"--David Ruelle, author of "Chance and Chaos" "This is an important book, one that should cause an epoch-making change in the way we think about mathematics.

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 Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics and The Blind Spot: Science and the Crisis of Uncertainty. (shrink)

Why absolute certainty is impossible in science In today's unpredictable and chaotic world, we look to science to provide certainty and answers—and often blame it when things go wrong. The Blind Spot reveals why our faith in scientific certainty is a dangerous illusion, and how only by embracing science's inherent ambiguities and paradoxes can we truly appreciate its beauty and harness its potential. Crackling with insights into our most perplexing contemporary dilemmas, from climate change to the global financial meltdown, this (...) book challenges our most sacredly held beliefs about science, technology, and progress. At the same time, it shows how the secret to better science can be found where we least expect it—in the uncertain, the ambiguous, and the inevitably unpredictable. William Byers explains why the subjective element in scientific inquiry is in fact what makes it so dynamic, and deftly balances the need for certainty and rigor in science with the equally important need for creativity, freedom, and downright wonder. Drawing on an array of fascinating examples—from Wall Street's overreliance on algorithms to provide certainty in uncertain markets, to undecidable problems in mathematics and computer science, to Georg Cantor's paradoxical but true assertion about infinity—Byers demonstrates how we can and must learn from the existence of blind spots in our scientific and mathematical understanding. The Blind Spot offers an entirely new way of thinking about science, one that highlights its strengths and limitations, its unrealized promise, and, above all, its unavoidable ambiguity. It also points to a more sophisticated approach to the most intractable problems of our time. (shrink)