Customer Reviews

The Art of Mathematics (Dover Books on Mathematics)

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This is really a wonderful book. It's one of the reasons I went to grad school.

Dr. King really does a great job of explaining what math is really all about. His analogy to sculpture is perfect: If you want to appreciate a statue, you can't stand too close or too far away. Those in liberal arts are standing too far away from math to see it. Those in engineering and the sciences are standing too close -- they use pieces of math as tools but never see the beauty of mathematics.

And I must respectfully disagree with Dr. Greenberg -- the Appel-Haken proof of the four color theorem really is a travesty. It's not a question of being ugly: it's invisible. It's not a proof at all -- it's an "argument from authority" and hence is inductive, not deductive.

I must be on a hot streak. Almost every book I've read lately is completely mistitled--a ploy by the pubishers that I find beyond annoying. This book has nothing to do with the 'art' of mathematics. The author tried in the beginning to alert the reader that this book was possibly anathema to mathematicians since it wasn't 'math' as they understand it; rather a book allegedly 'about math.' Whatever it is and whatever King set out to accomplish with it, he lost his way. The first third of the book seemed to be a roundabout description of the abstraction of numbers. That much was well-written and not terribly complicated. But then the trouble started. The last two-thirds of the book meander all over the place in an exceedingly poorly crafted and unfocused effort to explain....WHAT? King went from a vaguely technical book for the layman to a muddled concoction of memoir, diatribe, philosophical thesis, critique and on and on and on. It's a patchwork of opinion, criticism, and at times chronicle. A 'behind the scenes' account of academic immaturity and petty politics. And the 100 or so pages on aesthetics? Oh my god. What is this guy doing? He might have had an idea in his mind of what he wanted to say. Unfortunately whatever it was got lost....really LOST between his mind and mouth (or keyboard). The book really is a reflection on a couple of topics that obviously have been part and parcel of the author's career. He seems eager to distance himself from the 'riff-raff' of mathematicians, yet proves he is nothing but lost in the academic world he wants so much to scorn and rise above. King's style as a writer really isn't all that bad, but the book was simply entirely too unfocused and pieced together. In a way, you get a lot for your money here, with 2 separate books in one. Why this was published is a mystery because it sheds no light aesthetics or the politics of being a mathematics professor. And it sure as hell says nothing about art.

I happend to pick this book up by chance and enjoyed every page. I agreed with everything the author had to say. I suggest this book to anyone with ANY interest in Math. It is best suited for those trying to understand why other people enjoy/understand Mathematics so much

I requested the latest edition of this book from my local library but I got the 1992 edition; so my review is based on the 1992 edition.

I am very disappointed with this book and I could not go beyond page 62. And here is why.

The first 40+ pages are devoted to abstraction about the beauty of Mathematics with no solid examples. I read through the pages patiently, without gaining any insights into the beauty of Mathematics.

Having described that 'precision' is the hallmark of Pure Mathematics, the author goes on to state Peano's Axioms in Chapter 3. What an Axiom is is never explained before using that term.

Then the book states: "We also know by Axiom C that the successor of 1 is not 1." Axiom C, according to the book, states "There exists no natural number whose successor is 1."

How the statement of Axiom C leads to the statement that the successor of 1 is not 1 is not clear. The entire premise of Pure Mathematics, which according to the author is, p => q fails.

The reference should actually have been to Axiom B.

Then comes the climax of 'precision': The author says that we do not know that the successor of 2 is not 2 and goes on to state a Theorem which says "the successor of any natural number is different from the number." [Which 'number'?]

And the author says "the proof of this theorem is not difficult but when you try it you see you need a preliminary result. Namely, you need" and then states yet another theorem: "Different natural numbers have different successors"

Are these two 'Theorems' saying the same thing?

Let us pause here and consider what Axiom B is according to the author: "For each natural number there is exactly one other natural number called its successor."

The key words in this Axiom are: 'each', 'exactly', 'other', and 'successor'.

The Axiom states that for 'every' natural number, there is exactly one [meaning one and only one] successor which is 'other' than itself.

What are those two Theorems the author is talking about saying that is not said by this Axiom B? The author does not explain.

So much for 'precision'!

I stopped reading further.

Having read a lot of books by Bertrand Russell, I must say Russell must be turning in his grave at references to him in this book.

Dr. King may be a great mathematician, but that does not make him a great meta-mathematician, that is, one who knows what mathematics is about.

Beauty of Mathematics is realized in those mathematical instances that have elegance, that is, simple and appealing. When you see an elegant proof or an elegant statement, you must get the same feeling that you get when you take a cool drink on a hot day! It should be invigorating and should create the 'aha' feeling in you.

Granted that this book is intended for non-mathematicians and is, according to the author, descriptive rather than precise, it still should not purvey misleading information.

E.g., on p.43, he calls spherical geometry "Riemannian geometry" and then claims on p.44 that "actual space is not Euclidean but rather more nearly Riemannian." That is incorrect if we adopt his notion of "Riemannian." It becomes correct when one provides the correct definition of "Riemannian" found in any advanced text on the subject.

On p.73 he misstates the Gelfond-Schneider Theorem, writing ab when it should be a to the power b that is transcendental. And on p.72 he misspells Mahler as Maher.

I doubt that most laymen will be enlightened by his discussion of the law of signs on p.73ff. He does show that it follows easily from other algebraic laws, and if the layman finds those laws acceptable, then the law of signs must be accepted, unintuitive as it may be.

His discussion of Fermat's Last Theorem on p.89 was unfortunately written before Wiles corrected the initial error in his proof. He needs a second edition to update this.

On p.88ff he argues that Appel and Haken's solution to the Four Color Problem was ugly because they used a lengthy computer calculation that cannot be surveyed. I and other mathematicians thought it was quite beautiful the way they reduced the problem to a finite computation.

That's as far as I've been willing to read in this book.