Godel's Proof | 
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Product Description
"A little masterpiece of exegesis." -Nature "An excellent nontechnical account of the substance of Gödel's celebrated paper." -American Mathematical Society In 1931 Kurt Gödel published his fundamental paper, "On Formally Undecidable Propositions of -Principia Mathematica- and Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Gödel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences-perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times." However, few mathematicians of the time were equipped to understand the young scholar's complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel's proof. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject. Marking the 70th anniversary of the original publication of Gödel's Proof, New York University Press is proud to publish this special anniversary edition of one of its bestselling books. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.
Amazon.com Review
Gödel's incompleteness theorem--which showed that any robust mathematical system contains statements that are true yet unprovable within the system--is an anomaly in 20th-century mathematics. Its conclusions are as strange as they are profound, but, unlike other recent theorems of comparable importance, grasping the main steps of the proof requires little more than high school algebra and a bit of patience. Ernest Nagel and James Newman's original text was one of the first (and best) to bring Gödel's ideas to a mass audience. With brevity and clarity, the volume described the historical context that made Gödel's theorem so paradigm-shattering. Where the first edition fell down, however, was in the guts of the proof itself; the brevity that served so well in defining the problem made their rendering of Gödel's solution so dense as to be nearly indigestible. This reissuance of Nagel and Newman's classic has been vastly improved by the deft editing of Douglas Hofstadter, a protégé of Nagel's and himself a popularizer of Gödel's work. In the second edition, Hofstadter reworks significant sections of the book, clarifying and correcting here, adding necessary detail there. In the few instances in which his writing diverges from the spirit of the original, it is to emphasize the interplay between formal mathematical deduction and meta-mathematical reasoning--a subject explored in greater depth in Hofstadter's other delightful writings. --Clark Williams-Derry
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Customer Reviews:
• How I Understand Godel 29 April, 2007
Godel was able to construct a formula from the axioms of Principia Mathematica (PM, and related systems, due to Russell and Whitehead) that is (roughly) "There exists no proof for this formula". This IS the formula itself. now, we need to know if this is true or not. so we try to find a proof for it within PM. if it is possible to find the proof, that means the formula is correct. but the formula says you can't find it. if the proof cannot be found, that means the negation of the formula is correct, but then the formula tells you that you cannot find it, so the formula must be correct, not its negation. in another words, the formula is undecidable within PM. also meaning PM is incomplete. what is amazing now is, of course that is what the formula tells you. so we do know it is true eventhough we can't show it within PM.
note that some people like reviewer Paul Vjecsner, who has also posted his disagreements on other Godel-related books, still confuse mathematics and meta-mathematics. although i have described and mixed-up meta-mathematics meanings to the formula above, Godel's proof was completely mathematicized within PM. either Vjecsner didn't understand the proof or he underestimated the grand aim of PM and thus the significance of Godel's work in taking it apart. Vjecsner argues that Godel's proof is essentially a linguistic paradox that has been unreasonably translated into PM. but the proof doesn't need that. that explanation is only done to give the readers a vague glimpse of the proof in a meta-mathemtical level. the proof only shows that there is a formula which is decidable if its negation is decidable. if you then argue that this is an unacceptable formula, then you are making even bigger claim than Godel, namely that the system is inconsistent! but the problem for you is you can't prove that within PM, thus still showing that PM is incomplete. Godel's proof is only as meaningful as PM. if you poke hole at Godel, you are poking hole at PM, which is exactly what Godel wanted to prove.
the book gives you an outline of how Godel went about constructing that formula with the language of PM, how he made the proof number-theoretical, and many more details. of course reading Godel's original paper would still be nightmarishly difficult even for many mathematicians, so the book gives a very good 'Godel's proof for dummies'. so you want to know what that formula looks like? read this book.
- Reviewed by customer ID: A2L901XYBZ4RES
• A Simple Presentation Of A Complex Topic 12 August, 2008
Godel's proof represents a milestone in mathematical and philosophical thought. This book, annotated by the remarkable Douglas Hofstadter, presents Godel's ideas in a language that's readily understandable by the educated layperson. It is clear, concise, and fascinating.
- Reviewed by customer ID: A29J5BJMEZFD8Y
• Godel For People Such As We (who Are Familiar With A Little Theory Of Numbrets) 21 September, 2008
I ran into Godel back in about 1955 in "Scientific American". I did not understand that. Now 53 years later, and with some more understanding of the theory of numbers, I find this work to be a magnificent opening into Godel's world. AS WOMDERFUL read!.
- Reviewed by customer ID: AHGMM0GMK5DDY
• A Fine Essay Introducing The Basic Idea 22 October, 2008
Anyone interested in the foundation of modern math ought to be at least familiar with Godel. This is really a very readable short essay giving the general outline and the basic idea of this important proof. Even motivated high-school students will have no problem grasping it (assuming they leave their tv's and computer games long enough!)
- Reviewed by customer ID: A2MAP0L7CCKTD5
• Godel's Incompleteness Theorems Explained In Non-technical Language 20 September, 2007
There is no question in my mind that the most misunderstood mathematical theorems of all time are Godel's incompleteness theorems. In essence, they state that no system powerful enough to do basic arithmetic is complete. Meaning that there will always be statements that are true in the system but that can never be proven in that system. These results have been seized upon by people with many different agendas and used to argue conclusions as significant as the existence of God and that human intelligence is not simply the outward manifestation of neurotransmitters flowing from place to place. While this is all somewhat amusing, it is also disquieting, as the theorems cannot be used to conclusively justify such significant claims.
This book is one of the very first books where an attempt is made to explain Godel's theorems to the mathematical laity. In that sense, it is a success; the appropriate background is effectively put forward before the theorem and proof are explained. There is little in the way of formal mathematics and the bulk of the terminology is non-technical. If the people who use Godel's results to justify their extravagant claims were to read this book with an open mind, they would recognize the absurdity of their positions.
- Reviewed by customer ID: A1S3C5OFU508P3
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