13 of 13 people found the following review helpful:
3.0 out of 5 stars
A Tough Slog, August 31, 2006
This review is from: Beyond Reason: Eight Great Problems That Reveal the Limits of Science (Hardcover)
"Some of the integers z will appear inside their corresponding subsets f(z) and some won't. If we take the set of all integers z that are not members of f(z) and place them in a special set W, we can ask a very serious question about W: To what integer does W correspond under this scheme? If we write that integer as w, we can ask if w belongs to W. If w belonged to W, then w would not be a member of f(w) by the definition of W. But wait! This is a contradiction, since W=f(w). On the other hand, if w were not a member of W, then w must lie in f(w) (i.e., W), another contradiction" (p. 143).
If that paragraph makes you giddy with glee, then you'll love Beyond Reason by A.K. Dewdney. I found such paragraphs less delightful, but it does illustrate my recurrent experience with this book: the nagging feeling that just a little closer reading will unlock what he's trying to say. Dewdney mixes layman's prose with mathematical abstractions, and the resulting book is like a girl who won't say yes or no. She's just interesting enough to make the frustration feel worth it (at least for a while).
I'm not a mathematical wizard. While I scored very highly in reading comprehension and language skills on the ACT, my math scores barely passed the threshold of acceptability (think of an airplane chipping its undercarriage on the peak of a mountain). I passed high school algebra and geometry with tears and travail, and never wanted to know anything about calculus or trigonometry beyond how to spell them. Dewdney makes me wish I'd worked harder at it. Not only because I might have a shot at fully grasping the nature of the problems he presents, but also because he's opened my eyes to the importance of mathematics in understanding our universe. As he writes on page 2, "We still have to account for the amazing success of mathematics as a description of physical reality. The precision of so many theories of physical reality may hint at a deeper truth, that mathematics is a major structural foundation of our universe."
Thus is introduced the premise of these eight chapters, exploring what appear to be the mathematical barriers that hem us into this three-dimensional reality. Since we're on the topic of structure, the book is divided into two halves. Part One deals with the mathematical realities that tell us we will never (1) build a perpetual motion machine, (2) surpass the speed of light, (3) predict the behavior of quantum particles, or (4) predict the long-term behavior of dynamical systems (e.g., the weather). Part Two deals with more abstract mathematical limits, telling us that we will never (5) square the circle, (6) prove every true mathematical theorem, (7) compute the answer to every yes or no question, or (8) compute every mathematical problem.
At the end of each chapter, Dewdney includes a short section entitled "Is There a Way Around It?", exploring the theoretical possibilities of surmounting these apparently insurmountable barriers to knowledge. This is the caveat hanging over the book like the Damoclean sword: will it still be relevant a century or two from now, or will we have uncovered deeper and broader mathematics that will reveal the key to passing through these barriers?
I found the book's main fascination in the concept that mathematics describe the boundaries of the universe. In other words, the history of mathematical discovery has been like a man in an empty space slowly lighting candles to explore his surroundings. That is, until he lights a candle that reveals a doorless wall. Then he finds another, and another, and yet another. He realizes there is a profound structure surrounding him, but he can't see what the structure is. As for what lies beyond the structure...well, no amount of imagination can see through walls.
In summary, Beyond Reason is a tough slog unless you're mathematically sharp. However, it has given me a glimpse (however unsatisfactory my vision) of the boundaries on our existence, which is like getting to touch God's fingerprints. That alone is the worth the furrows left from scratching my head.
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13 of 16 people found the following review helpful:
4.0 out of 5 stars
Obviously for a Specific Audience, September 27, 2004
This review is from: Beyond Reason: Eight Great Problems That Reveal the Limits of Science (Hardcover)
Dewdney's book is very interesting, challenging, and is, admittedly, a tough read. You might need ready access to Google to look up some of the terms he uses. But for the most part, anyone who took high school math and physics should be able to understand it. I'd recommend it to college and graduate students, and math and physics instructors. It has its down points, but on the whole is a very intriguing look at some of the problems afflicting all our logic.
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1 of 1 people found the following review helpful:
4.0 out of 5 stars
limits of explanation, October 4, 2010
This review is from: Beyond Reason: Eight Great Problems That Reveal the Limits of Science (Hardcover)
I enjoyed this book, and did not find it unnecessarily difficult to understand. However, I should clarify that when I read a book like this, I assume from the start that there will be a limit to how well I will understand a specialized topic being explained in layman's terms. Therefore I am satisfied to get an overview and do not try too hard to "really understand" the details. I'm sure if I did I would be as frustrated as some of the other reviewers.
All the topics covered in this book are inherently challenging, and it is inevitable that non-specialist readers will not understand everything. The key question is whether the book sheds a reasonable amount of light on these difficult topics, and I think the answer is clearly yes.
Actually, the title of this book is somewhat misleading, as the book addresses different kinds of limits in science and math. One type of limit is a physical limit like the speed of light or the precision with which we can simultaneously measure a subatomic particle's position and momentum. Another type of limit is the inability of any algorithm to decide certain clearly defined propositions, or the fact that any mathematical system will either contain true theorems which cannot be proven or inconsistencies. To me only the second type can be called "the limits of science" (i.e. of science itself as opposed to physical limits predicted by science). Nonetheless, each topic chosen is well worth knowing to anyone who would like to be reasonably literate in science and math on the layman's level.
One can quibble over whether the author explained everything as clearly as possible. All of these topics have been addressed by other authors, so the confused reader can try a different book to see if there is a clearer explanation. I found the explanation of the Turing machine and the Undecidability Theorem clearer in this book than in the biography of Turing by Leavitt. But ultimately this is a matter of taste. Even if you don't like a particular explanation, you've at least been introduced to the topic.
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