For someone like me who stopped formally learning mathematics around trigonometry, this book was a remarkable next step in my math education. I finally understood what calculus actually is, and got a good introduction to Chaos Theory (hint: it's not really chaotic, it just appears to be.)

Highly recommended for people who want to understand math better, and how it is actually applied in the "real" world. ( )

The jacket markets this as short (true) and easy-to-read (ehhhh not so much). There's an awful lot of jargon here, and I think the audience for this is the liberal arts college student, rather than inquisitive lay person. It introduced some interesting terms (strange attractors!) but really its not a great choice for everyman. ( )

This book has a current impact in relation to an article published in the May 2009 issue of Physics Today, questioning the reality of everything except particles. Although many critical letters were published in subsequent issues, none of these related to the analogous subject of the reality of numbers and mathematics. Surely these do not have a material reality, but concepts like 'property of the universe' or 'law of nature' suggest alternatives.

The title and the subtitle attracted me to this book, but the author devoted only a few pages to the subject of reification (or thingification as he preferes to call it) as applied to numbers and mathematics in general. Although he does explain this as making a thing out of an abstracion, he does not take sides in this question or give any reasons that this concept applys to numbers or that numbers should not be thought of or should be thought of as a being a kind or reality. However the contents of the book as a whole are a good argument in favor of the latter since it is hard to understand how the facts of mathematics which he presents could be said to attain existance from any conceivable human experience. For example, he cites a theorm in number theory that involves numbers so large that they could not be written down even using all the matter in the universe. What human experience could this possibly be an abstraction of? On a more mundane level,it is easy to see how a child learns that 2 + 2 = 4 from playing with blocks, but it is hard to see how any other experience could lead to a different conclusion. Also when finding or proving theorms, most mathematicians have a sense of discovery rather than invention.

However, this book is good introduction to modern mathematics. After discussing the fundmentals, the author presents their application to calculus, differential equations, chaos theory and complexity theory. With regard to the latter, the author says that "the trick is to invent some kind of structure that retains" deterministic behavior while allowing random behavior. (Here he has a relapse in reverting to 'invention'.) Following some ideas presented by Stuart Kaufamnn, a researcher into complexity theory, in his books "Reinventing the Sacred" and "At Home in the Universe", I believe this structure already exists. We call it life. All that remains is to transfer that concept to the structure of fundamental particles such as quarks, as suggested indirectly by Frank Wilczek in his book "The Lightness of Being". This may be done in the next decade. Maybe we need to define and accept the concept that there are two kinds of reality.

The book concludes (like frosting on the cake) with discussions of the shapes formed by drops from a faucet, the dynamics of wild animal populations, and the relation of the Fibonacci numbers to the petals of flowers. In each of these topics mathematical concepts supply us with facts that we did not foresee on the basis of experience. ( )

This book is about math and for a math book to engage me it must be very good indeed. Yet even now as I write this review, what little I learned and understood by reading escapes my memory. Flipping through the book it's almost as if I've not read the book at all. At any rate after owning this book for several years I've finally read it and now it appears I have to read it again. I do like what Stewart has to say about research for research sake though.

"Mathematicians are forced to resort to written symbols and picture to describe their world -- even to each other. But symbols are no more that world than musical notation is music." p. ix

"The pursuance of safe research will impoverish us all. The really important breakthroughs are always unpredictable. It is their very unpredictability that makes them important: they change are world in ways we didn't see coming." -p. 29 ( )

Over the centuries, the collective minds of mathematicians have created their own universe. I don't know where it is situated -- I don't think that there is a ""where"" in any normal sense of the word -- but I assure you that this mathematical universe seems real enough when you're in it. And, not despite its peculiarities but because of them, the mental universe of mathematics has provided human beings with many of their deepest insights into the world around them.
I am going to take you sightseeing in that mathematical universe. I am going to try to equip you with a mathematician's eyes. And by so doing, I shall do my best to change the way you view your own world.