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Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries) Paperback – February 17, 2006
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“Gödel's torment and his genius. By the book's end, we understand well why Einstein would look forward to 'the privilege of walking home with Gödel,' and we can't help but wish that we'd been able to join them.”
- Brian Greene, author of The Elegant Universe and The Fabric of the Cosmos
“In this penetrating, accessible, and beautifully written book, Rebecca Goldstein explores not only the work of one of the greatest mathematicians but also the relation of the human mind to the world around it.”
- Alan Lightman, author of Einstein's Dreams
About the Author
Rebecca Goldstein is a MacArthur Fellow, a professor of philosophy, and the author of five novels and a collection of short stories. She lives in Cambridge, Massachusetts.
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This is a current, widespread dispute.
''What Gödel’s theorems prove, he argues, is that even in our most technical, rule-bound thinking—that is, mathematics—we are engaging in truth-discovering processes that can’t be reduced to the mechanical procedures programmed into computers.''
If mathematics is trusted (everyone does), Gödel's proof confirms human reason is more powerful than any possible computer! Wow!
''Notice that Penrose’s argument, in direct opposition to the postmodern interpretation of the previous paragraph, understands Gödel’s results to have left our mathematical knowledge largely intact. Gödel’s theorems don’t demonstrate the limits of the human mind, but rather the limits of computational models of the human mind (basically, models that reduce all thinking to rule-following).'' (25)
I A Platonist among the Positivists
II Hilbert and the Formalists
III The Proof of Incompleteness
IV Gödel’s Incompleteness
Goldstein covers Godel's incompleteness theorem and it's effect on mathematics. She highlights the philosophical issues. Shows the metamathematical implications are more profound than the mathematical ones. She includes some biography of Godel's life, but she seems focused on explaining Godel's motives and goals more than events.
Excellent description of Schlick and his Vienna circle, of which Godel was a member. The developments of the logical positivists and why they became so influential is explained. The milieu of post war Vienna is described as 'the research laboratory for world destruction'. (Page 69)
"The overall topic was the moral and intellectual death and decay of all that had come before, and the need to construct entirely new methodologies, forms and foundations." The old world died in 1914. They knew that and wanted to create a new one. They did and we are living with the result.
The fascinating thing was although the Vienna circle converted much of the intellectual world to logical positivism (reality is only what can be positively shown) Godel, even though a member, created a proof that they were wrong! Goldstein believes Godel's desire to disprove positivism led him to prove the 'incompleteness' of mathematics.
Clearly explains that the usual meaning of Godel's work is to justify relativism, which is not how Godel understood his work. Godel deeply believed his work demonstrated that the human mind can discern deeper mathematical truth than any formal system of mathematics can find. In other words, he proved -mathematically- that it is not possible for any computer program to find all the mathematical truths that are available to the human mind!
Where does that capacity come from? How can the human mind connect to the hidden, difficult, profound and amazing mathematical concepts that continue to match the physical world? How does the insight or 'intuition' of mathematical concepts enter the mind of the mathematician?
This is the theme of the book as Goldstein presents the isolation of Godel because of his rejection of positivism. She also connects his friendship with Einstein due to Einstein's same intellectual battle. His work is called the 'Theory of Relativity' and is used to justify a subjective world. Einstein disagreed. He believed the accurate name would be 'Theory of Invariance', totally the opposite! Two world class thinkers, whose ideas were twisted to mean the contrary to what they believed. How sad. Both were isolated and world famous.
Also shows how Godel's devotion to logic gave him an uncommon courage to believe what his research found. For example, (page 60) she relates "I found those little Bible studies published by the Jehovah's witnesses. . .These contained careful underlinings and marginalia in the logicians hand." Another author mentions that Godel believed in the resurrection. He also was devoted to the writings of Leibniz.
Chapter eleven starts with an excellent description of thee problem of certainty. What is "proof"? Reminds us of the well known point that all deductive reasoning (such as mathematics) must start with unproven beliefs. We hope to use beliefs that are clear, simple and agreed by everyone (therefore do not need proof).
(Page 123) "The resulting beliefs can feel intuitively obvious precisely because we are not prepared to face their real and suspect source in our own personal situations and egos." Euclid used only five. These five were trusted for thousands of years and yielded fantastic results. We now believe one of them (parallel postulate) is wrong!
Godel's work shows that mathematics requires various direct beliefs, "intuitions", that cannot be "proved" by mathematics. This was and is a shocking, deeply disturbing conclusion to those who understand the significance. David Hilbert, the leader of mathematics at the time, was angry when he saw Godel's work. It destroyed his life's work.
This book is an excellent introduction to Godel's work. I find it fascinating due to its effect on the intellectual world. If, as with other world changing ideas, (Copernicus, Newton, Einstein) it takes a century for them to be assimilated into the culture, we should be seeing an increased effect soon. What the effect will be is unknown. As Godel showed, deductive reason cannot learn all that the human mind can discern.
For me, the book was a jumping off point for a series of startling insights, after which the entire world suddenly made more sense. Gödel's theorem has sweeping implications not only for mathematics but for philosophy, physics, theology, reductionism and materialism. After reading this book I bought many more, not-nearly-as-accessible books on Gödel, randomness, Turing, incompleteness and various aspects of mathematics.
It's just not that often that a book opens a door that wide. A Princeton grad who knew the author recommended this and it's certainly one of the 10 most important books I've read in the last 10 years.
Professor Goldstein does provide a simplified explanation of Goedel's incompleteness theorems (there are 2), and a reference to Godel's Proof, by Nagel, Newman, and Hofstadter, which she cites as a fuller presentation of the theorems themselves. Professor Goldstein's presentation of the theorems was, for me, a very helpful introduction which I am very glad to have read. It gives the reader a broad, but shallow overview of the forest, which should keep the reader from getting lost among the trees when tackling the actual proof, if s/he even chooses to do so, and it gives sufficient understanding to satisfy probably the great majority of us.
Also, the biography of Goedel is interesting in itself and well worth reading.
Read this enjoyable and well-written book first, then decide whether you want to tackle Nagel, Newman, and Hofstadter. If you do, you will be better prepared for it.
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I found this book to be incredibly painful, academically, to read.Read more