How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics

By William Byers

Book Overview

To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results. Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure. The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory? Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself.

• Format: Hardcover
• Language: English
• ISBN: 0691127387
• ISBN13: 9780691127385
• Release Date: May 2007
• Publisher: Princeton University Press
• Length: 424 Pages
• Weight: 1.62 lbs.
• Dimensions: 9.5" x 1.5" x 6.3"

Customer Reviews

Ambiguity and paradox as inspirations for mathematicians???

Those of us who spent painful hours learning how to do "proofs" in geometry, or tried to keep in mind all the rules and procedures for solving polynomial expressions will probably exclaim "I Knew IT!" about a third of the way through the introduction. The author makes clear that he does not share that "Middle School" view of mathematics. In fact, it seems apparent that he considers that teaching approach responsible for the sorry state of mathematical knowledge in this society. Most of the book is an earnest attempt to "rescue" mathematics from the prevailing opinion that it is made up of well-defined processes and fully developed principles, with a list of known "problems" yet to be solved. The author makes clear that "doing math" is less like following blueprints and more like wandering in a garden, picking the prettiest flowers. As he makes his point, the non-mathematical reader will find insights into concepts and theories that were confusing, difficult, or just plain unknown. Readers who found T.S.Kuhn's "The Structure of Scientific Revolutions interesting and thought-provoking will enjoy this book. Those who are more comfortable with a view of mathematics and mathematicians as ruled by logic and devoid of emotion, will be challenged and disconcerted. All readers will come away with a much better understanding of the current "state of the art" of mathematics.

interesting book on the process of doing mathematics

The central thesis of How Mathematicians Think is that mathematics is more creative than algorithmic. The author describes mathematicians as dealing with ambiguity most of the time rather than simply adhering to a formula and plugging in numbers to get somewhere. The book is very interesting. The author covers some aspects of mathematics such as the concept of infinity and Cantor's work on the same. Although this book is about the creative process in mathematics don't expect to end up with a set of rules by which to create new mathematics. As there is no algorithmic approach to creativity in math there is no set of rules you can use to produce new math. I think this is as it should be, as no field as complex and profound as mathematics should devolve into a simple set of rules. I highly recommend this book to budding mathematicians and to lay people, like myself, who want a peek into the stuff of mathematical creativity.

Clear, Accessible Book on Philosophy of Mathematics

I've been looking for a book like this for years. It presents major issues in the philosophy of mathematics (e.g., what is mathematical truth?) in a clear manner and takes an unconventional view towards many of the big questions (e.g., is proof the essence of math?). You do need to be comfortable with basic algebra and geometry to follow most of the arguments, but it never delves into anything more complicated than basic ideas on complex numbers or simple calculus. The ideas make you think about more basic questions of epistemology. It's not light reading but it's not dry or too technical either.

How Mathematicians Think

This volume explains ambiguities, paradoxes and classic contradictions in higher mathematics from the scientific and philosophical perspectives. For instance, an ambiguity is a single idea perceived in 2 self-limiting but mutually exclusive or incompatible frames of reference. Irrational numbers are set forth i.e. the square root of two and Pi. Classic modern theories are set forth simply. For instance, string theory unites general relativity and modern quantum mechanics. i.e. the black hole and the big bang theory (expansion and contraction of the Universe) Nothing is a contradiction in that it is the negation of something. A presence implies an absence. Infinity implies an extended infinity or multiple infinities. One is the ultimate principle in life. In Probability and Statistical Inference, one is a limiting factor. i.e. 0 Generally, probabilities cannot be negative or greater than ONE. This book would be great for supplementing mathematics exams with theory questions to test knowledge of the essence of the subject matter.

Ubiquity of Ambiguity

Byers demonstrates the ubiquity of ambiguity, rather than of absolute certainty, in mathematics. It is easy to dismiss the contradiction in 0 (the nothing that is), because we have become so familiar with it. More people have trouble equating the infinite process indicated by 0.99999... with the integer captured in the symbol 1. Who could be confused about 'x + 2 = 5'? Students will be confused until they have absorbed the strange idea that before you solve the equation, 'x' represents any number, but afterwards only 3. Where is the difficulty in proving that the angles of a triangle add up to 2 right angles? Once you get the ideas to focus on one vertex and extend a side and draw a parallel, it becomes straight-forward to match up the angles. Byers structures his book around Andrew Wiles' metaphor of turning on the lights in unexplored rooms of a mansion for the long process of disproving Fermat's conjecture. In the introduction, he says "This book is written in the conviction that we need to talk about mathematics in a way that has a place for the darkness as well as the light and, especially, a place for the mysterious process whereby the light switch gets turned on." Exactly so, and well done!