Logic: A Very Short Introduction (Very Short Introductions)
by Graham Priest
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Graham Priest's Logic Intro. is well worth the purchase.
Graham Priest's Logic on Oxford Univ. Press I must say is superb.A short introduction that straddles the line between 'keeping to the basics of logic', but not really so.

The latter is strange to say, I must admit, but the point is that Priest touches upon the general basis of formal logics, but does not patronize the reader with cliche examples and content that can be found in several intro. books on logic.Rather, Priest introduces many important and contemporary problems that go on in logic (hence giving the reader a taste of-- not only technical aspects of logical systems-- but the philosophical issues that arise out of logics as well).Although this book does not necessarily go into extreme depth of particular issues, its survey of issues such as tense logic, existential import problems, and so forth are more than satisfactory.

In addition, Priest's style of writing is very clear, organized, and to the point.I must admit, Priest's writing is superb (as is evident in this work, as well as his other books and numerous logic/philosophy articles).In addition, Priest is quite possibly one of the most well-established and well known logicians of our day; so, the buyer should feel assured that he is more than qualified to write this book.

My advice is pick up this book! For the price---its a killer deal!

Also, I would recommend as 'follow-ups' to this book, Priest's Intro. to Non-Classical Logics and Susan Haack's Philosophy of Logics and her other work, Deviant Logics and Fuzzy Logic (in which Haack tries to tackle many philosophical problems that arise in classical/non-classical logics while defending the classical logics position).

Intro to Graham Priest's Logic
For its length and breadth, I'd recommend this little book to most people.I teach a course on Logic, and even I learned a bit from this short introduction.

The worry, however, is that Priest takes this book as an opportunity to push his own non-standard view on formal logic, which is why a better title for this book would be `A short Introduction to Priest's Logic'.

Most immediately evident is that Priest uses many of the chapters as a place to show how an argument for the existence of god.For example, much of the chapter on predicate logic is devoted to showing the fallacy inherent in the cosmological argument for the existence of god.His chapter on decision theory is in part devoted to showing how Pascal's wager goes wrong.

Probably even more subversive is that, in an introductory level logic book, Priest presents his own unorthodox solutions to paradoxes in logic.For example, to solve the problem of self-reference, he presents his own view regarding four valued logics, without even a word explaining that not only is this not the standard view, it's not even a very popular one.He also offers fuzzy logic as a solution of vagueness and sorites paradoxes.

Overall, I thoroughly enjoyed reading this book, and would offer it to any friend interested in a brief introduction to some advanced topics in logic.But I'll be sure to emphasize that much of what he offers for solutions is hardly accepted by mainstream analytic philosophy.Priest took advantage of this chance to write an introductory level text as an opportunity to push his own views, and anyone reading this should be aware of this fact before beginning.

Sloppy organization, irritating style, confusing content.
This short little book gives the impression of having been written by a lazy professor who wanted to pick up a few quid on the side, so he spent a couple of weekends adding some verbal fluffery to his already-potted lecture notes, and OUP let him get away with it.

How do I know he's lazy?Because he never explains anything.Here's how he puts it in the introduction:

"It is true that you will need the determination to master some symbolism that may be new to you;but this is a lot less than is required to have a basic grasp of any new language.And the perspicuity that the symbolism gives to difficult questions makes any trouble one may have in mastering it well worth it."

So in the second chapter he jumps into truth tables and begins to use symbolic logic to explain his arguments, and does so all the way through the rest of the book.Each of the chapters ends with its main idea given in symbolic notation, which I found very irritating and not at all helpful.He may as well have said, "Here are some Japanese ideograms for you to look at.If you're really smart, and stare at them long enough, they will begin to make sense to you.Tah, tah!"

By the way, "perspicuous", one of Prof. Priest's favorite words (another is "tendentious"), means "clear" or "lucid", neither of which applies to this book.But that is the book's second big problem:Prof. Priest's style.His chunks of symbolic logic are broken up by "chatty", breezy "discussions" that reminded me of Abbott and Costello routines.Here is an example:

"Consider the liar paradox and its cousin.Take the latter first.The sentence 'This sentence is true' was supposed to be an example of something that is neither true nor false.Let us suppose that this is so.Then, in particular, it is not true.But it, itself, says that it is true.So it must be false, contrary to our supposition that it is neither true nor false.We seem to have ended up in a contradiction."

And Who, exactly, was it that was on first, anyway?And why are we hearing about the Liar's Paradox in a Very Brief Introduction to Logic, unless it's just Prof. Priest showing off again?I'm sure he would be a real annoyance to have to sit next to on a long airplane ride.

The third big problem with this book is its organization.The last chapter, on the history of logic, should logically come first.Right?So we know where we're going.And if logic is really hard-wired into us, then the book should be organized logically, with the easy stuff first, and the hard stuff, like logical symbols, later.Right?But that takes time, and thought, and, well, WORK, and Prof. Priest wasn't up for WORK,at least on this book.

I don't know what the discussion of "possible worlds" is doing in a book on logic, except that Prof. Priest wants to show off again, rather than to teach.And why all the talk about God?Does Prof. Priest have a thing about God, as another reviewer suggests?This is a Very Short Introduction, not a book on the philosophy of religion, so there isn't much room, and what room there is should go to discuss LOGIC, NOT theology.

In short, OUP should find someone to rewrite this book so that it really IS a Very Short Introduction to Logic, rather than a mish-mash of symbols, philosophy of religion, cutesy discussions of paradoxes, and a tendentious lack of perspicuity.

Not recommended. ... Read more

Isbn: 0192893203
Sales Rank: 54131
Subjects:  1. Logic    2. Philosophical Logic    3. Philosophy   

An Introduction to Logic
by H. W. B. Joseph
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A classic exposition of logic
This is what philosophical logic used to look like before the linguistic philosophers got hold of it ;-). And the fact that the book is still in print is a very good sign.

H.W.B. Joseph was one of Oxford's leading philosophical lights in the late nineteenth and early twentieth centuries. This book, originally published in 1906 and republished in a revised second edition in 1916, is a classic exposition of the entire subject of logic. It's got everything you look for in a philosophy text: clarity and accuracy of exposition, breadth of scope, and a keen sense of the philosophical importance of every topic under discussion. I've had a battered copy of the second edition for many years now and I still refer to it fairly often.

The range of subjects will probably seem remarkable to the modern reader, who expects logic to have something to do with mathematics and set theory. There's not a truth table in sight; in fact, there's no discussion of formal, propositional logic at all. Instead there's discussion of e.g. the existence and nature of universals, the principles of the syllogism, the nature of the propositional judgment, the nature of causation (and whether it's reciprocating -- i.e., whether effects determine their causes too), the meaning of "explanation," and the nature of reasoning in mathematics and science.

Joseph's logic is a branch, not of mathematics, but of metaphysics. And one thing the reader will notice right away is that Joseph engages the metaphysicians of his day -- especially the neo-Hegelian idealists, like Bradley, Bosanquet, and Joachim. His criticisms are cogent and invariably well-taken. But in fact Joseph, a Platonist himself, is far closer to the "spirit" of the British idealists than he is to that of modern more or less analytic philosophy; my own view, at least, is that idealism could easily have taken up his criticisms and moved on.

At any rate, this is a fine, fine text, and when I noticed it was in print, I couldn't resist reviewing it. If you're interested in this subject, do yourself a favor and buy a copy. ... Read more

Isbn: 1889439177
Sales Rank: 720908
Subjects:  1. Logic    2. Philosophy   

An Introduction to Philosophical Logic
by A.C. Grayling
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Paperback (01 January, 1998)
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Philosophy Majors: Read This Before Tackling Logic Exercises
Lucid book on the philosophical implications of modern logic.

I wish my philosophy professors had assigned this book BEFORE they plunged us straight into rote drills in propositional and predicate logic. Grayling shows you what exactly all these sterile-seeming symbolic manipulations have to do with epistemology and metaphysics.

Also check out books by Graham Priest.

Highly recommended.
Enough centuries have passed that everyone should know this: If you want a clear and readable exposition of a difficult and complex subject, you ask a British philosopher.

In that grand tradition of solidity and soundness, A.C. Grayling here provides, without fanfare but with a good deal ofclarity and wit, a thoroughly reliable and lucidly intelligibleintroduction to logic as this topic is understood within the broad spectrumof analytic philosophy.

A standard textbook that is now in its thirdedition (with extensive revisions and additions by the author), this volumealso makes for useful reading by interested laypersons (who may also knowGrayling as the author of two excellent volumes in the _Past Masters_series, on Russell and Wittgenstein). It is highly recommended to anyoneseeking an accessible introduction to the field.

Grayling is alsorecommended as a master of what Brand Blanshard memorably called"philosophical style." The oracular pronouncements of the world'sNietzsches, Kierkegaards, Wittgensteins, and Ayn Rands usually get all theattention, but what really keeps the enterprise of philosophy going is themuch-underappreciated art and skill of writing fine expository prose. Inthat respect, this volume is a gem.

Very helpful and clear
I have found A.C.Grayling's _An Introduction to Philosophical Logic_ to be very helpful. It is a very non-technical and clear discussion of necessity, analyticity, a priority and other related concepts.

Sean Choihttp://www.freeyellow.com/members2/sean-choi/ ... Read more

Isbn: 0631199829
Sales Rank: 50761
Subjects:  1. Language and languages    2. Logic    3. Philosophy   

First-Order Logic
by Raymond M. Smullyan
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Paperback (30 January, 1995)
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Great stuff.
First, this isn't one of Smullyan's popular puzzle books- its a serious mathematics text. Second, don't use this as your first exposure to first-order logic (note the title doesnt say "Introduction to ...")- although logically self-contained, it requires some experience to appreciate what a neat little book this is.

It's not a general mathematical logic text- there is no model theory (beyond basic Skolem-Lowenheim), incompleteness, recursion theory, or set theory. It covers tableaux (this alone is worth the price of the book), Hilbert-style axiomatic systems (briefly), sequent systems, Gentzen's Hauptsatz and Extended Hauptsatz, Craig's and Beth's theorems, and more. But the heart of the book is completeness theorems, their proofs, and closely related material such as compactness and Herbrand-like theorems. Smullyan shows there are two main approaches to completeness (analytic vs. synthetic), breaks each into stages, provides nice abstracted formulations, and usually gives several different proofs of each result. The centerpiece is his "Fundamental Theorem of Quantification Theory", a theorem associating a truth-table tautology with every valid first-order sentence (check out the amazingly slick proof of completeness for the the Hilbert-style system that this provides). Similar constructions such as magic sets are also discussed. All this forms a much more extensive and illuminating look at completeness proofs than I've seen elsewhere.

The first-order logic used in the book has no equality and no function signs. There are few exercises, most of them simple. Smullyan writes clearly and with an appropriate amount of rigor (but its not as polished as his later books). Makes a great supplement to more general-purpose introductory mathematical logic books. If you haven't seen the tableau method yet buy this book immediately. Experienced readers will appreciate the sophisticated coverage of completeness proofs.

An Oddity But a Good-ity. Wait, that's terrible.
The reviewer from Illinois gave a very good characterization of Smullyan's style here:
"Smullyan has divorced logic from its roots: logics are simply recursively-defined sets of sentences and mappings, and that is that. No discussions, ala WvO Quine, on the history or linguistic difficulties of a concept, just definition and proof."
Readers familiar with Smullyan's enormous talent for popular exposition may be expecting the same herein: not so. This is very much for people who have attained what medical professionals call "mathematical maturity" (which is about as difficult to attain as zen, yet perhaps amounts to little more than the ability to read VCR instruction manuals). For example, the very first section is a wiz-bang treatment of trees (not the usual graph-theoretic ones), defined in the abstract/axiomatic fashion.
Of course, people who spend perhaps way too much of their time steeped in math are attracted to treatments of just this sort.
A structural characterization in terms of sets and mappings is much more meaningful, interesting, and aesthetically pleasing to those with these unusual inclinations (compulsions?) than a characterization framed significantly by historical motivation (please understand that I'm speaking roughly here). This is why I gave a positive review. A star was witheld for the selfish reason that I'm not sure I'll find much use for such an odd treatment of model theory, the topic for which I was seeking a more mainstream treatment when I purchased this. Regrets are nonetheless few: time spent reading Smullyan is never a waste.

a classic
I mainly bought this book because of the influence it has had on numerous modern-day logic texts. If you are unfamiliar with the tableaux method for structural proofs, then you will gain alot from reading this, as it provides a different perspective from the more popular Hilbert-system approach. Tableaux systems, of course, have been made popular because they are easy to program with a computer. Please see Gallier's "Logic for Computer Scientists" for more on this matter. ... Read more

Isbn: 0486683702
Sales Rank: 143966
Subjects:  1. First-order logic    2. Logic    3. Mathematics    4. Science/Mathematics    5. Mathematics / General   

The Laws of Thought (Great Books in Philosophy)
by George Boole, John Corcoran
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Boole's LT Breaks Ancient Mold , Founds Math Logic
.Aristotle and Boole are the two most original logicians before the era of modern logic. Aristotle presented the world's first system of logic.His system involves the standard three parts: first, a limited formalized predicational language; second, a formal method of step-by-step deductions for establishing validity of arguments having unlimited numbers of premisses; and third, an equally general method of countermodels for establishing invalidity. Boole's LAWS OF THOUGHT showed that logic is mathematical. Its stated aims were to refine, systematize, and complete the project started by Aristotle and, more ambitiously, to demonstrate the mathematical character of logic.His two-part system involves, first, a limited formalized equational language capable of expressing tautologies or "laws of thought", a breakthrough dramatically altering Aristotle's plan, and, second, a semi-formal method of derivation using equational reasoning totally absent from previous systematic logic.Boole's primary goals included construction of a method for generating solutions to sets of equations regarded as conditions on "unknowns", an unprecedented innovation with radical implications for the future development of logic. As for the third part of a system of logic, a method of establishing invalidity, surprisingly, Boole's book contains no systematic discussion of independence nor does it contain anything like a method of countermodels. Boole's LAWS OF THOUGHT set in motion forces that would lead to the ultimate fulfillment many of his goals including the establishment of mathematical logic.

NOW IS A GOOD TIME TO STUDY BOOLE.
The publication of The Laws of Thought in 1854 launched modern mathematical logic. The author George Boole (1815-1864) was already a celebrated mathematician specializing in what is known as analysis. If, as Aristotle (384-322 B.C.E.) tells us, we do not understand a thing until we see it growing from its beginning, then those who want to understand modern mathematical logic should begin with The Laws of Thought.There are many wonderful things about this book besides its historical importance. For one thing, the reader does not need to know any mathematical logic. There was none available to the audience for which it was written-even today a little basic algebra and a semester's worth of beginning logic is all that is required. For another thing, the book is exciting reading. The reader comes to feel through Boole's intense, serious, and sometimes labored writing that the birth of something very important is being witnessed. Of all the foundational writings concerning mathematical logic, this one is the most accessible to the nonexpert and it has the most to offer the nonexpert. The secondary literature on Boole is lively and growing, as can be seen from an excellent recent anthology (A BOOLE ANTHOLOGY by J.Gasser 2000) and a complete bibliography that is now available (Nambiar 2003).Boole's manuscripts on logic and philosophy, once nearly inaccessible, are now in print (Grattan-Guinness and Bornet 1997). This is a good time to start to study Boole.
It is true that Boole had written on logic before, but his earlier work did not attract much attention until after his reputation as a logician was established. Today he is known almost exclusively for his logic. In 1848 he published a short paper "The Calculus of Logic" (Boole 1848) and in 1847, at his own expense, he published a pamphlet The Mathematical Analysis of Logic (Boole1847). By the expression `mathematical analysis of logic' Boole did not mean to suggest that he was analyzing logic mathematically or that he was using mathematics to analyze logic. Rather his meaning was that he had found logic to be a new form of mathematics, not a form of philosophy as had been previously thought. More specifically, his point was that he had found logic to be a form of the branch of mathematics known as mathematical analysis, which includes algebra and calculus. (For a short description of this branch of mathematics, see the article "Mathematical Analysis" in the 1999 Cambridge Dictionary of Philosophy (Audi 1999, 540-41).
Although this book begins mathematical logic, it does not begin logical theory. The construction of logical theory begins, of course, with Aristotle whose logical writings were known and admired by Boole.In fact, Boole explicitly accepted Aristotle's logic as "a collection of scientific truths" (1854, 241) and he regarded himself as following in Aristotle's footsteps. He thought that he was supplying a unifying foundation for Aristotle's logic and that he was at the same time expanding the range of propositions and deductions that were formally treatable in logic. Boole thought that Aristotle's logic was "not a science but a collection of scientific truths, too incomplete to form a system of themselves, and not sufficiently fundamental to serve as the foundation upon which a perfect system may rest" (Boole 1854, 241). As has been pointed out by Grattan-Guinness (2003; Grattan-Guinness and Bornet 1997), in 1854 Boole was less impressed with Aristotle's achievement than he was in 1847. In "The mathematical analysis of logic" (Boole 1847) Aristotle's logic plays the leading role, but in The Laws of Thought (Boole 1854) it occupies only one chapter of the fifteen on logic. Even though Boole's view of Aristotle's achievement waned as Boole's own achievement evolved, Boole never found fault with anything that Aristotle did in logic, with Aristotle's positive doctrine. Boole's criticisms were all directed at what Aristotle did not do, with what Aristotle omitted doing.Aristotle was already fully aware that later logicians would criticize his omissions, but unfortunately he did not reveal what he thought those omissions might be (Aristotle, Sophistical Refutations, Ch. 34).
The new 2003 edition by Prometheus Books(ISBN 1-59102-089-1, Paper ...)contains an accessible 25-page introduction by a modern logician.

difficult, but a classic--and worth the effort.
Yes, this is the Boole of Boolean algebra.No, this is not a primer.But if you have any interest at all in intellectual history or where the tools of computer science came from, then you will find this book worth the effort. ... Read more

Isbn: 1591020891
Sales Rank: 514898
Subjects:  1. Epistemology    2. Logic    3. Logic, Symbolic and mathematic    4. Logic, Symbolic and mathematical    5. Mathematical And Symbolic Logic    6. Mathematics    7. Philosophy    8. Probabilities    9. Thought and thinking   

Principia Mathematica
by Alfred North Whitehead
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Hardcover (01 June, 1962)
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Editorial Review

Could it be true that Whitehead and Russell's Principia Mathematica is the most influential book written in the 20th century?Ask any mathematician or philosopher--or anyone who understands the impact these fields have had on modern thinking--and you'll get a short answer: yes. Their goal, to set mathematics on a firm logical foundation, was revolutionary, and their tools and rigor continue to influence modern professionals. Using Peano's symbolic logic, they formalized axioms and produced theorems (including the famous "1 + 1 = 2") in orderings, continuous functions, and other areas of mathematics.

Although the Principia is far from comprehensive, Whitehead and Russell's method and program captivate their readers. The audacity to hope to formalize all of mathematics logically was inspirational and helped to give great boosts to math and logical philosophy. Though Gödel proved in 1931 that any such program is doomed to incompleteness, the tools found in and developed from the three volumes helped build the atomic bomb and the Internet. It may not be summer-vacation reading (for most), but Principia Mathematica will reward the dedicated student with a deeper understanding of how we got here. --Rob Lightner ... Read more

Reviews (15)

ludicrously quixotic work
I have not read this book.I tried, having been fascinated by logic and mathematics since high school, but it has absolutely nothing to offer most people.in fact I find it hard to believe anyone has ever read this book.The 4 and 5 star reviews on this page should be taken as evidence there are some people out there with very different taste from mine, and I bet yours. In fact I have difficulty believing they are serious.

I think only a fanatic could enjoy reading this book, certainly not a budding mathematician.If you are attracted by a book that proves 1+1 = 2 somewhere after 100 pages, this is the book for you!

I admit I have been surprized before at what some people find interesting, but the idea that anyone would pay 5 or 6 hundred dollars for the set!the publishers seem to me to be sniffing glue.(I have a PhD in mathematics, a mathematical library costing thousands of dollars, and tried to read this work at Harvard as a young math student.)

To call this book influential, is to me really ridiculous, since I suspect few people have even looked at it in the last half of the 20th century, nor would want to do so at any length, in my opinion.

But don't take my word for it, go to your scientific library and check it out for yourself.You might like it, but I seriously doubt it.I did not intend to review this book, but some of the reviews here really defy belief.I could not let them pass without comment.

One must assume those reviewers here are serious who praise it, but I suggest almost no mathematics student need give it more than a passing look.The review that stated something like "if you do not already know you want this book, then you do not"is pretty accurate.

OK, a quick re reading of reviews here shows many of them say truthfully that this book is only appropriate fora very small group of readers.However I would suggest that group does not even include most mathematicians.The ones who like it are apparently philosophers, and some are the sort who resort to calling people stupid who disagree with them.

A Hallmark in the History of Mathematics and Philosophy.

Much nonsense has been said on the subject of the importance of Principia Mathematica by people ignorant of the history of mathematics and logic. Principia Mathematica together with Frege's Grundgesetze der Arithmetik is the book which gives birth to modern logic. It is absurd to assume that Russell and Whitehead intended their axiomatization of mathematics as a guide to learn the subject, as one reviewer thinks, in fact what they tried to show was that the whole of mathematics could be deduced from a small stock of premises and inference rules and using only notions of first order logic and set theory. In doing this they were following a trend in mathematical thought in the late XIX century, that of introducing more rigour to the subject, they intended to do this by demonstrating that the derivation of mathematics needed only logic (think of Weierstrass, Dedekind, Cantor, Frege). From a philosophical standpoint they also did it to rebut the intuitionist views of Kant and Poincare as well as certain opinions regarding truth coming from British Idealism (think of Bradley). Of course there are much more rigurous treatises on logic, but they would have been impossible without PM because PM was the first thorough treatment of this subject-matter and, indeed, the first book to use the modern day notation. As another reviewer pointed out, Godel's proof would've been impossible without Principia; someone first needed to show that you could reduce mathematics to logic to a great extent (Russell and Whitehead were aware that their treatment used certain axioms unprovable within the system, like the axiom of infinity, but were hopeful a solution would be found, Godel found it, it was a negative solution, there could be no complete system PM like). This book together with Frege's gave birth to modern logic, it gave a tremendous boost to research in set theory, it influenced the presentation of modern mathematics to the extent that every student has to learn about sets at the beginning of a mathematics course, it showed also the scope of the deductive powers of logic and axiomatic systems which made possible the revolution in computers and AI. It developed an influential and responsive philosophy of mathematics, perhaps the most influential of the XX century. In it Russell's superb theory of descriptions, a cornerstone in logic and philosophy, is applied with success. This theory is tremendously important in logic through its use of quantification to break up much more complex expressions revealing their true logical form. In philosophy it provided a theory which would prove immensely useful and important in epistemology, metaphysics and the philosophy of language. Russell's paradox ( regarding those sets of sets which are not members of themselves) is disposed through ramified type-theory, now obsolete in logic (though not in computer science), because, thanks to it, other ways to avoid the paradox were developed, think of Zermelo-Fraenkl or Ramsey's simple type theory. Carnap, Hilbert, Weiner, Ramsey, Quine, Wittgenstein, Turing, Tarski, Godel etc were, as thinkers, tremendously influenced by it. In short, this work is one of the greatest achievements in the history of thought, its importance for mathematics, logic, philosophy (linguistics also) and computer science is first rate, suffice to say that none of these studies would be as advanced as they are now, or as complex, or in the same direction were it not for Russell and Whitehead's groundbreaking scientific work. Of course, like Newton's Philosophia Naturalis Principia Mathematica it is now, because the subjects it initiated are today tremendously advanced, mostly of historical interest, however, for the philosophers at least, Russell's introduction still holds great philosophical interest and rigourous arguments helpful in the contemporary debate. For more details check out Ivor Grattan Guiness's great works on the history of mathematics, logic and set theory.

A spoiler!
The denouement in which we discover that the Vicarwas murdered by the Butler, in the Conservatory, with a Candlestick was weak. But the sex scenes, on pages 183 - 879 were the most sensitive yet erotic that I have ever read (except for page 1334 of the "Catalogue of Insects, Arachnids and Marsupialsvol XXIV").

Top work, Whitehead and Russell! I eagerly await volume 4. ... Read more

Isbn: 052106791X
Subjects:  1. General    2. Logic    3. Mathematics    4. Science/Mathematics    5. Mathematical foundations    6. Philosophy of mathematics   

On Formally Undecidable Propositions of Principia Mathematica and Related Systems
by Kurt Gödel
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Paperback (01 April, 1992)
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For the history buff
There are at many academic pathways to Godel's Incompleteness theorems: philosophy, computer science, and mathematics, to name a few. What does this presentation of the results - among the many others available - have to offer the wide readership it is likely to attract?

To begin with, it is a reprint of the first English translation of Godel's original published Incompleteness proofs. It's slim and it's cheap, and includes a good detailed and clear "introduction" which is really a less formal paraphrase of the results with some additional information and clarification provided. The real value, though, is in the details of the proofs. Godel's original proofs - unlike many of the easier, more modern equivalents - are constructive. It was Godel's intention to formulate a universally acceptable proof of the inadequacy of Hilbert's program (which sought to defend mathematics as science of consistent formal systems) and of Russell's and Whitehead's logistic philosophy in Principia Mathematica (which sought to reduce all of mathematics to a theory within Russell's formal logic). With his constructive proofs, Godel produced a document of historical relevance to each of the major early 20th century philosophies of math: Formalism, Logicism, and Intuitionism; criticising the first two positions with an argument compatible with the third. Within the same proofs, he also provides the basic definition of the (primitive) recursive functions. In this sense - in containing the first published results of recursive function theory - Godel's original proofs have significance in the history of computer science and mathematics proper. These original arguments, then, have a clear place history of mathematical logic beyond the simple statement of their results, and are well worth reading for their historical significance.

As has been mentioned by other reviewers, there is a lot of typos in the text, some of which aren't good for the integrity of the argument. But, to be honest, this is hardly a text to learn from - it's simply contains too many fine details. Rather, it would make a valuable addition to the library of anyone with sufficient interest in the history of formal logic to want a copy of the original argument, and who is mathematically competent enough to correct the numerous errors as they read through it. 5 stars for the combination of price and content - though an errata sheet from Dover would certainly be an improvement.

In addition to this publication, there are numerous good accounts of the Incompleteness theorems:

An excellent informal account can be found in "Godel's Proof" by Nagel and Newman. This is about right for the reader who wants to know the argument without having to worry about all of the formal technicalities. As technical accounts go, Smullyan's "Godel's Incompleteness theorem's" isn't bad, and it proves a slightly more comprehensive version of the results. Smullyan was an extraordinary expositor of mathematical logic: his account is both conceptually clear and insightful, and the theorems are approached in a unique stepwise fashion, building up to the main theorems. An interesting (though *difficult*) account of Incompleteness can be found in the "Syntax" chapter of Quine's "Mathematical Logic". This is remarkable for proceeding by simple syntactic considerations, without the use of any basic number theory. Lastly, there are countless good proofs of the Incompleteness theorems to be found in the numerous good introductions to recursive function theory (Rogers or Cutland are O.K.). For the mathematically inclined this approach is good, as it offers immediate access into some of the more interesting undecidability results that followed Godel's results.

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.

From the horse's mouth, 'le text'
Speaking not as a math specialist but one disposed to read a number of the popular explications of Godel's famous proof I can say that it was Godel's original text that did it for me. The reason is that it is the proof and not a lot of verbiage about the proof. And it is short and sweet. One problem is that the more common Turing Machine approach is actually 'easier', where Godel's approach is that of recursive functions which are more obscure, or at least less often discussed. If you can sort of glare at the recursive function issue and proceed with the basics of the proof it will stand out suddenly better than many of the popularizations. At least give it a try. ... Read more

Isbn: 0486669807
Sales Rank: 69660
Subjects:  1. Godel's theorem    2. Gèodel's theorem    3. Logic    4. Mathematics    5. Science/Mathematics    6. Mathematics / General   

Godel's Proof
by Ernest Nagel, James R. Newman, Douglas R. Hofstadter
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Editorial Review

Gödel's incompleteness theorem--which showed that any robust mathematical system contains statements that are true yet unprovable within the system--is an anomaly in 20th-century mathematics. Its conclusions are as strange as they are profound, but, unlike other recent theorems of comparable importance, grasping the main steps of the proof requires little more than high school algebra and a bit of patience. Ernest Nagel and James Newman's original text was one of the first (and best) to bring Gödel's ideas to a mass audience. With brevity and clarity, the volume described the historical context that made Gödel's theorem so paradigm-shattering. Where the first edition fell down, however, was in the guts of the proof itself; the brevity that served so well in defining the problem made their rendering of Gödel's solution so dense as to be nearly indigestible.

This reissuance of Nagel and Newman's classic has been vastly improved by the deft editing of Douglas Hofstadter, a protégé of Nagel's and himself a popularizer of Gödel's work. In the second edition, Hofstadter reworks significant sections of the book, clarifying and correcting here, adding necessary detail there. In the few instances in which his writing diverges from the spirit of the original, it is to emphasize the interplay between formal mathematical deduction and meta-mathematical reasoning--a subject explored in greater depth in Hofstadter's other delightful writings. --Clark Williams-Derry ... Read more

Reviews (26)

This Book Is One of the Reasons You Should Want to Never Die
I hope my review title was of a sufficient degree of hyperbole -- superlatives, after all, can lose their power if you run across too many of them.Anyway, the editorial review is entirely spot on and has more subject-matter content than my review; so see the editorial review to find what the book is about. But here I'll tell you how you will be rewarded by reading this book.Whether you came across this book quite purposefully -- and therefore know about the treat it ought to be -- or are a complete novice to the whole topic, I guarantee this book will fill you with treats.Though the last bit of philosophy of Hofstadter's new foreward I'm not sure I agree with, much of the rest of the foreward is itself filled with treats -- some of the same kind as the text proper and some of a quite different (poetic/sentimental) nature.Beyond the individual treats you will find sprinkled throughout, the book accomplishes its objectives admirably and one of those goals is making it all accessible to the mathematical/logic novice.In the cognitive arena, this book is one of the things that belongs to the set of things that you cannot conceive of ever permanently separating yourself from -- hence you have to live forever.(There are non-cognitive things, e.g. certain music instances/performances, that belong to the aforementioned set, but from the cognitive realm, this book absolutely belongs to the 'I gotta live forever because of this' set.)

Godel's Proof:A Precursor of a Modern Creation Theory
Godel's proof followed the discovery in the 1920's by linguists that 'empirical data are primarily symbolic.'This discovery distinguished 'signs' from 'symbols'.Signs are used by lower animals in their sign language whereas symbols are used only by humans to build rational human knowledge.Essentially, Godel's proof supports the panentheistic theory of God and the endless world that He creates. In such an endless world, human knowledge cannot be completed and immortals are thus reincarnated endlessly.

It's like "Brief History of Time" in Mathematics
It gives me the same feeling after reading "Brief History of Time". They both explain some very fundamental thing in Science in layman's term. But the difference from "Brief History of Time" is that I can fully understand what the authors are trying to convey.

The footnotes are very helpful in clarifing the terms and concepts used in the main body. I would suggest you not to skip those valuable footnotes.

The whole book is not hard to understand, although you may have trouble reading Section 7: Godel's Proofs. But just go slowly (don't pause in the middle, otherwise you may forget what a particular symbol means) and everything is fine. This Section is the most exciting part of the whole book.

As a Math Grad, this book makes clear to me some concepts that I was not so sure before. One of these corrected concepts is: Godel only ruled out the possibility of getting a proof of consistency within arithmetic. So there is still a hope (though quite unlikely) of finding the proof not representable in arithmetic. See the last section of the book for details. ... Read more

Isbn: 0814758169
Subjects:  1. General    2. Godel's theorem    3. Gèodel's theorem    4. Logic    5. Mathematical And Symbolic Logic    6. Mathematics    7. Science/Mathematics   

Godel's Incompleteness Theorems (Oxford Logic Guides, No 19)
by Raymond M. Smullyan
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Reviews (2)

Finally -- Straight Talk About Incompleteness!
Well. This is the book. Read this instead of, or before you read Goedel�s paper. Within 20 pages you will know the �trick� that Goedel used. It�s a beauty, but it is far easier to see it under Smullyan�s tutelage than by coming to the classic paper cold, since Goedel uses a more difficult scheme to achieve his ends. Much work has been done since 1931, and we get the benefit of the stripping-down to essentials that such as Tarski (and Smullyan himself) have contributed.

The book has much of interest to those who wish to pursue the subject of the incompleteness and/or consistency of mathematics, or to come at Goedel from a number of angles. For me, though, the first 3 chapters were enough. I just wanted to find out how K.G. did what he did. Now I know, and I know where to go if I need even more.

The exercises are helpful to keep you on track and test your understanding. They also contribute materially to the exposition. A stumbling-block for many readers will be the extremely abstract nature of the discussion, and the new notations and definitions that constantly come at one. Viewing numbers as strings and strings as numbers (and knowing when to switch from one view to another) will be confusing at first. This is the hard part: what Goedel did, in essence, is demonstrate that one can view proofs in two ways � as numbers, and as strings of characters. As in viewing an optical illusion, it is sometimes tough to hold the proper picture in mind.

Smullyan�s book �First-Order Logic� is enough preparation for this work. One must here, even more than there, keep straight the difference between the �proofs� that are part of the subject matter (and so are strings of characters), and the proofs we go through that verify facts about these strings. Before we started reading this book, of course, we had some informal sense that we were going to prove something about proofs. What we are REALLY doing, though, is proving something about �proofs�. You get the picture. Goedel must have been a lot of fun at parties.

Mainline Incompleteness with this Book!
I highly recommend this title because it supplys all the necessary proofs for a nuts and bolts understanding of incompleteness, including incompleteness proofs for Peano arithmetic and the unprovability ofconsistency.

This title is a difficult read but the only prerequisite isa familiarity of first-order logic equivalent to a one semester collegecourse.

A lot of the proofs are based on new material and are easier tounderstand than the original work by KG.

An added benefit is theexercises.They are not impossible and aid in one's understanding.

This book is well worth the work in demands. ... Read more

Isbn: 0195046722
Sales Rank: 362476
Subjects:  1. Godel's theorem    2. Gèodel's theorem    3. Logic    4. Mathematics    5. Philosophy Of Mathematics    6. Science/Mathematics    7. Mathematical logic   

Forever Undecided: A Puzzle Guide to Godel
by Raymond M. Smullyan
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Paperback (01 June, 1988)
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Reviews (3)

Where is an example?
In the every day practice of mathematics or viewing the works of others, I have never ran across or derived a self-referential statement of the kind that Goedel used in his proof of the incompleteness theorems. The appearance of these kinds of statements takes place only in the context of mathematical logic, and their construction is is somewhat artificial, involving the use of 'diagonalization'. It is for this reason that I don't find the incompleteness theorems in any way troubling for the "truth" of mathematics. If they kept arising in the everyday practice of mathematics, this would lend support to the incompleteness theorems. As such though there is no "empirical" support for them, and until they do arise they can be safely ignored.

The results of Goedel have been used to cast doubt on the "foundations" of mathematics and the "strong AI" problem. But they have also been used to support "strong AI", as it is felt that the existence of self-referential statements are an indication that a machine is conscious. All of these arguments are interesting, but they have yet to help in the practice of mathematics or in the construction of intelligent machines. In fact, too long an emphasis on these results has probably retarded the advancement of artificial intelligence research. The incompleteness theorems though have stimulated research in the field of 'automatic theorem proving' and in this respect they can be said to have had some value.

This book gives an overview of Goedel's incompleteness theorems and its corollaries from a "semi-popular" point of view, meaning that readers are expected to have some background in elementary logic as well as philosophy, in order to appreciate the contents. The author is eloquent and enthusiatic throughout the book, and this serves to make the book more palatable for the absolute beginner. It would be unfortunate maybe that readers will begin to doubt the "truth" of mathematics after reading this book, but that doubt could also be viewed as a virtue, in that it would motivate further thinking and research. But again, to this date there has been no self-referential statement that has appeared in the everyday practice of mathematics....that I know of anyway.

Brain candy in a logical form
Do you believe that you know what you believe? Is it possible to believe only those things that are true? These questions are but two of the many addressed in this book by Professor Smullyan, and the answers will surpriseyou!
With most of the material in the form of puzzles, and nearly allof them solved, it is possible for a logical novice to read and understandthe book. However, it is recommended that you find another source forexplanations of the standard logical operators and read it first. Eventhen, you should not be surprised to find yourselfrereading a solutionthinking that the answer simply cannot be correct. But keep trying, aseventually you will begin to understand, and come to appreciate some of theunresolvable consequences of mathematics, the most precise of disciplines.For while Professor Smullyan may not be logically consistent, he isconsistently logical. (This is of course a bad pun, but it is no worse thansome of the groaners in the book.)
Finally, as the following exampleshows:

"The following dialogue ensues between a student and histheology professor:

Student: If I believe that God exists, thenI will also believe that I will be saved?
Professor: It that is true,then God exists.
Student: If I believe that God exists, then will I besaved?
Professor: If God exists, then that is true.


Provethat if the professor is accurate and if the student believes theprofessor, then God must exist and the student will be saved"

You may also be given insight into one of the most fundamental questionsregarding human existence.

Published in Journal of RecreationalMathematics, reprinted with permission.

Unique method uses puzzles to explain Go:del's theorem
This is a great book for anyone wanting to understand Go:del's theorem,which basically proves that logic is inherently flawed.The book is veryreadable.Just make sure that you don't let your head explode. ... Read more

Isbn: 0192821962
Sales Rank: 1942613

Gödel, Escher, Bach: An Eternal Golden Braid
by Douglas R. Hofstadter
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Editorial 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

Reviews (203)

The science of self-reference and repetition
What do Godel, Escher and Bach have in common?They're demonstrations of self-referencing paradoxical behavior in 3 different fields: Music, Visual Art and Mathematics.If you can follow the ideas presented in Hofstadter's career building Pulitzer Prize winning work, consider yourself mentally stretched.

Godel's incompleteness theorum refers to itself in proving itself.(If it's true, it can't be.If it can't be true, it is)Escher's art is self-referencing - follow the waterfall down until you realize you are back at the top.Follow the stairs up and around a box, and get back to where you started.Same if you go down.Bach's music rises note by note until it's back where it started.

Attaching the same abstract idea behind each of these ideas is a fantastic synthesis.For some it will seem obvious.Others may consider it nonsense.I consider myself lucky to be in the group that considers themselves stretched.I don't get all of it, but enough to be glad I read it.

One of the great popular science books

This book richly deserved its Pulitzer prize.  It's one of the great pieces of popularscience writing and it's  remarkable that it has lost solittlein 25 years.  You learn the intricacies of Bach's music, of Godel's Incompleteness  theorem, Escher's drawings and DNA replication.Although his purpose seems to have been very general, with everythingtied together looselyas ideas for his future work in artificalintelligence, one could view this as abook about some parts ofcognitive psychology--  how the templates we inherit inour DNAcreate and interpret sounds  and images and theorems and how theseseemto relate to one another via the concepts of recursiveness,tangled hierarchies,and incompleteness.  It is mostly  thelack of these inference engines thataccounts for the fact thatto this day AI has still not been able to make a machine withthe brains of an ant(ie, go out into an arbitrarily complex world,recognize and deal with friend and foe, eat, reproduce, and stayout of the sunand rain and keep doing it for years).
Hisfollowup the next year with Daniel Dennet--`The Mind's I` complementsthis book nicely(see my review).

So one could say that thisis really a psychology text.  It  is about human  behaviorand reasoning-about why we thinkand act the way we do.  But(likeall such discussions until recently) none ofthe explanationsare really explanations.  Nobody at that time had muchunderstandingof  the mental mechanisms involved.  Like most 'explanations` ofbehavior, the comments here are often more interesting forwhat kinds of thingshe tries to use (and omits) than for theactual content.  As with all reasoningand explaining, art, math,music, etc, one now wants to know which of the brainsinferenceengines are activated.  This book and most books and AI  research were largely oblivious to such explanations until quite recently.

 Cognitiveand evolutionary psychology are still not evolvedenough to provide fullexplanations but an interesting start hasbeen made.  Boyer's  `ReligionExplained` is a good place tosee what a modern scientific explanation of  humanbehavior lookslike,and works on art, music and math are sure to appear soon. Pinker's`How the mind  Works` is a  good general survey. They do not explainall of intelligence or thinking but give an idea of how to start. See severalof the recent  texts(ie, 2004 onwards) with evolutionarypsychology in the titleor the web for further info.
Wenow recognize that the bases for art, music, math, philosophy,psychology, sociology, language and religion are found in theautomatic functioning of  templates or inference engines. This is why we canexpect similarities and puzzles and inconsistenciesor incompleteness and often,dead ends. The brain has no generalintelligence but numerous specializedmodules, each  of whichworks on certain aspects of  some problem and theresults arethen added, resulting in the feelings which lead to behavior. Hofstadter, like  everyone, can only generate or recognize explanationsthat areconsistent with the operations of his own inference engines,which  were evolved to deal with such things as resource accumulation,coalitions in small groups, social exchanges and the evaluationof the intentions of other persons. It is amazing they can producephilosophy and science, and not surprising that figuring outhow  they themselves work together to produce consciousness orchoice or spirituality is way beyond reach.

Hedoes not try to deal with the endlessly vexing issue of whetherthese correlations are out there in the world or in here in themind. Yes, weuse our templates, but why did we evolve  thoseand was there anotherpossibility?   Some will say this willall become clear when psychology andgenetics  are sufficientlyadvanced, while others say the same of physics andmathematicsor programming. And, did they all evolve from some  prototype enginein a precambrian invertebrate or did they come much later andfrom many sources?

It occurred to me that some of the mostcomplex products of human reasoning --superstring  theory andthe associated math--are recursive( in somenontrivial  sense)to quantum field theory, subatomic particle behavior and  theentire universe. Physics unites many areas of the most advanced  mathbecauseit needs  self consistent structures, but since we know math is logicallyproven to be  inescapably incomplete and math is a product of the mind, itseems reasonable  that there must be a sense in which the mind is incompletealso. We expectsince they use math that computers must be incomplete. We knowthatTuring's halting theorem for computation(we can not discover inadvancewhen a computer will stop) is logically equivalent toGodel's incompletenesstheorem.  It might follow that physicswill be incomplete as well and there willbe many physical lawsor phenomena that will never be compatible with orderivable fromthe others.  Or perhaps physics can be complete andselfconsistentin one universe but not in others

Just as he did not go veryfar into the many realms of psychology or  physics, neither did he venture farinto philosophy.  Perhaps the book could havebenefited greatly from anunderstanding of the infinitely subtlerelationships between language, thoughtand reality.  An acquaintance with  Wittgenstein would have helped immensely,especially his'Lectures on  the Foundations of Mathematics: Cambridge, 1939'edited  by Cora Diamond(1990).  It is better to get this onerather than theearlier `Remarks on the Foundations of Mathematics, Vol. 1` edited by RushRhees( as they are based on different setsof notes if you  are really  into ityou should get both).
AlthoughI've never seen anyone say so, W can be regarded as a pioneer incognitive psychology.  All of his  research was thought experimentsandintrospection  into the relations between  language, thoughtand reality. Perhaps  nobody ever approached his talent for describing the mind at work. The point is that Hofstadter istrying to  understand how the mind  works as apreliminary tomaking programs that work the same way(or at least get similarresults)so anyone who is interested in this book(or nearly any area ofphilosophy,language, psychology, or  intellectual  discourse) can look intoWwith great profit(but  be forewarned W may seem very  shallow,but if you jumpin you may never stop swimming)!
Just afterreading  this book I happened  to read  Wittgensteins ``Cultureand  Value``(published the  same  year(1980), but written decadesearlier), and,though it's his least interesting  book, I pickedout a few comments  that maybe regarded  as pertinent to muchof  this book and of course to a large part ofmodern intellectuallife.

 ``There  is no religious denomination  in whichthemisuse of metaphysical  expressions has been responsible for somuch sin asit has in mathematics.``  

 ``People  sayagain  and again that philosophydoesn't really progress, that  we are  still occupied with the samephilosophical problemsas were the Greeks.  But the people who say this don'tunderstandwhy is has to be so.  It is because our language has remained thesame  and keeps seducing us into asking the same  questions.  As long  as therecontinues to be a verb 'to be'  that looks as if it  functions  in the same wasas 'to eat' and 'todrink',  as long as we still  have  the adjectives'identical','true', 'false', 'possible', as long as we continue  totalk of  ariver of time, of an expanse  of space, etc., etc.,people  will keep stumblingover the same  puzzling  difficultiesand  find themselves staring at something which no explanation seems  capable of clearing up.  And what's more, thissatisfiesa longing for the transcendent, because, insofar as people think  they can see `the limits of  human understanding',  they believe of coursethat  they can see beyond  these.``
 
Wheneverone gets philosophical itis relevant to take a step back fromtime to time and see just what is reallygoing on.  Hofstadteris not a  philospher and he does not seem to take thatstep.  Incompleteness  seems well defined in math but what about elsewhere?  In what sense is music or  art or biology incomplete?  And exactly what willcount as a tangled  hierarchy, and recursiveness orself referencing in suchdifferent realms(and as W would say,such different language games)?   Its notreally so clear that the  recursiveness in art, music, biology and math are thesame sort of thing at all an, insofar as they are, what exactly thatmeans.What should count as  ``same` here?

H doesnot address these questions in any depth but one might  find them by far the most interesting theme of the book.  We are tantalized at  theseeming connections but do they mean anything?  Do they go to the core of  ourbeing(how the mind works)? Are they merelythe result of the use of  some of thesame templates by art,math, and music?  Do they relate  to the molecularstructureof  matter or to particle physics and  string theory?  Is ituseful toextend these analogies(or are  they homologies?)almostendlessly further intophilosophy, language,  psychology, biology(e.g.,not only the recursive natureof DNA,  RNA  and proteins, butthe many levels of feedback in the nucleus, cytoplasm,  intercellular,interorgan, intracerebral, exchange  of chemicals andgenes betweennucleus, mitochondria and chloroplasts  as well as with thebacteriaand  viruses that wander in out of  our bodies into other bodiesandother organisms  happily picking up and dropping off genesas they go--tangled,recursive,  hierarchical  and in some sense,incomplete).

Or, to take it further, one  might  findyet more connections between artand music, math and biology, computer programs, physics and chemistry andbiochemistry and add such  dimensions as color, geometric shapes, measurements,self organizing abilities, chaos, and other temporal, spatial or purelypsychological ways(emotions,  sensations, dreams etc).  There are many books inart, music,  math, biology,  psychology,physics and chemistry that alreadytouch upon these  themes butI think the most progress is being made incognitive psychology.The brain is highly recursive in many ways.   We conversewithourselves  internally and many times externally. The  schizophreniccommonly hears voices,  but they rarely say nice  things.

Oneis reminded ofthe cut-ups that William Burroughs  and ByronGysin  created.  They cut up booksor even newspapers  andstuck them back together  randomly.  There was usuallysome perverse kind of logic to the result showing  the hidden threads indiscourse.  Burroughs later did the same thing with films,with similar results.

Of course pursuing hidden relationships between seemingly unconnectedthings  quickly leads to numerology, pyrimidology and madness. One can findcodes or algorithms toconnect or derive anything from anything. Hofstadter doesnotgo  into this here but he mentions it in his next book, The Minds I(1981). I am reminded of string theory which has math so powerful it can probablyexplain any possible  universe and so it is verysuspect as  an explanation ofours. 

He suggest that incompleteness,tangled hierarchies etc may beresponsible  for the emergenceof higher phenomena which do not exist and cannotbe explainedat lower levels(eg, consciousness and in fact, everything)and seemsto be something of a holist( but in other places he seems  clearly behavioristor reductionist). You might say he is suggesting we look for the  explanationof emergence in the bizarre phenomena of the foundations of math,  rather thanin those in the foundations of physics. Given a universe where life ispossible, is it  notinevitably full of  recursiveness, tangled hierarchies,incompleteness etc. 

As H is well aware, Zen can be regarded as using  theseaspects of the  world to trick the mind into stopping-- at which point allrelationships become  irrelevant. However hewas  just starting in Zen at thetime so he does not go  veryfar with  it.  For those who want to go into itfurther, probablythe best and most readable recent books on Zen  are thevariousvolumes  by Osho. 

Its a pity he has not been  able to writeanother  book like this as there is  now a vast amount  ofinformation available about DNA and RNA, the inflationary theory of the  universe, quantumtheory, and the beautiful fusionof string  theory  and advanced math, whichcould greatly extendand  amplifiy  the themes of recursion, tangledness,hierarchies,and  incompleteness.  One could  make a good case that the basicstructure  of the universe has these  properties at its smallestandlargest scales.   Both quantum physics and string theoryhave  complex  sets oflaws  that appear tangled,nested, hierarchical and incomplete--  and so far noone can  unifythem, unless one  accepts string  theory on faith-but nobody cansolve string theory and physics, like mathematics  whichit mirrors (orexpresses?)may remain forever incomplete( Kaku's`Hyperspace` gives a summary upto 1994-see my review).

Itwas one of the few times he stuck  his neck outwhen he predictedthat  the future of AI would involve  recursive programs butareneural nets and fuzzy  logic recursive?   And do these relateat all to howthe brain works or to anything Wittgenstein hasto say about language andreality?  The diligent might want tolook at B.A.  Worthington's book--`SelfConsciousness and SelfReferencing:an interpretation of  Wittgenstein's Tractatus`.

Sincethis book appeared, mathematician Gregory Chaitin has mademajorextensions of incompleteness and alsodeveloped the amazing omeganumberdefining the limits of math(his  popular and tech bookseasy to find on the net and  his most recent  on omega-- Meta Math --appeared in 2005). 

Somereaders will find interesting avaguely similar book ``Labyrinth``  by PeterPesic (2000)  whichuses the  form of the triple fugue to link symbolicmathematicsto the  pursuit  of science.
He does not mention that Godelshowed that (if  the universe is rotating) time  travel is possible(ie,time isrecursive), nor that all theories of physics,  includingquantum  field theory,remain incomplete.  Also the highestproduct of  the  mind--Superstring Theoryis recursive to quantumfield theory and  the  behavior of particles and theentireuniverse. A good bit  of this was known in 1980 and Hofstadterwas aphysicist so it''s surprising it does not appear here. We know that the mostadvanced  physics and the most advancedmath fuse in superstring theory  andthis seems amazingly holistic. Physics must have the  self  consistentstructures of mathematicsbut as math is inescapably  incomplete  does it followthatphysics is also? And worse, as  math is a product of the mind is not themind forever incomplete  too?  Does this mean therewill always be  physicallaws or phenomena  that are not deriveablefrom(compatible with) the others orcan  physics be completeand self consistent in one universe(however we delimitor describethat) but inconsistent in others?  All these questions seem likelyto go on forever. 
 

GEB, garbage, pseudo-science??
"Gödel's theorem concerns a problem in "formal logic" and has nothing to do with human-cogno-something." says a reviewer, and concludes that H's treatment of G's theorem is "complete garbage"?!
Firstly, I have a problem with people who use insulting labels.
Secondly, it's even worse when they motivate this not with actual arguments, but by stating their (perceived) scientific status INSTEAD ("I am convinced anyone with a degree of mathematics will agree with me.") Truth be said, you can get a degree in math without even coming close to G's incompleteness theorem. Not only do I not agree with the reviewer, but I happen to think H's presentation is the best out there.
Thirdly, G's theorem is indeed one of formal logic. But to say that it has "nothing to do with human-cogno-something" is to beg the question against the very book you're reviewing. One of the main points of GEB is to explain how G's theorem could be relevant to cognitive science. The reviewer effectively disregards all the arguments presented in the book, and simply STATES that there is no connection :)
Finally, about the "alchemy and pseudo-science" part: GEB is not a science book (and is not presented as one). True, the author has a very distinguished scientific career, but GEB is a book written to popularize science, not to present new results to peers. Of course some ideas are far-fetched, poetic, speculative - that's exactly what I would expect from a book with the title of "GEB" :)

In conclusion, this type of review is pretty useless - I like critical, even negative reviews, but let them have some meat, not only poor style and truncated understanding :) ... Read more

Isbn: 0465026567
Subjects:  1. 1685-1750    2. Artificial Intelligence    3. Artificial Intelligence - General    4. Bach, Johann Sebastian,    5. General    6. Logic    7. Metamathematics    8. Philosophy    9. Speculative Philosophy    10. Symmetry   

Symbolic Logic and the Game of Logic
by Lewis Carroll
Average Customer Review:
Paperback (01 June, 1958)
list price: $11.95 -- our price: $9.56
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Editorial Review

Yes, this is the Lewis Carroll who wrote Alice in Wonderland, and these two works show the same quirky humor. Here you see Carroll the mathematician at his playful best. Don't let the title of the first work mislead you--this isn't about modern symbolic logic but about ways of expressing classical logic with symbols. It's loaded with amusing problems to delight any mathematical puzzler. In the second work he turns logic into a game played with diagrams and colored counters, giving you hundreds of challenging and witty syllogisms to solve. Great mind-stretching fun. ... Read more

Reviews (3)

Content 5 Stars, Format 1 Star - Averages out
Lewis Carroll's contents timeless and fundamental and should be interesting to anybody with interest in Symbolic Logics or even simple logical problem solving.

However, I am kicking myself for not looking at the published year and having been fooled by the snazzy cover into assuming that the presentation would be of contemporary standards. It is ditto reproduction of fonts and pagination from the original, which a contemporary reader would have a tough time keeping up. Just look at the Table of Contents befor you buy.

Reminds us that math can be FUN
Math is fun, but the rhetoric of most 'taught' (probably an overstatement) math (and, by extension, logic) is so incredibly dry that the forest is rarely seen for the bark on the trees. But here Carroll, with tongue lodgedfirmly in cheek, turns the rhetoric (and by extension, the way we thinkabout math problems) on its ear, and the result is an often incrediblyfunny approach to math and logic problems which stays with you andultimately worms its way into your quotidian. I'll also say that, as anatrociously poor student in high school, this book allowed me to ace theSATs, and then ten years later the GREs.

great book to teach logic to children
This book by Lewis Carroll is a wonderful source to learn the basics of logic in a funny and natural way. It can be used as a self-study guide or as a manual for educators teaching elements of logic to schoolchildren. It is very clear and consequent and gives the basic idea of propositionsand syllogisms. The theory is framed in an unusual game that makes it muchmore understandable.As always Carroll's examples are a little bitabsurdic but this is exactly what makes them humorous, attractive andinvolving. The book is also a great brain teaser for readers of all ages. Unfortunately it is not as well known as Alice in Wonderland but it hasbeen translated into many foreign languages. I widely used the Russaintranslation when teaching logic to schoolchildren in St. Petersburg,Russia. Currently being a doctoral student in the States I try to introduceit to my colleagues. ... Read more

Isbn: 0486204928
Subjects:  1. General    2. Logic    3. Logic, Symbolic and mathematical    4. Mathematical recreations    5. Puzzles    6. Science    7. Games / Puzzles   

Lewis Carroll: The Complete, Fully Illustrated Works, Deluxe Edition (Literary Classics)
by LEWIS CARROLL
Average Customer Review:
Leather Bound (30 August, 1995)
list price: $19.99 -- our price: $19.99
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Editorial Review

This beautiful, 868-page leather-bound volume contains a delightfulcollection of stories from one of history's most beloved children's authors.Lewis Carroll's stories are still as fresh and appealing as when they were firstpublished more than a century ago. John Tenniel's original illustrationsaccompany the Alice stories and bring to life the wildly popular characters sowell known to us all: the Mad Hatter, the White Rabbit, the Cheshire Cat, and apassel of others.

Carroll, one of 11 children, knows his audience well. His stories--clever,provocative, and bizarre--capture the imaginations of children worldwide. Thougha prolific storyteller from childhood, he went on to become a mathematician, a fact evidenced by the Tangled Tales serial, which contains amathematical equation in each installment.

Other stories included in this collection are "The Hunting of theSnark," which was composed backward, in a sense, when inspiration for thetale came by way of the last line; "Rhyme? And Reason?"; the Sylvieand Bruno books; and the original Alice story, "Alice's AdventuresUnderground," penned and illustrated in Carroll's own hand. Two never-before-printed poems, originally inscribed in two storybooks and presented asmementos to a little girl and boy, conclude this enchanting collection. ... Read more

Reviews (25)

Kinda disappointed
Of course, it is very nice to have pretty much every Carroll work you could want in one volume, however there is one major drawback here.While the blue leather binding and gold gilt text and page edges are quite nice, I found the actual pages to be a very low quality thin paper.Also, the margins are very small, cramming alot of text on each page, while the paper and print quality actually take away a minor of the illustration detail.Overall, not a bad collection, yet I think "The Complete Works of Lewis Carroll"thru B&N is superior even without the bells and whistles.

MK

THERE'S ONLY ALICE
The indeterminate way that Alice In Wonderland appeared could probably never happen again.Somehow the book would not have had the same ring to it if either of the other sisters had pestered Charles Dodgson for the tale to be written down.
The majority discover Alice via the Disney movie which combines elements of both "Alice In Wonderland" and "Through The Looking Glass",changing much along the way.
Thus Alice is only sat in the tree at the beginning in this version.
In the original tale called "Alice's Adventures Underground" some of the characters are missing such as the Cheshire Cat and the Mad Hatter.This is because Dodgson added them later when the book was published with a title change and John Tenniel's artwork
Alice is,today,a massive cottage industry,fuelled by the Internet
and collecting Alice goes three ways really as it adds Lewis Carroll and Alice Liddell.
To my mind the most fascinating area of all collecting

A Monumental Achievement
Outside of William Shakespeare and Miguel de Cervantes, I enjoy no author in any language more than I do Charles Lutwidge Dodgson, known to the world as Lewis Carroll.His bizarrely creative mind stands alone in world literature, and his maze-like stories (which can be interpreted by the reader in a variety of interesting ways) are both immensely enjoyable and intellecutally stimulating.These tales, from the immortal "Alice" books (translated and quoted from more often than any English language text besides Shakespeare and the King James Bible) to the early "Phantasmagoria," are largely accessible, succinct, and simple, yet driven by a striking undercurrent of delicious parody and delightful nonsense.In the age of Marx and Freud, men who attempted to explain everything, Carroll, more than anyone else, stood up for true art; his work (especially the twisted "Hunting of the Snark") defy conventional academic interpretation and application to the real world, thus embodying Nietzsche's sentiment that "we have art so that we don't go insane."His highly creative vocabulary puts his work on the cutting-edge, over 50 years before anyone had heard of his most visible successors, James Joyce, Vladimir Nabokov (who translated Carroll into Russian), et al.Perhaps it a testament to Carroll's greatness that his works, more than almost any others of the Victorian age, endure so visibly during the modern day; his patented blend of simplicity, wordplay, and, above all, imagination, may never again be seen, and thus ought to be treasured very highly indeed. ... Read more

Isbn: 0517147815
Subjects:  1. Classics    2. English, Irish, Scottish, Welsh    3. Literary Criticism    4. Sale Adult - Literature - Classics & Contemporary    5. Fiction / Classics   

Labyrinths of Reason : Paradox, Puzzles, and the Frailty of Knowledge
by WILLIAM POUNDSTONE
Average Customer Review:
Paperback (01 December, 1989)
list price: $11.16
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Editorial Review

We conceive of and describe the world in ways that usually work just fine, but in the far corners of the labyrinth of reason, our best intentions fold back on themselves, and we end up trapped in an intractable loop or tumbling down a chute of infinite regress. Labyrinths of Reason is a collection of classic philosophical thought experiments and other imponderables that push reason and language to their logical limits. Beyond just idle brainteasers, William Poundstone shows that these mental exercises have profound implications for such fields as cryptography, decision theory, subatomic physics, and computer programming. But most of all, they're good, clean philosophical fun! ... Read more

Reviews (9)

A good starter
I enjoyed this book, but one should realize the audience for this work.I was a novcie to logic when I read this book and felt quite satisfied with the content and structure.

Labytinths does not overwhelm the reader; it introduces a fair amout of classic logic problems, scenarios, etc.Many of the examples offered would serve well in a high school math class.

Great book on logic and the meaning of 'knowing'
I have to confess - I bought this book after reading Mr. Poundstone's book 'How would you move mount Fuji?' I probably never would have bought this book otherwise - and I'm glad I did.
So first of all - this is not a riddle book. Its a philosophy book trying to dismantle well know paradoxes in modern epistemology. It's not revolutionary in any way, but it collects its ideas from good sources and gives a nice and coherent view of the topic and the field.
I would recommend this book only to people willing to read slowly and think about what they read. The journey is not easy - but Mr. Poundstone would get you to your destination every time.

Brain Workout in a Nutshell
With this collection of paradoxes and intrigues, Poundstone gracely accomplished what he set out to do -- guiding the readers on an unforgettable journey through the many Labyrinths of Reason. Chapter by chapter, Poundstone took some famous paradox as a seed, and made it grew into a whole field of thought provoking ideas.

A great way to spend Spring Break if you're looking for some fun for your mind. ... Read more

Isbn: 0385242719
Subjects:  1. Knowledge, Theory of    2. Logic    3. Paradox    4. Philosophy    5. Puzzles    6. Science    7. Science / Philosophy & Social Aspects