Customer Reviews

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

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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.

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!

I CANNOT believe the intellectual insight described in this book.

I am a Mathemetician Master's school graduate from CSULB and USC and have been tested to have exceptional mathematic ability.
You would think with my background, I could breese through this book. (albeit, as a mathemetician, I might be bored reading the book!)
Quite the opposite. This book is NOT about Mathematics. It is about why SCIENTISTS (mathemeticians as an example) hate ambiguity so much that they "create" never-before-invented solutions to resolve that ambiguity. This book also explains the process of solving those problems.

This book explains why and how our human scientific itellect has evolved. AMBIGUITY is the impetus. SOLVING the ambiguity is the goal and "engine" for evolution.

Without the existence of ambiguity, contradiction, and paradox, we humans would never have raised our combined intellect beyond "nature" or "God".

To me, the "ah ha" teachings in this book MUST become a classroom experience for young scientific minds that leads them to look for ambiguities in their life and motivates them to solve them.

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.

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.

There is a vastness about mathematics that is daunting: Many works are identified as mathematical; mathematics has a long and colorful history, extending back to the dawn of civilization (and with "primitive" concepts that extend even into primate behavior and the behavior of other animals); and there are many people today, all over the world, who identify themselves as mathematicians, including teachers and university professors who dedicate their lives to mathematics. Thus, the task of considering "how mathematicians think" is a huge one.

It is little wonder, that in faced with this task, Prof. Byers focuses on ambiguity, contradictions and paradox. However, since the common perception of mathematics (apart from the mathematicians) is as a field of formal purity and certainty, such a viewpoint seems to present us with an unusual view of mathematics.

The central feature of Prof. Byers account is "duality", i.e. that ambiguity, for example, which he takes to mean seeing mathematics and mathematical ideas from multiple perspectives, cannot be separated from the "certainty" of mathematics. He insists that the dualities are inherent in mathematics, and in particular that we cannot just make a clean separation between the objective and the subjective. He sees this view as not only having consequences with respect to the way in which mathematician think, qua mathematicians, but also with respect to appreciating the history of mathematics, and in how we teach and study mathematics.

Mathematicians were challenged to an extraordinary degree in the twentieth century. The work in metamathematics, especially by Godel, raised the issue of the limitations of mathematics. The rise of computers, especially in the latter part of the twentieth century, began to attack basic assumptions of mathematicians, not the least of which is whether or not it is even a "human" activity. Can computers supersede people at mathematics? If Prof. Byers is correct, and mathematics is an inherently human and creative activity, the answer is ultimately: No. This is despite the fact that in certain formal ways, computers already far-exceed human capabilities.

Even in physics, the arguably most mathematically-oriented science, the twentieth century has seen the rude shocks of relativity and quantum mechanics. In fact, in part, Prof. Byers motivation to focus on "dualities" arises partly from the experience of physicists in creating and developing quantum mechanics.

This complex notion of duality impinges on us even when we consider human behavior and intelligence with respect to very simple and basic mathematical ideas. Thus, the concept of "zero" both represents "something" and "nothing".

Prof. Byers considers many dualities, and this succeeds in giving a fascinating portrait of mathematics as a human activity. His consideration of a historical viewpoint is also rather wonderful. However, as he emphasizes, one must be careful about a historical approach to mathematicians and mathematics, because there are so many possible "distractions" and details, that one is in danger of losing focus.

I found this book to be very challenging to some of my ideas, and I must say, his views do not always represent mine about mathematics. However, he lays his account out beautifully, and gives those of us who love mathematics a real intellectual treat. I think, too, that his work would be of interest to others besides mathematicians, in attempting to expose a "humane" edge, to an ancient science.

I read this title originally in 2007, and at the time enjoyed Byers unique treatment of ambiguity. From his work, I concluded that there was good ambiguity (where you "know" there is a problem and have a hunch at a solution) and bad ambiguity (where cluelessness prevails). Byers treatment of contradiction, paradox, and patterns went largely over my head. At the time it was an ok read....fast forward to this year. Last month I finished Howard Margolis' Patterns, Thinking, and Cognition A Theory of Judgment. While writing, I grabbed Byers from the shelf---and before I knew, four hours passed and I'd reread most of the book. Some math specialists have panned the book as simplistic, but I'm not a math specialist----I found Byers' insights have profound analogous impacts in areas other than math---like decision making and cognition. This is a very good book for the math review---and there is a fair amount---but viewed in the macro, Byers offers another perspective, another view into how we reason. Highly recommended.

Mathematics is a fascinating subject. I am not a mathematician, but deal enough with it in my chosen profession to be constantly amazed by how logical the application of mathematics to proving a theorem or analyzing an algorithm turns out to be. But wait ... is it really logical? Or does it merely seems so and what is actually happening is that the author of the said proof is using creative tricks and techniques from the mathematical tool box to somehow tie everything up with a nice red bow-tie? In this book, the author argues that mathematics is creative more than algorithmic, and that mathematicians use a good dose of ambiguity mixed with equal parts of contradiction and paradox to create mathematics. Now, I shall point out that the term "ambiguity" here does not mean vagueness, rather it refers to a central truth that is perceived in two self-consistent but mutually incompatible contexts. The author takes the reader on this journey of ambiguity, paradox and contradiction on the way to discovering a lot of interesting mathematics. There is a section on counting numbers and cardinality, complete with Hilbert's Infinity Hotel; there is an interesting section on how to approach geometry through Euclid's Elements, and so on. I don't suppose that this is the sort of book you would pick up for a plane ride -- contemplating the philosophy of mathematics at 35,000 feet is enough to induce stupor. But if you are interested in the field and still remember the Central Limit Theorem from Calculus-I or Series and Sequences from Calculus-II, then you will definitely enjoy this book. I know I did.

I would classify this book as a Mathematical Philosophy Book. The author definitely places Philosophy more than the hard-core Mathematics, so don't be disappointed if the reader's main goal is on Math.
Overall this book is a great book, but definitely not for the weaker math students. It brings you to the higher platform to look down on Math issues in a pensive way - ie "Switch on the light" à la Andrew Wiles.

This book should be best read not in sequential manner, because of the writing style of the author which is quite verbose.

Some chapters are very well written:
Chap 8: (Pg 363) Obstacles to Learning Mathematics : Many great ideas and truths are hidden behind the math theoretical structures, unfortunately in the university math profs emphasize more on structures and leave the poor students to find out the 'beauty' of truth themselves - because 'Beauty' is not tested in Exams :(
Chapter 4: Paradoxes of Infinity. The "Cantor Set" Construction example is very refreshing.
Chapter 5: on "Quotient Space" (X/R) is excellent.
I also like the Isomorphism ideas (Pg 216-217): as Isometry (in Geometry), homeomorphism (in Topology), besides various isomorphism in Groups, Rings, Fields...

In summary, this book will elevate your math philosophical thinking like a Mathematician.

I suspect that Prof. Byers is an excellent mathematics teacher and I very much enjoyed the snippets of mathematics in this book. However, most of the book was devoted to philosophy, which I found to be at least one of the following: (1) repetitive, (2) unoriginal, or (3) wrong. Repetitive for sure. Unoriginal in that he repeats many of the points made more eloquently and clearly by folks like Lakatos (though Byers does do a good job of giving credit where it is due). Wrong in the philosophy of mind, as in the section toward the end of the book where he tries to argue (a la Searle?) that machines can't think, and that computers might be able to write proofs but they can't do the inherently creative aspects of mathematics. It's very strange to me to run into a mathematician who holds these kinds of mystical views about minds, that they are not machines!

I feel like this would make a truly excellent 50 page book, with just a few of the key philosophical points clearly explained and illustrated with some of the excellent mathematical examples in this book. It could even be expanded -- but with more of Byers' mathematical illustrations, not his philosophical ramblings.

If I focused just on my favorite 50 pages of this book, it would get at least four stars; but the other 300 pages average it down to two.