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Gödel's incompleteness theorems. Proof sketch for Gödel's first incompleteness theorem, Mechanism (philosophy), Mathematical induction, Principia Mathematica, Zermelo¿Fraenkel set theory, Proof theory, Euclidean geometry, Diagonal lemma ... Read more

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Traditional logic as a part of philosophy is one of the oldest scientific disciplines and can be traced back to the Stoics and to Aristotle. Mathematical logic, however, is a relatively young discipline and arose from the endeavors of Peano, Frege, and others to create a logistic foundation for mathematics. It steadily developed during the twentieth century into a broad discipline with several sub-areas and numerous applications in mathematics, informatics, linguistics and philosophy.

This book treats the most important material in a concise and streamlined fashion. The third edition is a thorough and expanded revision of the former. Although the book is intended for use as a graduate text, the first three chapters can easily be read by undergraduates interested in mathematical logic. These initial chapters cover the material for an introductory course on mathematical logic, combined with applications of formalization techniques to set theory. Chapter 3 is partly of descriptive nature, providing a view towards algorithmic decision problems, automated theorem proving, non-standard models including non-standard analysis, and related topics.

The remaining chapters contain basic material on logic programming for logicians and computer scientists, model theory, recursion theory, Gödel’s Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text. Each section of the seven chapters ends with exercises some of which of importance for the text itself. There are hints to most of the exercises in a separate file Solution Hints to the Exercises which is not part of the book but is available from the author’s website.

Customer Reviews (2)

Complete and demanding
For a motivated student, the best way to learn logic - in my opinion - is to study this book of Prof. Rautenberg. Thanks to a clever path through the subject of logic, the maximum result is obtained with the minimum effort. Every theorem is stated in the most general form and each proof appears to be the best.
This is neither a quick introduction nor an easy book, rather a dense and complete introduction to logic, demanding time and good will for a rewarding result.

The best intro to logic to-date
This is the best introductory text in logic available. It only lacks a coverage of set theory, and I advise the author to include it as chapter 4 of the third edition. ... Read more

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The Fourth Edition of this long-established text retains all the key features of the previous editions, covering the basic topics of a solid first course in mathematical logic. This edition includes an extensive appendix on second-order logic, a section on set theory with urlements, and a section on the logic that results when we allow models with empty domains. The text contains numerous exercises and an appendix furnishes answers to many of them.Introduction to Mathematical Logic includes:opropositional logicofirst-order logicofirst-order number theory and the incompleteness and undecidability theorems of Gödel, Rosser, Church, and Tarskioaxiomatic set theoryotheory of computabilityThe study of mathematical logic, axiomatic set theory, and computability theory provides an understanding of the fundamental assumptions and proof techniques that form basis of mathematics. Logic and computability theory have also become indispensable tools in theoretical computer science, including artificial intelligence. Introduction to Mathematical Logic covers these topics in a clear, reader-friendly style that will be valued by anyone working in computer science as well as lecturers and researchers in mathematics, philosophy, and related fields. ... Read more

Customer Reviews (11)

A big mistake
Late in August, the text originally selected for my mathematical logic class became unavailable. On the basis of reviews only, I chose Mendelson's Introduction to Mathematical Logic as the replacement. A disasterous choice. There may be a page without a typo, but I don't expect to find it. The presentation is inconsistent in notation and focus. Concepts are confused and more difficult than they should be. Definitions are not wisely selected. This book reads like something that has been patched for four decades (since 1964). On the positive side it contains interesting supporting material and will be a valuable private source of ideas to the lecturer. Be sure to read sections from chapter 2 and 3 before selecting this as a text.

A must have....
This is a very useful and must have book for every graduate student in logic.Theory covers many fields(logic and computability) and has a lot of exercises (and also solutions to the tough ones)!!!

twisted pants unleashed on men
This is one of the more popular introductory textbooks on mathematical logic, with Enderton's being its biggest competitor. I prefer Mendelson's for its breadth of material and the choice of proofs he uses, which are generally the most intuitive (e.g. Kalmar's for the completeness of the propositional calculus). This is not to say that they are always constructive, as they many of them are in the older texts (e.g. Kleene, Introduction to Metamathemaitcs).

The exercises are thoughtfully chosen. There's a good range of difficulty and a good portion of the answers can be found in the back. Difficult questions are indicated to the reader.

Out of all the mathematical logic texts I have (which are quite a few in number), this is the most oft-referred-to.

Wonderful at the second glance.
Mendelson's Introduction to Mathematical Logic was the textbook for a logic-course I took a couple of years ago. At the time I did not like the book at all. It seemed too difficult and so typographically ugly that I thought I would never use it. Things have changed though. Now, I keep it close at hand on my desk and use it almost every day. Technical questions that used to require a trip to the library and several different books to answer, can usually be resolved by a look in Mendelson's book. It's wonderfully rich and clear! I still don't find everything easy but that's because the material isn't easy and so not something Mendelson can be blamed for. I do find the typography ugly and at times annoying, but that's a small price to pay for a presentation as rigorous and detailed as Mendelson's.
So in summary: it's not the ideal book for the complete newcomer, but once you get past the initial hurdle it's a must read.

Best reference in first step math logic
Mendelson reaches an optimal point between the concision of the expert reference, and the wideness requested to a introductory text. Not in vain it has been the text forced in the universities during forty years.
Nevertheless, I believe to have found an error in the demonstration that does of the theorem of the completeness of the Predicate calculus, in the part in which it tries to demonstrate that all logical truth is
a theorem of the system.
[...] ... Read more

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Retaining all the key features of the previous editions, Introduction to Mathematical Logic, Fifth Edition explores the principal topics of mathematical logic. It covers propositional logic, first-order logic, first-order number theory, axiomatic set theory, and the theory of computability. The text also discusses the major results of Gödel, Church, Kleene, Rosser, and Turing.

New to the Fifth Edition

• A new section covering basic ideas and results about nonstandard models of number theory
• A second appendix that introduces modal propositional logic
• An expanded bibliography
• Additional exercises and selected answers

This long-established text continues to expose students to natural proofs and set-theoretic methods. Only requiring some experience in abstract mathematical thinking, it offers enough material for either a one- or two-semester course on mathematical logic.

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People have always been interested in numbers, in particular the natural numbers. Of course, we all have an intuitive notion of what these numbers are. In the late 19th century mathematicians, such as Grassmann, Frege and Dedekind, gave definitions for these familiar objects. Since then the development of axiomatic schemes for arithmetic have played a fundamental role in a logical understanding of mathematics. There has been a need for some time for a monograph on the metamathematics of first-order arithmetic. The aim of the book by Hajek and Pudlak is to cover some of the most important results in the study of a first order theory of the natural numbers, called Peano arithmetic and its fragments (subtheories). The field is quite active, but only a small part of the results has been covered in monographs. This book is divided into three parts. In Part A, the authors develop parts of mathematics and logic in various fragments. Part B is devoted to incompleteness. Part C studies systems that have the induction schema restricted to bounded formulas (Bounded Arithmetic). One highlight of this section is the relation of provability to computational complexity. The study of formal systems for arithmetic is a prerequisite for understanding results such as Gödel's theorems. This book is intended for those who want to learn more about such systems and who want to follow current research in the field. The book contains a bibliography of approximately 1000 items. ... Read more

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Dr Gregory Chaitin, one of the world s leading mathematicians, is best known for his discovery of the remarkable number, a concrete example of irreducible complexity in pure mathematics which shows that mathematics is infinitely complex. In this volume, Chaitin discusses the evolution of these ideas, tracing them back to Leibniz and Borel as well as Gödel and Turing. This book contains 23 non-technical papers by Chaitin, his favorite tutorial and survey papers, including Chaitin's three Scientific American articles. These essays summarize a lifetime effort to use the notion of program-size complexity or algorithmic information content in order to shed further light on the fundamental work of Gödel and Turing on the limits of mathematical methods, both in logic and in computation. Chaitin argues here that his information-theoretic approach to metamathematics suggests a quasi-empirical view of mathematics that emphasizes the similarities rather than the differences between mathematics and physics. He also develops his own brand of digital philosophy, which views the entire universe as a giant computation, and speculates that perhaps everything is discrete software, everything is 0's and 1's.Chaitin's fundamental mathematical work will be of interest to philosophers concerned with the limits of knowledge and to physicists interested in the nature of complexity. ... Read more

Customer Reviews (1)

Very interesting

I bought this book, from a New Scientist review, for my husband who is extremely gifted in math, and not so much verbally.Reading is a chore.So when, out of all the Christmas books past, he pronounced this volume a really, really good book and read some every day - I knew it was extraordinary.Now I hope he finishes soon so I can start it! ... Read more

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This conference was devoted to fundamental questions raised by quantum mechanics, especially in quantum information theory. As has become customary in our series of conference in Växjö, we were glad to welcome a fruitful assembly of theoretical physicists, experimentalists, mathematicians and even philosophers interested in the foundations of probability and physics. This conference belongs to the series of Växjö conferences in foundations of quantum mechanics (especially probabilistic foundations) combined of two subseries, Foundations of Probability and Physics: 2000, 02, 04, 06, 08, and Quantum Theory: Reconsideration of Foundations: 2001, 03, 05, 07. We also mention the first Växjö conference: Bohmian mechanics 2000. This is definitely the longest series of conferences on foundations in the history of quantum mechanics.

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A masterly introduction to the life and thought of the man who transformed our conception of math forever.

Kurt Gödel is considered the greatest logician since Aristotle. His monumental theorem of incompleteness demonstrated that in every formal system of arithmetic there are true statements that nevertheless cannot be proved. The result was an upheaval that spread far beyond mathematics, challenging conceptions of the nature of the mind.

Rebecca Goldstein, a MacArthur-winning novelist and philosopher, explains the philosophical vision that inspired Gödel's mathematics, and reveals the ironic twist that led to radical misinterpretations of his theorems by the trendier intellectual fashions of the day, from positivism to postmodernism. Ironically, both he and his close friend Einstein felt themselves intellectual exiles, even as their work was cited as among the most important in twentieth-century thought. For Gödel , the sense of isolation would have tragic consequences.

This lucid and accessible study makes Gödel's theorem and its mindbending implications comprehensible to the general reader, while bringing this eccentric, tortured genius and his world to life.

About the series:Great Discoveries brings together renowned writers from diverse backgrounds to tell the stories of crucial scientific breakthroughs—the great discoveries that have gone on to transform our view of the world.Amazon.com Review
Kurt Gödel is often held up as an intellectual revolutionary whose incompleteness theorem helped tear down the notion that there was anything certain about the universe. Philosophy professor, novelist, and MacArthur Fellow Rebecca Goldstein reinterprets the evidence and restores to Gödel's famous idea the meaning he claimed he intended: that there is a mathematical truth--an objective certainty--underlying everything and existing independently of human thought. Gödel, Goldstein maintains, was an intellectual heir to Plato whose sense of alienation from the positivists and postmodernists of the 1940s was only ameliorated by his friendship with another intellectual giant, Albert Einstein. As Goldstein writes, "That his work, like Einstein's, has been interpreted as not only consistent with the revolt against objectivity but also as among its most compelling driving forces is ... more than a little ironic."

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

Customer Reviews (59)

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

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

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

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

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

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

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

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

[...]

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

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

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

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

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

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

This sentence is false.

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Gödels Incompleteness Theorems are among the most significant results in the foundation of mathematics. These results have a positive consequence: any system of axioms for mathematics that we recognize as correct can be properly extended by adding as a new axiom a formal statement expressing that the original system is consistent. This suggests that our mathematical knowledge is inexhaustible, an essentially philosophical topic to which this book is devoted.

Basic material in predicate logic, set theory and recursion theory is presented, leading to a proof of incompleteness theorems. The inexhaustibility of mathematical knowledge is treated based on the concept of transfinite progressions of theories as conceived by Turing and Feferman.

All concepts and results necessary to understand the arguments are introduced as needed, making the presentation self-contained and thorough. ... Read more

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A New York Times bestseller when it appeared in 1989, Roger Penrose's The Emperor's New Mind was universally hailed as a marvelous survey of modern physics as well as a brilliant reflection on the human mind, offering a new perspective on the scientific landscape and a visionary glimpse of the possible future of science. Now, in Shadows of the Mind, Penrose offers another exhilarating look at modern science as he mounts an even more powerful attack on artificial intelligence. But perhaps more important, in this volume he points the way to a new science, one that may eventually explain the physical basis of the human mind.

Penrose contends that some aspects of the human mind lie beyond computation. This is not a religious argument (that the mind is something other than physical) nor is it based on the brain's vast complexity (the weather is immensely complex, says Penrose, but it is still a computable thing, at least in theory). Instead, he provides powerful arguments to support his conclusion that there is something in the conscious activity of the brain that transcends computation--and will find no explanation in terms of present-day science. To illuminate what he believes this "something" might be, and to suggest where a new physics must proceed so that we may understand it, Penrose cuts a wide swathe through modern science, providing penetrating looks at everything from Turing computability and Godel's incompleteness, via Schrodinger's Cat and the Elitzur-Vaidman bomb-testing problem, to detailed microbiology. Of particular interest is Penrose's extensive examination of quantum mechanics, which introduces some new ideas that differ markedly from those advanced in The Emperor's New Mind, especially concerning the mysterious interface where classical and quantum physics meet. But perhaps the most interesting wrinkle in Shadows of the Mind is Penrose's excursion into microbiology, where he examines cytoskeletons and microtubules, minute substructures lying deep within the brain's neurons.(He argues that microtubules--not neurons--may indeed be the basic units of the brain, which, if nothing else, would dramatically increase the brain's computational power.) Furthermore, he contends that in consciousness some kind of global quantum state must take place across large areas of the brain, and that it within microtubules that these collective quantum effects are most likely to reside.

For physics to accommodate something that is as foreign to our current physical picture as is the phenomenon of consciousness, we must expect a profound change--one that alters the very underpinnings of our philosophical viewpoint as to the nature of reality. Shadows of the Mind provides an illuminating look at where these profound changes may take place and what our future understanding of the world may be. ... Read more

Customer Reviews (26)

Deep Debate on Mind Machine Problem
Shadows of the Mind is undoubtedly a less populist book than its predecessor "The Emperor's New Mind". It is also significantly more technical in places than the predecessor. Its purpose is to extend the Godel based arguments used in ENM in several directions. Firstly to attempt to address various criticisms of the central argument of the previous book and then to develop some new ones. Also there is a discussion of the application to Robotics. An example of this sort of discussion is whether any capability of a Robot to learn would undo any of the Robot restrictions deduced in his basic argument. After all learning (human or robotic) will imply going beyond previous restrictions and being aware of new facts.

So there is a subtle argument needed to continue to show that despite this, humans will come out on top. If you are interested in this kind of subtlety after reading ENM then this is the book for you.

In effect Penrose is right at the heart of the Mind-Machine debate in this book. I give an overview of this debate as follows:

We need to find a scientific theory of the Mind. So we can examine what kind of cognitive or thinking device it might be, recognising that it also thinks about Mathematics. For that we need a model of cognition sufficiently general: the Turing Machine model is available and generally considered to be that model - there are no obvious rivals. So one can focus on whether the Turing Machine model could really be a model for the Human Mathematical Mind. If the answer is "yes" we would conclude also: Robots could have Minds.

Penrose draws the conclusion about mathematical reasoning that:

G: "Human mathematicians are not using a knowably sound algorithm in order to ascertain mathematical truth".

This statement isn't quite the statement that the Mind is not a Turing Machine (algorithm here), and some critics have attempted to expose the gaps. In this book Penrose discusses several lines of argument to close the gap. A possible rebuttal might be: "could mathematicians just be using unsound algorithms" - the faulty machine argument. This is very close to questions in the foundations of mathematics itself - after all is this suggesting that mathematics itself is fundamentally unsound? If so where is that unsoundness? So Penrose comes round to the conclusion step by step and through 100 odd pages, that the statement G above implies that indeed the Mind is not a Turing Machine.

This latter conclusion however introduces another problem: if the Mind is not a Turing Machine / algorithm then what sort of (scientific) model exists for it? At the end of the book Penrose examines a generalisation of the Turing Machine model (called Oracle Machines and also due to Alan Turing) and determines that a statement similar to G also applies to that model class as well. Thus the story is left incomplete and I would be tempted to say that somewhere a model M exists of which we can deduce:

"Human mathematicians are using M to ascertain mathematical truth"

However Shadows of the Mind ends without a discovery of that model M. So maybe the answer lies in studying Quantum properties, or in other aspects of these obscure machine models? If you want to be able to study this question further Shadows needs to be studied (and "studied" is the word)!

An Effort to Discover Consciousness
"Shadows of the Mind" addresses first, all of the arguments Professor Penrose had to counter regarding the assertions he made in "The Emperor's New Mind." He then goes on to hint at an approach to the subject of consciousness, the great puzzle that science has heretofore been unable to tackle, and has therefore denied.

It provides an exceptional mental excursion into the questions surrounding the subject of consciousness, as well as the peculiar nature of matter.

Why is it that a physicist is the one to tackle this subject as opposed to a biologist?

Professor Penrose suggests that physicists may be in a better position to comprehend how matter really behaves than biologists are.

Read it for yourself to see whether he is right. I have no doubt about it!

Very Good
I got this book in a very good condition. My book arrived just in time. Price was low.
I am very much satisfied buying this book from amazon.com.

MIND MYSTERY and QUANTUM MECHANICS
I advise that students in any discipline: Mathematics, Physics, Chemistry,
Biology, ....... should be aware and even become involved in studying Quantum Mechanics (QM) right away after their B.SC. degree. That is what I had the chance to be introduced to this fantastic branch of physics on 1949. It is time to realise and be convinced that using the very simple
quantum mechanical models one can realise results in excellent agreement
with experiments, results that are impossible to obtain by using classical
physics. That is why the books of ROGER PENROSE are of tremendous treasure
in a world not familiar of this branch that the MIGHTY GOD have put in their hands to explain this universe He has created.

No other book tackles this subject so clearly
Just opening this book to a random page and reading that page - sets one's mind on fire.

The basic thread running throughtout the book is that of 'what is computable and what is not'. The process of 'Understanding' as humans know it - Penrose argues - is NON-COMPUTABLE. He provides brilliant examples of how computers can 'solve' any problem - without 'understanding' what they are solving (e.g. DeepThought and the simple chess move which stumped it).

This theme in itself would make this a worthwhile read. However - this book offers further gems from Quantum Physics - with perhaps the simplest and best explanation of lesser known quantum paradoxes such as the 'delayed choice' experiments. Godel's theorem is also dealt with lucidly.

Few authors can tackle the issue of 'mind and conciousness' without stepping into some mystical/unscientific goo. Penrose stays scientific - and works from facts and well known experiments.

I do not know of any other book that tackles this subject so clearly - and in such an exciting fashion. From my perspective - this clearly deserves 5 stars. ... Read more