- Buy 2, save 50% on 1. Offered by Amazon.com. Shop items
- Amazon Business: Make the most of your Amazon Business account with exclusive tools and savings. Login now
- Business Prime : For Fast, FREE shipping, premium procurement benefits, and member-only offers on Amazon Business. Try Business Prime free.
Other Sellers on Amazon
+ $4.53 shipping
+ $3.99 shipping
+ $5.99 shipping
Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries) Paperback – February 17, 2006
Enhance your purchase
Enter your mobile number or email address below and we'll send you a link to download the free Kindle App. Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required.
To get the free app, enter your mobile phone number.
Frequently bought together
Special offers and product promotions
― 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
- Publisher : W. W. Norton & Company; Reprint edition (February 17, 2006)
- Language : English
- Paperback : 224 pages
- ISBN-10 : 0393327604
- ISBN-13 : 978-0393327601
- Item Weight : 9.1 ounces
- Dimensions : 5.4 x 0.9 x 8 inches
- Best Sellers Rank: #162,687 in Books (See Top 100 in Books)
- Customer Reviews:
Top reviews from the United States
There was a problem filtering reviews right now. Please try again later.
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.
Top reviews from other countries
1. The Vienna Circle and Godels place within it. There are a references to differing philosophical schools without explaining what they are and their relevance. This part is overlong.
2. The Theory itself - an attempt to explain in lay terms isnt entirely successful. Its not surprising as its a complex theory but part 1 could have been reduced and this worked on more
3. Godels adult life. An anecdotal skim through Godels social interactions. Could have been omitted to be honest,
(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.)
「ヒルベルト・プロジェクト」の経緯について勉強になったことと（「数学者は絶対に『不可知／Ignorabimus』を許してはならない」ってヒルベルト宣言は凄まじいなあ）、著者のシンパシーがかなり濃厚に伝わることに四つ星を進呈するが、ゲーデル哲学を知りたい方はゲーデル本人の未発表論文『Modern Development of the Foundations of Mathematics in Light of Philosophy』（原文英語）を探してお読みになることをお勧めする。私は本書よりそちら論文の方に感動した。