Incompleteness: The Proof and Paradox of Kurt Godel

by Rebecca Goldstein

"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.

Genres: Philosophy, Mathematics, Biography, Science, Nonfiction, History, Logic

296 pages, Hardcover

First published January 1, 2005

About the author

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.

After earning her Ph.D. she returned to her alma mater, where she taught courses in philosophy of science, philosophy of mind, philosophy of psychology, the rationalists, the empiricists, and the ancient Greeks. It was some time during her tenure at Barnard that, quite to her own surprise, she used a summer vacation to write her first novel, The Mind-Body Problem. As she described it,

"To me the process is still mysterious. I had just come through a very emotional time, having not only become a mother but having also lost my father, whom I adored. In the course of grieving for my father and glorying in my daughter, I found that the very formal, very precise questions I had been trained to analyze weren’t gripping me the way they once had. Suddenly, I was asking the most `unprofessional’ sorts of questions (I would have snickered at them as a graduate student), such as how does all this philosophy I’ve studied help me to deal with the brute contingencies of life? How does it relate to life as it’s really lived? I wanted to confront such questions in my writing, and I wanted to confront them in a way that would insert `real life’ intimately into the intellectual struggle. In short I wanted to write a philosophically motivated novel."

The Mind-Body Problem was published by Random House and went on to become a critical and popular success.

More novels followed: The Late-Summer Passion of a Woman of Mind; The Dark Sister, which received the Whiting Writer’s Award, Mazel, which received the 1995 National Jewish Book Award and the 1995 Edward Lewis Wallant Award; and Properties of Light: A Novel of Love, Betrayal, and Quantum Physics. Her book of short stories, Strange Attractors, received a National Jewish Book Honor Award. Her 2005 book Incompleteness: The Proof and Paradox of Kurt Gödel, was featured in articles in The New Yorker and The New York Times, received numerous favorable reviews, and was named one of the best books of the year by Discover magazine, the Chicago Tribune, and the New York Sun. Goldstein’s most recent published book is, Betraying Spinoza: The Renegade Jew who Gave Us Modernity, published in May 2006, and winner of the 2006 Koret International Jewish Book Award in Jewish Thought. Her new novel, Thirty-Six Arguments for the Existence of God: A Work of Fiction, will be published by Pantheon Books.

In 1996 Goldstein became a MacArthur Fellow, receiving the prize which is popularly known as the “Genius Award.” In awarding her the prize, the MacArthur Foundation described her work in the following words:

"Rebecca Goldstein is a writer whose novels and short stories dramatize the concerns of philosophy without sacrificing the demands of imaginative storytelling. Her books tell a compelling story as they describe with wit, compassion and originality the interaction of mind and heart. In her fiction her characters confront problems of faith: religious faith and faith in an ability to comprehend the mysteries of the physical world as complementary to moral and emotional states of being. Goldstein’s writings emerge as brilliant arguments for the belief that fiction in our time may be the best vehicle for involving readers in questions of morality and existence."

Goldstein is married to linguist and author Steven Pinker. She lives in Boston and in Truro, Massachusetts.

Reviews

[BlackOxford · Rating 4 out of 5]

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 quantum theory. People unaccountably will always do things which are bad for them. And my socks will continue to disappear in the dryer. There is, in other words, a fundamental continuity, a necessary humility, in life that will never be interrupted by the latest technology from Apple or Trump’s most recent tweet.

Goldstein appreciates the cultural import of Gödel’s Proof. In an age rocked by the counter-intuitive implications of things like Relativity Theory and Quantum Mechanics, which present paradoxes that seem resolvable by further thought, the Incompleteness Theorem is even more of a scandal. It exists in logic not in observation. It will remain in force no matter what else we learn about the world. As Goldstein says, “Gödel’s theorems, then, appear to be that rarest of rare creatures: mathematical truths that also address themselves—however ambiguously and controversially—to the central question of the humanities: what is involved in our being human?”

Incompleteness is a leveller. It applies to the deepest thinker, the wealthiest entrepreneur, the most powerful politician, as well as to any random cog in the modern economic machine and to those who have been rejected by it. It is the modern form of the ancient Christian doctrine of Original Sin, formulated so forcefully by the great saint, Augustine of Hippo.

Just as Original Sin, Incompleteness affects us all. We inherit it, not through our genes, but through our memes. Incompleteness comes packaged in language itself. To engage the world through language is to enter the domain of Incompleteness, and therefore of profound doubt. And just as Augustine said in his religious idiom, Godel has restated the situation in his: There is no escape. The user of language is trapped and is incapable of extricating himself from an existence of rational error - about himself as well as the world around him.

But unlike Augustine, Gödel doesn’t presume he has a solution. Augustine withholds his assent to radical doubt, which he neutralises through ‘faith.’ Clearly Augustine cannot stomach the intellectual humiliation of not having a way forward, of not mitigating the debilitating effects of the human condition. Like many before him and since, Augustine fills the intellectual vacuum with the magic of a divine saviour, the guarantor of the ultimate rationality of human and other life on Earth. For Augustine Christ is the deus ex machina who is capable of correcting, literally remaking, flawed human nature into something reliable. And if other people don’t ‘get it,’ he feels entirely justified in literally throwing them to the lions.

Augustine, of course, merely demonstrates the extent to which the basic human flaw can make us crazy. That his need for and presumption of a beneficent saviour is part of his Original Sin is something which doesn’t occur to him. His solution is actually the greatest delusion produced by his fundamental insight. He neurotically invents in order to avoid his own logic, and then projects his neurosis onto the world as a defect which must be eliminated. He is the first Christian terrorist.

Gödel has no such delusion and therefore accepts the bleakness of our prospects. What Godel allows us to see, however, is that mathematics is a genre of poetry with its own arbitrary, but still rather satisfying, conventions. It is something to be done for its own sake, not because it leads anywhere else (the Princeton Institute for Advanced Studies of which Godel, along with Albert Einstein, was a founding member was created on the principle of “the usefulness of useless knowledge”).

Prospects, bleak or not, - whether spiritual or material - have nothing to do with the matter, therefore. Mathematics, like the rest of poetry, is important in the continuous present. It doesn’t save us but it passes the time rather pleasantly. Gödel was no materialist or relativist, however. For my money he was more spiritual than Augustine, as well as more committed to the idea of truth. He knew there was something permanently beyond human reach. As a committed Platonist, he considered this to be the abstract realm of numbers, which exist quite independently of human thought about them.

Numbers for Gödel are eternal and impassive, that is, there is nothing we can do to affect their existence. They call to us from elsewhere, much like Augustine’s God. The principle difference however is that numbers make no absolute demands and pass no judgments. They exist for our comfort and edification not for remaking us as something we’re not. And very few have felt compelled to use violence to defend number theory.

Goldstein makes an apposite observation: “Paranoia isn’t the abandonment of rationality. Rather, it is rationality run amuck, the inventive search for explanations turned relentless.” Augustine is an example of rationality run amok. This certainly is the heart of Gödel’s riposte to Augustine’s and all other religious arrogance.

[Matt · Rating 4 out of 5]

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 late 1920s, the emigration to Princeton (the Institute for Advanced Study), where Gödel became friends with Albert Einstein, and finally his personal incompleteness and his tragic death, apparently brought on by self starvation in 1978.

There is, of course, a chapter devoted to Gödel’s proof of the incompleteness of formal systems, which is not entirely without mathematics and formulas. The proof itself is rather simplified and no great mathematical knowledge is required. Logical understanding is very helpful, however!

As opposed to most of the participants in the Viennese circle, Gödel was not a positivist, but rather held it with Plato: He had an axiom by which he looked at the world: nothing that happens in it is due to accident or stupidity. Gödel believed in an abstract reality, and “that the truths of mathematics are independent of any human activities, such as the construction of formal systems — with their axioms, definitions, rules of inference, and proofs.” The positivists, on the other hand, believed in the creation of the (meta-) mathematical reality by man alone. Everything outside this reality, all meta-mathematics, was meaningless and of no importance to them. Since Gödel also acted in this circle of positivists, there was reason for many to believe that his proof of incompleteness was a success story of positivism, but, according to Goldstein, "could not be further from the truth."

For me the most exciting topic was the confrontation of Gödel and Ludwig Wittgenstein, who was venerated by the Viennese circle in an almost mystical way.

Wittgenstein’s Tractatus Logico-Philosophicus was apparently read and read again in the circle for months, as if it were a holy book. Gödel never acknowledged Wittgenstein’s work, and Wittgenstein was “adamant in denying the possibility of a proof such as Gödel’s.” The characters of the two could not be more different. Ironically, however, one can conceive the final statement of the Tractatus as a kind of incompleteness theorem: “Whereof one cannot speak, thereof one must be silent.” Unfortunately, Gödel was anything but ironically inclined, and perhaps he did not see this, so that one can almost speak of a meta-irony here.

Gödel’s other important relationship was the warm and close friendship with Albert Einstein. The two of them were very dissimilar (Einstein was 27 years older than Gödel), but they understood each other well and appreciated each other extremely. Einstein confessed that “he only went to his office to have the privilege of walking home each day with the logician, the two great minds of the twentieth century able to share, at least for a while, their intellectual exile with one another.” After Einstein’s death in 1955 Gödel became very lonely indeed. The final chapter shows the incessant descent of this brilliant thinker quite impressively, as I find.

Apart from the formal parts of the third chapter, this is a very easy to grasp and recommendable reading for all those who want to make themselves more familiar epistemology, history of science, meta-mathematics and logic.


This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

[Szplug · Rating 4 out of 5]

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 elevated mathematical logic to a respectable berth at the table; who produced several earth-shaking proofs that performed the almost unheard-of double duty of having mathematical and metamathematical implications; the dear friend of Einstein who presented the latter, on his 70th birthday, with mind-bending solutions to his field equations; the paranoid recluse who, seeing conspiracies everywhere, eventually died from starvation when his wife, being in the hospital, was no longer available to prepare his food and thus left him wide open to his pervasive fear of being poisoned by ill-wishers and positivist conspirators.

Goldstein quite clearly harbors a fondness and admiration for the eccentric logician, whom she once saw in person at a Princeton house party, and she does an excellent job of situating him within his time period, his academic milieu, his long tenure at the Princeton-adjunct Institute for Advanced Studies and, especially, in describing both his two Incompleteness Theorems—having first outlined his graduate student dissertation on Completeness—and explaining the immense impact they had upon the mathematical, scientific, and philosophic world once they had interpenetrated these disciplines of the mind.

I've read other reviews that complain about the amount of time she devotes to Wittgenstein, who is said to be peripheral to the wonders that Gödel created, but I think she made a smart choice. Not only does the fiery and charismatic Wittgenstein add some missing color to the proceedings, but Goldstein makes an IMO apt analogy between the thought of the philosopher during his Tractatus Logico-Philosophicus period and Gödel's Theorems. That the latter proclaims first that in a formal mathematical system of assumed consistency there will exist a statement that is both true and unprovable; and second that said formal system's consistency cannot be proved from within itself, is, as Goldstein argues, from the same mental territory that Wittgenstein drew from in his early thought. The TLP end statement Whereof one cannot speak, one must pass over in silence hints that the silence contains all of the important things; abstractions that inhabit the same Platonic ideality that Gödel believed existed. Two brilliant Viennese, one a Platonist amidst a sea of Positivists, the other—well, a sui generis explorer on the roiling seas of language.

If, in the end, the reader finds he hasn't actually discovered much that Gödel actually said, it is perhaps because Goldstein found it difficult to shoehorn what was available—and relevant—into a brief biography, and one more concerned with his ideas than the actual man himself; and, with this end in mind, I enjoyed her presentation from start to finish. I do plan to read Nagel's Gödel's Proof , because the idea has always been stimulating for me, even when I only understood it in a very general sense. At a time when Hilbert and Russell, at the apex of the Positivist surge, were attempting to chain mathematics within the bounds of a formal human rationalism, Gödel proved how futile it was for man to think he could tame infinity. This illusion has recurrently manifested itself as a product to be manufactured in the mental workshops, the principal theme of Leszek Kołakowski in his slim masterpiece Metaphysical Horror —though to the genial Pole, the shops themselves must perforce endeavor to continue operating. To finite man the infinite is an awesome, disturbing, and chaotic beast, forever mocking human aspirations and advances with its eternities and paradoxes and circularities. We have always been able to intuit it, yet without restraining it and fixing it into place, and the quiet genius showed that this cannot be effected. What implications does this bring to the existence of God, of a Platonic world of abstract ideals, of all that the mind can conjure but never empirically locate? It's a source of endless fascination for me—and, apparently, for Gödel as well; so at least I've got that in common with the great man.

[Darwin8u · Rating 5 out of 5]

A nice micro-history of Kurt Godel and his incompleteness theories. One of the better books in this excellent 'Great Discoveries' series.

[Brad Lyerla · Rating 5 out of 5]

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 accessible to a casual reader of science and philosophy like me.

Godel's personal story is interesting. He was not a Jew, but had many colleagues who were. Yet, he failed to take a stance against the Nazis, instead choosing to continue his work even as Hitler's policies forced the Universities of Germany and Austria to purge Jewish faculty members. It is unclear how much he knew about the worst atrocities perpetrated by the Nazis. Later, Godel immigrated to the US and became a close friend and frequent companion of Albert Einstein in Princeton, NJ. Godel struggled with mental illness and, ultimately, it contributed to his death.

I may bestir myself to write a full review of INCOMPLETENESS eventually. In the meantime, I recommend the book highly and I am deeply impressed with Goldstein. I look forward to reading more of her stuff.

[Nelson Zagalo · Rating 5 out of 5]

Kurt Gödel foi uma das mentes brilhantes da ciência do século XX, existindo quem o compare a Aristóteles, a Einstein ou Heisenberg, mas ao contrário destes, e apesar do seu inabalável contributo, nunca conseguiu alcançar o mesmo patamar de respeitabilidade pública. Rebecca Goldstein procura neste livro colmatar esse problema. Para o fazer, traça a história de vida de Gödel, aproveitando a sua veia romancista para nos envolver empaticamente com vários personagens históricos, e enquanto o faz dá conta do contexto científico com que nos conduz até ao âmago demonstrativo dos dois teoremas da incompletude de Gödel. Gostaria de dizer que é um livro acessível, porque foi para isso que Goldstein trabalhou, e admito que fez um trabalho notável, mas ainda assim não é facilmente acessível a todos, talvez por que tal não seja possível para algo que constitui em si mesmo a complexidade primordial da lógica. Contudo Goldstein consegue com este livro tornar clara a relevância dos teoremas e só por isso vale completamente a sua leitura.

Continua no blog: https://virtual-illusion.blogspot.com...

[Mat · Rating 3 out of 5]

اما هر خطایی ناشی از عامل های بیرونی است(از قبیل عاطفه، تعلیم و تربیت)،خرد به خودی خود خطا نمی کند.

[Kyle York · Rating 3 out of 5]

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, she outlines Godel's relationship with Einstein, both intellectual and personal, which turns out to be rather significant.

Objective topics covered in this book range from the nature of mathematical reality, to the nature of time, to the nature of the mind.

In addition to all this, she gives an account of Godel's personal life and a picture of who he was as a person. The picture that she paints is tragic, warm, and very eccentric. By the end of the book, I found myself as touched by these accounts of Godel's life as by any novel.

Goldstein was the perfect person to write this book-- because of her lucidity in regards to mathematical logic, science, and analytic philosophy, and because of her experience as a novelist and capacity for wit, humor, and sympathy for the human experience.

If you are like me and unfamiliar with but interested in Godel, you will never look at the world exactly the same way after reading this book. I highly recommend it.

(On a side note, I found it interesting that this book was listed under Science/Mathematics and not Philosophy. Telling of the current cultural atmosphere, no?)

[Collin Winter · Rating 2 out of 5]

"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 Gödel's proofs were before she ever unpacks the proofs themselves. Once she does set to explaining them, she makes the same mistake many authors attempting to popularize math and science make by simplifying too much and skipping steps in the name of not overwhelming the reader. What's left is paradoxically difficult to follow, as it is riven with caveats like "we won't be rigorous" and "this isn't really what Gödel did".

I would not recommend this book to anyone seeking to learn more about Gödel's work on incompleteness. However, if you're looking for a book about Wittgenstein, Gödel, and philosophical circles in 1920s and 30s Vienna, this may be the book for you.

[Liquidlasagna · Rating 1 out of 5]

Amazon Review

Bad history, bad math

Pretentious and sloppy, filled with mistakes and repetitions. See the AMS review on-line for some details. Terrible.

And shame on S. Pinker and B. Greene for their glowing cover reviews. Were they swayed by the adoring references to them in the text?

---

Review from someone at Stanford

Goldstein bases her story of the development of the incompleteness theorems on the supposed fact that – in contrast to the views of the Vienna Circle – Gödel was already a confirmed Platonist.

The philosophy of Platonic realism in mathematics holds that abstract objects such as numbers, points, sets and functions have an objective, immutable existence independent of the observer, and that the task of the mathematician is to establish truths about this reality.

It is Goldstein’s conceit that Gödel fell in love with Platonism in 1925 in an introductory course on the history of philosophy. According to her, this put him at complete odds with the logical positivists when he attended their meetings.

‘Gödel’s audacious ambition to arrive at a mathematical conclusion that would be a metamathematical result supporting mathematical realism was precisely what yielded the incompleteness theorems.’

Goldstein claims that by 1928 this ambition had driven him to begin work on the proof of the first incompleteness theorem, ‘which he interpreted as disproving a central tenet of the Vienna Circle . . . He had used mathematical logic, beloved of the logical positivists, to wreak havoc on the positivist antimetaphysical position.’

In addition, her view is that Gödel’s theorems were designed to refute the formalist program of David Hilbert, according to which mathematics is nothing but an arbitrary human creation, ‘a game played according to certain simple rules with meaningless marks on paper’.

Wrong, wrong, wrong! But we can see how Goldstein was misled. There is no doubt that the mature Gödel was a mathematical Platonist, as is attested by some of his published and unpublished papers from the 1940s onwards, and by his many communications with colleagues.

True, Gödel himself said here and there that he held these views since his student days. There is considerable evidence, however, that Gödel was by no means fixed in his philosophical views prior to 1940.

For example, in a lecture he gave to the Mathematical Association of America in 1933 he made a strongly anti-Platonist statement about the axioms of set theory (proposed by some as a foundation for all of mathematics), namely that ‘if interpreted as meaningful statements, [these axioms] necessarily presuppose a kind of Platonism, which cannot satisfy any critical mind and which does not even produce the conviction that they are consistent.’

Goldstein uses the word ‘metamathematics’ mistakenly throughout, taking it to refer to the philosophy of mathematics, by loose analogy with ‘metaphysics’. (Not only that, she uses ‘meta’ as a neologizing modifier with abandon, viz. ‘metaquestion’, ‘metaconviction’, ‘metalight’, ‘metaovertones’, etc.)

This is not at all the sense in which it has been used by logicians since the 1920s, when it was introduced by Hilbert. In this usage, metamathematics is a branch of mathematics: it studies the syntax and semantics of formal languages and axiomatic systems set up to model informal mathematical reasoning.

In that sense, Gödel’s theorems are a contribution to metamathematics, but not in Goldstein’s sense.

Finally, Hilbert never said that mathematics is nothing but ‘a game played according to certain simple rules with meaningless marks on paper’, a view incorrectly ascribed to him by Eric Temple Bell, a mathematician but a very careless historian of mathematics (as well as an author of science-fiction under the pseudonym, John Taine).

Hilbert, the most versatile and influential mathematician of the late 19th and early 20th centuries, certainly thought that mathematical concepts are determined by axiomatic systems, but he did not deny that those concepts had meaning; on the contrary, in his practice he took mathematics fully at face value.

....

As Gödel explained for an article by Hao Wang about his achievements, the incompleteness theorems came out of his attempt in 1930 to contribute to Hilbert’s program by providing a consistency proof of an axiomatic system for analysis.

Whatever his philosophical views may have been at the time, his motivation had nothing to do with undermining logical positivism or formalism.

To understand Gödel’s intervention, and in what way Goldstein has misconstrued it, we need to take account of some more logical terminology....

Nor do the incompleteness theorems in and of themselves support mathematical Platonism.

....

Moreover, on closer examination, it is not the truth of mathematical statements in general that is at issue, but only the truth of number-theoretic statements of a very special form
.
In other words, not only is Goldstein mistaken about Gödel’s philosophical motivations for proving his incompleteness theorems, she is also wrong about their supposed philosophical consequences.

She has, perhaps, been misled by the fact that Gödel himself thought they had such consequences, though not until much later, during the 1950s.

He made such claims only in unpublished lectures and essays which, following his death, were retrieved from his Nachlass and published in his Collected Works.

Towards the end of a 1951 lecture, ‘Some basic theorems on the foundations of mathematics and their implications’, he says of his arguments in favour of Platonism: ‘Of course I do not claim that the foregoing considerations amount to a real proof of this view about the nature of mathematics. The most I could assert would be to have disproved the nominalistic view, which considers mathematics to consist solely in syntactic conventions and their logical consequences.’

Between 1953 and 1959, he elaborated this claim in an essay entitled ‘Is mathematics syntax of language?’ Gödel prepared six versions of the essay, but in the end was not satisfied with any of them and did not submit it. He wrote to the editor: ‘It is easy to allege very weighty and striking arguments in favor of my views, but a complete elucidation of the situation turned out to be more difficult than I had anticipated.’

All this is marked by Gödel’s extreme caution about publishing his philosophical convictions, when he could not arrive at completely unassailable arguments in their favor.

In addition to Goldstein’s mistaken view that Gödel was out to refute Hilbert’s program, much, too, is made in her book of Gödel’s supposed ‘decades-smoldering resentment’ toward Wittgenstein as the ‘idol’ of the Vienna Circle.

Despite Wittgenstein’s major influence on the Circle through the Tractatus, he was not a positivist nor one of its members, and he and Gödel never met or corresponded.

It is generally agreed that Wittgenstein, in his 1967 Remarks on the Foundations of Mathematics, seriously misunderstood the content of Gödel’s theorems.

When this was brought to Gödel’s attention a few years later, he became quite annoyed, but for his part he could not appreciate Wittgenstein’s questioning of their philosophical significance. The two were talking past each other, not at ‘loggerheads’, as Goldstein would have it.

....

Those who are fascinated by Gödel’s theorems, and the general idea of limits to what we can know—may still hunger for a more universal view of their possible significance. But they should not be satisfied with Goldstein’s ‘vast and messy’ goulash, hers is not a recipe for true understanding.

Solomon Feferman

[Stephie Williams · Rating 3 out of 5]

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 Einstein and Godel, or what can be gleamed by the little information that relates to this relationship. It moves on to Godel’s early life, his life at university and within the Vienna Circle (a philosophical club of the logical positive school of thought). After this the problems with the foundations of mathematics and its formalism is presented, starting with Frege, moving on to Russell and Whitehead, and finishing up with Hilbert and his formalist program. Then comes Godel’s incompleteness theorems. Finally, it covers the responses to his theorems, both his and others’, and the latter stages of his life.

The following are some of the comments I made while reading the book. Page numbers are in brackets [] from the W. W. Norton & Company paperback edition from 2005.

[24] “Though one might not guess it from this terse statement [from an The Encylopedia of Philosophys article] of them, the [Godel theorems] . . . are extraordinary for (among other reasons) how much they have to say . . . they range far beyond their narrow formal domain [mathematical logic], addressing such vast and messy issues as the nature of truth and knowledge and certainty . . . Godel’s theorems have also seemed to have important things to say about what our minds could—and could not—be.” (italics hers) I do not think that Godel’s theorems have that much to say outside of their domain; although, others certainly seem to think so. And, they say preciously little on the human mind, if anything. Even its effects on mathematics proper is limited. Most mathematicians happily go about their business without a thought about the two theorems.

[26] Goldstein makes a similar claim here: “Godel’s theorems, then, appear to be that rarest of rare creatures: mathematical truths that also address themselves—however ambiguously and controversially—to the central questions of the humanities: what is involved in our being human . . . Though there is disagreement about precisely how much, and precisely what, they say, there is no doubt that they say an awful lot and that what they say extends beyond mathematics, certainly into metamathematics and perhaps beyond.” At least she hedges her claim here, but what they say about being human is beyond me, except that they were produced by a human mind.

[32] Footnote 6 mentions “Godel’s hostility to the theory of evolution” with its “chance and randomness.” I think it is somewhat of misnomer that changes in the bases of DNA are random. The changes themselves are not random in a physical sense. This is because these changes have exact causes, such as chemical, ultraviolet radiation, or cosmic rays. The randomness comes in from appearances only, or if like, possibly which base gets mutated.

[39] At the end of a quote from William Barrett’s book Irrational Man it reads “. . . since mathematics has no self-subsistent reality independent of human activity that mathematician carry on.” This goes against everything Godel believed in about mathematics. This is that mathematics has an existence outside the human mind in some sort of Platonic realm.

[139] She characterizes Cantor’s continuum hypothesis thusly: “Cantor hypothesized that there is no infinite set that intervenes between the set of natural numbers and the set of real numbers; that is, there no set that has a higher ordinality than the natural numbers and a lower ordinality than the real numbers.” This is wrong. It is cardinality, not ordinality that Cantor was talking about. It makes me wonder if she can commit one (big) blunder, could she be committing further blunders when it comes to her description of the incompleteness theorems, like her previously statement that Godel did not use numbers in his proofs [23]. I am not certain here, but is not Godel numbering using numbers? She could have been saying that it can be stated without using numbers rather than the actual proof used no numbers. When I came across the mistake on page 139 I immediately check her acknowledgments. I could not find it was reviewed and commented on by anyone known to be competent in mathematics, while she did consult others for their comments. If she had consulted some one of this caliber she might have avoided this mistake as well as others that may lurk within the book.

[200] It is claimed by John Lucas [and others] that Godel’s theorem shows that minds are not machines because no machine could devise Godel’s proof. Since Godel himself proved his theorem within a formal system would that not indicate that a computer could be programmed to produce his proof. This may not apply to Turing’s uncomputability theorem. While no computer could decide if a particular algorithm would stop, no human could decide this either.

[217] Godel claims mathematical realism on the basis, not only from his own theorem, but from the undecidability of Cantor’s continuum hypothesis under the current axioms of set theory and the as yet unproved Goldbach conjecture. But, I ask: If mathematical truth lies in some Platonic realm, what about mathematical falsities? Would not they have to reside there too? If they did would that not be contradiction.*

[232] In footnote 8 talking about Godel’s argument that a dictatorship could arise from the United States Constitution, she states: “Unfortunately, Morgenstern’s account, and so all others that derive from it, omits mention of the precise constitutional flaw.” I guess I will have to give up the hope that I will ever discovered what Godel thought he saw there.

As far as Goldstein’s mathematical explanation of Godel’s theorems are concerned, I think she did a fairly good job, but the mistake when explaining Cantor’s continuum hypothesis makes me wonder. I thought a far better description was done by Douglas Hofstadter in his Godel, Escher, Bach. This being said, her biographical information was informative for me and intriguing. Godel was certainly an interesting character, as well as a very smart man; his proof is quite ingenious. So my overall rating would be mixed. On the proof itself I would call it fair. Depicting the man, Godel, it was real good, and the importance of the theorem outside of mathematics and philosophy of mathematics I could not agree with her coverage, but to be fair she did not take sides as to whose interpretations, if any, might be correct.

If you are interested in Godel’s life or are unfamiliar with his theorems this book should be of interest to you. In addition when it comes to how people both in mathematics and without view the theorems you would probably also be interested as well. I would just caution to be a little wary of some of her information.

* I just posted a blog on Mathematics called "Can Mathematics Be Constructed?" which partially includes a more fuller discussion of mathematical realism @ https://aquestionersjourney.wordpress...

[Rossdavidh · Rating 3 out of 5]

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 Wittgenstein, a much more famous thinker than Goedel in his time and a major player in this book's story, is not nearly so influential now).

For all that, we know relatively little about Goedel's life. Rebecca Goldstein attempts here to fill in the gaps as best we can. Her basic thesis, which she does a fair job of demonstrating, is that Goedel's famous theorems, in his opinion, mean roughly the opposite of what most people took them to mean.

To summarize greatly, Goedel's Theorems state that every sufficiently powerful system of arithmetic thought contains well-formed statements that are true, but unprovable. In addition, no sufficiently powerful system of arithmetic thought can prove its own consistency. To put it another way, systems of arithmetic thought can be either:
1) so crippled that they cannot be used for much because they can't say much
2) able to say a lot, but some of it is unable to be proved (or disproved), and anyway cannot prove that they are consistent

I am, of course, grotesquely abbreviating, and therefore leaving out much which is necessary. But then, the summary of Goedel's theorems may perhaps have been more influential than the full content.

Goedel was a young man in Vienna between the world wars, a time of enormous intellectual ferment. The dominant thinking of the time was Logical Positivism, and more generally the rejection of Platonic ideals.

Goldstein asserts that Goedel was himself a devoted (even devout) Platonist, who saw his own work as demonstrating that there is a realm of absolute truth, which exists independent of any (necessarily flawed) system of thought we use to get at it. This ran counter to much of 20th century thought, which saw all truths as relative.

One wonders if this wasn't all simply a rejection of "absolute", because the term "absolutism" had become associated with a political system in which all power is held by the central executive. Einstein's theory of Relativity actually takes as its starting point the tenet that the speed of light in a vacuum is absolute. What would its reception have been if it had been called the Theory of Absolutism? More to the point, there was never any chance of it being interpreted that way. It was similar with Goedel's Theorems. Goedel saw them as proving that Platonic ideals are more fundamental and important than human-created systems of thought. Almost all of the brightest minds of Goedel's time took his work to mean much the opposite, that all systems are incomplete and therefore absolute truth cannot exist.

The first half of the book is a fascinating look at a very private man, whose behavior almost caricatures the stereotypical absent-minded professor. Einstein may have been the only person who was able to relate to Goedel at all levels (both were German speakers who were driven from Europe by fascism, became famous for theoretical work cited far beyond its intended field, and ended their years at the Institute for Advanced Study in Princeton). After Einstein's death, Goedel became increasingly isolated, and eventually paranoid, ending his life in a state of near-starvation brought on by a paranoid fear of poisoning.

In the end, what does Goldstein's book tell us about Goedel that we didn't know before? I think it tells us that the way in which even the most brilliant discoveries and original thinking are interpreted is controlled as much by the zeitgeist into which it emerges, as by the intentions of the creator or even the nature of the discovery itself. It reminds us that being a genius is no protection against being fatally neurotic. And it points out to us that thinking as creative, original, and far-reaching as Goedel's may take a century or more for the rest of the world to fully come to terms with. This book brings us one step closer to doing that.

[Matthew · Rating 2 out of 5]

Suck it, postmodernists.

[Jeff · Rating 3 out of 5]

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 & author hadn't noticed something was copied from one section to another rather than moved. Another quibble of mine: intending a positive connotation for prolix as a modifier of these theorems, specifically meant to imply that despite the few words needed to write them, they affect an astounding variety of intellectual disciplines.

On the positive side, i actually enjoyed much of the "history" and "biography" bits as they fleshed out the time and places covered by Logicomix.

I'm following Goldstein's implied advice and reading Nagel and Newman's Gödel's Proof as soon as it's delivered to my doorstep.

[Davin · Rating 3 out of 5]

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 on technical subjects. (See David Foster Wallace's book on Cantor and Infinity for the same "Great Discoveries" Series if you want read more about the difficulties of writing a book like this.) Still, I wanted a little more. Since these proofs are crux of the book, this was disappointing.

2. The book spends surprising amount of time on Wittgenstein but he seems tangential at best to the story. It's interesting to read about him, but it felt tacked on.

[Megan Lawson · Rating 2 out of 5]

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 and his work. Godel was sorely neglected in the first part of the book about his discoveries.

[john · Rating 5 out of 5]

Math is fun :)

[Keith Akers · Rating 4 out of 5]

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; it's not just understanding the proof that is hard, but just figuring out What It All Means. Goedel didn't know (or didn't tell us), and we're still trying to figure it out today. There is so much more to reality than science and logic can explain, and we don't need metaphysics to see this -- all we need, actually, is mathematics. Go Plato! This is what the author conveys so well.

I read this book and "A World Without Time" by Palle Yourgrau at the same time. They are both quite good books and, as written by academic philosophers, generally mitigate my general negative opinion of academic philosophy. If you are interested in Goedel's ideas about "time travel" then read the book by Palle Yourgrau. If you are more interested in the proof itself, read this book, which goes into more detail. But really you should read both, because unlike some philosophers from Austria that I could name (ahem! cough, cough!), they actually take the time to try to explain things to you.

What I liked most about this book was the anecdotes about Goedel and those around him. She gives a fairly complete account of a really interesting anecdote, which I will have to blog about at some point, concerning Goedel's becoming an American citizen. Goedel indignantly protested that the constitution had a contradiction in it that would allow a dictator to take over! Also, the accounts of the personal relationships between the people in the Vienna Circle really helped me to understand the very different ideas which they and Goedel were respectively trying to articulate.

The main negative of the book, which is also paradoxically a strong positive, is its treatment of Wittgenstein. This book is fair towards both Wittgenstein and Goedel, but makes a lot more sense out of Wittgenstein than I think he deserves. I will spare you the comparisons with livestock agriculture and the waste products thereof. What bothers me about Wittgenstein is his condescension and failure to explain things -- being deliberately enigmatic. Yeah, sure, he might be a genius, but why should I read someone who clearly doesn't want to talk to me, or apparently anyone else? This seems like the philosophical equivalent of the medieval practice of self-flagellation.

On the other hand, the author actually makes more sense out of Wittgenstein than anyone else I've heard, and the anecdotes about Wittgenstein are helpful in describing the intellectual scene around the Vienna Circle. So paradoxically, I now feel more sympathy with Wittgenstein than I did before. But not agreement with W. -- I'm with Goedel on this one.

Wittgenstein rejected Goedel's proof, and this book makes it fairly clear that Wittgenstein never really understood it and somehow wanted to dodge the conclusions with condescending statements about having, somehow, transcended it all. But what is more amazing than that Wittgenstein rejected Goedel, was that Goedel, a master logician, who should be the hero of all the "analytic" philosophers in the U. S. A. -- since he proved something really significant about logic and mathematics that rivals or exceeds Aristotle -- is hardly even regarded as a philosopher at all, a fact which reveals the shallowness of modern academic philosophy.

I found the explanation of "Goedel's proof" of the incompleteness of mathematics (actually two proofs, as it turns out) to be quite accessible. However, I should warn you, I went to graduate school in philosophy, and took one logic class in which Goedel's proof was discussed. Unfortunately, it was not the proof that I wanted to learn about, the incompleteness of mathematics, but the completeness and consistency of what the author calls "limpid logic," a nice turn of phrase. I think that this is going to be over a lot of people's heads. But even if it is, it will at least convey what Goedel's proof means, which is actually in some ways harder than following the formal proof itself, although that's hard enough.

[Bakunin · Rating 4 out of 5]

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 was quite hard for mrs. Goldstein to show. She did a good job despite this difficulties and hints at an early life crisis at the age of 5 when Gödel realized he was smarter than his parents. What does one do in such a situation? How does a child handle that type of uncertainty? One of Goldsteins theories is that Gödels whole life was a reaction to this insight and an attempt to find stable mathematical ground given the uncertainty of the world.

Much has been written about Kurt Gödel and his incompleteness theorem. It is a mathematical theorem which has managed to enthrall even those persons (myself included) who are not mathematicians and has therefore begun to live a life of its own. Wikipedia defines the axiom (or rather axioms as there are two) in the following way:

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers. For any such consistent formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.

I am to this day not quite sure what this means but I interpret it as saying there is something magical to human intuition which cannot be boiled down to the pure rules of a system. You cannot prove a system is true within the system and like the Austrian economists say, you cannot find an objective value to things which can only be evaluated subjectively by an individual. It was also interesting to learn that Gödel was a mathematical platonist, i.e. one who believes in that mathematics exists in reality and are not a system created by humans. Why else would you be able to use math in physics (which explore how the real universe works)?

A delightful read and I recommend it to anyone interested in the history!

[Domhnall · Rating 5 out of 5]

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 need to be aware of the way others not only use but also misrepresent them. For example, William Barrett, in a famous book which I greatly enjoyed: Irrational Man: A Study in Existentialist Philosophy, 1962, concluded from his account of Godel that "mathematics has no self-subsistent reality independent of the human activity that mathematicians carry on." Godel believed pretty much the direct opposite of this fashionable assertion.

This is a very widespread problem. Goldstein gives another example in the way many serious minded people assume from Einstein's Theory of Relativity that there is no absolute reality, since everything depends on the subjective point of view of an observor. On the contrary, Einstein was quite satisfied that the purpose of science is to obtain an accurate account of a reality that is authentic and independent of any observor.

How is the general reader to avoid being sucked into false and sometimes dishonest positions based on misrepresentation? Paradox plays a major role in the story of this book. For a general reader like myself, one paradox might arise if we are asked to rely on Goldstein as an authority and to reject the opinions of other authorities, not least Wittgenstein. The solution cannot possibly be to elevate her in that way - instead, she invites us to join the debate about Godel and stop observing passively from the sidelines. Godel will start to be important to us when we start using his ideas in our own thinking and when we can do that in a credible way: based on understanding and not preconceptions.

[Aberdeen · Rating 4 out of 5]

Nerd alert!! I’m a little scared to write this because I don’t fully grasp it yet and I’ll probably explain things wrong, but how else do you learn? So here we go. This review has two sections: First, an overview of what this book is about (mostly for myself, to see if I can explain it, and for my lovely followers who tend not to be the mathematics types). And second, an overview of what I thought of the book itself, which will appeal more to people who have read or are considering reading this book. Here’s to deepening our wonder! 

What On Earth Is This About?

I first heard about Gödel’s incompleteness theorem through a Veritasium YouTube video whose description begins: “There is a hole at the bottom of math, a hole that means we will never know anything with certainty.” I have claimed as my life motto a Venn diagram I saw once with one circle holding “science” and one “art” and in the center “wonder.” I am fascinated by the intersection of different disciplines and how they move us toward wonder. And this theorem, which I'll explain more in a minute, is the glorious epitome of that: math and philosophy harmonizing and creating an opera that has shaken both fields with its majesty and unexpectedness.

Though the details of the proof are difficult, the overall strategy is—happily!—almost simplicity itself. Simple but strange, as one would expect of a proof that draws so close to the edge of self-contradiction, proving that there are true arithmetical propositions that are not provable.

So, Gödel’s incompleteness theorem: A basic summary, in Goldstein’s words, is this: “A system rich enough to contain arithmetic cannot be both consistent and complete.” This is referring to mathematical systems (like your basic Euclidean geometry which is built upon rules, or axioms, like: parallel rules never touch). Consistency means there are no contradictions (so if Euclidean geometry is consistent, it is not possible for parallel lines both to touch and never touch), and completeness means you can prove this consistency in the system (so given all the basic axioms in Euclidean geometry, like parallel lines cannot touch, can you prove that all those principles must be true?).

Here's why people were arguing about this in the first place. In the early 1900s philosophers known as logical positivists were trying to eliminate any pesky paradoxes and ambiguities in thinking and language. The best way to do this, they thought, was to get rid of metaphysics altogether since it can’t be proven empirically (i.e., through our senses). Parallel to them in mathematics, a group known as the formalists had a similar goal of eliminating paradoxes and all such messiness from math. They wanted to do this by reducing all arithmetic to symbol games. Math is not about external realities, like numbers, that exist, they argued. It’s just a manipulation of the rules we set. Two plus two doesn’t equal four because “two” really exists out there and we have found a way to describe it and how it works. Two plus two equals four because we have determined that’s what two and four mean and how they’re going to work in this system.

Gödel, however, was neither a logical positivist or a formalist. He was a Platonist. He believed that there were actual external realities to which our language and our logic conform. We do not invent systems, we discover them. 

If you treat numbers like this, however, you can run into some messy problems. Consider the famous self-referential paradox of set theory: If a set contains all sets that do not contain themselves, it cannot contain itself, otherwise it should be in the set. But if it does not contain itself, then it is a set that does not contain itself and therefore should be in the set. 

The formalists hated these conundrums and wanted a formal proof that would get rid of paradoxes like those in set theory. Here's where completeness comes in. If this proof could be “drained of the descriptive content” (i.e., take away the idea that symbols for numbers actually stand for numbers that exist and have real properties whether we have a symbol for them or not) and if, using only the rules of logic and arithmetic, this proof could show that a mathematical system was both consistent and complete, then we wouldn't have to deal with any of the paradoxes that come when you treat numbers like independently existing entities. 

Then along came Gödel, motivated by the belief that you could not treat numbers the way the formalists wanted to, that numbers did exist themselves. His famous theorem proved that this proof the formalists desired is unprovable. Specifically, it proved that there are true statements that are unprovable in a consistent system, which means it’s incomplete—the system can’t tell you everything you need to know about itself. So if you want consistency, you can't have at the same time completeness. As Goldstein says, 

The second incompleteness theorem put formalism in an impossible bind: the formalist incentive was to banish the opacity of the nature of the thing in itself (space, numbers, sets) for the transparency of formal systems. But it's of the highest priority that a formal system—drained of the descriptive content that would, so long as the axioms were truly descriptive, ensure its consistency—be proved consistent. This can only be done by going outside the formal system and making an appeal to intuitions that can't themselves be formalized.

And how did Gödel prove this? This is the most exciting part. I can't begin to get into the details—not even Goldstein does—but the basic idea is this. Gödel created Gödel numbering, where logical propositions and numbers have their own special number. Then when you start manipulating those numbers with mathematics, you not only get actual mathematical statements ( like 2 + 2 = 4 ) but those mathematical statements simultaneously map onto what Goldstein calls “metamathematical statements” or philosophical statements about the nature of math. 

A very simplistic and kind of inaccurate but hopefully helpful example: If the Gödel number of 2 means, say, “this number is provable,” then in Gödel numbering “2 + 2 = 4” would saying something about the mathematical, numerical nature of 2 (namely, that added to itself you get four) but it would also be saying something about numbers being provable (in addition to whatever philosophical statement +, =, and 4 map onto). 

Goldstein uses different metaphors to help explain the genius of this method: Gödel numbering, she says, “allows some propositions to engage in an interesting sort of double-speak, saying something arithmetical and also commenting on their own situation within the formal system, saying whether they're provable;” and it is “basically the idea of encoding, which allows you to move back and forth between the original propositions and the code.”

Think Shakespeare's A Midsummer Night's Dream, a play within a play, how the lines the characters are speaking in the internal play also have meaning about the larger play as a whole. Only now it’s mathematical statements—equations, numbers—that are lines in the internal play, with double meanings that say something about the play as a whole—in this case, the nature of mathematics.

And by the nature of mathematics, I mean those questions the logical positivists wrestled with: Can we know things that are not empirically provable? Is math about actual numbers that we do not control or is it setting up the parameters of a game and making sure we follow them, using symbols we all agree on? Somewhat insanely, Gödel’s theorem answers those questions about mathematics through mathematics. 

I found Goldstein’s description about how the proof leads us to these big questions very helpful: 

The most straightforward way of understanding intuitions is that they are given to us by the nature of things; again intuition is seen as the a priori analogue to sense perception, a direct form of apprehension. So Gödel’s conclusion, in having something to say about the feasibility (or lack thereof) of eliminating all appeals to intuitions from mathematics might also have a thing or two to say about the actual existence of mathematical objects, like numbers and sets. In other words, the adequacy of formal systems—their consistency and completeness—is linked with the question of the ultimate eliminability of intuitions, which is linked with the question of the ultimate eliminability of mathematical reality, which is the defining question of mathematical realism, or Platonism. It is because of these linkages that Gödel’s conclusions about the limits of formal systems have so much to say.

This brings up what for me was the most interesting part of the book: the idea of mathematical intuition. Goldstein sets this up best: 

Sense perception allows us to make contact with what's out there in physical reality. What is the bedrock of mathematical knowledge? Is there something like sense perception in mathematics? Do mathematical intuitions constitute this bedrock?...Mathematical proofs must start from somewhere. Often proofs start with conclusions from other proofs and then the deduce further conclusions from these. But not everything can be proved, otherwise how can we get off the ground? There must be, in mathematics just as in empirical knowledge, the “given.” Given to us through what means? Mathematical intuition is often thought of as the a priori analogue to sensory perception.

An example of these intuitions could be, for example, the idea that parallel lines will never touch. The problem with these intuitions is that they can lead to paradoxes, like that set of all sets that do not contain themselves. Hence why Hilbert and the formalists wanted to be rid of them and have just the pure “game” of math, only holding onto whatever is provable within the rules of the system. As Gödel showed, however, that's not very much. In fact, that doesn't include the foundations upon which the rest of the mathematical system is built. 

So if there are true statements that are unprovable, these unprovable true statements must come to us through intuition. And how does intuition operate except by “bumping into” an external reality? And so Gödel’s proof gets us into the deepest of philosophical territory: What is real? How do we know it? Do our senses encompass all there is? What about our reasoning?

And gosh, if the fact that a mathematical proof not only raises those questions but has something to say about them doesn't excite you, then maybe you need a fresh dose of wonder. 

The Book:

I really enjoyed the style of this book. It's part biography, part basic mathematical, logical, and philosophical explanation. There's enough biography to understand why Gödel was so interested in the question of completeness and consistency and how he had the unique temperament to pursue it in the mind-bending way he did. There is also very helpful context for the philosophical and mathematic landscape of the time, setting up why this question was so pressing and what other leading intellectual figures (namely the Vienna Circle, Wittgenstein, and David Hilbert) had to say about it. 

Intellectual discoveries, they're all epic adventure stories. So you have to have a firm grasp of the world-building, know who your nemeses are, get an inside look at the foibles and geniuses of your hero, and, most of all, understand the magic system which is, in this case, a mishmash of philosophy (specifically, Platonism and positivism) and mathematics (specifically, a bit of set theory and logic). 

Goldstein is a wonderful narrator on this breath-taking journey. My one critique is that she veers a bit too close to hagiography at times. She does a good job arguing for why Gödel was such a genius, and since she was a student at Princeton when he was there, I understand her personal connection to him. But she could have toned down the superlatives and instead provided a longer list of examples of the other discoveries and inventions that his theorem has birthed. 

I'm a theology major with a pretty solid undergrad foundation in philosophy and an amateur’s interest in mathematics (the highest I got was AP Calculus in high school). It required a close read and intense focus, and it helped that I've watched some YouTube videos about this theorem before. There were definitely some technical components that went over my head but overall this read was not only accessible but exciting. 

At one point Goldstein describes Gödel’s theorem as having “heart-stopping beauty,” and I felt that. Her description of it enthralled me—you can tell she's a novelist but she also has the philosophical chops to pull this off. It's a marvelous combination and quite appropriate for the subject matter, a theorem that unites two different fields. 

I'm a Christian, and this book led me to worship. Part of what excites me about this theorem is that it points to external, objective reality. (Or at least, it can be interpreted to point to that—a big part of Goldstein's argument is that Gödel’s peers, critics, and followers have consistently misinterpreted him here.) Mathematics, so Gödel’s theorem seems to say, is not simply the “semantic-free mechanical processes of mindless symbol-manipulation.” It is bumping up against something real, even though we cannot prove it in a formal system. 

Of course you can't deduce from that the existence (or non-existence) of God. That's part of the whole point of what's so revolutionary about Gödel’s theorem: “Whereof we cannot formalize, thereof we still can know” or, a la early Wittgenstein, “We cannot speak the unspeakable truths, but they exist.” The most important things in life, questions about God and morality and the meaning of life, cannot be spoken of in transparent linguistic systems, packaged in logical proofs, or formalized in equations beyond doubt. But just as we somehow know that two plus two equals four and can build civilizations around that fact and the others that follow from it, we can have high confidence in beliefs about God, ethics, and human nature and build lives upon them. 

I do not want to collapse the distinctions between mathematics/logic/science and philosophy/theology/ethics. But I am saying that this discovery of Gödel’s—that there are things that are true but unprovable, that the limits of our reasoning do not necessitate disbelief in what is beyond reason—bolsters my faith in God. It is another example that to believe in the existence of a Master Creator and Sustainer like the Judeo-Christian God—a God who exists outside of us, a reality independent of us that we do not create but discover—is not unreasonable. Of course, for some people Gödel’s proof has persuaded them of the opposite. Fair enough. Such is the problem with messy, unprovable metastatements. This is exactly the kind of thing that Wittgenstein and Hilbert, in their own ways, tried to avoid. And it is exactly the kind of thing that Gödel showed us we cannot escape.

[Carly · Rating 3 out of 5]

loved all the history/relationships between professors and how the context of world war 2 affected some of the greatest mathematical minds of the time

[Aarthi Ramesh · Rating 5 out of 5]

An amazing book that gives insight into the person, the milieu (the 1920s in Vienna), the philosophies that mathematicians debated on and an accessible explanation of his proofs (the writing is accessible and clear, can't say the same for my understanding)

[Brian · Rating 3 out of 5]

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 (fully 1/2 the book) until I noticed she's a professor of philosophy. (Though I suppose that gives her a slightly higher allotment of words beginning with meta-, I still think she nonetheless far exceeded it!)


Sadly, I still don't think I grasp the divide among the Platonists, formalists, Sophists etc. to whom she refers...so when she seems to argue that his philosophical views were the driving force behind his mathematical work as well as his failure to navigate the politics of academia, I was bored. I'll also excuse her giving in to the temptation of injecting herself into the narrative (we hear several times of her one encounter with Gödel at a garden party), but she clearly need play no role in the text itself.


The discussion of Gödel's incompleteness proofs themselves, however, was fun. The meta-s nearly disappeared completely, and we got down to actual proof synopses and the reactions within academia. So I enjoyed the second half of the book, probably inflating my final rating a bit.


I do appreciate some of the references she gives, and look forward to reading more into his proofs themselves. I'm sure they're pretty inaccessible to me, but there are some popular treatments of them that should be perfect. Of course, I can't write about Gödel without suggesting that you read Gödel, Escher Bach: An Eternal Golden Braid by Hofstadter (how clever he is, including that title).

[Jr · Rating 2 out of 5]

Seems to have a major misconception.

I got it as a gift and picked it up to see if I wanted to read. Actually, based on the subject I thought I might want to recommend to a book club I am part of because it's a subject I am very interested in and would make for a good discussion.

However, diving in at chapter ii (which is well into the book) there was a discussion about postulates (or axioms) and intuition. That axioms are necessary for a mathematical system because you need a foundation to build on, but the discussion was all about whether you could know that the axioms were true in some more fundamental system -- that you had to arrive at that determination intuitively, inherently it is impossible to prove and there is no way to validate it for sure (but that real life and looking at examples could provide some clues but could also be misleading). I think this totally misunderstands the idea of axioms and mathematical systems more generally.

Axioms are assumed to be true within the mathematical system. Mathematical systems may be more useful (practical) if they bear more resemblance to the "real world" but there is no assertion or need for them to be. An obvious example is the case of Euclidean and non-Euclidean geometries. Both are very interesting and useful systems but they have axioms that contradict each other.

Also, it seems deceptive that one of the prominent blurbs on the book is by the author's husband.

[Branko Nikovski · Rating 5 out of 5]

The greatest logician since Aristotle. - A sentence that explains Kurt Godel's polymath's abilities. His proof of the incompleteness of formal systems created an enormous turbulence in Vienna Circle [the biggest intellectual place from late 1920s till now] ; an young unknown logician that confuted Wittgenstein's philosophy [the first philosophy before the Philosophical Investigations] , Hilbert's axiomatic consistency and the desire to put the Logic in the world's pedestal. The chapter with the explanation of his theorems and paradoxes is fundamental for every logician interested in Godel's Numbers and their application not only in formal systems , but in socio-dynamical determined situations.
Later , us Nazi refugees , his friendship with Albert Einstein and Oskar Morgenstern [economist and friend with Von Neumann with a great contribute for Game Theory] is a fundamental aspect for the popularity of Princeton University ; Einstein once said that doesn't care anymore for his equations , his only work privilege is to take walks with Godel and listening the profound monologues from this genius.

[Brett · Rating 3 out of 5]

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. Wittgenstein is set up as an intellectual foil against which Gödel prevails, and the story that she recounts is, though not unique within the world of mathematics (or philosophy, for that matter), is highly selective (without acknowledging the fact) and, unfortunately, mischaracterizes that which is not within it's primary purview. One may glean an accurate (though vague) understanding of Gödel from this book, but not, I'm afraid, of the greater intellectual tradition whence it sprang. And not of Wittgenstein.

[Greg Talbot · Rating 2 out of 5]

Focuses largely on the man, Kurt Godel, not so much the implications of the incompleteess theorem.

No doubt, Kurt Godel lived an interesting life. He was in the exemplary Vienna Circle, friends with Einstein, and is generally considered one of the most important mathematicians of the 20th century. However, his inwardness, strangeness and general disinterest in human connection does not make for a particularly interesting story.

If you're interested in the ideas of Godel, I don't think this is the best part to start. Maybe it's just the difficulty of understanding the material, but I thought the biography to be uninspiring and the math concepts to be very complicated.

[Ed · Rating 4 out of 5]

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 this argument bracing and largely convincing. Overall the book is a joy to read.

[Christopher · Rating 2 out of 5]

This book attempted to be a biography, a history of an idea, and a history of an intellectual time all at the once, and it passed with average colors. This is the kind of book where nothing went terribly wrong, but nothing went terribly right either. So while I don't really have any major problems with it, it remains stubbornly "OK."

Also note: Godel really needed to read some Chesterton, even though he would have hated it.