Product Description
The question "What is truth?" is a question that Ralph McNeil, a down-to-earth engineering student, has never before had cause to ponder. But when he encounters a famous theorem (Gödel's 'Incompleteness Theorem') which poses a seemingly unsolvable puzzle about the very meaning of truth itself, he becomes convinced against all the prevailing wisdom that the theorem must be wrong, despite the fact that it has been accepted as correct for over seventy-five years. He sets out on a journey to prove it wrong, a journey that becomes a quest to seek out the truth about truth itself. As he pursues his mission, we see into the heart and mind of Kurt Gödel, the enigmatic character who first posed the theorem - a man who won acclaim as the greatest logician of his time, but a man tortured by his own deep mystical convictions, convictions that would eventually drive him to the brink of insanity. Fact and fiction, past and present are intertwined in a compelling story of love, intrigue, deception and death that leads to a startling conclusion that will change forever the way the world understands the concept of truth. ... Read more

Customer Reviews (1)

A COMPLETE MISUNDERSTANDING OF GÖDEL'S THEOREM
I must confess this was a pleasant read although I became annoyed by the absurd claims the author and publisher makes about Gödel's theorem by means of the novel's main character, the student Ralph McNeil.

Contrary to what he claims in the preface, the author is essentially misguided about Gödel's famous result and his contention to have disproved it is ridiculous. There are errors even in his exposition of the theorem.

So, the reader may really enjoy the novel but should not take the mathematical claims of its main character too seriously. My advise: take it as complete fiction and enjoy it as such.

... Read more

Product Description
The famous theory of undecidable sentences created by Kurt Godel in 1931 is presented as clearly and as rigorously as possible. Introductory explanations beginning with the necessary facts of arithmetic of integers and progressing to the theory of representability of arithmetical functions and relations in the system (S) prepare the reader for the systematic exposition of the theory of Godel which is taken up in the final chapter and the appendix. ... Read more

Product Description
This book constitutes the joint refereed proceedings of the 17th International Workshop on Computer Science Logic, CSL 2003, held as the 12th Annual Conference of the EACSL and of the 8th Kurt Gödel Colloquium, KGC 2003 in Vienna, Austria, in August 2003. The 30 revised full papers presented together with abstracts of 9 invited presentations were carefully reviewed and selected from a total of 112 submissions. All current aspects of computer science logic are addressed ranging from mathematical logic and logical foundations to the application of logics in various computing aspects. ... Read more

Product Description
This is a book about the philosophy of time, and in particular the philosophy of the great logician Kurt Godel (1906-1978). It evaluates Godel's attempt to show that Einstein has not so much explained time as explained it away. Unlike recent more technical studies, it focuses on the reality of time. The book explores Godel's conception of time, existence, and truth with special reference to Plato, Aristotle, Kant, and Frege. In the light of this investigation an attempt is made to shed light on such issues as the precise sense in which Godel believed in the possibility of time travel, the relationship of the reality of time to the objectivity of temporal becoming, and the significance of time for human existence.This is a book about the philosophy of time, and in particular the philosophy of the great logician Kurt Godel (1906-1978).It evaluates Godel's attempt to show that Einstein has not so much explained time as explained it away.Unlike recent more technical studies, it focuses on the reality of time.The book explores Godel's conception of time, existence, and truth with special reference to Plato, Aristotle, Kant, and Frege.In the light of this investigation an attempt is made to shed light on such issues as the precise sense in which Godel believed in the possibility of time travel, the relationship of the reality of time to the objectivity of temporal becoming, and the significance of time for human existence. ... Read more

Product Description
This book constitutes the refereed proceedings of the 5th Kurt Gödel Colloquium on Computational Logic and Proof Theory, KGC '97, held in Vienna, Austria, in August 1997. The volume presents 20 revised full papers selected from 38 submitted papers. Also included are seven invited contributions by leading experts in the area. The book documents interdisciplinary work done in the area of computer science and mathematical logics by combining research on provability, analysis of proofs, proof search, and complexity. ... Read more

Product Description
Der österreichische Mathematiker Kurt Gödel (1906-1978) ist einer der herausragendsten Logiker des 20. Jahrhunderts. Seine Resultate sind von allerhöchster Bedeutung für die Mathematik, und zunehmend werden auch die Auswirkungen auf unser modernes Weltbild sichtbar. Mit dem Erscheinen von Douglas Hofstadters Buch ,Gödel, Escher, Bach wurde Gödels Werk einem breiten Publikum bekannt gemacht. John Dawson zeichnet in dieser Biographie das Bild eines Mannes, dessen Werk allgemein für abstrus gehalten wurde und dessen Leben Elemente der Rationalität und der Psychopathologie vereinigt. Die lang erwartete Gödel-Biographie ergründet den Mythos des Mathematikers, indem sie gleichermaßen seine Ideen und Arbeit verständlich aufbereitet, als auch die Person zu erfassen vermag. "... Nur sehr wenige Mathematiker haben einen ähnlichen Bekanntheitsgrad wie Gödel erreicht. Dawsons neu erschienenes Werk, das den ideengeschichtlichen und biographischen Hintergrund wie kein anderes zuvor ausleuchtet, wird daher nicht nur für die zahlreichen Gödel-Forscher und Gödel-Freaks von Interesse sein. Es beschreibt eine Entdeckungsfahrt, die in ihrer unerbittlichen Konsequenz den Vorstößen von Polarforschern in Packeis und Finsternis um nichts nachsteht." FAZ ... Read more

Product Description
This volume contains the proceedings of the conference Logical Foundations of Mathematics, Computer Science, and Physics-Kurt Gödel's Legacy, held in Gödel's birthplace, Brno, Czech Republic, on the 90th anniversary of his birth. The wide and continuing importance of Gödel's work in the logical foundations of mathematics, computer science, and physics is confirmed by the broad range of speakers who participated in making this gathering a scientific event.

Because of the continued interest and value of the work as a source and inspiration to younger researchers, this volume is being reissued as a co-publication of the Association for Symbolic Logic and A K Peters, Ltd. ... Read more

Product Description
The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory by Kurt GödelKurt Gödel (1906-1978) was a logician, mathematician and philosopher. He is regarded as one of the most significant logicians of all time, who's work has had immense impact upon scientific and philosophical thinking in the 20th century.Gödel is best known for his three theorems, which set forth and explain the foundations of mathematics.This book provides the proof for the third of his three theorems.Included is a new foreword by Richard Laver, Professor of Mathematics at the University of Colorado at Boulder, who explains in simple, non-technical terms the basic, underlying ideas behind Gödel's theorems and proofs and why they are relevant and important. ... Read more

Product Description
This brief text assists students in understanding Godel's philosophy and thinking so that they can more fully engage in useful, intelligent class dialogue and improve their understanding of course content. Part of the "Wadsworth Philosophers Series," (which will eventually consist of approximately 100 titles, each focusing on a single "thinker" from ancient times to the present), ON GÖDEL is written by a philosopher deeply versed in the philosophy of this key thinker. Like other books in the series, this concise book offers sufficient insight into the thinking of a notable philosopher better enabling students to engage in the reading and to discuss the material in class and on paper. ... Read more

Customer Reviews (3)

Q:How is this book on Godel's Incompleteness Theorem like Chinese-American food?
John V. Karavitis Answer: They both leave you hungry for more after you're done with them.

Seriously, what is it with the editors at Wadsworth?The typos, grammatical errors, and broken English in this work on Godel, his life, philosophy, and infamous theorems on completeness and incompleteness, left me nonplussed.Granted, Jaakko Hintikka knows his stuff, and provides the reader with four pages of suggested readings/references at the end.However, the explanations of Godel's ideas are not clear, and, Heaven forbid you lack a degree in Mathematics, there are some parts of this where it gets a bit intimidating.John V. KaravitisAlso, this work only hit 70 pages.Come on, people!A good editor would have either demanded a re-write or handed this assignment over to a more competent expert in the field.It's one thing to know something;the true test is, can you teach it to someone who knows absolutely nothing about your field?Jaakko Hintikka does his best, but, like the title line of this review says.... there was a bit of "incompleteness" in his work on Godel's Theorems of Incompleteness.

Oh well, three stars out of five.A good, quick review, some interesting new ideas, but don't let the math lingo scare you - this is only 70 pages long.It will be over soon enough.John V. Karavitis

Too many typos
I'm surprised that the previous review did not comment on the number of typographical errors; as far as I can see there is only one edition. The typos range from the merely distracting, to places where sentences become gibberish. Based on content, I'd give this at least 4 stars, but I found it too difficult to read. It's a nice complement to *Goedel's Proof* by Nagel and Newman--N & N give a much clearer exposition of Goedel's work, but Hintikka brings up a number of points I have not seen elsewhere (warning: many of the points raised can't be fully understood without referring to other works that treat them in more depth, unless you already have a strong background in mathematical foundations and logic).

Very interesting
This is a very interesting introduction to the thought and life of a great mathemetician and sometime philosopher.Hintikka has a clear writing style that helps with some difficult material and has special ability in making complicated math seem not so daunting.An excellent overview of the life of Godel. ... Read more

Product Description
It is a widely known but insufficiently appreciated fact that Albert Einstein and Kurt Goedel were best friends for the last decade and a half of Einstein's life. They walked home together from Princeton's Institute for Advanced Study every day; they shared ideas about physics, philosophy, politics, and the lost world of German-Austrian science in which they had grown up. What is not widely known is that in 1949 Goedel made a remarkable discovery: there exist possible worlds described by the theory of relativity in which time, as we ordinarily understand it, does notexist.He added a philosophical argument that demonstrates, by Goedel's lights, that as a consequence, time does not exist in our world either.If Goedel is right, Einstein has not just explained time; he has explained it away.

Without committing himself to Goedel's philosophical interpretation of his discovery, Einstein acknowledged that his friend had made an important contribution to the theory of relativity, a contribution that he admitted raised new and disturbing questions about what remains of time in his own theory. Physicists since Einstein have tried without success to find an error in Goedel's physics or a missing element in relativity itself that would rule out the applicability of Goedel's results. Philosophers, for the most part, have been silent.

_A World Without Time_, addressed to experts and non experts alike, brings to life the sheer intellectual drama of the companionship of Goedel and Einstein, and places their discoveries -- which can only be measured on a millennial scale -- in the context of the great and disturbing intellectual movements of the twentieth century -- in physics, mathematics, logic, philosophy, and the arts. It contains, as well, a poignant and intimate account of the friendship between these two thinkers, each put on the shelf by the scientific fashions of their day -- and ours --and attempts to rescue from undeserved obscurity the work Goedel did, inspired by Einstein, which made clear for the first time the truly revolutionary nature of the theory of relativity, which to this day is hardly recognized. ... Read more

Customer Reviews (35)

Worth the time ...
I think topic is fascinating even if discussion about <> comes in second half of the book. I liked the balance between geniuses' biography and their ideas and paradoxes, despite the difficulties in understanding Godel's proofs.
Also read Rebecca's book and found extremely naïve the idea that authors should of have collaborated. It's like asking" Why didn't Einstein and Godel write a book together so we can get all their ideas in one single read?"

Bah. Hard to read if you don't have a degree in math
I didn't finish the book.Could be my own mistake because I thought I would be able to digest the book better.I think it really helps if you have a degree in math and know all/most the people from more than 100 years ago better.Other than that, I would say this book is a waste of money.And I don't know why you arrived here in the first place, I was intrigued by the idea that there could be a universe without time(I got a feeling heaven is something like that).Well I'm still seeking as I was before reading the book:-).

Time is real because it is an ideality
Palle Yourgrau writes (page 11) of "Godel`s foolproof method for evading a rendezvous": "He would carefully arrange a precise location in space and time for the projected meeting. With these coordinates in place, he confided to friends, he had achieved certainty as to where not to be when the appointed time arrived. Yet this method had its limitations. Finding himself trapped at an unavoidable institute tea, he negotiated the territory between guests, noted the mathematician Paul Halmos in his memoirs, with maximum attention to the goal of avoiding any possibility of physical contact."

We are to believe such a shy and sly man when he asserts that time is an illusion, a mere ideality? Yourgrau thinks so, but he has been fooled. It is only that time escapes our attention, like Godel, when we focus on the formal details that Godel projects. Godel`s approach is valid as far as he is able to defeat formality, but I will argue that it is not valid to defeat time itself.

Yourgrau tells us that Godel was a philosophy-loving Platonist, and writes (page 23) on Platonists: "who like Plato believed in the objective, independent existence of ideal, disembodied forms, of which the natural numbers are a paradigm." Here truth discovered objectively becomes conflated with existence, while the person that discovers truth fades in importance. Godel would have been better served by resolving his issues with the "Dutch anti-Platonist, intuitionist mathematician L.E.J. Brouwer," who also visited Vienna as Yourgrau (page 29) indicates.

Perhaps it was positivism that polarized Godel, and pushed him deeper into Platonist philosophy rather than finding an opportunity in Brouwer`s visit? Godel was to adopt a path to defend mathematical intuition, albeit a path that erred in its ultimate treatment of time (in my view). Yourgrau (page 28) writes: "Positivism, a particularly severe brand of intellectual minimalism - a spirit that thrived in Godel`s Vienna - is an antiphilosophical philosophy dedicated to the belief that most of what has passed for deep metaphysical thinking over the centuries is nothing more than confusion based on an inadequate understanding of language, which, through artifice, leads the mind by the nose in all the wrong directions. Godel did not share the positivists credo that philosophy begins and ends with an analysis of language and its limitations, nor Wittgensteinian`s doctrine that the subject matter of traditional philosophy, as opposed to that of physical science, is precisely that which cannot be expressed in language."

Yourgrau tells us that it was positivism that was behind the drive to formalize mathematics. Positivism rejected the intuitions that Kant described, intuitions that were found a-priori to empiricism and science. Yourgrau (page 29 -30) writes: "what gave it [science] its logical twist were recent efforts by Frege, Russell, Hilbert, and others to develop logic both as an instrument that served to formalize the physical sciences - and thus to assist in their policing - and as a new branch of mathematics that was simultaneously a foundation for the rest of mathematics."

Yourgrau (page 30) writes: "As Frege`s former student Carnap put it, mathematics is not a genuine language that can express thoughts but rather the logical syntax of language. This was a doctrine that Godel, the true heir of Frege, would spend the rest of his working life to defeat."

Yourgrau (page 40) tells us that Einstein`s thinking impacted the Vienna Circle, as well as Godel: "It was precisely the hegemony of positivism, Godel wrote later, that allowed the members of the circle to mistake Einstein for an ally and to underestimate the difficulty of rendering mathematics empirically acceptable by reconstructing it as a system for formal manipulation of signs. Einstein himself would awaken the positivists from their misconceptions about the ultimate relationship between his thoughts and theirs. And Godel, in short order, would surprise everyone by striking a fatal blow to the most rigorous attempt to reconstitute mathematics as a formal theory of signs."

Yourgrau (page 53) writes: "Godel`s incompleteness theorem of 1931 began innocently, as an attempt not to refute but to fulfill Hilbert`s program. Hilbert`s idea was to safeguard mathematics from hidden contradictions by replacing the intuitive mathematics of each mathematical domain with a system of axioms written in pure formula language that, although having a standard semantic interpretation, could be manipulated according to mechanical rules of pure syntax."

But surprise, surprise, surprise, Yourgrau (page 57-58) writes: "What Godel proved is that mathematical truth is not reducible to (formal or mechanical) proof. Syntax cannot supplant semantics. The leitmotif of the twentieth century, it turns out, stands in need of revision. Mechanical rules cannot obviate the need for meaning, and what gives us access to meaning, namely intuition, cannot be dispensed with even in mathematics, indeed, even in arithmetic. This was the first nail in Hilbert`s coffin. The second nail was not long in coming. Godel soon proved his second incompleteness theorem, which demonstrated, with yet further irony, that if a given system of axioms for arithmetic were in fact consistent, then it could not be proved consistent by the system itself."

And so it was that Godel destroyed an absolute faith in formalism, and left positivism in crisis. Yourgrau (page 106) tells us that Einstein and Godel were united against positivism: "At the heart of Godel and Einstein`s opposition to positivism was their unfashionable realism, their reluctance to make ontology, the theory of what is, subservient to epistemology, the theory of what can be known. At the bottom, the positivist mentality consists in deriving ontology from epistemology."

Both Einstein and Godel found themselves in Princeton, a refuge that became available prior to the turmoil of the second world war. Yourgrau writes of their growing friendship. I can only speculate that their best achievements had passed.

Godel became interested in time, Yourgrau (page 115) writes: "In his contribution to the Einstein volume, Godel would construct a world model for the equations of general relativity whose geometry was so extreme that the temporal component of the resulting space-time structure could not reasonably be seen as representing intuitive time. Einstein had already succeeded, in the theory of relativity, in bringing about the geometrization of physics. What Godel did was to construct a limit case for the relativistic geometrization of time."

The Godel universe implied that time travel was possible in a very fast spaceship. Yourgrau (page 116) writes, "if time travel is possible, time itself is not." Godel thought that he discovered intuitive time to be unreal, and Yourgrau seems to agree with Godel`s conclusion. But what Godel demonstrated was the impossibility to formalize time and make it an abstract dimension in the Godel universe that is suitably elevated to Plato`s ideal realm (again my view). Time itself escapes the formal, and Godel`s Platonist philosophy is only now finding itself weak compared to Brouwer`s intuitionism. It is formal time that is unreal! Or time is real because it is an ideality that can reveal a privileged reference frame enough to ponder the Godel universe!

Godel spent his life defeating formality that only pretends to mimic the intuitive, so he should have anticipated my objection. Yourgrau (page 128) figures as much and writes: "The question that remains is whether this intuitive concept [of time] can be captured by the formal methods of relativity." Nevertheless, Godel`s (and Yourgrau`s) "dialectical dance with time" remains unconvincing to me. Indeed, the time taken for the dialectical dance is not to be ignored.

The bigger mistake is to remove the intuitive from a perceived objectivity, even going so far as to refer to Edmond Husserl to justify this removal. In Husserl`s phenomenology, objectivity is itself to be purified before arriving at a transcendental subjectivity. Yourgrau (page 171) unwittingly reveals Godel`s misconception: "What Godel found valuable in Husserl, however, was a turn to the thinking subject, the source of cognition, which was meant not as an alternative to objectivism, but rather as an account of how what is objective is given to us." But it remains true that Husserl`s phenomenology is better grafted onto Brouwer`s intuitionism, than Godel`s Platonist philosophy.

A World Without Time
A WORLD WITHOUT TIME

When the name Albert Einstein is mentioned, there is name recognition and perhaps, from high school physics E=mc2 flashes before our eyes.Albert Einstein is as common as the days of the week.

When the name Kurt Gödel is mentioned, most people would have a difficult time placing him.Not only was Gödel the greatest logician of recent centuries, he was Einstein's closest friend at Princeton's Institute for Advanced Studies.

Palle Yourgrau's A WORLD WITHOUT TIME explores this friendship.In their discussions and arguments the two men tossed about a wide range of subjects -mathematics, physics, logic, philosophy, religion, time travel, theory of relativity, etc.This is what this book of 210 pages offers the reader.

Gödel is known in scientific circles for his incompleteness theorem which dealt with mathematics.There is no simple explanation of his theorem.

On a very human level, he found Walt Disney animation films fascinating.Of course, Einstein had no interest in such trivial things.

On the issue of religion, Einstein was born a Jew, but would embrace the philosophy of Benedict De Spinoza, the 17th century philosopher.Gödel was born a Lutheran and embraced the philosophy of Gottfried Wilhelm Leibniz, another 17th century philosopher who was the direct opposite of Spinoza.The one philosopher which Einstein and Gödel both embraced was Emmanuel Kant who influenced both men in their discourses and writings.

When it came down to American politics, these two Europeans shared a different world view. Einstein was a supporter of Adlai Stevenson and Gödel supported Dwight Eisenhower.Einstein felt that his friend had lost touch with reality.Einstein did not think a military general should be president of a republic.

Overall, colleagues at Princeton did not understand the friendship of Einstein and Gödel since they were so different.Yourgrau explores these differences and the ensuing bond which bridged ideological gaps to form a friendship which benefited both men and the world.

Gödel's concept of time is explored in these pages.For Gödel time is a reality which had no relevance.

Time was curves, closed loops, where the past, present and future all existed together.
If the past is still a part of present reality, then the passing of days, months and years fade into these time curves.

Based on this time theory, Gödel and Einstein are still taking their walks and bouncing off each other their ideological frames of reality. Their discourses are still boisterous as the overflowing beer steins at the local German beer hall.The fellows at Princeton are still scratching their heads at these two European gentlemen displaced by the rise of Nazism.

Of course in Gödel's time reality, Nazism is still plotting its plan for globe dominance on the Third Reich wall graph, but unknown to them, their plan will never see fruition.For even their thousand years has no relevance to past, present and future in Gödel's world.


This is a well written book.Without some scientific background the reader may find some passages incomprehensible, but overall it is a fascinating tale of two men who happened to be two of the greatest thinkers of the 20th century as well as being best friends.Regardless of this scientific talk and other small faults, WORLD WITHOUT TIME is definitely worth the curious reader's time.

A World Without TIme
Reviewed by: Stephen J. Hage

Two of the twentieth century's greatest thinkers, Kurt Gödel and Albert Einstein became friends in 1942. This book describes how intimate that friendship was and how they influenced each other's thinking.

For people interested in pure mathematics the name Kurt Gödel is as famous as Albert Einstein.His incompleteness theorem shattered the efforts of Bertrand Russell and Alfred North Whithead in their attempt to write the definitive tome to end all tomes on mathematics, the Principia Mathematica.The stated goal of the Principia was to tie together everything that was known about mathematics so that it would, once and for all, be complete.Gödel's incompleteness theorem not only demonstrated but proved, mathematically, that such a task was not only folly, it was impossible.

For people interested in physics and relativity, Yourgrau reveals how Gödel brought his formidable mathematical skills to bear on Einstein's work to show that time, as we understand it simply doesn't exist.He deals with the metaphysical and philosophical implications of that in ways that are both lucid and satisfying.But, more than that, he offers insight into the personal aspects of the two men in ways that humanize them by elucidating not only what they did but who they were.Here's an example:

"Physically they were opposites.Gödel, thin to the point of emaciation, hid his spectral body even in the heat of summer in overcoat and scarf.Gaunt, harrowed, and haunted, peering through thick glasses like an owl from another dimension, he could not fail to arouse suspicion.Early in life he had come to the conclusion that the less food one ate the better.This dubious insight he carried out with ruthless consistency, unencumbered by the excess baggage of common sense, a faculty he approached life without.

Einstein, in contrast, whose sanity was never in question, was as satisfied by a good sausage as by a good theorem.He had a taste for solid German cooking, which he consumed with relish, topped off by his omnipresent pipe....Late in life he was the proud owner of a respectable professorial paunch."

Even more satisfying, Yourgaru offers four pages of archival photographs of Gödel with members of his family and other scientific luminaries like Albert Einstein, Rudolph Ladenburg, J. Robert Oppenheimer and Eugene Wigner.

Gödel's insights had philosophical implications that were and are deep and strong but he was vilified by the philosophical establishment.

The book is valuable because it provides insight into the man, his friends and his thoughts on so many different levels.

For people interested in such things this book is a must read.
... Read more

Product Description
What happens when the century's greatest logician meets thecentury's greatest physicist ? In the case of Kurt Gödel andAlbert Einstein the result is Gödel's revolutionary new world models forrelativitytheory.

In the 'Gödel Universe' the philosophical fantasy of 'timetravel' becomes a scientific reality. For Gödel,however, the reality of time travel signals the unrealityof time. If Gödel is right, the real meaning of the Einstein revolution remained, for half a century, a secret.Now, half a century after Gödel met Einstein, the realmeaning of time travel in the Gödel Universecan at last be revealed. ... Read more

Customer Reviews (5)

Philosophy of Time
This book is incredible.I am reading it twice over.That's how great a book I think it is.What I find so intriguing about it is that the author, Palle Yourgrau, writes a very scholarly exposition about how Godel viewed time to be ideal rather than real against the backdrop of Einstein's concept of the universe as a four dimensional manifold.It really brings to light what the great genius, Einstein, thought about time and whether time travel is possible in the Godel universe.

Why Philosophers shouldn't be allowed to write popular Science books
This is a truly appalling book.If you decide to base your knowledge of relativity, time, Godel or Godel's work on this book then you justly will be humiliated by the first person who has even a passing grasp of the issue.

Firstly, it is clear that Yourgrau doesn't understand Godel's famous theorems.Whilst patronizing the reader about how it is OK not to understand this theorem he shows his own shallow grasp.Godel's theorem essentially says in any mathematical system there are infinitely more statements that cannot be "decided" as to whether they are true or false based on the axioms.In fact if you pick a statement at random is has a probability zero of being decidable.So as for it showing that somehow computers cannot prove things that humans can, well clearly this claim is false.

Secondly there are glaring historical errors.The people who invented the mathematical basis for computers are Turing, Church and Von Neuman.Godel did not invent recursive functions, Church did.We have discussions on people who were proven wrong about the nature of space and time literally hundreds of years ago - Kant and his assertions that space must a priori only be Euclidean and 3 dimensional was being proven wrong even as he wrote.Only a philosopher would bring him up today.

Then finally we get to the nub of Yourgrau's claims.Godel came up with a very specific solution that has closed worldlines.What does this mean?It means that that if you follow any particles path long enough it comes back to the same point - ie same spot in space and crucially also in time.So A isn't "before" B and B isn't "before" A.Surely this means that there is no time?No it doesn't, it means that rather than a worldline being like the line it is like a circle.But surely it means that causality is violated because A cannot cause B if it is not before B!No, it is a universe that is Acausal, ie there are no causes which is not the sameas being non-causal - which is when causality is violated.But maybe this could have a relevence to our universe?Well no, because we KNOW - and have done for over 50 years - that OUR universe is expanding which is not allowed in Godel's solution.This is the REAL reason Godel's solution has been ignored because it falls at the first test when compared with the real world.Apparently ignorant of this fact, Yourgrau attempts to use "analogy" to show "relevence", that the differences are "accidental" - they aren't but never mind - and so Godel's solution "could" be relevent.Well that sort of illogical, ignorant pseudo-argument make cut the mustard in Philosophy but it is laughable in Maths and Physics.

In short one of the worst popular science books around.If you buy this book then it will give some him money to carry on writing, so do yourself - and the general public - a favour; buy Goldstein's "Incompleteness" and Simon Singh's "Big Bang" and learn from people who have bothered to have at least a passing grasp of their topic.I wish I could give it no stars.

Intriguing but a hard read
Although ostensibly this book is about Godel's solutions to Einstein's General Theory of Relativity and his consequent view of time, it is in fact broader than that. This could be instead considered a book on the philosophy of time with consideration of the GTR and using Godel's views on time and the GTR as a common thread.

As such, it is a difficult read. It will require of someone a fairly strong comprehension of philosophy and metaphysics, as well as (at least) a conceptual grasp of GTR, formal logic and Godel's Incompleteness Theorem, and some understanding of Cantor's transfinite numbers.

The philosophical references run from Plato and Aristotle to Kant and McTaggart. It is wide-ranging in it's coverage of differing views and how they relate to Godel's own views on time. Topics such as "The Epistemology of Potential Infinity" and "Frege and the Decontextualization of Thought" are representative of the depth of this book. Discussions of potential vs. actual infinities, truth in relation to time, the ontology of space-time diagrams, and so on, can leave one reeling.

In addition, comparisons between Godel's Incompleteness Theorem and his approach to his solutions to GTR are made, in hopes of elucidating Godel's ingenuity in finding unexpected solutions to various formalisms.

So, this book really attempts to go far beyond it's title. I think it generally succeeds, however, I find some of the organization of the book annoying, and I wish it were layed out differently. (However, in fairness, if it were my task I have no idea how I would go about reorganizing such complex material - it just seems that it could be done). The book is very heavily referenced (sometimes excessively it seems), and practically every page has at least two or three quotes. I think this tends to make the overall flow of writing less than natural, but it is a matter of style.

Finally, the appendix on Zeno's paradoxes is, to me, very well done and worth half the price of the book right there. Yourgrau clarifies in particular the issue of limits vs infinite sums, and all-in-all provides compelling reasons for claiming that Zeno's paradoxes remain unsolved despite some claims to the contrary.

Underground classic?
This absolutely riveting book delivers at two levels: first, it can serve for any intelligent reader as an exhilirating introduction to most of the central questions of philosophy--questions concerning time, truth, death, thought, free will, and infinity, to mention just some of the ground covered. Second (and this is what makes it such an astonishing book) it offers, in the context of its luminously clear and helpful discussions of these issues, a series of original and profound philosophical results. Many of these insights in effect open a new chapter in the domains they deal with (see, for example, Yourgrau's powerfully argued rejection of the standard way of understanding both existence (represented in logic these days by the 'existential quantifer') and infinity (esp., the sum of an infinite convergent series). In reading the book you are given a front row seat onto all of this--some of the newest, most surprising, and most fundamental work in philosophy. Those coming to the book from physics (i.e.the 'Einstein' end of things) will, I suspect, be refreshed by the idea of a harmonious and fruitful relationship between philosophy and science that infuses the book (after all, Einstein and Godel talked philosophy AND physics). In short, an extraordinary book not to be missed.

challenging but rewarding
This is a difficult philosophy book, but there are a number of interesting ideas worth thinking about for those who are interested in the philosophy of time, specifically in what physics seems to tell us about the nature oftime. This book is actually an expanded edition of Yourgrau's (1991) book,The Disappearance of Time. I think it's a bit disingenuous to issue what ispretty much the same book under a new title. Godel Meets Einstein has onenew chapter (definitely worth reading) and two new appendicies. ... Read more

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Kurt Godel was one of the most outstanding logicians of the 20th century and a giant in the field.This book is part of a five volume set that makes available all of Godels writings. The first three volumes, already published consists of the papers and essays of Godel. The final two voulmes of the set deal with Godel's correspondence with his contemporary mathematicians, this fifth volume consists of material from correspondents from H-Z. ... Read more

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This digital document is an article from First Things: A Monthly Journal of Religion and Public Life, published by Institute on Religion and Public Life on August 1, 2010. The length of the article is 4004 words. The page length shown above is based on a typical 300-word page. The article is delivered in HTML format and is available immediately after purchase. You can view it with any web browser.

Citation Details
Title: The God of the mathematicians: David P. Goldman explores the religious beliefs that guided Kurt Godel's revolutionary ideas.(Report)
Author: David P. Goldman
Publication: First Things: A Monthly Journal of Religion and Public Life (Magazine/Journal)
Date: August 1, 2010
Publisher: Institute on Religion and Public Life
Issue: 205Page: 45(6)

Article Type: Report

Distributed by Gale, a part of Cengage Learning ... Read more

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Death And Anti-Death, Volume 6: Thirty Years After Kurt Gödel (1906-1978)[Charles Tandy, Ph.D., Editor] [ISBN 978-1-934297-03-2]------Volume 6, as indicated by the anthology's subtitle, is in honor of Kurt Gödel (1906-1978). The chapters do not necessarily mention him. The chapters (by professional philosophers and other professional scholars) are directed to issues related to death, life extension, and anti-death, broadly construed. Most of the contributions consist of scholarship unique to this volume.As was the case with all previous volumes in the Death And Anti-Death Series By Ria University Press, the anthology includes an Index as well as an Abstracts section that serves as an extended table of contents. (Volume 6 also includes a BRIEF COMMUNICATIONS section.) The ten chapters are entitled as follows:------> 1. Life And Death Economics: A Dialogueby Giorgio Baruchello and Valerio Lintner (pages 33-52)------> 2. Charles Hartshorne by Daniel A. Dombrowski (pages 53-78)------> 3. Choosing Death in Cases of Anorexia Nervosa - Should We Ever Let People Die From Anorexia? PART II by Simona Giordano (pages 79-100) ------> 4. The Ethics Of Enhancement by Bill Grote and William Grey (pages 101-126)------> 5.Cosmology And Theology by John Leslie (pages 127-156)------> 6.Positive Logicality: The Development Of Normative Reasonby J. R. Lucas (pages 157-222)------> 7. The Basic Ideas Of Conformal Cyclic Cosmology by Roger Penrose (pages 223-242) ------> 8.Deconstructing Deathism: Personal Immortality As A Desirable Outcome by R. Michael Perry (pages 243-264)------> 9. What Mary Knows: Actual Mentality, Possible Paradigms, Imperative Tasks by Charles Tandy (pages 265-284)------> 10.The Future Of Scientific Simulations: From Artificial Life To Artificial Cosmogenesis by Clément Vidal (pages 285-318) ... Read more

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Berto’s highly readable and lucid guide introduces students and the interested reader to Gödel’s celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments.

• Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters
• Discusses interpretations of the Theorem made by celebrated contemporary thinkers
• Sheds light on the wider extra-mathematical and philosophical implications of Gödel’s theories
• Written in an accessible, non-technical style

Customer Reviews (2)

Enhanced my understanding
I've often wondered how professional film critics can figure out how much of their reaction to a movie is due to its intrinsic merits and how much is based on the mood they happened to be in when they saw it. I find myself in an analogous situation as I write this review.I read _Godel, Escher, and Bach_ as a college freshman when it first came out, I've read Franzen's book on Godel, and I've been exposed to Godel's arguments on several other occasions during my career, but reading _There's Something about Godel_ has left me with a much better understanding of the Incompleteness Theorem than I previously had.I don't know for sure how much of this is really due to Berto's skill as a thinker and a writer and how much is due to those past influences finally sinking in, but to counterbalance some of the criticism he's received from others, I'm willing to give him the credit.He impressed me as a very clear writer, being patient without being tedious.(In that last aspect, this book didn't feel at all like a pop math book, a genre which professional mathematicians typically find boring.)Even the chapter on Wittgenstein, which I anticipated hating, was surprisingly tolerable, though I don't think I'm in any danger of becoming a fan of either Wittgenstein or paraconsistent logic.

Two final points:

(1)This is an English translation of an Italian book, and I presume Berto is a native Italian, but the English in this book is just fine--not at all stilted.
(2)Wiley has come out with some books with really ugly printing lately, even uglier than an average print-on-demand book.There's no such problem with this book, though.

A perpetuation of a fundamental falsehood
This is not my first review of a book on Gödel, and therefore I want to respond freshly, by considering this book in particular.

In view of the numerous attacks I was subjected to for not accepting "Gödel's incompleteness theorem", the main subject of this book, I could be taking comfort in author Berto's attention to Wittgenstein as a critic of Gödel's theorem. At least I have that prominent figure sharing my negative attitude, with Dr. Berto giving Wittgenstein some credence (p.213): "an audacious rethinking...may nowadays vindicate some of Wittgenstein's 'outrageous claims' on Gödel's Theorem, too swiftly dismissed by commentators who dogmatically took the logic of Russell and Frege as the One True Logic".

My view of Wittgenstein, though, is not the best, since he seems never to substantiate his flamboyant declarations, and Dr. Berto is really fully committed to Gödel, describing him, in keeping with today's adulation, as "the logician of the [last] millennium" (p.189).

In my eyes Gödel is nothing of the sort. He contrived an impossibly elaborately symbolized "proof" of a simple sentence, and perhaps still more preposterously he equivocated that sentence with a mathematical one. That sentence, famous by now, is

(1) "THIS SENTENCE IS UNPROVABLE" (in the logical system in which it occurs).

The sentence is modeled on the ancient Liar paradox, stating

(2) "THIS SENTENCE IS FALSE".

If (2) is true then, by its content, it is false; and if it is false then, again by its content, it is true.

Now, Dr. Berto cites logician Alfred Tarski as showing that the truth of a sentence cannot be defined in the language of the sentence (similarly to the above unprovability), quoting his statement (p155): "'Snow is white' is true...if and only if snow is white". The single quotes delimit such a sentence, the rest quoted is to belong to the "metalanguage". But why the distinction? Because the Liar, sentence (2), couldn't be resolved for over two millennia. However, its resolution is very simple. By the truth of a sentence, expressed in the same language, we mean what it says. The Liar, like any sentence, is meant to be true, but illegitimately says it is false; i.e. it harbors an implicit contradiction, same as if a sentence explicitly declares something both true and false. (Allow me to mention I treat this and other issues in my book "On proof...")

But as a result of the Liar and other paradoxes, all internally contradictory, logicians instead devised various hierarchies, of languages, types, systems, by which the contradictory results would be eliminated by placing them in other hierarchies. One such attempt was made by David Hilbert (p.39ff), who famously proposed "formal systems" of logical symbols lacking meaning and "interpreted" afterward, delegating the logical process to a separate system of "metamathematics". This is fundamentally the basis of Gödel's arguments.

He made his above sentence (1) an "interpretation" of (giving meaning to) certain symbols that are part of a formal system, thereafter arguing that his proof is achieved via separate metamathematics. But is there such a distinction between a logical system and a proof regarding it? Logic in general consists of principles to be followed, such as the "laws of thought", referred to in the book (p.7) by the "Principle of Bivalence" ("all sentences are either true or false") and the "Law of Non-Contradiction" (a sentence cannot be both true and false). Accordingly, if such logic is applied to sentences, it is applied within the system. Now consider sentence (1), the one alleged proved.

As Dr. Berto offers the proof in one case (p.50), "Suppose [(1)] is provable. Then, given what it says, it is false, since it claims not to be provable... Therefore [(1)] is not provable in [the system]... But if [(1)] is not provable, then [(1)] is what it claims to be; therefore, it's a true sentence". But wait! How did we reach this conclusion? By finding the provability of [(1)] a contradiction, thereby PROVING within the system the sentence true! But by thus proving it, we again contradict it! The upshot is, sentence [(1)] ends up a paradox.

As indicated, the finding of paradoxes in logical or mathematical systems is thought intolerable, and therefore great efforts have been expended to make the systems paradox-free, largely by devising mentioned hierarchies. But the paradoxes are no reflection on the systems; they are contained within the particular sentences or descriptions concerned. Exceptions can be paradoxes concerning whole theories, like Cantorian set theory. I can't go into more detail here, adding only a remark again on allegations that Gödel's theorem applies to just mentioned mathematics.

"Gödel numbering" is an impressively intricate system of assigning numbers to constituents of "formal systems", like the symbols for sentence (1). It is then alleged that (1) also applies to some of mathematics. But however imposing the numerical names of the linguistic constituents, of the symbols, they don't change the contents behind them. I.e., to change the content of sentence (1) into a numerical one commits the fallacy of equivocation.
... Read more

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Chapters: Austrian Logicians, Kurt Gödel, Ludwig Wittgenstein, Karl Popper, Friedrich Von Hayek, Paul Feyerabend, Ernst Mach, Alfred Schütz, Rudolf Steiner, Otto Neurath, Franz Brentano, Hans Köchler, Vienna Circle, Ludwig Von Mises, Martin Buber, Anton Günther, Ivan Illich, André Gorz, Otto Weininger, Roland Benedikter, List of German-Language Philosophers, Alexius Meinong, Erik Von Kuehnelt-Leddihn, Karl Leonhard Reinhold, Leopold Kohr, Nathan Birnbaum, Martin A. Hainz, Walter Johannes Stein, Hans Kelsen, German Philosophy, Wolfgang Stegmüller, Ernst Mally, Martin Balluch, Ingo Zechner, Nachman Krochmal, Othmar Spann, Wilhelm Jerusalem, Otto Bauer, Gustav Bergmann, Christian Von Ehrenfels, Herbert Feigl, Helene Von Druskowitz, Victor Kraft, Ferdinand Ebner, Friedrich Waismann, Johann Nepomuk Ehrlich, Theodor Gomperz, Richard Hönigswald, Gerhard Streminger, Ludwig Landgrebe, Heinrich Gomperz, Adam Tanner, Johann Heinrich Loewe, Richard Wahle, Johann Heinrich Pabst, Hubert Schleichert, Olga Hahn-Neurath, Alois Riehl, Kurt Rudolf Fischer, Gustav Kafka, Rudolf Maria Holzapfel, Thomas Resch, Robert Von Zimmermann, Oskar Ewald, Alfred Kastil. Source: Wikipedia. Pages: 391. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt:Ernst Mally (11 October 1879 - 8 March 1944) was an Austrian philosopher affiliated with the so-called Graz School of phenomenology . A pupil of Alexius Meinong , he was one of the founders of deontic logic and is mainly known for his contributions in that field of research.Life Mally was born in the town of Kranj (German : Krainburg ) in the Duchy of Carniola , Austria-Hungary (now in Slovenia ). His father was of Slovene origin, but identified himself with Austrian German culture (he also Germanized the orthography of his surname, originally spelled Mali, a common Slovene surname of Upper Carniola ). After his... ... Read more

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Chapters: People From Brno, Kurt Gödel, Gregor Mendel, Ernst Mach, Adolf Loos, Milan Kundera, Leoš Janáček, Bohumil Hrabal, Jana Novotná, Erich Wolfgang Korngold, Lucie Šafářová, Lukáš Rosol, Ondřej Sekora, Fredy Perlman, Churches of Brno, Peter Sís, Pavel Haas, Karel Abrahám, Eugen Von Böhm-Bawerk, Hc Kometa Brno, Brno Exhibition Centre, Zdeněk Ščasný, Mahen Theatre, Starobrno Brewery, Volejbal Brno, Heinrich Wilhelm Ernst, 1. Fc Brno, Masaryk University, Ernst Weiss, Dina Babbitt, Michal Březina, Baron Franz Von Pillersdorf, Veveří Castle, Brno-Tuřany Airport, Vinzenz Eduard Milde, Villa Tugendhat, Josef Bulva, František Graus, Vítězslava Kaprálová, Jiří Hudec, David Vincour, Heinrich Wawra Von Fernsee, Ondřej Hotárek, Rc Bystrc, Roman Catholic Diocese of Brno, Georg Joseph Kamel, Osvald Chlubna, Ferdinand Ritter Von Hebra, Brno University of Technology, Jan Polák, Jaroslav Jiřík, Špilberk Castle, Jan Graubner, Jan Stejskal, Nella Simaová, Maria Restituta, Maria Jeritza, Emanuel Von Friedrichsthal, List of People From Brno, Israel Bruna, Radúz Činčera, George Placzek, Helena Dvořáková, Jan Trefulka, Zdeněk Blatný, Brno City, Libuše Šafránková, Hans-Adam I, Prince of Liechtenstein, Zdenka Podkapová, Nikola Višňová, Leopold Janauschek, Luboš Kalouda, Josef Berg, Hana Černá, Wilhelm Elsner, David Kostelecký, Zdeněk Svoboda, Rc Dragon Brno, Libor Došek, Viktor Kaplan, Hans Müller-Einigen, Fred Iltis, Leo Marian Vodička, Tereza Kerndlová, Dagmar Havlová, Ada Kuchařová, Jan Kotěra, Emil Redlich, Zdeněk Neubauer, Hugo Haas, Brno Hlavní Nádraží, Treaty of Brno, Palace Moravia, Michal Hašek, Emil Zinner, Igor Ardašev, Michal Kolomazník, Julius Brach, Max Maretzek, František Halas, Richard Von Schaukal, Hellmuth Karasek, Bohuslav Fuchs, Robert Kron, Heinrich Wilhelm Schott, René Wagner, Tomáš Julínek, International Performers Competition Brno, Ondřej Liška, Milan Uhde, Brno-Bohunice, Tomáš Mica, Petr Čoupek, Karel Lang, Jana Galiková, Jaroslav Borák, Leopold Adametz, ... ... Read more