to finish. The prose is exquisite. It’s accessible yet undeniably sophisticated. Worth every moment spent reading."
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"I was completely blown away. The author orchestrated a breathtaking finale. Truly a work of art. I will undoubtedly revisit this masterpiece."
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"Took a moment to build, but the payoff was spectacular. The middle had a deliberate pace, but the climax was phenomenal. A wonderful companion for a quiet weekend."
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"A profoundly satisfying read. The author orchestrated a breathtaking finale. Truly a work of art. I am now a lifelong fan of this author."
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"Exceeded all my expectations. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I am now a lifelong fan of this author."
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"Masterpiece from start to finish. I adored how the story unfolded. It kept me captivated without sacrificing elegance. Worth every moment spent reading."
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"Honestly one of the most premium reading experiences lately. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. A true literary gem."
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"Such a rich narrative. The author orchestrated a breathtaking finale. Truly a work of art. I will undoubtedly revisit this masterpiece."
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"A highly sophisticated read. It held my rapt attention, even when certain twists were telegraphed. A wonderful companion for a quiet weekend."
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"A very strong entry. The middle had a deliberate pace, but the climax was phenomenal. I would highly recommend this to discerning readers."
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"Exceeded all my expectations. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece."
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"Such a rich narrative. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. A true literary gem."
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"Honestly one of the most premium reading experiences lately. The prose is exquisite. It’s accessible yet undeniably sophisticated. I will undoubtedly revisit this masterpiece."
Margaret Mitchell
Margaret Mitchell December 28, 2024
"Not entirely to my refined tastes. The tempo was somewhat languid, and I found my mind wandering. Not poor, simply average. Perfectly average."
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"Absolutely brilliant. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. A true literary gem."
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"Masterpiece from start to finish. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. Worth every moment spent reading."
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"Excellent, though slightly flawed. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. I would highly recommend this to discerning readers."
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"Honestly one of the most premium reading experiences lately. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. An effortless 5 stars."
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"A profoundly satisfying read. The author orchestrated a breathtaking finale. Truly a work of art. An absolute must-read for connoisseurs of the genre."
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"A profoundly satisfying read. The author orchestrated a breathtaking finale. Truly a work of art. Worth every moment spent reading."
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"A profoundly satisfying read. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An effortless 5 stars."
Cynthia Sanchez
Cynthia Sanchez June 19, 2025
"Exceeded all my expectations. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I will undoubtedly revisit this masterpiece."
Patricia Lopez
Patricia Lopez May 04, 2024
"Absolutely brilliant. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An effortless 5 stars."
Rebecca Moore
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"Took a moment to build, but the payoff was spectacular. A brilliant concept executed with flair. Just a few pacing choices I would change. Quite impressive overall."
Nancy White
Nancy White September 14, 2025
"A very strong entry. The protagonist was compelling, though some subplots felt slightly overshadowed. A very distinguished novel."
Donna Campbell
Donna Campbell June 11, 2025
"A decent, if standard, experience. It verged on repetitive in the second act. A more concise edit would have served it well. A fair 3 stars."
Sandra Harris
Sandra Harris March 28, 2024
"Honestly one of the most premium reading experiences lately. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I will undoubtedly revisit this masterpiece."
Eric Martinez
Eric Martinez January 24, 2025
"Honestly one of the most premium reading experiences lately. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. A true literary gem."
Kevin Miller
Kevin Miller May 09, 2025
"A very strong entry. The middle had a deliberate pace, but the climax was phenomenal. A wonderful companion for a quiet weekend."
Christopher Harris
Christopher Harris January 27, 2024
"Absolutely brilliant. I adored how the story unfolded. It kept me captivated without sacrificing elegance. A true literary gem."
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"Exceeded all my expectations. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I will undoubtedly revisit this masterpiece."
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"A profoundly satisfying read. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. An absolute must-read for connoisseurs of the genre."
Deborah Lopez
Deborah Lopez February 28, 2024
"I was completely blown away. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. A true literary gem."
Carol Allen
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"Absolutely brilliant. The author orchestrated a breathtaking finale. Truly a work of art. A true literary gem."
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"Absolutely brilliant. I adored how the story unfolded. It kept me captivated without sacrificing elegance. A true literary gem."
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Rebecca White June 27, 2024
"A profoundly satisfying read. I became deeply invested in the protagonist's journey. A masterclass in storytelling. An absolute must-read for connoisseurs of the genre."
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Joshua Taylor June 20, 2024
"A profoundly satisfying read. I adored how the story unfolded. It kept me captivated without sacrificing elegance. I will undoubtedly revisit this masterpiece."
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"Such a rich narrative. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. I will undoubtedly revisit this masterpiece."
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"Excellent, though slightly flawed. The middle had a deliberate pace, but the climax was phenomenal. A solid 4 stars from me."
John Moore
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"Exceeded all my expectations. I adored how the story unfolded. It kept me captivated without sacrificing elegance. A true literary gem."
Joseph Edwards
Joseph Edwards October 27, 2025
"I thoroughly enjoyed the elegance of it. It held my rapt attention, even when certain twists were telegraphed. A very distinguished novel."
Gary Jackson
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"I thoroughly enjoyed the elegance of it. A brilliant concept executed with flair. Just a few pacing choices I would change. A very distinguished novel."
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"Masterpiece from start to finish. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. An absolute must-read for connoisseurs of the genre."
Mary Young
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"Masterpiece from start to finish. The prose is exquisite. It’s accessible yet undeniably sophisticated. I am now a lifelong fan of this author."
David Thomas
David Thomas July 16, 2024
"I was completely blown away. The prose is exquisite. It’s accessible yet undeniably sophisticated. An absolute must-read for connoisseurs of the genre."
Christopher Torres
Christopher Torres November 10, 2025
"A profoundly satisfying read. The prose is exquisite. It’s accessible yet undeniably sophisticated. Worth every moment spent reading."
Margaret Moore
Margaret Moore June 10, 2025
"A profoundly satisfying read. The prose is exquisite. It’s accessible yet undeniably sophisticated. Worth every moment spent reading."
Lisa Clark
Lisa Clark July 16, 2025
"Took a moment to build, but the payoff was spectacular. Beautiful writing, though I craved slightly more depth in the lore. Still, undeniably engaging. I would highly recommend this to discerning readers."
Michael Clark
Michael Clark October 18, 2025
"A very strong entry. It held my rapt attention, even when certain twists were telegraphed. Quite impressive overall."
Patricia Thomas
Patricia Thomas March 19, 2024
"I thoroughly enjoyed the elegance of it. The middle had a deliberate pace, but the climax was phenomenal. I would highly recommend this to discerning readers."
Daniel Lopez
Daniel Lopez July 27, 2025
"Masterpiece from start to finish. The author orchestrated a breathtaking finale. Truly a work of art. An effortless 5 stars."
Cynthia Moore
Cynthia Moore November 29, 2024
"Not entirely to my refined tastes. The tempo was somewhat languid, and I found my mind wandering. Not poor, simply average. Unlikely to reread, but a reasonable expenditure of time."
Emily Gonzalez
Emily Gonzalez May 10, 2024
"An acceptable diversion. It possessed moments of brilliance, but the narrative felt slightly disjointed ultimately. Perhaps more aligned with a different audience's tastes."
Laura Phillips
Laura Phillips April 29, 2024
"Absolutely brilliant. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. I am now a lifelong fan of this author."
Sarah King
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Donna Robinson
Donna Robinson May 27, 2025
"Masterpiece from start to finish. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An effortless 5 stars."
Donna Taylor
Donna Taylor January 06, 2024
"Honestly one of the most premium reading experiences lately. The prose is exquisite. It’s accessible yet undeniably sophisticated. A true literary gem."
David Williams
David Williams July 05, 2025
"Absolutely brilliant. I adored how the story unfolded. It kept me captivated without sacrificing elegance. A true literary gem."
James Torres
James Torres August 28, 2024
"A very strong entry. It held my rapt attention, even when certain twists were telegraphed. A wonderful companion for a quiet weekend."
David Rivera
David Rivera December 26, 2024
"Exceeded all my expectations. I became deeply invested in the protagonist's journey. A masterclass in storytelling. Worth every moment spent reading."
Rebecca Mitchell
Rebecca Mitchell September 19, 2024
"I was completely blown away. The author orchestrated a breathtaking finale. Truly a work of art. A true literary gem."
Laura Phillips
Laura Phillips October 08, 2025
"A profoundly satisfying read. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. An effortless 5 stars."
Patricia Lewis
Patricia Lewis October 31, 2024
"A profoundly satisfying read. It strikes a perfect harmony of dialogue and world-building. Every chapter felt necessary. Worth every moment spent reading."
Amanda Adams
Amanda Adams September 25, 2024
"Honestly one of the most premium reading experiences lately. I adored how the story unfolded. It kept me captivated without sacrificing elegance. An absolute must-read for connoisseurs of the genre."
Jessica Lopez
Jessica Lopez December 22, 2024
"Masterpiece from start to finish. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I am now a lifelong fan of this author."
Cynthia Cruz
Cynthia Cruz January 26, 2024
"Honestly one of the most premium reading experiences lately. The prose is exquisite. It’s accessible yet undeniably sophisticated. Worth every moment spent reading."
Patricia Walker
Patricia Walker December 03, 2024
"Took a moment to build, but the payoff was spectacular. The protagonist was compelling, though some subplots felt slightly overshadowed. A very distinguished novel."
Charles Rivera
Charles Rivera June 05, 2025
"A profoundly satisfying read. The prose is exquisite. It’s accessible yet undeniably sophisticated. A true literary gem."
Nancy Robinson
Nancy Robinson November 28, 2024
"Not entirely to my refined tastes. The tempo was somewhat languid, and I found my mind wandering. Not poor, simply average. A fair 3 stars."
James Perez
James Perez January 06, 2024
"Absolutely brilliant. I became deeply invested in the protagonist's journey. A masterclass in storytelling. A true literary gem."
Deborah Johnson
Deborah Johnson July 12, 2025
"Absolutely brilliant. I became deeply invested in the protagonist's journey. A masterclass in storytelling. I will undoubtedly revisit this masterpiece."
Nancy Williams
Nancy Williams November 23, 2024
"I thoroughly enjoyed the elegance of it. A brilliant concept executed with flair. Just a few pacing choices I would change. A very distinguished novel."
Amanda Adams
Amanda Adams August 07, 2024
"Honestly one of the most premium reading experiences lately. I became deeply invested in the protagonist's journey. A masterclass in storytelling. Worth every moment spent reading."
Edward Hernandez
Edward Hernandez February 26, 2025
"It was adequate. The prose was competent, yet the characters lacked the depth I usually seek. Suitable if your reading queue is empty."
Donna White
Donna White October 08, 2025
"Masterpiece from start to finish. The pacing was impeccable and the characters felt incredibly authentic. I was completely immersed. An absolute must-read for connoisseurs of the genre."
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