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Incompleteness: The Proof and Paradox of Kurt Gödel

(Great Discoveries)

3.96  ·  Rating details ·  1,295 ratings  ·  124 reviews
"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.
Paperback, 224 pages
Published February 17th 2006 by W. W. Norton Company (first published 2005)
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Dec 07, 2018 rated it really liked it
Gödel’s Riposte to Augustine

I find an unexpected comfort in Gödel’s Proof of Incompleteness in mathematics - essentially that we have no good reason to believe that even arithmetic has a solid logical foundation. To me the implication is that no matter how much we learn, we will still be wrong. Not because we don’t know everything, but because what we do know is fundamentally uncertain. We are not unsure only about mathematics. Physics for example will always exhibit paradoxes like those of
I’ve always been fascinated by Kurt Gödel and his incompleteness theorems. While Douglas Hofstadter did a fine job in explaining the latter in his book Gödel, Escher, Bach , and also in a video lecture, there’s hardly any biographical/personal information about the human behind the mathematician here to be found. That’s where Rebecca Goldstein jumps in. Her book focuses on the life of the “greatest logician since Aristotle”. About his time at the Vienna Circle (a.k.a. the Schlick-Group) in the ...more
Jan 21, 2011 rated it really liked it
The more I think about language, the more it amazes me that people ever understand each other at all.

Fucking Gödel.

The above (pictured with a rueful smile and head shake) succinctly summarizes my feelings for the incomparable Kurt Gödel—the greatest logician since Aristotle, as Rebecca Goldstein makes sure to iterate several times—the quiet and unassuming genius whose steel-trap mind could capture those ethereal abstract truths and convert them into human language constructs; who single-handedly
Brad Lyerla
Nov 18, 2013 rated it it was amazing  ·  review of another edition
INCOMPLETENESS is an excellent book about an intellectually elusive subject. Kurt Godel's fame was established by his proof of something called "the Incompleteness Theorem." His proof employed formal logic to establish a basic truth about mathematics. Namely, that in closed systems, there will be true statements that cannot be proved. Until Godel's proof, many leading mathematicians assumed the opposite was true. This is a challenging subject to write about, but Goldstein makes it easily ...more
Mεδ Rεδħα
Nov 24, 2018 rated it it was amazing
"…the precocious Gödel grasped the limits of parental omniscience at about the age of five. It would be comforting, in the presence of such a shattering conclusion, especially when it’s reinforced by serious illness a few years later, to derive the following additional conclusion…the grownups around me may be a sorry lot, but luckily I don’t need to depend on them. I can figure out everything for myself. The world is thoroughly logical and so is my mind—a perfect fit.

Quite possibly the young
Kyle York
Mar 20, 2013 rated it it was amazing
What a wonderful book. Goldstein not only lays out Godel's famous theorems in relatively understandable terms for the layman (an accomplishment in itself,) but provides an original, funny, and lucid account of the intellectual atmosphere in which these theorems arose. She discussed Godel's relation to the Logical Positivists and Formalists, which sheds great light upon the meaning of his discoveries. She also dispels the postmodernists mythologies about what Godel's theorems mean. In addition, ...more
Jul 13, 2008 rated it it was ok
Shelves: real-worldy
Suck it, postmodernists.
Jan 11, 2018 rated it liked it  ·  review of another edition
Recommends it for: Gödel-girls-and-boys
I'm going to reread the sections specifically about Gödel's incompleteness theorems because i'd really like to be able to speak about them without misrepresenting them one of these days. You could call it a New Year resolution if you wanted to.

I don't know how to rate this book because i'm so incapable of rating Goldstein's ability to convey the mathematical ideas. I can say that i thought i read many sentences more than once ... but in completely different sections of the book, as if the editor
Stephie Williams
Apr 30, 2014 rated it liked it
I may have read this book previously, but I could not remember doing so. This review replaces the very short review I had previously made.

In this book Rebecca Goldstein sets out to explain Kurt Godel’s life, including his incompleteness theorems. She first sets the stage in an environmental context, both personal and mathematical. Then comes her explanation of Godel’s theorems. And finally, the later stages of his life.

The book starts our interestingly enough with the relationship between
Keith Akers
May 21, 2010 rated it really liked it
This is a great book to learn more about "Goedel's Proof" (actually two proofs, or actually three proofs if you count his Ph. D. thesis on predicate calculus). The incompleteness of mathematics is an astounding concept -- it's so astounding, that you are left breathless, not even sure what the whole thing means. Does this mean that God exists? Actually, Goedel himself toyed with variations of the ontological proof. The incompleteness of mathematics is just really hard to wrap your brain around; ...more
Apr 24, 2016 rated it it was amazing
Shelves: philosophy
This book is succinct, accessible and well constructed. Godel's Incompleteness Theorems are so significant in the history of ideas that it is essential to have a decent grasp of just what they are and why they mattered and this book supplies that need for general readers. It gives a good enough explanation of Godel's findings and deals with the reactions of other major names to his theories, which sheds interesting light on their work too.

We need to grasp Godel's theories accurately because we
Collin Winter
"Incompleteness" is less about Gödel's actual incompleteness theorems -- the proofs and their specific mathematical legacy -- than it is about the philosophical environment those theorems were developed in. Put another way, this is a book less about Gödel and more about Gödel and Wittgenstein, or perhaps more accurately, about Wittgenstein and Gödel.

This is a book that prefers to tell rather than show: Goldstein spends 160 pages telling the reader how amazing and important and revolutionary
Apr 23, 2007 rated it liked it
Well written and a good picture of Godel, his work, philosophy and the times he lived in. There would be more starts up there but for 2 reasons:

1 The book goes through thumbnail sketches of Godel's famous proofs and then a more involved version, but even after the more detailed explanation I still felt like I had only scratched the surface of it. Some of the things asserted about the process of Godel numbering seemed almost magical as a result. This is a tough balancing act for any popular take
Jun 28, 2019 rated it really liked it
Rebeccca Goldstein is an American philosopher and writer. This book managed to dive into who Gödel was and what his views on mathematics really meant. I was intrigued by this myth of a man; a man who seemed only to get more quiet as he grew older. He never seemed to argue against the popular trends of the day yet was quite certain of his own opinion. In fact the last great philosopher he admired was Leibniz. I was interested in how Gödel was formed as a person and given his infamous secrecy this ...more
Jun 08, 2014 rated it liked it  ·  review of another edition
Shelves: purple
David Foster Wallace (RIP) once referred to Kurt Goedel, the subject of this book, as mathematics' Prince of Darkness. Douglas Hofstadter gave his Incompleteness Theorems (1 and 2) a central role in his book “Goedel, Escher, Bach”. Goedel's ideas are so central to 20th century thinking that it is likely that Einstein (for many years until his death Goedel's closest friend) was the only person he ever met who he was not, eventually, to become more famous and influential than (for example ...more
Apr 16, 2014 rated it liked it  ·  review of another edition
This book is serviceable in being a readable introduction to Gödel's famous proofs and a general (very general) understanding of the intellectual and historical circumstances within which it grew. While I do not have the capabilities to judge whether she provides a completely faithful representation of Gödel and his proofs, it is within my power to deem her handling of Wittgenstein, whom she spends a great deal of pages discussing, to be sorely unsympathetic, uncharitable, and misrepresentative. ...more
Peter Flom
Sep 02, 2012 rated it it was amazing
Shelves: math
This is a great book about a strange and brilliant man. It's hard to think of a writer better qualified to write it than Rebecca Goldstein. Full review:
Feb 27, 2009 rated it liked it  ·  review of another edition
A persuasive interpretation of Godel as mathematical Platonist rather than as harbinger of modernist anxiety.
Daniel S
Dec 30, 2013 rated it really liked it
Shelves: favorites
Sad at the end. So sad.
Jonathan Fretheim
Mar 31, 2011 rated it liked it
Gödel died a paranoid and lonely man. This book is a bummer.
Feb 14, 2016 rated it it was amazing
This book is perfectly balanced between the historical and analytical. The last line in the book was worth all the effort and shoots the book up several layers of appreciation.
Mar 15, 2007 rated it really liked it
A great book about this most genius mathematical mind of the last 100 years. Rebecca Goldstein actually provides a simplified version of the proof of the Incompleteness Theorem.
Megan Lawson
Aug 04, 2018 rated it it was ok
I very much enjoyed the second half of this book - in which there was a discussion (though I wish more mathematically and logically minded) of Godel's Incompleteness Theorems as well as stories of his life and interactions at Princeton.

However I did not enjoy the first half of the book much at all. It felt like it was a 150 page set up to what the philosophical world was like that Godel was walking into. I didn't need that and didn't feel like it did much to move along my understanding of Godel
Nov 29, 2018 rated it really liked it
Goldstein, a novelist as well as a philosopher, writes engagingly. The story she tells about Kurt Gödel has all the drama of a tragic novel. As well as the book is written, I still had trouble getting my mind around the Gödel's Incompleteness Theorems. Goldstein's central argument is that the cultural appropriation of Gödel's work, as well as of Einstein's, has misrepresented them as icons of post-modernism. As someone who has tasted at least a few sips of the post-modernist kool-aid I found ...more
Jul 21, 2015 rated it liked it
Shelves: math
Godel's incompleteness theorem proves that, in any consistent formal mathematical system sufficiently complex to be useful (for instance, the one we use, with our pluses and minuses and our ten digits), there must exist certain arithmetical statements which are true, but impossible to prove. The gist is that it's impossible to remove all intuitive statements--for instance, that a=a--from mathematics, without rendering our system inconsistent, i.e. broken. This was, and still is, a huge deal for ...more
What a fascinating and under appreciated man.
Oct 30, 2018 rated it really liked it
Coming from a background with limited mathematical and philosophical knowledge, I thought the author did an admirable job of creating an easy to understand narrative behind the intellectual climate in which Gödel came up with his theorems. I struggled to understand the more technical elements of the proof and will definitely have to go back to it some day to fully appreciate the beauty of it.

What makes the book really enjoyable are the themes that the author skillfully builds on throughout the
Jamie Collins
Probably this is a really good book, but it’s intended for a more erudite reader. It was largely over my head. It's part of the Norton "Great Discoveries" series, and the other two books in that series I've read (on Marie Curie and Ignac Semmelweis) were more accessible than this one.

The introduction is nice - she talks about the creation of the Institute for Advanced study in Princeton, and the unlikely friendship between Gödel and Einstein. Then she launches into a discussion of the Vienna
May 21, 2010 rated it really liked it
This added to my understanding of philosophy more than mathematics, but the book was primarily a biography of Gödel, so the fact that a reader can actually take away some philosophical insights from a biography makes this book unique.

The description of Gödel’s proof was short and well written, but it was only about 25 pages. I can say I have less of a misunderstanding of the proof, but I can’t say I understood Goldstein’s explanation. If you’re looking for a good explanation, this probably isn’t
Mar 16, 2011 rated it liked it  ·  review of another edition
I found half quite interesting

Perhaps it's narrow-minded of me, but I didn't really care too much about Gödel's upbringing and his (non-proof-based) philosophical views. It is good to cover early history in a biography, but the focus should have been on his incompleteness work. Goldstein spends far too much time on his philosophical views (to the point of feeling quite redundant to me) and how they contrasted with other leading thinkers of the time. I found her focus on this topic inexplicable
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Rebecca Newberger Goldstein grew up in White Plains, New York, and graduated summa cum laude from Barnard College, receiving the Montague Prize for Excellence in Philosophy, and immediately went on to graduate work at Princeton University, receiving her Ph.D. in philosophy. While in graduate school she was awarded a National Science Foundation Fellowship and a Whiting Foundation Fellowship.


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“The necessary incompleteness of even our formal systems of thought demonstrates that there is no nonshifting foundation on which any system rests. All truths — even those that had seemed so certain as to be immune to the very possibility of revision — are essentially manufactured. Indeed the very notion of the objectively true is a socially constructed myth. Our knowing minds are not embedded in truth. Rather the entire notion of truth is embedded in our minds, which are themselves the unwitting lackeys of organizational forms of influence.” 10 likes
“Einstein’s and Gödel’s metaconvictions were addressed to the question of whether their respective fields are descriptions of an objective reality—existing independent of our thinking of it—or, rather, are subjective human projections, socially shared intellectual constructs.” 1 likes
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