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Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries) Paperback – February 17, 2006

by Rebecca Goldstein (Author)
4.4 4.4 out of 5 stars 282 ratings
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"A gem…An unforgettable account of one of the great moments in the history of human thought." ―Steven Pinker

Probing the life and work of Kurt Gödel, Incompleteness indelibly portrays the tortured genius whose vision rocked the stability of mathematical reasoning―and brought him to the edge of madness.

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  1. Part of series
    Great Discoveries
  2. Print length
    300 pages
  3. Language
    English
  4. Publisher
    W. W. Norton & Company
  5. Publication date
    February 17, 2006
  6. Dimensions
    5.4 x 0.9 x 8 inches
  7. ISBN-10
    0393327604
  8. ISBN-13
    978-0393327601
  9. See all details
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Editorial Reviews

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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.

Product details

  • Publisher ‏ : ‎ W. W. Norton & Company; Reprint edition (February 17, 2006)
  • Language ‏ : ‎ English
  • Paperback ‏ : ‎ 300 pages
  • ISBN-10 ‏ : ‎ 0393327604
  • ISBN-13 ‏ : ‎ 978-0393327601
  • Item Weight ‏ : ‎ 9.6 ounces
  • Dimensions ‏ : ‎ 5.4 x 0.9 x 8 inches
  • Customer Reviews:
    4.4 4.4 out of 5 stars 282 ratings

About the author

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Rebecca Goldstein

Rebecca Goldstein

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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.

Customer reviews

4.4 out of 5 stars
282 global ratings

Customers say

Customers find the writing style good, readable, and beautifully formulated. They also describe the biography as interesting, pertinent, and wrapped in a fascinating tale of genius.

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Writing styleBiographyElegance
32 customers mention "Writing style"23 positive9 negative

Customers find the writing style good, unexpectedly readable, and beautifully formulated. They also describe the author as engaging, lucid, and informative. Readers mention the book is clear and deals with personalities as well as mathematics.

"...But for anyone, mathematics devotee or not, who is interested in a readable, clear, and enjoyable study of Gödel’s life and influence, Goldstein’s..." Read more

"...The author is very good, too, at using her philosophical training at putting everyone in philosophical boxes -- platonists, positivists,..." Read more

"...Although my experience suggests that the theorems are being suppressed...." Read more

"...This book is an excellent introduction to Godel's work. I find it fascinating due to its effect on the intellectual world...." Read more

17 customers mention "Biography"17 positive0 negative

Customers find the biography interesting and pertinent. They say it's wrapped in a fascinating tale of genius. Readers also mention the book weaves deep knowledge, personal experience, and great storytelling. They say it provides a nice historical context to the meaning and importance of Godel's theories.

"...This is a remarkable book – the story of a mathematical genius, a disturbed mind, a “Platonist” about mathematics in an age that was anti-Platonist..." Read more

"...Goldstein’s descriptions of the Vienna Circle are pretty fascinating, and her recollections and the recollections of her colleagues at Princeton..." Read more

"...I find the theorems a source of optimism and a clear, present and immensely important antidote for authoritarian dependency...." Read more

"...In my opinion, Rebecca Goldstein is one of the most interesting and pertinent among those contemporary philosophers who write for the wider public...." Read more

3 customers mention "Elegance"3 positive0 negative

Customers find the book elegant. They say it paints a lovely portrait of a genius who triumphs but sadly ends.

"...de force on Kurt Gödel, his two main "theorems", and a painless, elegant treatment of well-known self-referential paradoxes in philosophy in general..." Read more

"...Second, it paints a lovely portrait of a genuis who triumphs but sadly ends his life in despair...." Read more

"...She still gets his story across in a comprehensible and appealing manner such that you cannot help but appreciate the man for who and what he was:..." Read more

Top reviews from the United States

Daniel Putman
5.0 out of 5 stars Engaging and highly enjoyable biography of a singular man
Reviewed in the United States on July 30, 2024
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For readers who are interested (or not interested) in the actual mathematics of Gödel’s incompleteness theorem, it is worth knowing that roughly twenty pages are given over in this book to actually going over the math of the theorem. But for anyone, mathematics devotee or not, who is interested in a readable, clear, and enjoyable study of Gödel’s life and influence, Goldstein’s biography is excellent. She covers Gödel’s background, the personality of this unusual man, and those who influenced him. She does an excellent job showing how his ideas were related to and opposed to those of the Vienna Circle, the group of philosophers, scientists, and mathematicians in Vienna who had a powerful influence on the direction of 20th century philosophy. She also does an exceptional job at showing how Gödel related to Wittgenstein, perhaps the single most important philosopher of the twentieth century. (Some background in philosophy would be useful but not critical.) But what I found most satisfying about the book is Goldstein’s explanation of Gödel’s long friendship with Einstein at the Institute for Advanced Study and how the death of Einstein destroyed the only person that Gödel could call a friend. This is a remarkable book – the story of a mathematical genius, a disturbed mind, a “Platonist” about mathematics in an age that was anti-Platonist to the core, and a man who stood out for his mind among so many other brilliant thinkers in the 20th century. And it is written in an engaging style, a fact not expected about the life of an isolated mathematician who did little of what we would be call exciting things in his life.

While a little background in philosophy would be useful, it is not necessary and I recommend this completely enjoyable biography in the strongest terms.
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Martin
4.0 out of 5 stars Like, Maybe, Godel Just Had Colitis
Reviewed in the United States on December 26, 2018
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I’ve always liked Rebecca Goldstein’s novels, which deal with the problems of finding romantic love for people working in the hard sciences -- math, physics, and philosophy. The truth is that it’s really hard for people to find romantic love in these fields, but Goldstein is at least a cheerleader for it. So, when I heard that Goldstein had written a non-fiction book about Godel, a historic thinker whose contributions I’ve long thought about, well, the book piqued my interest. Generally, I think that if a book of this nature gets you thinking about things, then it’s mainly successful. And mainly this book is. Goldstein’s descriptions of the Vienna Circle are pretty fascinating, and her recollections and the recollections of her colleagues at Princeton and the Institute for Advanced Study about Godel are illuminating, too. Godel in all of this still comes out as pretty much a cipher, being overshadowed by the stronger personalities of Einstein, Wittgenstein, Russell, even Oppenheimer. Apparently, that’s just the way he was, and maybe there isn’t all that much to say about him personally, but the author does do some digging. The author is very good, too, at using her philosophical training at putting everyone in philosophical boxes -- platonists, positivists, empiricists -- if you’re into that sort of thing, but where the book falls down is on the explanation of the proof itself. Granted, the author admits that her mathematical skills maybe were somewhat better when she was still in graduate school. Still, the easy stuff of the proof is the encoding of mathematical statements by prime number factorization and the general idea that the proof is premised on the liar’s paradox. However, she glosses over the more difficult and central elements of the proof as to how there is necessarily a propositional function that can decide whether a godel number of a statement has a property that makes the statement provable, and then the diagonalization lemma -- well, let’s just say she punts on that. This appears to me to be fairly problematic. If someone is continually gushing about how brilliant the proofs are, and what a major contribution the incompleteness proofs are to the history of human and mathematical thought, and yet really can’t explain precisely how the proof works, then it appears that maybe someone is just sucking up to the authorities that be. If the author doesn’t really understand it, then how many experts in the world really do, and how sure can we be that there isn’t some problem in it somewhere.

One has to wonder how important the incompleteness proofs will still be a hundred or two hundred years from now. Goldstein appears to recount how both Russell and Hilbert resisted the conclusions of incompleteness, but Russell was overwhelmed by the force of personality of Wittgenstein, and Hilbert couldn’t always get his own program back on track by herding in the cats. If the heart of a logical proof is itself a paradox of logic, as the liar’s paradox is, then one has to wonder about its conclusions. This is a little like premising a proof of arithmetic on ‘zero divided by zero’, where you then might be able to prove almost anything. Russell sounds in the book that he wanted to limit this sort of problem by the theory of types, but perhaps there should just be some more general prohibition against introducing paradoxes into mathematical proofs, and then to see where you can go from there. Godel is commonly compared to Escher, who built logically impossible buildings, which may be interesting, but perhaps it is more important to understand ways to build actual buildings -- or build proofs -- that architecturally won’t fall down in the real world. When I was growing up, all of the proofs in geometry class appeared to be pretty straightforward -- you could generally follow the logic and understand why they had to be true. Perhaps there was always some over-preaching about ‘finding a neighborhood around a point’ or ‘taking the limit as a value goes to zero’, but you could usually understand what they were getting at. Yet these days everything is more complicated and less immediately verifiable: take the four color map theorem, which was evidently only ‘proved’ by a computer enumerating a vast number of special configurations (even though earlier proofs from the nineteenth century looked pretty good), and then Wiles’ proof of Fermat’s Last Theorem that is premised on the Taniyama-Shimura Conjecture relating elliptical curves and modular forms. Maybe someday seventh graders will understand how elliptical curves relate to modular forms, but if everything is so complicated, then how do we really know what is true.

Then there’s the question of what it all means. Even if there is no proof, the idea it embodies still resonates. Goldstein appears to want to limit the impact of Godel’s incompleteness theorems to just what it says about the theory of the arithmetic of natural numbers, and wants to reject the implications used by modernism, existentialism, and anti-intellectualism that makes everything relative to man and downplays the power of the rational. But she still wants to draw on her own philosophical training to bring in its relevance to the old academic debates on the theory of mind, and whether a computer can ever think the way a person does. Undoubtedly, Godel’s work has had implications, or at least established some benchmarks, that are relevant today to the fields of computer science, coding theory, and algorithmic complexity. But it appears to me to be fairly pointless to argue about whether computers will ever think like people, particularly as new varieties of neural networks continue to develop; these philosophical problems may help point the way, but they don’t really contribute to the technical solutions, where only time will tell, and machines may never be able to do everything exactly the way a human can, even if they are able eventually to pass a Turing Test. And I guess the author would disagree with me if I tried to draw implications from the incompleteness theorems to the more rigorous theories of physics, which she doesn’t consider, where incompleteness might be read to say that the entire movement to unify the laws of physics is pointless because each theory -- gravity, quantum mechanics, electromagnetism -- simply stands independently on its own, is incomplete, not necessarily consistent with the other theories, but simply co-existent. This problem may have originated with Einstein, pondering in his old age at what the author implies is the turkey farm of the Institute of Advanced Study, and how his unified field theory -- continued by others with theories of everything and spontaneous symmetry breaking -- is all leading us astray. Suppose we just draw the implication that the theories of physics don’t unify, that the universe just continues to diversify with more and more unrelated phenomena, that ‘all laws are local’, which is really all that is implied by incompleteness.
41 people found this helpful
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Attachment+Logic=Mind
5.0 out of 5 stars Great Read
Reviewed in the United States on April 24, 2021
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An excellent book. For me personally, the single most helpful piece of information was: 'logicians are in denial of Godel's theorems today'. I was so relieved because my preliminary literature review suggested that logicians were not interested in the theorems. I am very interested in them, and the truth theoretic properties of model theory (Twersky). Although my experience suggests that the theorems are being suppressed. That is, when contacting with the executor of Godel's papers to discuss them, he directed me to read one [or both\] of two books he recommended. Then if I wished to speak with him, I could contact him again. So, I got the book that was not over $200.00 in price. In it I read that the theorems should be discussed 'only with professional philosophers and logicians. The material is too sophisticated for the inexperienced non-professionals. The theorems have been used to justify entirely unrelated positions about the world.' I was instantly reminded of Wittgenstein's statement that 'anything can be misunderstood'. In addition, freedon of speech anyone? Goldstien's book is superb. She treats Godel with sympathy, as a human being, mano a mano. Not some ubermenchen. My undergraduate logic professor worked closely with Godel and had only kind things to say. I think it is dreadful that his work is not treated more appropriately, as Goldstein clearly treats Godel. Yes, he had flaws. I've not met a human being without any, myself at the top of the list. His work is importamt. It should not be suppressed. Thank you, Ms. Goldstein. I studied Godel under a professor who worked closely with him, and clearly found his work important. As do I. Not for the same reasons as others, perhaps. I find the theorems a source of optimism and a clear, present and immensely important antidote for authoritarian dependency. I do not think the theorems justify lawless thinking. Rather, I think the theorems supprt locating authority clearly upon the shoulders of thoughtful human beings. Socrates was sentenced to death for teaching youth how to think. Godel is in a fashion sentenced to death professionally.
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Top reviews from other countries

George Jost
1.0 out of 5 stars Title shouldn't include the word proof
Reviewed in Canada on June 26, 2022
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Gödel's life was certainly interesting but the title suggests the author will actually discuss his ideas which she doesn't seem to do. She probably has indeed read his proof but she might as well have titled the book what Gödel had for lunch and why the Nazis hated him. I expected her to discuss at some level how he proved the actual theirem and on a kindle, it's a nightmare grom hell reading irrelevant details. What's sad is, she probably understands how his proof works and probably could have written the book I expected
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Ransen Owen
5.0 out of 5 stars One of the best non fiction books I've read this year.
Reviewed in the United Kingdom on June 1, 2019
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The flow was good, the subject was interesting and well explained. I'd even describe it as a "page turner".

(I skipped the 10 or so pages which are supposed to explain the proof, I find that either you are an expert in a given field or the simplified explanation is pointless.)
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Gregory Connor
5.0 out of 5 stars A wonderful book about the philosophy of logic and Godel/Wittgenstein
Reviewed in the United Kingdom on January 21, 2013
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I was familiar with Godel but knew little about Wittgenstein before reading this short but enchanting book about their mid-20th century clash of ideas. Two very interesting characters-- so different but each brilliant in his own way. This book also provides clear insights into Godel's key contributions to mathematical logic, for the layman like myself.
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Kiri
4.0 out of 5 stars An intersting account of some very deep ideas.
Reviewed in the United Kingdom on November 3, 2021
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Roughly speaking, Godel's theorem tells us that there are true statements in. mathematics that cannot be proved. This put an end to Hilbert's dream of finding some axiomatic system for the whole of Maths, where every true statement could be proved. In this book Godel's life is described from his days in Vienna to his time at Princeton. Particularly interesting is his interaction with Einstein. They used two walk together to work at the institute in Princeton This part is easy to read and well-written but the author then makes a valiant effort to explain the ideas that Godel used to prove his results. Of course, not easy reading but very worthwhile.
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V. H. Svoboda
5.0 out of 5 stars Five Stars
Reviewed in the United Kingdom on September 19, 2017
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Well written and informed story of an eccentric brain
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