Product Description
"Experts in mathematical logic will find this book of engrossinginterest. For mere philosphers it will have a different fascination: inseeing how the achievements of a genius can seem to him to provide afirm foundation for a species of Platonism and the conviction of thesuperiority of minds over computers, and at the same time can encouragehim to favour a quasi-Leibnizian speculative metaphysics and theology.Hao Wang records and assesses the whole with an expert and balancedreasonableness." -- Sir Peter F. Strawson, Magdalen College, Oxford

Hao Wang (1921-1995) was one of the few confidants of the greatmathematician and logician Kurt Gödel. A Logical Journey isa continuation of Wang's Reflections on Gödel and alsoelaborates on discussions contained in From Mathematics toPhilosophy. A decade in preparation, it contains important andunfamiliar insights into Gödel's views on a wide range of issues,from Platonism and the nature of logic, to minds and machines, theexistence of God, and positivism and phenomenology. The impact ofGödel's theorem on twentieth-century thought is on par with that ofEinstein's theory of relativity, Heisenberg's uncertainty principle, orKeynesian economics. These previously unpublished intimate and informalconversations, however, bring to light and amplify Gödel's othermajor contributions to logic and philosophy. They reveal that there ismuch more in Gödel's philosophy of mathematics than is commonlybelieved, and more in his philosophy than his philosophy of mathematics.Wang writes that "it is even possible that his quite informal andloosely structured conversations with me, which I am freely using inthis book, will turn out to be the fullest existing expression of thediverse components of his inadequately articulated general philosophy."The first two chapters are devoted to Gödel's life and mentaldevelopment. In the chapters that follow, Wang illustrates the quest foroverarching solutions and grand unifications of knowledge and action inGödel's written speculations on God and an afterlife. He gives thebackground and a chronological summary of the conversations, considersGödel's comments on philosophies and philosophers (his support ofHusserl's phenomenology and his digressions on Kant and Wittgenstein),and his attempt to demonstrate the superiority of the mind's power overbrains and machines. Three chapters are tied together by what Wangperceives to be Gödel's governing ideal of philosophy: an exacttheory in which mathematics and Newtonian physics serve as a model forphilosophy or metaphysics. Finally, in an epilog Wang sketches his ownapproach to philosophy in contrast to his interpretation of Gödel'soutlook. ... Read more

Customer Reviews (4)

Like Isaac Newton, a mystical side to another great mathematician
Reading this book is discovering something entirely new about Kurt Gödel.It is the same revelation I had when I read the theological works of Isaac Newton.With partial exception to Laplace, the great mathematicians were theologians.Of course, Gödel's reflections fall into the category of "natural theology" or, if you wish, "metaphysics;" nevertheless, it reveals a unity between mathematical innovation and theological thinking.I cannot recommend this book too highly.There is an isomorphism in arguments for God's existence and arguments about infinity in mathematics (I include formal logic and its metatheory under this same rubric of "mathematics").For example, mathematical induction resorts to infinity in its argument when it employs the method of recursion.Gödel's famous incompleteness theorem also employs recursion when he applies Gödel numbering (modeling symbols and formulas in arithmetic with prime numbers).The common theme of noncontradiction in logic and its analogue of dividing by zero in number theory finds its analogue here with metaphysical assertions about the problem of infinite regress when we decline to posit an ultimate or infinite power grounding the entire order of things.This ultimate ground or referent warrants the appellation of "God."Because part of the order of things involves "personality," I would add that this warrants positing "God" as "personal," not "impersonal" as understood by the heathen.Gödel goes through these arguments and much more.His concept of consciousness as a unity sounds like he was influenced by Kant's notion of the transcendental unity of apperception or it could be an original thought.It is better to read one great book by a great mind such as this than a hundred books by mediocre minds.

Hao Wang, Unsung Hero
Wang's presentation of Godel brings the supergenius mathematical logician within the reach of people who are neither logicians nor mathematicians ... at least occasionally. "Godel, Escher and Bach," a previous best-seller effort, didn't manage to do that. I never thought I could or would stay with a book I comprehended so little. It was like digging through a 5 gallon drum of sunflower seeds in search of a cupful of sesame seeds that I could digest and metabolize. But I couldn't stop! Every time I found one of those sesame seeds I could understand and maybe even use to help me understand something else, I got a rush of motivation to keep on reading, in hopes there would be at least one more such sesame seed! The reason was Wang's delivery, based on his very way of being. He is a smart, trained mathematical logician himself who grew up in a contrasting philosophical culture [featuring Chinese nontheistic assumptions] and he managed to become as humble and honest and open minded and open hearted an individual as I have yet encountered in person or on the printed page. His use of self disclosure ... an au currant recommended practice among scientist science writers ... demonstrates a Goldilocks model for others to follow: not too much -- no egotistical tangents, and not too little -- he is remarkably clear about his own assumptons, biases and prejudices. Even if you don't care much about understanding Godel, the book is worth reading to get acquainted with Hao Wang.

The end of books: the pinnacle of knowledge
:The ideas expressed in this book are at least 100 years ahead of their time.Godel wasn't just friends with Einstein, he was (and is) widely regarded as "the greatest logician since Aristotle" (Oppenheimer said that, Aristotle was the father of logic).Einstein said that the only reason he showed up for work at the IAS in Princeton in his last years was so he could walk home with Godel.In his spare time, Godel was the first person in the world to show how Einstein's equations allowed for the possibility of time travel.He did this, not to show how to travel through time, but to show that time has no real existence, it is instead a consequence of the way in which our minds are organized.


:So much for the pedigree, here's some ideas from the book: the existence of an immortal soul can and will be proved scientifically, computers can never be conscious, and mathematical theorems have an existence every bit as real as the chair you are sitting in.


:I was an agnostic before I read this book.Now I know that "mind" and "soul" are just two words for the same thing.Godel is the smartest man that ever lived, and this book contains some of his most interesting ideas in a (reasonably) accessible form.Don't expect to understand more than 10% of it the first time you read it, I have been reading it for years and understand maybe a quarter of it.

Meet Gödel the philosopher
Many mathematicians know about Gödel's famous theorem.But very few know about Gödel the man. Through this book, we come to know the man, especially Gödel the philosopher.

Through this book we find out that althoughGödel and Einstein were close friends, Gödel, unlike Einstein, shunnedpublic debate.He held philosophical views which he knew would be verycontroversial if he were to publicize them, and he greatly dislikedpublshing anything he could not prove rigorously.Accoringly, heinstructed his biographer to publish these viewpoints only after his death.

This book contains hundreds of quotations from Gödel's conversationswith the author.Fortunately, the author left in quotations that he hesaid he did not understand, trusting that others might.

Here are a fewquotes:

"Consciousness is connected with one unity. A machine iscomposed of parts."

"The brain is a computing machine connectedwith a spirit."

"Materialism is false."

"Our totalreality and total existence are beautiful and meaningful . . . . We shouldjudge reality by the little which we truly know of it. Since that partwhich conceptually we know fully turns out to be so beautiful, the realworld of which we know so little should also be beautiful. Life may bemiserable for seventy years and happy for a million years: the short periodof misery may even be necessary for the whole."

If you find Gödel'stheorem interesting, I hope you will read this book and found out moreabout the man behind the theorem. ... Read more

Product Description
First English translation of revolutionary paper (1931) that established that even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. It is thus uncertain that the basic axioms of arithmetic will not give rise to contradictions. Introduction by R. B. Braithwaite.
... Read more

Customer Reviews (13)

Mathematical Rationalismhas limits
It is very hard to find faults in what may be the most famous proof of the 20th century.
For those not familiar with the Russell-Whitehead Principia Mathematica notation
this is a very hard book. I had the benefit of the Kac-Ulam explanation.
I didfind what might be problems with this proof.
1) One is the reliance on number theory proofs about prime numbers that are assumed true
in the Gödelization of the primes coding of the mathematical axioms.
2) The second is the assumption that the axioms statements represent the minimal
representation of such a system of axioms.
Both are slim if none chances, but ones the Gödel doesn't consider.
Information theory was after this time where we discovered that a system of symbols can indeed at times be more efficiently coded.
The best example of this seems to be Gray code compared to ordinary binary number code ( a number theory code
like Gödel's prime code) where less turns out to be more in information terms.
The theory of primes suffers from the new doctrine of strings that says
that infinite scales don't exist in the "real" world: that a maximum and a minimum
of measure are fixed parts of our reality. This kind of assumption can't be "proved"
but is an axiom of a system of a mathematical sort and is counter to the Euclidean proof of an infinite number of primes.
Primes already discovered by use of computers are much bigger than the numbers of ordinary physics, but
we are already reaching the Turing"stopping" problem in finding new ones.
Some people equate in algorithmic information theory andnumber theory
the stopping problem with Infinity. That point of view of people like G. J. Chaitin
is itself an unproved assumption. So the metamathematics used in the proof itself may be unprovable propositions.
If so, then the proof based on such propositions can't itself be true.
This argument in no way takes away from the greatness of Gödel and his unique genius
as shown by this line of reasoning.

The following is a dissenting view
As indicated in two other reviews of mine here, my comprehension of Goedel's work is opposite to the general one. My marking three stars regardless for this book is motivated by his extensive influence, but also by his fair admission later in life that his thesis could amount to hocus-pocus.

Indeed, I see it as one of the prominent mistakes in logical history, and I shall endeavor to explain as best I can. It should suffice to consider his Section 1, an outline of his proposed proof.

Although that section is brief, it already foreshadows an oppressingly complex logical symbolism for statements that in my view can be made much clearer using ordinary language. The symbolism, to be sure, is intended to establish a formal language, whose meaning is to be decided separately. This will be seen one of the problems.

For now, let me give the principal statement Goedel contended to be true but undecidable (neither provable nor disprovable):

"This statement is unprovable."

He symbolized it (p.40) as: "~Bew[R(n);n]". Font limitations made me slightly change it; the tilde "~" means "not", "Bew" is a German abbreviation for "provable", and within brackets "R(n)" says "Statement n" and "n" stands for the full statement.

Goedel proceeds: "...supposing...~Bew[R(n);n] were provable, it would also be correct; but that means...that...~Bew[R(n);n] would hold good, in contradiction to our initial assumption. If, on the contrary, the negation of ~Bew[R(n);n] were provable, then [its provability] would hold good. ~Bew[R(n);n] would thus be provable [in contradiction to the unprovability it states], which again is impossible." (I corrected some errors within brackets.)

So since both ~Bew[R(n);n] and its negation are unprovable, it is undecidable, and Goedel continues (p.41): "...it follows at once that ~Bew[R(n);n] is correct, since...certainly unprovable (because undecidable). So the proposition which is undecidable in the system...turns out to be decided by metamathematical considerations."

"Metamathematical", in excusing the contradiction, designates the above formal system void of assigned meaning, whereas the statement discussed is to have meaning. Not quite a lucid argument. Overlooked, furthermore, is a contradiction using the same reasoning as in the preceding.

Coupled with the preceding finding that ~Bew[R(n);n] CANNOT be proved unprovable (for if so proved, it would be contradicted), can in contradiction be that it CAN be proved unprovable. For if it were instead provable, it would again be contradicted. The statement in question thus becomes a paradox, rather than true, similar to paradoxes like the "liar", mentioned by Goedel (p.40).

He strangely adds to it the footnote: "Every epistemological [paradox] can likewise be used for a similar undecidability proof." The "liar", however, is, like all paradoxes, not a true statement, as required, but one harboring a contradiction. (I deal in my book with, and offer solutions to, paradoxes more fully, including Goedel's resulting one, without naming him.)

There occurs, further, another huge blunder in the alleged proof. The undecidability is said to apply to some of mathematics; in the above formula, ~Bew[R(n);n], the "n" refers to a number, with this justification by Goedel (p.38): "For metamathematical purposes it is naturally immaterial what objects are taken as basic signs, and we propose to use natural numbers for them." Adding (p.39): "Metamathematical concepts and propositions thereby become concepts and propositions concerning natural numbers..."

How so? In one breath he proposes using natural numbers as immaterial signs, and in the next breath the material concerns natural numbers!

The fallaciousness can indeed be made clear by considering our statement, ~Bew[R(n);n], interpreted as "This statement is unprovable." As noted, in ~Bew[R(n);n] the "n", now a number, is to name the whole statement, inside which it is also used in "Statement n..." But whether or not the statement is named by a number, the point is that the name must refer to the intended content of the statement to correspondingly function, not to the usual number possibly represented. Therefore the statement, or anything else similarly used, has nothing to do with numbers, or mathematics generally.

Gödel's proof of the inadequacy of formalism
Gödel proves that a formal system containing arithmetic must be incomplete (i.e. incapable of proving all true statements). The proof consists in creating a statement that says "this statement cannot be proved", for then it follows that either this this statement can be proved and we have proved something false, or it cannot be proved but it is still true. In either case our formal system is flawed. This is in a way an instance of the liar paradox, which was of course well know long before, but no-one had expected it to materialise inside a seemingly sensible formal system. Gödel shows that it does by means of his arithmetisation trick that enables the system to speak about itself. All symbols in the system's alphabet is given a unique number. Then all formulas in the system is assigned the following number: the product of all the factors (n:th prime)^(n:th symbol in the formula). By unique prime factorisation one can recreate the formula from its number. Sequences of formulas---proofs in particular---can be coded by the same method. We can now express the relation "x is a proof of y" inside the formal system. This relation takes two arguments: x*, the Gödel number for the sequence of formulas x, and y*, the Gödel number for the formula y. Inside the formal system it is a perfectly well defined and finite problem to decide whether x is a proof of y, as is quite plausible, although Gödel has to work hard with his recursion theory to prove this strictly. Now that we can express "x is a proof of y" we can also express "x is a proof of y(z)", i.e. a relation that takes three arguments: x*, y*, z*, the Gödel numbers for a sequence x of formulas, a formula y with a free variable, and a formula z. Thus we can also express "there exists no x such that x is a proof of y(z)". In particular, we can send in y* for z, and the statement becomes: "there exists no x such that x is a proof of y(y*)". This expression has one free variable, y. Call it F(y). F(y) is a formula in our formal system, so it has a Gödel number, say F*. Now we can formulate the statement "this statement cannot be proved" inside our formal system as follows: "F(F*)"="there exists no x such that x is a proof of F(F*)"="F(F*) cannot be proved". So if our formal system is consistent (i.e. does not prove false things) then we must accept that it cannot prove F(F*), but then F(F*) is true, so our formal system is incomplete.

One of the Best Books You Should Never Read
Godel's incompleteness theorem's are without a doubt genious.However, this day in age, no logician actually reads Godel's original work unless they are only interested in the historical aspect of it.Godel himself is not a very good writer.If you want to study Godel's incompleteness theorems there are other books out there that prove his theorems in a much more refined, shorter, and easier fasion.

Unbelievable theorem
Reading through the reviews of self-proclaimed math geniuses (see some of the below unhelpful reviews for examples) is hardly edifying, so I feel compelled to lend a hand. Here are a few comments about this publication:

First, the introduction does a poor job in explicating the theory. I suppose it gives you the basic idea, but this is hardly the first account of the theory one should read. Brathwaite does not connect all of the dots, and it will take a long time to figure out how the proof works from his intro, if you can do it all. (And that's not a challenge or insult; it simply isn't that well written.)

Second, forget about wading through Godel's proof on your own. The reviewer who claimed to do so with two years of algebra and a really good dictionary is simply lying. You do not wade through difficult theorems in mathematical logic without the appropriate tools. And the appropriate tools include having done similar but simpler proofs on your own and having a solid background in mathematical logic. Without this background, it doesn't matter whether you have the ability to be a mathematics professor at Princeton or place top five in the Putnam - you simply will not understand the proof in a rigorous manner. By all means, take a look at it to get a general feel for what's going on, but if you want a semi-technical account read Smullyan's "Godel's Incompleteness Theorems."

Third, as one reviewer pointed out, there are multiple errors in this printing of the proof. This makes what was a tall task virtually impossible.

So what did Godel do that was so interesting?
He proved that there were certain arithmetical statements about whole numbers that were not provable but true. (This was important because it shattered the widely held belief that if you stated a problem in mathematics clearly enough you would be able to determine whether it was true or false. Godel showed this isn't always the case. As an aside, simpler mathematical systems have been shown complete; that is to say, they can answer any well formed question.)
So, how can something be true but unprovable?
The sentence Godel constructed said this, more or less: I am not provable. This statement, if true, is not provable. If it is provable it's false, and correct systems (systems that do not prove false statements) cannot prove false statements. Therefore, it must not be provable. But then it's saying something true, and thus it's true but unprovable. Now, I'm simplifying and being sloppy, and you need to know about the difference between mathematical statements and metamathematical statements, but in a nutshell that's the thrust of his first theorem.

The other interesting aspect of his proof is that he constructed a statement that referred to itself indirectly. Russell, in Principia Mathematica - the work that contains the arithmetical system that served as the model for the arithmetical system in Godel's proof - created a "Theory of Types" which did not allow statements to mention themselves. But the sentence "I am not provable" references itself so it would seem that I've erred. But in fact I haven't; I just didn't fully explain how that sentence worked. (I know you were worried, if for just an instant.) Where was I . . . Godel created a sentence which referred to itself indirectly. The sentenced said, "Sentences with such and such characteristics are unprovable." It so happened that a sentence with such characteristics was itself. Thus, it referred to itself, but only indirectly and not in violation of the "Theory of Types."

All of my blathering, I hope, has impressed on you . . .
1) That this proof is worth understanding.
2) That you shouldn't believe anyone who tells you they worked through and understood the proof without having a signficant background in mathematical logic and the history of the proof. If you don't understand certain basic features of Principia Mathematica you're not going to grasp fully his proof.
3) That you should get an introductory account. Nagle's "Godel's Proof" is excellent and easy to understand. Smullyan's "Godel's Incompleteness Theorems" is more difficult, but not impossible and amounts to what would serve as the textbook of a solid mathematical logic course or two at an elite university.
4) That you shouldn't buy this work if you're hoping to work through his proof, unless of course you have the requisite training. Brain power is not enough.

... Read more

Product Description
"A gem. . . . An unforgettable account ofone of the great moments in the history of human thought."—Steven PinkerProbing 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. 4 illustrationsAmazon.com Review
Kurt Gödel is often held up as an intellectual revolutionary whose incompleteness theorem helped tear down the notion that there was anything certain about the universe. Philosophy professor, novelist, and MacArthur Fellow Rebecca Goldstein reinterprets the evidence and restores to Gödel's famous idea the meaning he claimed he intended: that there is a mathematical truth--an objective certainty--underlying everything and existing independently of human thought. Gödel, Goldstein maintains, was an intellectual heir to Plato whose sense of alienation from the positivists and postmodernists of the 1940s was only ameliorated by his friendship with another intellectual giant, Albert Einstein. As Goldstein writes, "That his work, like Einstein's, has been interpreted as not only consistent with the revolt against objectivity but also as among its most compelling driving forces is ... more than a little ironic."

This and other paradoxes of Gödel's life are woven throughout Incompleteness, with biographical details taking something of a back seat to the philosophical and mathematical underpinnings of his theories. As an introduction to one of the three most profound scientific insights of the 20th century (the other two being Einstein's relativity and Heisenberg's uncertainty principle), Incompleteness is accessible, yet intellectually rigorous. Goldstein succeeds admirably in retiring inaccurate interpretations of Gödel's ideas. --Therese Littleton ... Read more

Customer Reviews (59)

Gentle introduction to Godel
This book sets out to understand the incompleteness theorem through Godel's nature and intellectual relationships. In doing so, we get some excellent philosophical insights situating Godel's theorem with Einstein's relativity theory, Hilbert's formalism, Wittgenstein's philosophy of language and Turing's decidability theorems. The proof itself takes one chapter midway through the book. It's a general self-contained introduction that just gives a concise overview. The last chapter is pretty light and concerns the times of Godel in Princeton. Overall, this book is quite a tour de force to tie up all the loose ends to understand this very important theorem.

Incompleteness: The Proof and Paradox of Kurt Gödel
Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries) by Rebecca Goldstein tells the story of Kurt Gödel, one of the greatest mathematicians and logicians of modern times. Gödel's theorem suggesting that all mathematical truths or, for that matter, logic, cannot be known, defined or proved, rocked the scientific community of the 20 th century and the theorem significantly challenged all modern thought.

The author of Incompleteness, Rebecca Goldstein, has taught philosophy at various top U.S. universities. She is also the author of eight books. Incompleteness is a nonfiction account of Gödel's life and accomplishments. The book portrays a man of great, albeit tortured, genius.

Goldstein opens the book not by focusing on Gödel's revolutionary theorems but by exploring his intriguing friendship with Albert Einstein. On the first page we are shown an image of two men walking serenely together, ".hands clasped behind their backs, quietly speaking". As we read on we see that these two men on their daily walk are Albert Einstein and Kurt Gödel. These gentlemen used to take daily walks together at Princeton University . Einstein, the physicist, and Gödel, the mathematician, were in many respects, very different but they understood and respected each other, establishing a warm and close friendship.

The author relates the relationship between the two men to illustrate how they both saw themselves as exiles from their native Austria . Their intellect has greatly impacted human thought and at the same time, this same intellect isolated them from others. With the presentation of the Einstein/Gödel friendship, Goldstein sets up her thesis that those we consider to be "geniuses" have internal struggles with their intellect and may very well be alone and isolated.

The book then takes us through Gödel's life, from his entrance into the University of Vienna at the age of eighteen to his fascination with Platonism, the invitation to join a group of distinguished philosophers known as Vienna Circle and development of his incompleteness theorems. We are shown how Gödel's passions not only led him to his groundbreaking work, but also brought him to the brink of madness, which, in turn led to his tragic end.

Incompleteness-The Proof and Paradox of Kurt Gödel may be a daunting read for some but the strength of the work lies in the compelling look into the revolutionary mathematician's personal side of Gödel and a clear explanation of Gödel's "incompleteness" theorems.

The general reader may not be able to grasp the concepts of Incompleteness but those knowledgeable in the field of mathematics and philosophy should find this work very informative and enjoyable.

[...]

Brief and Engaging Book on Gödel
This book centers on the irony that Gödel's own philosophical interpretation of his work (which indeed may have driven his efforts to begin with) was in complete opposition to how it was most commonly interpreted by others.

Gödel was a Platonist, believing that the mind was able to make contact with absolute mathematical reality.Given that he was an attending member of the Vienna circle in the 1920's, which was the locus of logical positivism, many assumed he was of like mind, believing there was no truth beyond what man could empirically discover.Gödel's extreme reluctance to speak or write on his views helped make this misunderstanding possible.Indeed, the incompleteness theorems have often been co-opted by sloppy post-modernists (along with relativity theory and the uncertainty principle) in making the case for truth relativism.They would focus on the conclusion that we can't construct formal systems (large enough to at least encompass arithmetic) which are both complete and provably consistent and treat this as revealing a limitation in our ability to reach absolute truth.Gödel believed the actual lesson was that the human mind can and does perceive truth beyond the capability of formal systems (equivalently, algorithmic computing machines).

This book does a nice job in the treatment of the ideas as well as the biography.

A Most Important Read
Goldstein, does a masterful job describing the life and the work of the greatest logician to ever live. Ironically the genius and logical perfection exuded by Gödel is in the end matched by the equilibrium of the universe- he becomes completely illogical and insane.

Goldstein writes with a piercing passion and pointed savvy that I envy. He deep appreciation for the mind of the great logician bleeds all the way through the entire read. Gödel's incompleteness theorem took formalistic logic and arithmetic in a time when it was getting ready to announce its supreme dominance and perfection to the world and turned it on its head. Gödel proved that logic and arithmetic will forever be incomplete within themselves. In other words, logic and arithmetic will never take the place of human reasoning or mathematical truth. Man is not machine.

This all started with Russell's paradox which is the proposition

This sentence is false.

Known as the liar's paradox, Russell's paradox has a very strange quality about it. The "false" part applies to the whole sentence and its subject simultaneously. Thus if you seek to give the sentence a true or false value we run into immediate problems.

Is the proposition is false then it cant be false within itself and so it isn't false it must be true. This means that it is self contradictory.

But then again if the proposition is true then it isn't' false; another contradiction. Russell's paradox wins no matter what. There is something very special about negations indeed.


This book is monumental not simply because Goldstein can write like a demon on a mission but because Gödel's life and accomplishment is timeless. His theorem is crystal clear and logically flawless-- one of it's, if not "the" strangest and most ironically paradoxical qualities.

If you have any interest in philosophy at all- read this book. Its a must. Not.

Excellent
Among the interesting byproducts of feminism and the admission, commencing in 1970, of women to places like Princeton are overall more interesting and "cultured" readings of analytic philosophy and mathematics, before that male ghettos.

Goldstein, who studied logic and philosophy at Princeton (and who used vignettes from her experience in "The Mind-Body Problem", a novel) met Goedel, and understands the technical details of his work thoroughly. She does a better job, in fact, than Ernest Nagel did in 1968 because she makes emotional connections that exist in mathematical work but which mathematicians often don't like to talk about.

Nagel did say something about Goedel's "intellectual symphony", but Goldstein, unlike Goedel, did deeper research into Goedel's biography, snooping for example around the Mercer County courthouse for records of his US citizenship application.

She reveals the plight of the hyper-intelligent and why we have tenure, since guys like Kurt Goedel and John "A Beautiful Mind" Nash are snuffed out in the so-called "real world": once Einstein passed on, Goedel, we learn, had nobody to talk to.

Interestingly, we get no Pop-feminist nonsense and boo-hoo-ing about Goedel's wife and her loneliness, having married a truly weird individual. Mature women know today what my Mom knew: you make your bed and you lie in it, and any marriage is a unique contract. Gretel Karplus, Adorno's wife, was far more intelligent than Mrs. Goedel but she buried the possibility of being an Arendt or a Weil in service to Teddy and was shattered by his unexpected death. Likewise, Goedel's wife seems to have gotten what she wanted and what many women would kill for: a quiet husband and a house on Linden Lane.

Goldstein's "philosophy of mathematics" is nuanced. Unlike some feminist philosophers she makes no attempt to reduce the subject-matter to some sort of Freudianism. At the same time, she knows that "what we think about when we think about math" comes as do other inputs: by way of meat.

This is an *aufhebung* worthy in its own workyday way of an Aristotle or an Aquinas, because a sharper bifurcation and reification renders lifeless the terms on either side of the cut. Just as Aristotle realized that there are Forms but always instantiated, and just as Aquinas applied this insight to religion, Goldstein manages to hold together the apparently opposing thoughts, that mathematical realities are independent of our thought...but have no existence *that we know of* outside our embodied thought. They are the closest thing we have to noumena manifesting as phenomena.

As a dialectical thinker, Rebecca Goldstein knows how negation works in embodied space. By trying to make themselves over into things, "thinking machines", the Positivists transformed themselves, as she shows, from a sought objectivity into its reverse; this was also C. S. Lewis' insight, in his novel That Hideous Strength, in which the Logical Positivists of Belbury turn out to be merely Satanists, of a sort, in a word, chumps who bow down to wood and stone, having emptied themselves of the capacity for thought through a nihilistic metaphysics.

The problem with this gesture is that (as Adorno pointed out), the categories themselves are in motion so that at the end all we "know" is that:

(1) Logical Positivism imprisoned the scientific subject within a barrage of sense-data, without explaining how sense data organizes itself.

(2) Formalism in mathematics simply denies that anything exists outside a formal system in a relationship of containing. Fearful of either benign or else vicious circles, it refuses to do mathematical philosophy.

(2) First rate minds (Goedel and Wittgenstein) wanted no part of this malarkey.

As the Austrian philosopher Gustav Bergmann pointed out, Logical Postivism's denial was a perverse sort of metaphysics. In the middle of its denial, Goedel upped the ante by discovering that the paradox of the Liar has a metaphysical implication asregards the capacities of formal systems, versus that of human beings. Goedel stood outside the machine (the formal system) and derived an indirect existence proof of truths unprovable within the machine, such that if they were incorporated as axioms, new unprovable truths would appear, and this is why today we almost never anthropomorphise computers: whereas the pronoun for a ship was she, the pronoun for computer is it (and, the adjectives are not printable).

Parenthetically, I was glad to see Goldstein mention Gustav Bergmann, a relatively minor member of the Vienna Circle, since he'd self-marginalized by moving to the Midwest, that black hole, and teaching at the University of Iowa. Bergmann gave a talk at my university in which he pronounced a Goedelian commitment to the continued existence of ontology and its truth, saying he'd die in a ditch to defend it. At this time, in 1970, Goedel was invisible and people were unaware that he felt and thought pretty much the same as Bergmann.

Does Goedel's proof have metaphysical import? Goldstein rejects what she calls the postmodern interpretation, which she re-presents as the argument that (1) mathematics is undecidable ergo (or, as First Gravedigger says in Hamlet, argal) (2) there is no "truth", only "stories".

Of course, neither Derrida nor my fat pal Adorno make this argument. Indeed, there's quite a lot of metaphysical speculation and conviction in Derrida; for example, arche-writing is an ontological analysis of meaning which, ontologically and Kantian-metaphysically rejects doing ontology with received categories of writing and speech. Derrida was merely unconvinced that the only reine vernunft on tap is mathematically expressible as opposed to using natural language.

But this is a minor aporia on Goldstein's part, caused I think by the fact that during her studies at Princeton, "deconstruction" was fashionable and usable in a sloppy way unlike mathematics.

There are many popular books on mathematics that overstress fascinating and sexy details about the biological mathematicians. While the current rage for this, sparked by the movie A Beautiful Mind, might help to get math geeks laid, a mathematical biography should balance the math and the meat, and even more than Sylvia Nasar's book eponymous to the movie, Incompleteness does this. ... Read more

Product Description
Kurt Gödel was indisputably one of the greatest thinkers of our time, and in this first extended treatment of his life and work, Hao Wang, who was in close contact with Gödel in his last years, brings out the full subtlety of Gödel's ideas and their connection with grand themes in the history of mathematics and philosophy. The subjects he covers include the completeness of elementary logic, the limits of formalization, the problem of evidence, the concept of set, the philosophy of mathematics, time and relativity theory, metaphysics and religion, as well as general ideas on philosophy as a worldview.

Hao Wang is Professor of Logic at the Rockefeller University. ... Read more

Customer Reviews (2)

Wang Exposes Godel's Great Predictions.
On Pages 1 and 2, Wang tells us that Godel, the master of the incomplete, suggests the possibility of philosophy as an exact theory emerging within the next hundred years or even sooner.There will be, he believes, scientific disproofs of what he calls' mechanism in biology' and of the proposition that 'there is no mind separate from matter'; moreover he thinks it practically certain that the 'physical laws, in their observable consequences, have a finite limit of precision.In his conversations, he recommends the important project of finding what might be called a 'rational religion.'

I conclude that exact philosophy already exists because theological statements are being proven, even though the ultimate truth will always be incomplate.This prediction means that the scientific method cannot be used to prove worlds, which is a box in which we live.Thus, universe cannot be measured without measure standards. So the universe is relativistic and can never be known exactly.I also agree with Godel that mechanisms will never be found in living things.This is why US medical care is so bad.I agree with Godel that minds will never be without bodies because only organizations exist in Nature.I also agree with Godel that a rational religion is coming because theological statements are being proven.

Since no one else has reviewed this I will.
Wang has been an important source in compiling information on Godel and bringing it to public attention. This volume contains a variety of material about Godel- biographical facts, personal recollections, chronologies, Godel's philosophical ideas, the impact and historical setting of his mathematical work, his relationship with Einstein, comparisons to other prominent intellectuals, and more. It assumes a basic understanding of Godel's theorems. The bulk of the book is a presentation of some of Godel's (largely unpublished) philosophical activity. There is also quite a bit on Wang's own views as he contrasts them with Godel's. Some of these sections require more background in philosophy than most students of mathematics possess (myself included).

Wang supplies lots of interesting historical and biographical material as well. The 75 page chronology of Godel's life and work is very informative. Contains 11 photographs of Godel and company. The book ends with some useful commentary on selected publications of Godel. If you're looking just for a biography get Dawson's excellent book, but anyone seriously interested in Godel will want this as well. ... Read more

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"A compelling biography of the eccentric genius."- Discover.

Kurt Gödel was an intellectual giant. His Incompleteness Theorem turned not only mathematics but also the whole world of science and philosophy on its head. Shattering hopes that logic would, in the end, allow us a complete understanding of the universe, Gödel's theorem also raised many provocative questions: What are the limits of rational thought? Can we ever fully understand the machines we build? Or the inner workings of our own minds? How should mathematicians proceed in the absence of complete certainty about their results? Equally legendary were Gödel's eccentricities, his close friendship with Albert Einstein, and his paranoid fear of germs that eventually led to his death from self-starvation. Now, in the first book for a general audience on this strange and brilliant thinker, John Casti and Werner DePauli bring the legend to life. ... Read more

Customer Reviews (15)

Almost excellent, but not quite. (4.25 stars)
Even in books that I greatly enjoy and inevitably recommend, there are always at least a few problems. If/when I write a review of such a book, I find it difficult not to focus mainly on the apparent gaffs or simple imperfections (my wife says I'm just grouchy, I don't think so, but perhaps she's right). Through the first seven chapters, `Gödel: A life of Logic' was well enough presented that the text had given me nothing to grumble about. The exposition of humankind's greatest logician, both his curious and quirky personal life and his stunning accomplishments in mathematical logic and metamathematics, was highly enjoyable and, so far as I could discern, highly accurate. So good as to essentially overpower the shortcomings of the last three chapters.

Ah, but now for the afore-mentioned grumbling (I'll skip a small issue with the introduction of complexity in chapter nine). The abrupt little final chapter (10), a philosophical summation, struck me as being something of a disintegration. "Gödel's mathematical philosophy was resolutely Platonistic," record Casti and DePauli, and in this they are obviously correct. But now it gets rather muddled and I expect Godel, a lover of accuracy, would have done a little grumbling himself had he read the following:
[Godel held] "a Platonic view of mathematical objects, no doubt about it. For the Platonist, objects are thus intuitively presented. By way of contrast, an Intuitionist or Constructivist considers them inventions of the human mind . . . For Godel . . . we find a typically Platonist intertwining of an objective concept of reality with a kind of extrasensory perception of abstract, Platonic ideas." (so far, so good, but--) "Interestingly, Godel also had a Intuitionist streak in his mostly Platonic view of mathematics. Formalists and Platonists are diametrically opposed in their view on the question of [mathematical reality]." The authors have already sketched out this philosophical opposition, but how are they categorizing Godel now? "Although Gödel's mathematical philosophy was Intuitionist, his logical methods were Formalistic . . . This is classical Platonism in its purest form."

Okay, some of this is salvageable, but what a muddle. Had the authors just thrown in the towel? And this is followed by a brief whimsy about Gödel's similarity to "Copernicus, Darwin, and Freud," on the grounds that they all undercut our fantasies of human omnipotence. Godel, of course, would absolutely have found 2/3 of those associations (exempting Copernicus) problematic on various fronts, and I have to wonder--who but the most delusional of the intellectually handicapped have ever seriously expected anything that could accurately be described as human omnipotence? No one in the Eastern or Western intellectual histories comes to mind, and not even a potency-obsessive psychopath like Hitler was quite that crazy.

So that's my grumble, I may have failed but I tried to keep it concise. If we can set aside the last fading paragraphs, and a sentence here or there, the book was quite good. The expositions on Gödel's relationship to the Vienna Circle, as well as on Godel Numbering and how it has enabled analysis, are well presented. The exposition on computation and Artificial Intelligence is mostly accurate and succinct, but it too fades in its conclusions (perhaps this is well enough explained as basic intellectual dispassion on the part of the authors).

Un understandable overview of Godel and his completeness theorem
The main result of Godel's Completeness Theorem is that in arithmetic, there are true statements that can never be proven to be true in the system of arithmetic. Using this as a base system, this means that in any system equal to or greater than arithmetic in complexity, there will be true statements that cannot be proven to be true in the system. This result has been used by many people to argue for or against many things.
I have seen it used to argue for the existence of God.

"According to Godel's theorem, there are things that are true that cannot be proven to be true within the system of human thought. God is one such thing, therefore God exists."

I have seen it used to argue against the possibility of artificial intelligence (AI).

"According to Godel's theorem, there are things that are true that cannot be proven to be true within the system of programmable human thought. Humans take advantage of these unprovable truths, which makes intelligence. Since this advantage can never be programmed, AI is impossible."

I have suggested on more than one occasion that the people making these arguments need to spend more time studying both logic and what Gödel really concluded. For example, they could read this book.
It presents a brief biography of Kurt Gödel. In his later years he was quite eccentric and reclusive, however in his early years he apparently was quite a ladies man. Certainly Gödel was a genius; Albert Einstein himself openly expressed his admiration for Godel's intelligence. I was pleased to see the authors spend as much time as they did describing Gödel in his earlier years. So many other commentators spend so much time on his social difficulties that his achievements become overshadowed.
A complete explanation of his main results is also expressed in terminology that almost everyone can understand. There are few formulas; simple algebra is all that is needed to understand all of the mathematical symbolism used in the book. If I was teaching a course in popular mathematics, it would have to include Godel's Completeness Theorem and this is the book I would select for that section.

An abridged version of Hofstadter's book.
Author shows a great skill in Chapter Two and Three to explain a crux of famous theorem in a very succinct language without using mathematical terms. Also a short biography of Godel's strange life explains: why he died of paranoia; why he hated Austria; why he was suffering a guilt of not producing enough academic result in Princeton.
As the author acknowledges, many metaphors used in this book overlaps with the Douglas Hofstadter's Pulitzer-winning book. However, as many other books for past twenty years, the author presents a theorem in a way that is easily misinterpreted. In p11, he says "Essentially, what Godel showed is that no kind of mathematics is ever going to be comprehensive enough to express fully the everyday notion of truth." And then, the author spends a great deal of pages on AI and computer.
As far as I know, Godel's theorem mentions nothing about "truth in daily life" or computer. Godel's theorem applies only in a strictly circumscribed sphere, i.e., first-order logic. For example, Euclidean geometry is not imcomplete, and the higher-order logic doesn't produce Russel-type paradox. So, what we cannot speak of we must pass over in silence.
Also, author asserts in p71 that Wittgenstein's shift to "sociocultural position" later in his life, but he failed to mention that Wittgenstein did describe his thought about Godel's theorem in his "Remarks on the Foundations of Mathematics".

Biography: no --Look at his great theorm: YES!
I got to look at the book at a bookstore before I bought it so I knew I wasn't getting a biography. This book is a look at his theorem with comments about his life thrown in to put the work into some human context.For a thurough description of the theorem with a gentle human touch this is the book for you.Casti et al. does a great job of making tough ideas readable.If you want to know more about the theorem that turned mathematics on its head this is it.Not perfect (less talk about cake :-) ) but fun, readable, educational, A shame it is out of print.

Not the real Gödel ?
Sorry, but this book was somewhat a disappointment for me. The authors for the most part keep personal life and work of Gödel separated, instead of seeing them as a unity. A biography has to be the best of both worlds in my opinion. That's what makes the work of a biographical writer a difficult task. Maybe one of the two authors did the biographical part, the other one the mathematical ? And of course, everything about Gödel is great, brillant and alltogether grand. I am missing a critical view on his lifestyle and his view on music e.g.. Appearently the author of the biographical part was so in awe of Gödel, that he didn't dare to critisize anything about Gödel. Ironic, since Gödel stands for the idea, that you are allowed and even have the obligation to question everything to get to the bottom of the truth of things.
I am still waiting for the real biography of Kurt Gödel. ... Read more

Product Description
This groundbreaking Pulitzer Prize-winning book sets the standard for interdisciplinary writing, exploring the patterns and symbols in the thinking of mathematician Kurt Godel, artist M.C. Escher, and composer Johann Sebastian Bach.Amazon.com Review
Twenty years after it topped the bestseller charts, Douglas R. Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid is still something of a marvel. Besides being a profound and entertaining meditation on human thought and creativity, this book looks at the surprising points of contact between the music of Bach, the artwork of Escher, and the mathematics of Gödel. It also looks at the prospects for computers and artificial intelligence (AI) for mimicking human thought. For the general reader and the computer techie alike, this book still sets a standard for thinking about the future of computers and their relation to the way we think.

Hofstadter's great achievement in Gödel, Escher, Bach was making abstruse mathematical topics (like undecidability, recursion, and 'strange loops') accessible and remarkably entertaining. Borrowing a page from Lewis Carroll (who might well have been a fan of this book), each chapter presents dialogue between the Tortoise and Achilles, as well as other characters who dramatize concepts discussed later in more detail. Allusions to Bach's music (centering on his Musical Offering) and Escher's continually paradoxical artwork are plentiful here. This more approachable material lets the author delve into serious number theory (concentrating on the ramifications of Gödel's Theorem of Incompleteness) while stopping along the way to ponder the work of a host of other mathematicians, artists, and thinkers.

The world has moved on since 1979, of course. The book predicted that computers probably won't ever beat humans in chess, though Deep Blue beat Garry Kasparov in 1997. And the vinyl record, which serves for some of Hofstadter's best analogies, is now left to collectors. Sections on recursion and the graphs of certain functions from physics look tantalizing, like the fractals of recent chaos theory. And AI has moved on, of course, with mixed results. Yet Gödel, Escher, Bach remains a remarkable achievement. Its intellectual range and ability to let us visualize difficult mathematical concepts help make it one of this century's best for anyone who's interested in computers and their potential for real intelligence. --Richard Dragan

Topics Covered: J.S. Bach, M.C. Escher, Kurt Gödel: biographical information and work, artificial intelligence (AI) history and theories, strange loops and tangled hierarchies, formal and informal systems, number theory, form in mathematics, figure and ground, consistency, completeness, Euclidean and non-Euclidean geometry, recursive structures, theories of meaning, propositional calculus, typographical number theory, Zen and mathematics, levels of description and computers; theory of mind: neurons, minds and thoughts; undecidability; self-reference and self-representation; Turing test for machine intelligence. ... Read more

Customer Reviews (267)

A Little Bit Overrated, but Good
This was a fine book.I think it could have been all said in half the number of pages and been a little more substantial.Very thought provoking, and Hofstadter has a nice, original contribution that he makes toward mind-body philosophy.More than that, I think he makes some very important and objective observations about reality.

I have to suspect that he was awarded the Pulitzer Prize for the original contribution or for political reasons.Rarely, would a book of this type be given a Pulitzer unless the book pushed a socio-political ideology.If the latter is the case, I would say that people of a Marxist/Wundtian sentiment fell in love with Hofstadter's thoughts on consciousness as a strange loop.I speculate that this book would not have acquired the attention that it has if it were not embraced by the "non-solipsist community."

It is a book worth reading.If you think that man is an animal, you will love this book.If you are a solipsist, you will enjoy this book, finding new avenues to consider.I think this is a must read for anyone interested in the connection of all things in the world, be your disposition scientist, sociologist or musician.

Better on the re read
I read this book 12 years ago and it's even better today.I've studied more about life itself, not just the ideas about life that proliferate this book and found they match up.I particularly enjoyed the section on Buddhism...as going beyond Buddha.Of course this could be applied to any idea or theology.To go beyond...well that requires imagination.
I just wanted to thoroughly endorse this book!I could not read it all at once, however.I tended to skip around, and put pieces together to create a whole...it took time but in a sense I enjoyed it more that way.

Elitist, Confused, Show-off
The Emperor is naked!

The author has tried to 'show-off' his knowledge of various subjects.
He could have still done it - but with less words.

The people who want to to rate this book high are trying to belong to a group who think they are intelligent.
One of these elitist should at least try writing an abridged version.

A verbose waste of time, as you follow the author in an endless loop as he chases his tail around.

And finally...
I believe that if someone cannot be precise and concise about what they are saying, they themselves don't have a clue.
If that is the case, then we must be idiots to be listening to their rhetoric.

Out of the Matrix
Hofstadter employs the concepts of strange loops, recursion, computer systems, and artificial intelligence as metaphors to explain epistemology. The great insight of this book is that the truth or proof of a system is not contained within the system, but outside the system. Inside any system, self-referencing loops create the illusion of truth because each element can be supported by references to other elements within the system, so long as the system consistently and uniformly follows its own laws. The system acquires meaning as we gain relation to it, interact with it, and build our lives around it.

Similarly, in the movie "The Matrix," the matrix itself represents a masterful system of self-referencing elements that follow the laws of the system with uniformity and precision, but it is all an illusion, a lie. Inside the Matrix, one cannot verify whether the system itself is authentic. Each element references another element within the matrix. The laws of gravity function properly, the physical addresses are correct, the maps serve you well, and your favorite restaurant has "really good noodles." You build your life around it, your emotional and intellectual constructs around it, and it becomes part of you. From within the Matrix, one cannot challenge and test the critical assumptions upon which the matrix is built. As Neo learns, reaching outside the Matrix removes him from his life of ignorant comfort to the startling reality that he has been living a lie. On a darkling plain at the edge of a precipice, Morpheus wakes Neo to the terrifying reality: "Welcome to the desert of the real."

Good, but not great
I've read this book, on and off, over the better part of the last 4 years.It is certainly a fascinating read, the kind of book that effortlessly discusses everything from computers to art, from paradoxes to physics, from philosophy to tortoises, and seemingly everything in between.

However, the reason I don't give it a 5/5 is that I didn't think that it all ever really coalesced into any singular idea.Other reviewers see it as a profound meditation on the concept of consciousness or something.To me, it just seemed like a hodge-podge of neat stuff.Not that there's anything wrong with that, but I think if you go in expecting a revolutionary work, you might be a bit disappointed.

Still, it's such a hefty book, its easy to recommend as a sort of "desert island" kind of book.It's not just the thickness of the book, but the sheer density on each page.Most chapters contain logic games and stuff which you really should try out if you want to get the most out of the book.It's not the kind of book that you can just zip right through.Taken purely as a value proposition it's hard to beat the amount of insight and entertainment this book provides. ... Read more

Product Description
Kurt Gödel (1906-1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposium celebrating Gödel's centennial as well as papers from a 2004 symposium. Proof theory, set theory, philosophy of mathematics, and the editing of Gödel's writings are among the topics covered. Several chapters discuss his intellectual development and his relation to predecessors and contemporaries such as Hilbert, Carnap, and Herbrand. Others consider his views on justification in set theory in light of more recent work and contemporary echoes of his incompleteness theorems and the concept of constructible set. ... Read more

Product Description
This authoritative biography of Kurt Gödel relates the life of this most important logician of our time to the development of the field. Gödel’s seminal achievements that changed the perception and foundations of mathematics are explained in the context of his life from turn of the century Austria to the Institute for Advanced Study in Princeton. ... Read more

Customer Reviews (4)

Good Biography, a bit heavy on the math
This book has a kind of interesting way of doing a biography. The subject, Godel, is one of the pre-eminent mathematicians of the twentieth century. This biography, written by a mathematician spends a good bit of time on the math that Godel was doing as well as the story of his life.

Chapter III, for instance is a capsule history of the development of logic to 1928. This is to give background to the mathematical world as it existed when Godel was starting his work. In particular it discusses the open problems in mathematics that David Hilbert proposed in 1900.Godel resolved the second of these problems.

Coupled with his genius in mathematics, Godel also had serious psychological problems. He eventually died of starvation because he was convinced that the food he was getting had been poisoned and refused to eat. Dr. Dawson has written a compasionate and understanding biography, even if the mathematics gets just a bit deep once in a while.

Excellent.
An excellent biography of Godel. Examines his personal life and mathematical work in an integrated manner. Dawson is thorough, well-researched, and shows a command of the mathematics involved. He provides the most accurate picture available of the real Godel- in contrast to the anecdotal, 'crazy-genius' stories you see elsewhere. This is not a popular account of Godel's work, so the reader will need an understanding of fundamental mathematical logic and Godel's theorem to appreciate much of the book. But Dawson does provide a lot of history of mathematical logic, including a great chapter on developments up to 1928 that could stand by itself. The appendix provides a chronology, genealogy, and "biographical vignettes" of other important logicians.

The definitive biography of Kurt Godel
Knowing what went on in the mind of Kurt Godel will forever be unattainable.Nonetheless, John Dawson comes as close as possible to understanding what made Godel click.

Having catalogued Godel's works and personal papers, Dawson saw aspects of Godel's life that perhaps no one short of his wife had seen.

The book is a fascinating jaunt through the through the lives of one of the greatest minds of the 20th century.What is also interesting is Godel's interaction with personalities such as Einstein and Van Neumann.

While the mathematics is often abstract, as can be expected, Logical Dilemmas is a mesmerizing read.

By a Mathematician for Mathematicians
Writing a biography of anyone is difficult.How can a writer, no matter how talented, really claim to understand someone well enough to give an overview of his life?When the subject is a genius like Kurt Godel, whose name is known by few and whose work is really understood by even less, the job must be even more difficult.Fortunately, people like Mr. Dawson are will to give it a shot and he succeeds fairly well.

In putting together this biography, Mr. Dawson has the advantage of being mathematician.Additionally, he has the advantage of being the mathematician who catalogued Godel's papers after his death.This gives him a lot of insight into Godel that other writers cannot have and he weaves quotations from these papers into the biography very well.Mr. Dawson's is a well-documented and logical biography that is short on conjecture and long on footnotes.In brief, it is a biography about a mathematician clearly written by a mathematician.This is both its strength and its weakness.

Actually, I like the purely biographical sections of this book very much.The biographical information is clear and informative, though a bit dry in the academic style favored by mathematicians and scientists.Fortunately, having lived and worked among these people, I am comfortable with this style.More importantly, I feel like I have a better idea now of who Godel was and what he was like from reading this book.His focus on his work, his relationship with his family and friends (particularly his wife) and his ultimate decent into mental illness are much more in focus for me now.

On the other hand, the sections that deal with Godel's mathematics are much more difficult to take.The discussion of mathematics in this book goes far beyond what most people are going to be able to handle.I fear the average reader even with a decent math background who comes across this book will drop it as soon as the mathematics starts and that is unfortunate. (I am always looking for books to promote math even among non-mathematicians.This one does not do it.)A reader who can handle the math, however, will find this book revealing. ... Read more

Product Description
In the late 1950s, with mind-brain identity theories no longer dominant in philosophy of mind scientific materialists turned to functionalism, the view that the identity of any mental state depends on its function in the cognitive system of which it is a part. The philosopher Hilary Putnam was one of the primary architects of functionalism and was the first to propose computational functionalism, which views the human mind as a computer or an information processor. But in the early 1970s Putnam began to have doubts about functionalism, and in his masterwork Representation and Reality (MIT Press, 1988) he advanced four powerful arguments against his own doctrine of computational functionalism. In Gödel, Putnam, and Functionalism, Jeff Buechner systematically examines Putnam's arguments against functionalism and contends that they are unsuccessful.

Putnam's first argument uses Gödel's incompleteness theorems to refute the view that there is a computational description of human reasoning and rationality; his second, the "triviality argument," demonstrates that any computational description can be attributed to any physical system; his third, the multi-realization argument, shows that there are infinitely many computational realizations of an arbitrary intentional state; his fourth argument shows that there cannot be local computational reductions because there is no computable partitioning of the infinity of computational realizations of an arbitrary intentional state into a single package or a small set of packages (equivalence classes). Buechner analyzes these arguments and the important inferential connections among them—for example, the use of both the Gödel and triviality arguments in the argument against local computational reductions—and argues that none of Putnam's four arguments succeeds in refuting functionalism. Gödel, Putnam, and Functionalism will inspire renewed discussion of Representation and Reality and will reconfirm it as a major work. ... Read more

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Kurt Gödel, together with Bertrand Russell, is the most important name in logic, and in the foundations and philosophy of mathematics of this century. However, unlike Russel, Gödel the mathematician published very little apart from his well-known writings in logic, metamathematics and set theory. Fortunately, Gödel the philosopher, who devoted more years of his life to philosophy than to technical investigation, wrote hundreds of pages on the philosophy of mathematics, as well as on other fields of philosophy. It was only possible to learn more about his philosophical works after the opening of his literary estate at Princeton a decade ago. The goal of this book is to make available to the scholarly public solid reconstructions and editions of two of the most important essays which Gödel wrote on the philosophy of mathematics. The book is divided into two parts. The first provides the reader with an incisive historico-philosophical introduction to Gödel's technical results and philosophical ideas. Written by the Editor, this introductory apparatus is not only devoted to the manuscripts themselves but also to the philosophical context in which they were written. The second contains two of Gödel's most important and fascinating unpublished essays: 1) the Gibbs Lecture ("Some basic theorems on the foundations of mathematics and their philosophical implications", 1951); and 2) two of the six versions of the essay which Gödel wrote for the Carnap volume of the Schilpp series The Library of Living Philosophers ("Is mathematics syntax of language?", 1953-1959).

Customer Reviews (2)

Brilliance in obscurity
Godel has shown that truth is not identical to provability, and reality is not identical to consistency as imagined by logical positivists who tend to be rather easy-going about metaphysics. That is the view he challenged, or let us say instead, eliminated. He in fact worked on philosophy for a very long time, trying to construct a world view consistent with his mathematical results. Even before he obtained the celebrated incompleteness theorems, he says that he had a philosophical motivation and that he never looked at the Vienna Circle with much sympathy. In view of his mathematical results, he considered the realist implications of his work most likely among others. Rodriquez argues in the book that although many philosophers "borrowed" their most important ideas from Godel, Godel's realist philosophy does not seem to follow directly from his mathematical theorems. He could have argued against realism starting from the same logical premises. In particular, Godel believed in a deity and the existence of mathematical objects resolutely. It was hard to give up on these ideas no matter what he found in metamathematics.

He did, however, show that conventionalism had fatal flaws. Not simply minor flaws that could be patched later, but flaws which erased the unsatisfying metaphysics of conventionalism. He argued against Carnap and others but was hesitant to publish his papers, because he found his philosophical work incomplete, being a perfectionist he wanted to be absolutely sure he had the final answer. That is one of the reasons his philosophy remained in obscurity. The other is that no other Godel came.

In these essays he argues for a metaphysics in which mathematical objects exist in a realm of their own, and their perception can be achieved only through indirect means, by mathematical intuition which operates on a kind of data that is not the same as sense data. He also offered a disjunctive proposition which claimed that either the mind is infinitely more powerful than a computer, or there are such propositions that are absolutely unsolvable by a computer. He contends that both possibilities will frustrate the materialists, for either the human mind is not mechanical, or there is a realm of abstract objects. Throughout the essays, the most persistent idea is that mathematics cannot be reduced to syntax, and that mathematical truth is not our creation: it exists independently.

These ideas should be taken seriously because they originate from the same person who has shaken the world of 20th century mathematics with his foundational results.

Rodriquez exposes this dramatic turn of events in the first half of 20th century in a vibrant and rigorous fashion including Godel's completeness theorem and incompleteness theorems. The first half of the book is dedicated to laying out the background for Godel's two unpublished philosophical essays. In this part, Rodriquez summarizes the history of developments which have led to Godel's philosophical work and the philosophical views of Godel and his contemporaries. A very detailed and scholarly analysis of Godel's philosophy, including a lucid description of his ontology and epistemology are being presented. One wonders how much work must have gone into deciphering Godel's work and establishing the relationships which Rodriquez uncovered. The second part of the book contains the essays, edited from Godel's unpublished manuscripts. The essays are most striking, and they are written in an almost mathematically precise language and they are dense with argument in every sentence. The first essay is the script of his famous Gibbs lecture, which he read to an audience only once. The second essay is a response to Carnap's "The Logical Syntax of Language". Both essays have been revised significantly by Godel over the years, and Rodriquez does his best to provide a complete account of Godel's work.

I think, then I live! This is what Godel revealed.
I read this book in Japanese translation. Godel mentioned gvitalismh only once in it. But I think this word is what he wanted to say most about the incompleteness theorem. Mathematicians learned the form of the theorem, but ignored the meaning of it. This book is a precious clue to answer the question: What is mathematics?@I am now trying to depicting mathematics from the start of thinking to the end. This book is a reliable navigator for me. ... Read more

Product Description
In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter?  Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic. ... Read more

Customer Reviews (4)

An Introduction to Godel's Theorems
I have read much of the 2007 edition and have found it very illuminating and helpful.However, Peter Smith says on his website that there now exists a fourth2009 reprinting with many significant corrections and additions.Amazon's website description of the book mentions only the first printing.I have asked Amazon to confirm that the book it is now selling is this fourth corrected reprinting, but, alas, without success. Does anyone know whether Amazon is currently selling the fourth reprinting? An Introduction to Gödel's Theorems (Cambridge Introductions to Philosophy)

Good Book, Wrong Title
This is a terrific book that the reader can learn a lot from. The author presents Gödel's theorems - in fact, he provides many different proofs of the theorems - along with various strenghtenings and weakenings of the main results. In the many historical and conceptual asides the author does a great job of explaining the significance of Gödel's theorems and of directing the reader's attention to the big picture.

However, I don't think the book is a good *introduction* to Gödel's theorems. A student approaching these theorems for the first time will be overwhelmed by the amount of information here. Even more problematic is the author's adoption of a rather informal way of writing. This does make the book very readable but I think would frustrate the beginning student who needs a precise grasp of new concepts. For example, I don't think a student innocent of primitive recursive functions would be able to grasp how they work from the chapter here. This problem is further compounded by the lack of exercises.

In sum, the book is highly recommended for anyone looking to deepen and broaden their understanding of Gödel's theorems. However, I think that anyone who hasn't already seen a rigorous presentation of those theorems might find the book frustrating.

the best thing out there
For a couple of decades now, students who had completed their first logic class and dabbled in a little bit of metatheory (perhaps soundness and completeness) were forced to avail themselves of Boolos and Jeffrey's (fourth edition with Burgess) "Computability and Logic." Unfortunately, the third edition presented much of the material in too brief a manner, resulting in a big jump from lower level logic to the material covered. The fourth edition is much longer, but no more easier to teach to talented undergraduates.More recently, Epstein's book on computability was an improvement in this regard, but its logical coverage was much less.

Smith's book should now be the canonical text.First, the discussion and proofs are astoundingly clear to students who haven't done much logic beyond their first class.Pick any topic from B & J and Smith, for example primitive recursiveness, the tie between p.r. axiomatizability and axiomatizability via Craig's theorem, etc. and the discussion and proofs in Smith will be clearer, more accessible, and more clearly tied to the other relevant concepts.Second, the coverage is exactly what is needed to understand both theorems and the most important consequences and extensions.Third, the way he ties the disparate topics together (for example the informal proofs through Chapter 5 and their rigorizaiton through Chapter 18) is just fantastic.This is really important for helping the reader develop a deeper understanding of things.If you just pile theorem upon theorem it's easy for the reader accept them as true without developing any logical insight and appreciation of the landscape.

I don't know if Cambridge would allow this, but in the next edition they should seriously think about adding exercise sections like B & J and Epstein.If they did, I think this book would eclipse the other two for classroom uses.

It's not just for students, either.A colleague and I were arguing about something and we picked up Smith's work rathern than either of Smullyan's to figure out a point relevant to the debate.I find that my grasp of the relevant proofs is much cleaer for reading Smith (my colleague is much, much better at logic than me, but with Smith's help I won the debate).

It is both extraordinary and a cause for celebration when someone can combine in a logic text this level of coverage, rigor, accessibility, and funness of read.I don't think there is a precedent actually.In short, Smith's work is a service to Lady Philosophy.Joe Bob says check it out.

An introduction to Gödel's theorems by Peter Smith
Very rigurous but simultaneously very understandable (...for this kind of theorems that have many technical dificulties is much to say...) I enjoy (very much) to read this book(I took it to the beach during my vacations!.) ... Read more

Product Description
Kurt Gödel (1906-1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory and stronger systems, and the consistency of the axiom of choice and the continuum hypothesis.He is also noted for his work on constructivity, the decision problem, the foundations of computation theory, unusual cosmological models, and for the strong individuality of his writings on the philosophy of mathematics. The Collected Works is a landmark resource that draws together a lifetime of creative accomplishment.The first two volumes were devoted to Gödel's publications in full (both in the original and translation).This third volume features a wide selection of unpublished articles and lecture texts found in Gödel's Nachlass, documents that enlarge considerably our appreciation of his scientific and philosophical thought and add a great deal to our understanding of his motivations.Continuing the format of the earlier volumes, the present volume includes introductory notes that provide extensive explanatory and historical commentary on each of the papers, English translations of material originally written in German (some transcribed from Gabelsberger shorthand), and a complete bibliography.A succeeding volume is to contain a comprehensive selection of Gödel's scientific correspondence and a complete inventory of his Nachlass.The books are designed to be accessible and useful to as wide an audience as possible without sacrificing scientific or historical accuracy.The only complete edition available in English, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science. ... Read more

Customer Reviews (1)

Godel's unpublished philosphy
For anyone interseted in Godel's thought, this book is absolutely wonderful.Also for anyone interested in Platonism and how one can be a platonist after the crisis in math, this is a good thing to read. Moveover, Godel was sort of a freek-job and didn't like to publish stuffabout his personal philosphic views, so you won't get the real deal if youonly read the stuff he published.Much like his homie Einstein, Godelspent the last chunk of his life plugging away at a unified theory,Einstein's was reletivity, Godel's was metaphysics.Really good stuff.Sure, you can read that Godel, Escher, Bach stuff, but then you are onlylearning what the man wants you to know.You gots ta get the real dealfrom the source.Word. ... Read more

Product Description
Kurt Gödel was the most outstanding logician of the twentieth century, famous for his work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computation theory, as well as for the strong individuality of his writings on the philosophy of mathematics. Less well-known is his discovery of unusual cosmological models for Einstein's equations, permitting "time-travel" into the past.
This second volume of a comprehensive edition of Gödel's works collects together all his publications from 1938 to 1974. Together with Volume I (Publications 1929-1936), it makes available for the first time in a single source all of his previously published work. Continuing the format established in the earlier volume, the present text includes introductory notes that provide extensive explanatory and historical commentary on each of the papers, a facing English translation of the one German original, and a complete bibliography. Succeeding volumes are to contain unpublished manuscripts, lectures, correspondence, and extracts from the notebooks.
Collected Works is designed to be accessible and useful to as wide an audience as possible without sacrificing scientific or historical accuracy. The only complete edition available in English, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science. These volumes will also interest scientists and all others who wish to be acquainted with one of the great minds of the twentieth century. ... Read more

Customer Reviews (1)

Excellent material that fits lots of class uses
A summary of his statement on p. 125 on "Russell's Mathematical Logic" describes the "vicious circle principle: forbids a certain kind of circularity which is made responsible for the paradoxes. The fallacy in these, so it is contended, consists in the circumstance that one defines (or tacitly assumes) totalities, whose existence would entail the existence of certain new elements of the same totality, namely elements definable only in terms of the whole totality." This led to the formulation of a principle which says that "no totality can contain members definable only in terms of this totality, or members involving or presupposing this totality." (The vicious circle principle). (Also a "not applying to itself principle to keep the vicious circle principle from applying to itself p. 126

In describing Russell's theory of types he says, "The paradoxes are avoided by the theory of simple types which is combined with the theory of simple orders - a "ramified hierarchy""

Godel argues that the vicious circle principle is false rather than that classical mathematics is false.

p. 202 "A remark about the relationship between relativity theory and idealistic philosophy (1949a) (Note that this view supports my usual presentations in class on this!)

"The argument runs as follows:Change becomes possible only through the lapse of time. The existence of an objective lapse of time 4, however, means (or, at least, is equivalent to the fact) that reality consists of an infinity of layers of "now"

p. 203 which come into existence successively. But, if simultaneity is something relative in the sense just explained, reality cannot be split up into such layers in an objectively determined way. Each observer has his own set of "nows", and none of these various systems of layers can claim the prerogative of representing the objective lapse of time. 5" ... Read more

Customer Reviews (16)

Worth reading, but not the last word
Telling people not to read "Gödel, Escher, Bach" is like telling people not to read Harry Potter. This book has become so much of a cultural touchstone that everyone should read it, just to see what the hype is about. In fact, Hofstadter has written a very careful exploration of the nature of consciousness. Now, I don't find the questions that he raises, of self-reference or the consistency of systems of axioms, very interesting, but reading "Gödel, Escher, Bach" reminded me of all the problems that *do* interest me. In that respect, the book's "negative space" had a very deep influence on me, and, since one of the book's themes is negative and positive space, I like to think that my reading is in a spirit that Hofstadter would approve of.

If you're interested in theories of consciousness or Gödel's theorem, this book may appeal to you, especially if you appreciate a playful treatment of these topics and don't mind the author's long-windedness. Now, I was not enthusiastic to begin with, so I don't feel qualified to comment on the flaws in the book's argument. But I will recommend 3 other books on related topics which complement "Gödel, Escher, Bach" nicely, even for more sympathetic readers.



A more accessible, better contextualized, and more enlightening treatment of Gödel's Incompleteness Theorem can be found in "Logicomix" by Doxiadis and Papadimitriou.

Descartes' theory of consciousness assumes, like Hofstadter's, that consciousness is an individual (rather than a social) phenomenon. Descartes' approach is axiomatic without taking into account problems of self-reference. (You might say that Hofstadter tries to update Descartes for the post-Gödel era, which forces Hofstadter to place less faith in deductive reasoning.) You might read the "Discourse on Method" or the "Meditations on First Philosophy". Those books are both much shorter and much wittier than "Gödel, Escher, Bach".

Finally, Ramachandran has interesting things to say on the subject of consciousness and sees self-reference as only one of several defining characteristics of consciousness. He makes some remarks to that effect at the end of "Phantoms in the Brain".

Couldn't be more satisfied
I got the book at a very cheap price assuming that it's too old and it's in a pretty bad condition... but then the book arrived way sooner than I expected, and it was not as bad as I assumed it would be. It's almost like I ordered a new book with the classic smell and feel on it.

One for the Proverbial Deserted Island
If there was ever a book to take to an eternal isolated spot (or even prison), this is it.You will never be bored, even if you do figure it all out - it is mental gymnastics, brain sex and worth every bit of the Pulitzer it got.

No other word for it: Amazing.
It is quite likely that the hardest question I've ever been asked is, "What's that book about?" This book manages to discuss, coherently, cohesively, and interestingly, everything from molecular biology to quantum physics to computer science to music theory to philosophy to advanced mathematics to Elizabethan literature and beyond. Reading this will definitely change the way you see the world, and if you read one book this entire year, this should probably be it. VERY highly recommended.

Excellent book
As far as the layout and design of the book go, I find this piece to be particularly structured in a way that one studying abstract and modern mathematics might find appealing. It gives specific axioms for use with each topic and in doing so defines more than just what the topic might imply. As the content goes, for those taking an introduction course in abstract algebra, this book may be slightly heavy and unwieldy, however, for those well-learned in some of its background material, this book is enjoyable and pleasurable to read. The author even makes use of antecdotes to enforce his topics. Overall, this book has been one of the most pleasurable assigned readings I have endured. ... Read more

Product Description
Gödel's modal ontological argument is the centrepiece of an extensive examination of intensional logic. First, classical type theory is presented semantically, tableau rules for it are introduced, and the Prawitz/Takahashi completeness proof is given. Then modal machinery is added, semantically and through tableau rules, to produce a modified version of Montague/Gallin intensional logic. Extensionality, rigidity, equality, identity, and definite descriptions are investigated. Finally, various ontological proofs for the existence of God are discussed informally, and the Gödel argument is fully formalized. Objections to the Gödel argument are examined, including one due to Howard Sobel showing Gödel's assumptions are so strong that the modal logic collapses. It is shown that this argument depends critically on whether properties are understood intensionally or extensionally.
Parts of the book are mathematical, parts philosophical. A reader interested in (modal) type theory can safely skip ontological issues, just as one interested in Gödel's argument can omit the more mathematical portions, such as the completeness proof for tableaus. There should be something for everybody (and perhaps everything for somebody). ... Read more

Product Description
This volume is a translation of the book Gödel, written in Japanese by Gaisi Takeuti, a distinguished proof theorist. The core of the book comprises a memoir of K Gödel, Takeuti's personal recollections, and his interpretation of Gödel's attitudes towards mathematical logic. It also contains Takeuti's recollection of association with some other famous logicians. Everything in the book is original, as the author adheres to his own experiences and interpretations. There is also an article on Hilbert's second problem as well as on the author's fundamental conjecture about second order logic. ... Read more

Product Description

Product Description
Kurt Gödel (1906-1978) was the most outstanding logician of the twentieth century, noted for Gödel's theorem, a hallmark of modern mathematics.The Collected Works will include both published and unpublished writings, in three or more volumes.The first two volumes will consist essentially of Gödel's published works (both in the original and translation), and the third volume will feature unpublished articles, lectures, and selections from his lecture courses, correspondence, and scientific notebooks.All volumes will contain extensive introductory notes to the work as a whole and to individual articles and other material, commenting upon their contents and placing them within a historical framework.This long-awaited project is of great significance to logicians, mathematicians, philosophers and historians. ... Read more

Customer Reviews (1)

The Horror!
I thought I should post a brief note to prevent other potential buyers from being misled by the editorial reviews the way I was._Choice_ says that the editors "deserve the highest praise for the design of the edition."_Mind_ says it is "beautifully produced"._The Journal of Symbolic Logic_ calls it "beautifully prepared"._Zentralblatt_ says that it "was published in a very nice and careful way".

It is reasonable to interpret these statements as referring, at least in part, to the physical appearance of the book; under that interpretation, it is my opinion that these statements couldn't be more wrong.It looks like the volume was printed in the cheapest way possible, and the page layout is amateurish at best.

It is a pity that Oxford University Press hasn't seen fit to publish an edition whose physical beauty is commensurate with the beauty of Godel's ideas. ... Read more